Research Article Mathematical Modelling of the Inhibitory Role ...

Research ArticleMathematical Modelling of the Inhibitory Role of RegulatoryT Cells in Tumor Immune Response

Zhongtao Yang12 Cuihong Yang1 Yueping Dong 1 and Yasuhiro Takeuchi3

1School of Mathematics and Statistics Central China Normal University Wuhan 430079 China2School of Finance and Mathematics Huainan Normal University Huainan 232038 China3College of Science and Engineering Aoyama Gakuin University Sagamihara 252-5258 Japan

Correspondence should be addressed to Yueping Dong dongyueping0531gmailcom

Received 15 May 2020 Accepted 3 July 2020 Published 12 August 2020

Guest Editor Songbai Guo

Copyright copy 2020 Zhongtao Yang et al )is is an open access article distributed under the Creative Commons AttributionLicense which permits unrestricted use distribution and reproduction in any medium provided the original work isproperly cited

)e immune system against tumors acts through a complex dynamical process showing a dual role On the one hand the immunesystem can activate some immune cells to kill tumor cells (TCs) such as cytotoxic T lymphocytes (CTLs) and natural killer cells(NKs) but on the other hand more evidence shows that some immune cells can help tumor escape such as regulatory T cells(Tregs) In this paper we propose a tumor immune interaction model based on Tregs-mediated tumor immune escapemechanism When helper Tcellsrsquo (HTCs) stimulation rate by the presence of identified tumor antigens is below critical value thecoexistence (tumor and immune) equilibrium is always stable in its existence region When HTCs stimulation rate is higher thanthe critical value the inhibition rate of effector cells (ECs) by Tregs can destabilize the coexistence equilibrium and cause Hopfbifurcations and produce a limit cycle )is model shows that Tregs might play a crucial role in triggering the tumor immuneescape Furthermore we introduce the adoptive cellular immunotherapy (ACI) and monoclonal antibody immunotherapy (MAI)as the treatment to boost the immune system to fight against tumors )e numerical results show that ACI can control TCs morewhile MAI can delay the inhibitory effect of Tregs on ECs )e result also shows that the combination of both immunotherapiescan control TCs and reduce the inhibitory effect of Tregs better than a single immunotherapy can control

1 Introduction

Tumors can be benignant (not cancerous) premalignant(precancerous) and malignant (cancerous) Every yearmillions of people suffer with cancer and die from thisdisease throughout the world [1] It is important to un-derstand tumorrsquos mechanisms of establishment and de-struction cell-mediate immunity with cytotoxicT lymphocytes (CTLs) and natural killer cells (NKs)generally called effector cells (ECs) that are cytotoxic totumor cells (TCs) and play a basic role in immune responseagainst tumors [2 3] Moreover efficient antitumor im-munity requires the action of helper T cells (HTCs) whichcan directly activate naive CD8+ T cells to differentiate intoCTLs [4ndash6] Recently it has been reported that regulatoryT cells (Tregs) can inhibit CTLs and promote the escape ofTCs [7] Tregs suppress immune cells and when the war

between Tcells and infection is over the Tregs signal to stop[8] Cancer immunotherapy fights against cancer bystrengthening the bodyrsquos immune system but the involve-ment of Tregs inhibits the immune response and turns offthe anticancer effect Tregs inhibition is important in thedynamics of the tumor immune system which is one mo-tivation of this work

Adoptive T cell immunotherapy (ACI) as a commonimmunotherapy involves injecting adoptive T cells directlyinto tumor patients [9ndash11] Its advantages are good de-struction of tumor and persistence while its disadvantagesare serious toxic and side effects )e monoclonal antibodyimmunotherapy (MAI) is the immune checkpoint inhibitor[12 13] which has the advantage of removing the sup-pression state of the immune system and restoring theimmune function of the body to TCs and the disadvantage ofhaving serious immune-related adverse reactions

HindawiComplexityVolume 2020 Article ID 4834165 21 pageshttpsdoiorg10115520204834165

Tregs have become an important target in tumor im-munotherapy because of their contribution to tumor im-mune escape Cytotoxic T lymphocyte antigen 4 (CTLA-4) isa marker that is expressed on the surface of activated T cellsand transmits inhibitory signals in the immune response[14ndash16] Blocking CTLA-4 can reduce the inhibitory activityof Tregs and the anti-CTLA-4 humanized monoclonalantibody Ipilimumab and Tremelimumab are used to treatadvanced melanoma and malignant mesothelioma re-spectively [17] Similar to CTLA-4 programmed death re-ceptor 1 (PD-1) can also promote the activation anddevelopment of Tregs [18ndash21] Blocking PD-1 can preventthe development of Tregs and prevent the conversion ofHTCs into Tregs [22] Currently OPDIVO (Nivolumab) ananti-PD-1 monoclonal antibody has been approved by theUS FDA for the treatment of melanoma renal cell carci-noma and non-small cell lung cancer [23ndash25] Establish-ment of a mathematical model to study the immunotherapyon the reduction of Tregs inhibition has both theoretical andpractical significance

In order to describe the mechanisms of hostrsquos ownimmune response to against TCs various types of mathe-matical models have been proposed [26ndash43] )e modellingof the tumor immune system described by ordinary dif-ferential equations (ODEs) has a long history which can betraced back to the classic research of Stepanova in 1980 [26]In 1994 Kuznetsov et al established the famous two-di-mensional ODEs model postulating that tumor growthfollows the Logistic growth pattern )ey evaluated theparameters of the model by fitting experimental data frommice [27] In 2003 Stolongo-Costa et al assumed that TCsfollows the exponential growth pattern and constructed atwo-dimensional ODEs model )ey analyzed the basicproperties of the model and provided conditions for stabilityof the tumor-free equilibrium explaining its epidemiologicalsignificance [28] In 2004 Galach simplified Kuznetsovrsquossystem to account for the effect of immune delay on thetumor immune system [29] In 2014 Dong et al constructeda three-dimensional ODEs model focusing on the effects ofHTCs on the tumor immune system [4]

In 1998 Kirschner and Panetta generalized Kuznet-sovndashTaylor model and illustrated the dynamics betweenTCs ECs and IL-2 )ey firstly introduced ACI into theirmodel which can explain both short-term tumor oscillationsin tumor sizes as well as long-term tumor relapse [11] In2003 in order to study the role of cytokine therapy in theactivation of the immune system Stolongo-Costa et alintroduced cycle therapy term Fcos2ωt and established acycle immunotherapy model )ey obtained some thresh-olds of the frequency and intensity of immunotherapy [28]In 2006 de Pillis et al constructed the six-dimensionalODEs model to investigate the effects of combined che-motherapy and immunotherapy on tumor control )eybriefly analyzed the nature of the model and discussed theoptimal treatment using optimal control theory [30] In2008 Bunimovich-Mendrazitsky et al established a pulseddifferential equation model with Bacillus Calmette-Guerintumor immunotherapy )ey obtained the critical thresholdand pulse frequency of BCG injection dose that could

successfully treat superficial bladder cancer [31] In 2012Wilson and Levy established a mathematical model con-taining Tregs)ey studied the absence of treatment vaccinetreatment anti-TGF treatment and combination vaccineand anti-TGF treatment as well as sensitivity analysis ofsome important parameters [8] In 2018 Radunskaya et alestablished a mathematical model with blood spleen andtumor compartments to study PD-L1 inhibitors in the roleof tumor immunotherapy )e model was used to fit pa-rameters with the experimental data)e results showed thatincreasing the resistance of PD-L1 doses can greatly improvethe clearance rate of tumor [32]

)is paper investigates the role of Tregs in the tumorimmune system )erefore we incorporate the fourthpopulation of Tregs into the previous system in [8] For themathematical simplicity a bilinear term also has been usedto describe the interactions between immune response andtumor To our knowledge HTCs can recognize TCs andpromote the growth of ECs And ECs can provide directprotective immunity by attacking TCs When there are moreHTCs and ECs in order to maintain immune homeostasisthe body will produce corresponding Tregs to suppress ECsand Tregs originating from both HTCs and ECs )en weestablish a four-dimensional ODEs model described asbelow

dT(t)

dt aT(t)(1 minus bT(t)) minus nE(t)T(t)

dE(t)

dt pE(t)H(t) minus qR(t)E(t) minus d1E(t)

dH(t)

dt s2 + k2T(t)H(t) minus d2H(t)

dR(t)

dt r1E(t) + r2H(t) minus d3R(t)

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(1)

where T(t) E(t) H(t) and R(t) represent the populationsof TCs ECs HTCs and Tregs respectively )e firstequation describes the rate change of TCs population )etumor follows logistic growth dynamics with growth rate aand the maximum capacity is 1b n represents the loss rateof TCs by ECs interaction)e second equation describes therate change of the ECs population d1 is the mortality rate ofECs p is the activation rate of ECs by HTCs and q is theinhibition rate of Tregs on ECs )e third equation describesthe rate change of the HTCs population s2 is birth rate ofHTCs produced in the bone marrow and HTCs have anatural lifespan of an average 1d2 days k2 is HTCs stim-ulation rate by the presence of identified tumor antigens)efourth equation gives the rate change of the Tregs pop-ulation r1 and r2 are the activation rates of Tregs by ECs andHTCs respectively d3 represents per capita decay rate ofTregs A diagram of the various interactions between thesecell populations is shown in Figure 1

We nondimensionalize model (1) by taking the followingscaling

2 Complexity

t τ

nT0 T(t) T0x(τ) E(t) E0y(τ) H(t)

H0z(τ) R(t) R0u(τ) α a

nT0 β bT0

ρ p

n ω2

k2

n θ

qR0

nT0 δ1

d1

nT0 δ2

d2

nT0 δ3

d3

nT0 σ2

s2

nT0H0 c1

r1

nR0 c2

r2

nR0

(2)

and we choose the scaling T0 E0 H0 R0 106 Byreplacing τ by t we obtain the following scaled model

dx(t)

dt αx(t)(1 minus βx(t)) minus x(t)y(t)

dy(t)

dt ρy(t)z(t) minus θy(t)u(t) minus δ1y(t)

dz(t)

dt σ2 + ω2x(t)z(t) minus δ2z(t)

du(t)

dt c1y(t) + c2z(t) minus δ3u(t)

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(3)

with initial conditions

x(0) x0 ge 0

y(0) y0 ge 0

z(0) z0 ge 0

u(0) u0 ge 0

(4)

Here x y z and u denote the dimensionless densities ofTCs ECs HTCs and Tregs populations respectively

2 Model Analysis

21 Well Posedness of Model (3) )e following propositionestablishes the well posedness of model (3) with initialconditions (4)

Proposition 1 9e solutions (x(t) y(t) z(t) u(t)) ofmodel (3) with initial conditions (4) are existent unique andnonnegative on the interval [0 +infin)

Proof Since the right-hand side of model (3) is completelycontinuous and locally Lipschitz on the interval [0 +infin)there exists a constant δ gt 0 such that the solutions(x(t) y(t) z(t) u(t)) of model (3) with initial conditions(4) are existent and unique on the interval [0 δ) where0lt δ le +infin [44]

From model (3) with initial conditions (4) we obtain

x(t) x0e1113938

t

0(α(1minus β(s))minus y(s))ds ge 0

y(t) y0e1113938

t

0ρz(s)minus θu(s)minus δ1( )ds ge 0

z(t) z0e1113938

t

0ω2x(τ)minus δ2( )dτ

+ σ2 1113946t

0e1113938

t

sω2x(τ)minus δ2( )dτ

ds ge 0

u(t) u0eminusδ3t

+ 1113946t

0c1y(s) + c2z(s)( 1113857e

minusδ3(tminus s)ds ge 0

(5)

It is obvious to see that the solutions(x(t) y(t) z(t) u(t)) of model (3) with initial conditions(4) are nonnegative for all tge 0 )e proof is complete

22 Existence of Equilibria In this section we will study theexistence of various equilibria for system (3) We set(dxdt) 0 (dydt) 0 (dzdt) 0 and (dudt) 0 insystem (3) and we have

αx(1 minus βx) minus xy 0

ρyz minus θyu minus δ1y 0

σ2 + ω2xz minus δ2z 0

c1y + c2z minus δ3u 0

⎧⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎩

(6)

Putting x 0 andy 0 yields the tumor-free and ECs-free equilibrium namely

P0 x0 y0 z0 u0( 1113857 0 0σ2δ2

c2σ2δ2δ3

1113888 1113889 (7)

which always existsPutting x 0 andyne 0 yields the tumor-free equilib-

rium namely

P1 x1 y1 z1 u1( 1113857 0δ3 ρσ2 minus δ1δ2( 1113857 minus c2θσ2

θc1δ2σ2δ2

ρσ2 minus δ1δ2

δ2θ1113888 1113889

(8)

TCs

HTCsECs

Tregs

nET

aT (1 ndash bT)

k2TH

pEH

r2Hr1E

qRE

d2Hd1E

d3R

S2

Promotion

Inhibition

Figure 1 A diagram of interactions among the different cellpopulations in model (1)

Complexity 3

which exists when ρgt δ1δ2σ2 and θlt δ3(ρσ2minusδ1δ2)c2σ2 ≜ θ1

Putting x 1β yields the tumor-dominant equilibriumnamely

P2 x2 y2 z2 u2( 1113857 1β

0βσ2

βδ2 minus ω2

βc2σ2δ3 βδ2 minus ω2( 1113857

1113888 1113889

(9)

which exists when ω2 lt βδ2 ≜ ωlowast2 Putting xne 0 andxne 1β and eliminating y z and u in

(6) we have

F(x) Ax2

+ Bx + C 0 (10)

where A minusαβθc1ω2 lt 0 B θαc1(ωlowast2 + ω2) + δ1δ3ω2 gt 0

andC δ3(ρσ2 minus δ1δ2) minus θ(αc1 δ2 + c2σ2))en we have the coexistence equilibrium

Plowast

xlowast ylowast zlowast ulowast

( 1113857 xlowast α 1 minus βx

lowast( 1113857

σ2δ2 minus ω2x

lowast1113888

middotαc1 1 minus βxlowast( 1113857 δ2 minus ω2x

lowast( 1113857 + c2σ2δ3 δ2 minus ω2x

lowast( 11138571113889

(11)

which exists when xlowast lt μ ≜ min (1β) (δ2ω2)1113864 1113865 where xlowast isthe positive root of (10)

Below we consider the existence condition of Plowast

Case (1) Cge 0 ie θleδ3(ρσ2 minusδ1δ2)αc1δ2 + c2σ2 ≜ θ0Since F(1β) θαβc1δ2 + δ1δ3ω2 + Cββgt 0 and F(δ2ω2) θαc1δ2 + δ1δ2δ3 + Cgt 0 we have 1βlt xlowast

and δ2ω2 lt xlowast which contradicts with xlowast lt μ HencePlowast does not exist in this caseCase (2) Clt 0 ie θgt θ0

Let Δ B2 minus4AC 4αβc1θω2(δ3(ρσ2 minusδ1δ2)minus θ(αc1δ2+c2σ2)) + (θαc1(ωlowast2 +ω2) +δ1δ3ω2)

2 [θαc1(ω2 minusωlowast2 ) +δ1δ3ω2]

2 +4αβθc1ω2σ2(ρ δ3 minusθc2)If Δlt 0 then (10) does not have a positive root Hence

Plowast does not exist in this caseIf Δ 0 then (10) has one positive root xlowast where xlowast

θαc1(ωlowast2 +ω2)+δ1δ3ω22αβθc1ω2gtωlowast2 +ω22βω2 12((δ2ω2)+(1β))geμ which contradicts with xlowastltμ Hence Plowast

does not exist in this caseIf Δgt 0 then (10) has two positive roots where xlowast+

θαc1(ωlowast2 + ω2) + δ1δ3ω2 +Δ

radic2αβθc1ω2 and xlowastminus θαc1

(ωlowast2 + ω2) + δ1δ3ω2 minusΔ

radic2αβθc1ω2 Since xlowast+ gtθαc1 (ωlowast2 +

ω2) +δ1δ3ω22αβθc1ω2gtωlowast2 +ω22βω2 (12)((δ2ω2)+ (1β))ge μ which contradicts with xlowast ltμ If xlowastminus ltμ ie F(1β)

(δ1δ3ω2 +βδ2 (ρσ2 minusδ1δ2) minusθβc2σ2β)gt0rArrθlt(δ1δ3ω2+

βδ3 (ρσ2 minusδ1δ2)βc2σ2) ≜ θ2 and F(δ2ω2) σ2(ρδ3 minusθc2)gt0rArrθltρδ3c2≜θ3 there exists a unique coexistence equi-librium Plowast (xlowastylowastzlowast ulowast) (xlowast α(1minusβxlowast) σ2δ2 minusω2x

lowast αc1(1minusβxlowast)(δ2 minusω2xlowast) + c2σ2δ3(δ2 minusω2x

lowast))where xlowast xlowastminus

)e existence conditions for each equilibrium are givenin Table 1

23 Stability of Equilibria In order to investigate the localstability of the above equilibria P0 P1 P2 andPlowast of system(3) we linearize the system and obtain Jacobian matrix ateach equilibrium P(x y z u)

J(P)

α minus 2αβx minus y minusx 0 0

0 ρz minus θu minus δ1 ρy minusθy

ω2z 0 ω2x minus δ2 0

0 c1 c2 minusδ3

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠

(12)

)e corresponding characteristic equation is

det(J(P) minus λI)

α minus 2αβx minus y minus λ minusx 0 0

0 ρz minus θu minus δ1 minus λ ρy minusθy

ω2z 0 ω2x minus δ2 minus λ 0

0 c1 c2 minusδ3 minus λ

1113868111386811138681113868111386811138681113868111386811138681113868111386811138681113868111386811138681113868111386811138681113868

1113868111386811138681113868111386811138681113868111386811138681113868111386811138681113868111386811138681113868111386811138681113868

0 (13)

Theorem 1 System (3) always has one tumor-free and ECs-free equilibrium P0 which is unstable

Proof At P0 characteristic (13) becomes

(λ minus α) λ + δ2( 1113857 λ + δ3( 1113857 λ +θc2σ2 + δ1δ2δ3 minus ρσ2δ3

δ2δ31113888 1113889 0

(14)

It can be seen that one eigenvalue α is positive Hence P0is unstable

Theorem 2 System (3) has one tumor-free equilibrium P1when ρgt δ1δ2σ2 and θlt θ1 which is locally asymptoticallystable (LAS) if the inequality θlt θ0 holds


λ minusθ αc1δ2 + c2σ2( 1113857 minus δ3 ρσ2 minus δ1δ2( 1113857

θc1δ21113888 1113889

λ + δ2( 1113857 λ2 + δ3λ + θc1y11113872 1113873 0

(15)

4 Complexity

)en one root of characteristic equation is minusδ2 lt 0 It iseasily noted that as δ3 gt 0 θc1y1 gt 0 so λ

2 + δ3λ + θc1y1 0has solutions with negative real parts )erefore P1 is LASwhen θ(αc1δ2 + c2σ2) minus δ3(ρσ2 minus δ1δ2)θc1δ2 lt 0 that isθlt θ0

Theorem 3 System (3) has one tumor-dominant equilib-rium P2 when ω2 ltωlowast2 which is LAS if the inequalityθgt βδ3(ρσ2 minus δ1δ2) + δ1δ3ω2βc2σ2 equiv θ2 holds


(λ + α) λ + δ3( 1113857 λ minusω2 minus ωlowast2

β1113888 1113889 λ minus ρz2 minus θu2 minus δ1( 11138571113858 1113859 0

(16)

)e three roots of characteristic equation areλ1 minusαlt 0 λ2 minusδ3 lt 0 and λ3 ω2 minus ωlowast2 βlt 0 ifω2 ltωlowast2 So P2 is LAS when λ4 ρz2 minus θu2 minus δ1 ρβσ2δ3 minus

δ1δ3(ωlowast2 minus ω2) minus θβc2σ2δ3(ωlowast2 minus ω2)lt 0 which is equiva-lent to θgt θ2

At Plowast characteristic (13) becomes

λ4 + A1λ3

+ A2λ2

+ A3λ + A4 0 (17)

where

A1 αβxlowast

+ δ3 + δ2 minus ω2xlowast gt 0

A2 θc1ylowast

+ αβ δ2 minus ω2xlowast

( 1113857xlowast

+ αβxlowast

+ δ2 minus ω2xlowast

( 1113857δ3 gt 0

A3 θc1 αβxlowast


( 1113857ylowast

+ αβδ3 δ2 minus ω2xlowast

( 1113857xlowast

+ ρω2xlowastylowastzlowast gt 0

A4 θαβc1 δ2 minus ω2xlowast

( 1113857xlowastylowast

+ ρδ3 minus θc2( 1113857ω2xlowastylowastzlowast

(18)

Note that δ2 gtω2xlowast and θltmin θ2 θ31113864 1113865 are necessary for

the existence of Plowast then we have A4 gt 0 By theRouthndashHurwitz criterion the roots of (17) have only neg-ative real parts if and only if

A1 gt 0 A2 gt 0 A3 gt 0 A4 gt 0 A1A2

minus A3 gt 0 A1 A2A3 minus A1A4( 1113857 minus A23 gt 0

(19)

Hence we obtain the sufficient conditions for stability ofPlowast

(H1) δ2 gtω2xlowast θltmin θ2 θ31113864 1113865

(H2) A1A2 minus A3 θc1 αβxlowast


( 1113857ylowast


( 1113857xlowast

minus ρxlowastylowastzlowast gt 0

(H3) A1 A2A3 minus A1A4( 1113857 minus A23 B1θ + B2 gt 0

(20)

where

B1 αβxlowast


( 11138572δ3 + αβx

lowast+ δ2 minus ω2x

lowast1113872 1113873δ23minus ρω2x

lowastylowastzlowast αβx


lowast( 1113857c1y

lowast

minus αβxlowast


+ δ3( 11138572 αβc1 δ2 minus ω2x

lowast( 1113857x

lowastylowast

(

minusω2c2xlowastylowastzlowast1113857

B2 αβxlowast


( 11138572δ3 + αβx


lowast1113872 1113873δ231113960

minusρω2xlowastylowastzlowast αβδ3 δ2 minus ω2x

lowast( 1113857x

lowast+ ρω2x

lowastylowastzlowast

( 11138571113859

minus αβxlowast


+ δ3( 11138572ρω2δ3x


(21)

Theorem 4 System (3) has a unique coexistence equilibriumPlowast when ω2 leωlowast2 θ0 lt θ lt θ2 or ω2 gtωlowast2 θ0 lt θlt θ3 holdand Plowast is LAS when conditions (20) are satisfied

We summarize the above results in Table 1 whereωlowast2 βδ2 θ0 δ3(ρσ2 minus δ1δ2)αc1δ2 + c2σ2θ1 δ3(ρσ2 minus δ1δ2)c2σ2θ2 βδ3(ρσ2 minus δ1δ2) + δ1δ3ω2βc2σ2 and θ3 ρδ3c2

24 Numerical Simulations In this section we choose somesuitable parameters in (3) to simulate numerically thetheoretical conclusions obtained in the previous sections byusing the Matlab software package MATCONT [45] Weselect the following parametersrsquo set [4]

α 1636 β 0002 δ1 03743 σ2 038

δ2 0055 ρ 048 c1 015 c2 02 δ3 025

(22)

Note that ωlowast2 βδ2 000011 By calculations we have

Table 1 )e existence and stability conditions of each equilibrium

Existence conditions Stability conditionsP0 Always UnstableP1 θlt θ1 θlt θ0P2 ω2 ltωlowast2 θgt θ2

Plowast θgt θ0(i) ω2 leωlowast2 θlt θ2 lt θ3(ii) ω2 gtωlowast2 θlt θ3 lt θ2

1113896 (20)

Complexity 5

θ0 δ3 ρσ2 minus δ1δ2( 1113857

αc1δ2 + c2σ2 0452

θ1 δ3 ρσ2 minus δ1δ2( 1113857

c2σ2 05323

θ3 ρδ3c2

06

θ2 βδ3 ρσ2 minus δ1δ3( 1113857 + δ1δ3ω2

βc2σ2 615625ω2 + 05323

(23)

And we find the stability region of Plowast (xlowast ylowast zlowast ulowast)

(see Figure 2) Plowast is LAS in regions I and II Plowast is unstable inregion III

Let us denote the point in θ minus ω2 plane as Qi (θω2)

Case (a) we choose a point Q1 (048 00001) in theregion I then system (3) has one interior equilibrium

Plowast1 (776449 138195 80448 7265) (24)

)e eigenvalues of Jacobian matrix of (12) are

minus 04623 minus 0049151 minus 001991 minus 0378i and

minus 001991 + 0378i(25)

so Plowast1 is stable as shown in Figure 3(a)Case (b) we choose a point Q2 (0455 00004) in theregion II then system (3) has one interior equilibrium

Plowast2 (284485 162669 705506 662006) (26)


minus 00931 + 0301i minus 00931 minus 0301i minus 00633

+ 00188i and minus 00633 minus 00188i(27)

so Plowast2 is stable as shown in Figure 3(b)Case (c) we choose a point Q3 (048 0000111) inthe region II then system (3) has one interiorequilibrium

Plowast3 (72569 139855 8094 7314) (28)


minus0458 minus 00491 minus 00132 + 0383i and minus 00132 minus 0383i

(29)

so Plowast3 is stable as shown in Figure 3(c)Case (d) we choose a point Q4 (048 00004) in theregion III then system (3) has one interiorequilibrium

Plowast4 (26365 154973 854816 776837) (30)


minus04328 minus 00497 00509 + 0416i and 00509 minus 0416i

(31)

so Plowast4 is unstable as shown in Figure 3(d)Below we perform numerically bifurcation analysis of x

against θ for different values of ω2

Case (1) we choose ω2 00001ltωlowast2 000011 (seeFigure 4(a))We obtain the following result

Proposition 2 When 0lt θlt θ0 P0 exists and is unstable P1exists and is LAS P2 exists and is unstable and Plowast does notexist When θ0 lt θlt θ1 P0 exists and is unstable P1 exists andis unstable P2 exists and is unstable and Plowast exists and is LASWhen θ1 lt θ lt θ2 P0 exists and is unstable P1 does not existP2 exists and is unstable and Plowast exists and is LAS Whenθgt θ2 P0 exists and is unstable P1 does not exist P2 existsand is LAS and Plowast does not exist

Case (2) we choose ω2 00004gtωlowast2 000011 (seeFigure 4(b))

We obtain the following result

Proposition 3 When 0lt θlt θ0 P0 exists and is unstable P1exists and is LAS and Plowast does not exist When θ0 lt θlt θ4where θ4 04657 is a Hopf bifurcation P0 exists and isunstable P1 exists and is LAS and Plowast exists and is LASWhen θ4 lt θlt θ1 P0 exists and is unstable P1 exists and isunstable and Plowast exists and is unstable When θ1 lt θlt θ3 P0exists and is unstable P1 does not exist and Plowast exists and isunstable When θgt θ3 P0 exists and is unstable P1 does notexist and Plowast does not exist

Consider the case where HTCs stimulation rate (ω2) islow by the presence of identified tumor antigens (ω2 ltωlowast2 )When the inhibition rate of Tregs to ECs θ is lower than θ0the solution of system (3) approaches P1 implying that ECscan still effectively remove TCs When θ0 lt θlt θ2 the so-lution of system (3) approaches Plowast showing the coexistenceof TCs and immune cells which means that the patient cansurvive with tumors When θgt θ2 TCs escape the control ofthe immune system and develop into malignant tumors

Next consider the case where HTCs stimulation rate (ω2)is high by the presence of identified tumor antigens(ω2 gtωlowast2 ) When θlt θ0 the solution of system (3) ap-proaches P1 implying that TCs can be effectively removed byECs When θ0 lt θlt θ4 the solution of system (3) approachesPlowast implying that TCs can still be controlled by the immunesystemWhen θgt θ4 the system has a Hopf bifurcation pointand induces a limit cycle (see Figure 3(e)) In the biologicalsense it can be understood that the number of TCs presentsa periodic change

6 Complexity

0452 05 05323 06 065

Stable

I

IIIIIUnstable

Stable

Hopf bifurcation curve

times10ndash3

ω2

0

02

04

06

08

1

ωlowast2θ2

θ1 θ3 θ

Figure 2 Stability region of Plowast and Hopf bifurcation curve in θ minus ω2 parameter plane

10ndash1

100

101

102

103

Coun

t

x

y

z

u

0 100 200 300 400 500 600

t

(a)

x

y

z

u

0

2

4

6

8

Coun

t

0 100 200 300 400 500 600

t

(b)

10ndash1

100

101

102

103

x

y

z

u

0 100 200 300 400 500 600t

Coun

t

(c)

10ndash2

10ndash1

100

101

102

103

Coun

t

0 100 200 300 400 500 600

tx

y

z

u

(d)

084 086 08809 092 094

minus10

1minus2

minus1

0

1

2

3

uy

x

(e)

Figure 3 )e other parameter values are given in (22) (a) Plowast1 (77644 1381 8044 7265) is LAS θ 048 andω2 00001 (b)Plowast2 (2844 1626 7055 662) is LAS θ 0455 andω2 00004 (c) Plowast3 (72569 1398 8094 7314) is LAS θ 048 andω2 0000111(d) Plowast4 (26365 1549 8548 7768) is unstable θ 048 andω2 00004 (e) )e 3D phase portrait depicts tumor cell ECs and Tregsθ 048 andω2 00004

Complexity 7

3 Treatment Model

In order to investigate well the effect of Tregs in tumorimmune response under the treatment we follow the wayin [11] to introduce the constant treatment term s1 intothe second equation of (1) Since CTLA-4 and PD-1 caninhibit the development of Tregs and prevent thetransformation of HTCs into Tregs [12] the effect ofCTLA4 or PD-1 on Tregs can be considered )en weshall establish a five-dimensional ODEs model describedas below

dT(t)


dE(t)

dt s1 + pE(t)H(t) minus qR(t)E(t) minus d1E(t)

dH(t)


dR(t)

dt r1E(t) + r2H(t) minus d3R minus mR(t)W(t)

dW(t)

dt s3 minus d4W(t)

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(32)

where W(t) represents the concentration of monoclonalantibody in human body at time t s1 represents the treat-ment term of introducing LAK and TIL into the region oftumor localization m represents the inhibition rate ofmonoclonal antibody on Tregs s3 represents the amount ofmonoclonal antibody entering human body at time t and d4represents the attenuation coefficient of monoclonalantibody

We scale those new parameters in model (32) as follows

W(t) W0v(τ) σ1 s1

nE0T0 ξ

mW0

nT0 σ3

s3

nW0T0 δ4

d4

nT0

(33)

and we choose the scaling W0 106 By replacing τ by t weobtain the following scaled model with treatment

dx(t)


dy(t)

dt σ1 + ρy(t)z(t) minus θy(t)u(t) minus δ1y(t)

dz(t)


du(t)

dt c1y(t) + c2z(t) minus δ3u(t) minus ξu(t)v(t)

dv(t)

dt σ3 minus δ4v(t)

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(34)


x(0) x0 ge 0 y(0) y0 ge 0 z(0) z0 ge 0 u(0)

u0 ge 0 v(0) v0 ge 0(35)

Here v denotes the dimensionless concentration ofmonoclonal antibody

31 Model Analysis By a similar proof of Proposition 1 wecan obtain the well posedness of model (34) with initialconditions (35) as follows

Proposition 4 9e solutions (x(t) y(t) z(t) u(t)) ofmodel (34) with initial conditions (35) are existent uniqueand nonnegative on the interval [0 +infin)

0 02 0452 05323 05938 0650

100

200

300

400

500

600

P0

P1

P2

PlowastX

θ0 θ2θ1 θ

(a)

P0 06506053230200

135

50

100

1375

150

H

0452 04657

X

θ0 θ3θ1θ4 θ

Plowast

P1

(b)

Figure 4)e bifurcation diagrams of x with respect to θ for different ω2 00001 and 00004 respectively)e stable state is represented bythe blue curve and the unstable one corresponds to the red curve )e equilibria are P0(0 0 z0 u0) P1(0 y1 z1 u1) P2(500 0 z2 u2) andPlowast(xlowast ylowast zlowast ulowast) of system (3) (in order to distinguish P0 from P1 we move P1 up a little bit) (a) ω2 00001 (b) ω2 00004

8 Complexity

Next we discuss the effect of three types of immuno-therapy on tumors such as

Case I ACI model (σ1 gt 0 σ3 0)Case II MAI model (σ1 0 σ3 gt 0)Case III combined immunotherapy model(σ1 gt 0 σ3 gt 0)

We set dxdt 0 dydt 0 dzdt 0 dudt

0 anddvdt 0 in (34) and we haveαx(1 minus βx) minus xy 0

ρyz minus θyu minus δ1y + σ1 0


c1y + c2z minus δ3u minus ξuv 0

σ3 minus δ4v 0

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎩

(36)

311 ACI Model When σ1 gt 0 and σ3 0 we put v 0 and(36) becomes





⎧⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎩

(37)

Putting x 0 yields the tumor-free equilibrium namely

E0 x0 y0 z0 u0( 1113857 0 y0σ2δ2

c1δ2y0 + c2σ2

δ2δ31113888 1113889

(38)

where y0 minusB +B2 minus 4AC

radic2A A θc1δ2 gt 0 B δ1δ2δ3

+ θc2σ2 minus ρσ2δ3 andC minus σ1δ2δ3 lt 0 E0 always existsPutting xne 0 and eliminating y z and u in (37) we

have

F(x) C3x3

+ C2x2

+ C1x + C0

C3 θα2β2c1ω2 gt 0

C2 minusθα2βc1ω2 minus αβ θαc1 βδ2 + ω2( 1113857 + δ1δ3ω21113858 1113859lt 0

C1 θα 2αβc1δ2 + αc1ω2 + βc2σ2( 1113857

+ αδ3 δ1ω2 minus β ρσ2 minus δ1δ2( 11138571113858 1113859 minus σ1ω2δ3

C0 σ1δ2δ3 + αδ3 ρσ2 minus δ1δ2( 1113857 minus θα αc1δ2 + c2σ2( 1113857

(39)

Case (a) if F(0)F(μ) C0F(μ)lt 0 (whereμ min (1β) (δ2ω2)1113864 1113865 due to F(+infin)gt 0 andF(minusinfin)lt 0 then (39) has one positive root xlowast in theinterval [0 μ] )erefore system (34) has a uniquecoexistence equilibrium

Elowast



lowast( 1113857

σ2δ2 minus ω2x

lowast1113888

αc1 1 minus βxlowast( 1113857 δ2 minus ω2xlowast( 1113857 + c2σ2

δ3 δ2 minus ω2xlowast( 1113857

1113889

(40)

We show the existence conditions of Elowast in thefollowing

Case (1) C0 gt 0 F(μ)lt 0Since C0 gt 0 we have θlt σ1δ2δ3 + αδ3(ρσ2 minus δ1δ2)α(αc1δ2 + c2σ2)≜ θ5 Since F(μ)lt 0 we have thefollowing two casesCase where 1βlt δ2ω2 ie ω2 lt βδ2 Since μ 1βwe have F(μ) δ3σ1(βδ2 minus ω2)βgt 0 )at is acontradictionCase where 1βgt δ2ω2 ie ω2 gt βδ2 Since μ δ2ω2we have F(μ) ασ2(βδ2 minus ω2)(c2θ minus δ3ρ)ω2 lt 0implying θgt ρδ3c2 θ3 If θ5 ge θ3 ie σ1δ2δ3+αδ3(ρσ2 minus δ1δ2)α(αc1δ2 + c2σ2)ge ρδ3c2 we haveσ1 ge α(αρc1 + δ1c2)c2 there exists ElowastCase (2) C0 lt 0 F(μ)gt 0Since C0 lt 0 we have θgt θ5 Since F(μ)gt 0 we havethe following two casesCase where 1βlt δ2ω2 ie ω2 lt βδ2 Since μ 1βwe have F(μ) δ3σ1(βδ2 minus ω2)βgt 0 there exists ElowastCase where 1βgt δ2ω2 ie ω2 gt βδ2 Since μ δ2ω2we have F(μ) ασ2(βδ2 minus ω2)(c2θ minus δ3ρ)ω2 gt 0implying θlt θ3 If θ3 ge θ5 ie ρδ3c2 ge σ1δ2δ3+αδ3(ρσ2 minus δ1δ2)α(αc1δ2 + c2σ2) we have σ1 le α(αρc1 + δ1c2)c2 there exists Elowast

Case (b) If F(0)F(μ) C0F(μ)gt 0 then (39) can havetwo positive roots xlowast1 and xlowast2 in the interval [0 μ]where xlowast1 and xlowast2 are the roots of (39) )ereforesystem (34) can have two coexistence equilibria

Elowast1 x

lowast1 ylowast1 zlowast1 ulowast1( 1113857 x

lowast1 α 1 minus βx

lowast1( 1113857

σ2δ2 minus ω2x

lowast1

1113888

αc1 1 minus βxlowast1( 1113857 δ2 minus ω2xlowast1( 1113857 + c2σ2

δ3 δ2 minus ω2xlowast1( 1113857

1113889

(41)

Elowast2 x



lowast2( 1113857

σ2δ2 minus ω2x

lowast2

1113888



1113889

(42)

In the following we show the existence condition of Elowast1and Elowast2

Complexity 9

Differentiating (39) we have

Fprime(x) 3C3x2

+ 2C2x + C1 0 (43)

where C3 gt 0 and C2 lt 0 If C1 gt 0 and Δ≜C22 minus 3C1C3 gt 0

then (43) has two positive roots x+1 minusC2 minus

Δ

radic3C3 and

x+2 minusC2 +

Δ

radic3C3 And if C1 lt 0 then (43) has one pos-

itive root x+2 minusC2 +

Δ

radic3C3

Case (1) C0 gt 0 F(μ)gt 0Since C0 gt 0 we have θlt θ5 Since F(μ)gt 0 we have thefollowing two casesCase where 1βlt δ2ω2 ie ω2 lt βδ2 Since μ 1β wehave F(μ) δ3σ1(βδ2 minus ω2)βgt 0 If F(x+

2 )lt 0 thenthere exist Elowast1 and Elowast2 Case where 1βgt δ2ω2 ie ω2 gt βδ2 Since μ δ2ω2we have F(μ) ασ2(βδ2 minus ω2)(c2θ minus δ3ρ)ω2 gt 0 im-plying θlt θ3 )en we have θ ltmin θ3 θ51113864 1113865 IfF(x+

2 )lt 0 then there exist Elowast1 and Elowast2 Case (2) C0 lt 0 F(μ)lt 0

Since C0 lt 0 we have θgt θ5 Since F(μ)gt 0 we have thefollowing two cases

Case where 1βlt δ2ω2 ie ω2 lt βδ2 Since μ 1β wehave F(μ) δ3σ1(βδ2 minus ω2)βgt 0 )at is a contradiction

Case where 1βgt δ2ω2 ieω2 gt βδ2 Since μ δ2ω2 wehave F(μ) ασ2(βδ2 minus ω2)(c2θ minus δ3ρ)ω2 lt 0 implyingθgt θ3 )en we have θgtmax θ3 θ51113864 1113865 If F(x+

1 )gt 0 thenthere exist Elowast1 and Elowast2

Next we show the stability of the above two equilibria E0and Elowast of system (34) with v 0 We linearize the systemand obtain the characteristic equation whose expression isthe same as (13)

Theorem 5 System (34) with v 0 always has one tumor-free equilibrium E0 which is LAS if the inequality θlt θ5holds

Proof At E0 (x0 y0 z0 u0) (0 y0 σ2δ2 c1δ2y0 +

c2σ2δ2δ3) the characteristic equation becomes

λ minus α + y0( 1113857 λ + δ2( 1113857 λ2 +σ1y0

+ δ31113888 1113889λ + θc1y0 +σ1δ3y0

1113888 1113889 0

(44)

and one root of characteristic equation is minusδ2 lt 0 It is easilynoted that as σ1y0 + δ3 gt 0 θc1y1 + σ1δ3y0 gt 0 so thesolutions of λ2 + ((σ1y0) + δ3)λ + θc1y1 + (σ1δ3y0) 0have always negative real parts )erefore E0 is LAS if andonly if α minus y0 2αA + B minus

B2 minus 4AC

radic2Alt 0 that is if and

only if θlt θ5At Elowast (xlowast ylowast zlowast ulowast) characteristic (13) becomes

λ4 + B1λ3

+ B2λ2

+ B3λ + B4 0 (45)

where

B1 αβxlowast

+ δ3 +σ1ylowast

+σ2zlowastgt 0

B2 αβxlowast

+σ2zlowast

1113874 1113875σ1ylowast

+ δ31113888 1113889 + αβxlowastσ2zlowast

+ δ3σ1ylowast

+ θc1ylowast gt 0

B3 θc1ylowast

+ δ3σ1ylowast

1113888 1113889 αβxlowast

+σ2zlowast

1113874 1113875 + αβxlowastσ2zlowast

δ3 +σ1ylowast

1113888 1113889


B4 αβxlowastσ2zlowast

θc1ylowast

+ δ3σ1ylowast

1113888 1113889 + ρδ3 minus θc2( 1113857ω2xlowastylowastzlowast

(46)

By the RouthndashHurwitz criterion the roots of (45) havenegative real parts if and only if B1 gt 0 B2 gt 0 B3 gt 0

B4 gt 0 B1B2 minus B3 gt 0 andB1(B2B3 minus B1B4) minus B23 gt 0

Hence we obtain the sufficient condition for the stabilityof Plowast

(H4) B4 αβxlowastσ2zlowast

θc1ylowast

+ δ3σ1ylowast

1113888 1113889

+ ρδ3 minus θc2( 1113857ω2xlowastylowastzlowast gt 0

(H5) B1B2 minus B3 K1θ + K2 gt 0

(H6) B1 B2B3 minus B1B4( 1113857 minus B23 B3 K1θ + K2( 1113857 minus B

21B4

(47)

whereK1 σ1 + δ3y

lowast( 1113857c1

K2 δ3 +σ1ylowast

+σ2zlowast

1113888 1113889 α2β2xlowast2 + αβxlowast

+σ2zlowast

1113874 1113875σ1ylowast

+ δ31113888 11138891113890

+ αβxlowastσ2zlowast

1113891 + δ3σ1ylowast

σ1ylowast

+ δ31113888 1113889 minus ρω2xlowastylowastzlowast

K3 K1 αβxlowast

+σ2zlowast

1113874 1113875c1ylowast

K4 K2c1ylowast

+ K1σ1δ3ylowast

1113888 1113889 αβxlowast

+σ2zlowast

1113874 11138751113890 1113891

+ K11113890αβxlowastσ2zlowast

δ3 +σ1ylowast

minus αβxlowast

+ δ3 +σ1ylowast

+σ2zlowast

1113888 1113889

2

c1ylowast⎛⎝ ⎞⎠

+ ρ + c2( 1113857ω2xlowastylowastzlowast1113891

K5 K2 αβxlowast

+σ2zlowast

1113874 1113875σ1δ3ylowast

minus αβxlowastσ2zlowast

K2 δ3 +σ1ylowast

1113888 11138891113890

minus αβxlowast

+ δ3 +σ1ylowast

+σ2zlowast

1113888 1113889

2σ1δ3ylowast

⎤⎦ + K2 minus δ3( 1113857ρω2xlowastylowastzlowast

(48)

10 Complexity

Theorem 6 System (34) with v 0 has a coexistenceequilibrium Elowast when ω2 gt βδ2 σ1 ge α(αρc1 + δ1c2)c2

θ3 lt θlt θ5 or ω2 gt βδ2 σ1 le α(αρc1 + δ1c2)c2 θ5 lt θlt θ3 orω2 lt βδ2 θgt θ5 hold Further Elowast is LAS when condition (47)is satisfied

312 MAI Model When σ1 0 σ3 gt 0 (36) becomesαx(1 minus βx) minus xy 0




σ3 minus δ4v 0

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎩

(49)

Equation (49) can be simplified as followsαx(1 minus βx) minus xy 0




⎧⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎩

(50)

where δ31 δ3δ4 + ξσ3δ4)e analysis of (50) is similar to that of (12) and the

equilibria of (34) can be obtained as follows

F0 0 0σ2δ2

c2σ2δ2δ31

σ3δ4

1113888 1113889

0δ31 ρσ2 minus δ1δ2( 1113857 minus θc2σ2

θc1δ2σ2δ2

ρσ2 minus δ1δ2

θδ2σ3δ4

1113888 1113889

F2 1β

0βσ2

βδ2 minus ω2


σ3δ4

1113888 1113889

Flowast

xlowast α 1 minus βx

lowast( 1113857

σ2δ2 minus ω2x

lowast1113888



σ3δ4

1113889

(51)

Here xlowast isin (0 μ) and satisfies the equationa2x

2 + a1x + a0 0 where coefficients are defined as a2

minus θαβc1ω2 lt 0 a1 θαc1(βδ2 + ω2) + δ1δ31ω2 gt 0 and a0

δ31(ρσ2 minus δ1δ2) minus θ(α c1δ2 + c2σ2))e Jacobian matrix of system (34) at any equilibrium

F(x y z u v) is as follows

J(F)

α minus 2αβx minus y minusx 0 0 0

0 ρz minus θu minus δ1 ρy minusθy 0

ω2z 0 ω2x minus δ2 0 0

0 c1 c2 minusδ3 minus ξv ξu

0 0 0 0 minusδ4

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠




0 c1 c2 minusδ31 ξu

0 0 0 0 minusδ4

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠

(52)


|J(F) minus λI| 0 (53)

Substituting δ31 for δ3 in (12) we have

|J(F) minus λI| λ + δ4( 1113857|J(P) minus λI| 0 (54)

)us the stability analysis of F is similar to that of PSubstituting δ31 for δ3 in (20) to get new conditions for

the stability of Flowast(h1) A4 θαβc1 δ2 minus ω2x

lowast( 1113857x

lowastylowast


(h2) A1A2 minus A3 θc1 αβxlowast


( 1113857ylowast


( 1113857xlowast


(h3) A1 A2A3 minus A1A4( 1113857 minus A23 B1θ + B2 gt 0

(55)

where

B1 αβxlowast


( 11138572δ31 + αβx


lowast( 1113857δ2311113872

minus ρω2xlowastylowastzlowast1113857 αβx


lowast( 1113857c1y

lowast

minus αβxlowast


+ δ31( 11138572 αβc1 δ2 minus ω2x

lowast( 1113857x

lowastylowast

(


B2 αβxlowast


( 11138572δ31 + αβx


lowast( 1113857δ2311113872

minus ρω2xlowastylowastzlowast1113857 αβδ31 δ2 minus ω2x

lowast( 1113857x

lowast+ ρω2x


( 1113857

minus αβxlowast


+ δ31( 11138572ρω2δ31x


(56)

We set

Complexity 11

θ70 δ31 ρσ2 minus δ1δ2( 1113857

αc1δ2 + c2σ2

θ71 δ31 ρσ2 minus δ1δ2( 1113857

c2σ2


βc2σ2

θ73 ρδ31c2

(57)

and we obtain the following results

Theorem 7 System (34) always has one tumor-free and ECs-free equilibrium F0 which is unstable

Theorem 8 System (34) has one tumor-free equilibrium F1when ρgt δ1δ2σ2 and θlt θ71 which is LAS if the inequalityθlt θ70 holds

Theorem 9 System (34) has one tumor-dominant equilib-rium F2 when ω2 ltωlowast2 which is LAS if the inequality θgt θ72holds

Theorem 10 System (34) has a unique coexistence equi-librium Flowast when ω2 ltωlowast2 θ0 lt θlt θ2 or ω2 gtωlowast2 θ0 lt θlt θ3hold Further Flowast is LAS when condition (55) is satisfied

313 Combined Immunotherapy Model Whenσ1 gt 0 and σ3 gt 0 (36) can be simplified as follows





⎧⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎩

(58)



N0 x0 y0 z0 u0 v0( 1113857 0 y0σ2δ2

c1δ2y0 + c2σ2

δ2δ31σ3δ4

1113888 1113889

(59)


radic2A with A θc1δ2 gt 0

B δ31(δ1δ2 minus ρσ2) + θc2σ2 andC minusσ1 δ2δ31 lt 0

Nlowast

xlowast ylowast zlowast ulowast vlowast


lowast( 1113857

σ2δ2 minus ω2x

lowast1113888



σ3δ4

1113889

(60)

where xlowast isin (0 μ) and satisfies the equation c3x3 + c2x

2 +

c1x + c0 0 with coefficients

c3 θα2β2c1ω2 gt 0

c2 minusθα2βc1ω2 minus αβ θαc1 βδ2 + ω2( 1113857 + δ1δ31ω21113858 1113859lt 0

c1 θα 2αβc1δ2 + αc1ω2 + βc2σ2( 1113857 + αδ31 δ1ω21113858

minus β ρσ2 minus δ1δ2( 11138571113859 minus σ1ω2δ31

c0 σ1δ2δ31 + αδ31 ρσ2 minus δ1δ2( 1113857 minus θα αc1δ2 + c2σ2( 1113857

(61)

For system (34) at N0 (x0 y0 z0 u0 v0) Jacobianmatrix is expressed as follows

J N0( 1113857

α minus y0 0 0 0 0

0 minusσ1y0

ρy0 minusθy0 0

ω2z0 0 minusδ2 0 0

0 c1 c2 minusδ31 ξu0

0 0 0 0 minusδ4

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠

(62)


J N0( 1113857 minus λI1113868111386811138681113868

1113868111386811138681113868 λ + δ4( 1113857 J E0( 1113857 minus λI1113868111386811138681113868

1113868111386811138681113868 0 (63)

)us the stability analysis of N0 is similar to that of E0Set θ8 αδ31(ρσ2 minus δ1δ2) + σ1δ2δ31α(αc1δ2 + σ2c2) andwe can get the following result

Theorem 11 System (34) always has one tumor-free equi-librium N0 which is LAS if the inequality θlt θ8 holds

For system (34) at Nlowast (xlowast ylowast zlowast ulowast vlowast) Jacobianmatrix is given as follows

J Nlowast

( 1113857

minusαβxlowast minusxlowast 0 0 0

0 minusσ1ylowast

ρylowast minusθylowast 0

ω2zlowast 0 ω2x

lowast minus δ2 0 0

0 c1 c2 minusδ31 ξulowast

0 0 0 0 minusδ4

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠

(64)

9e corresponding characteristic equation is

J Nlowast

( 1113857 minus λI1113868111386811138681113868

1113868111386811138681113868 λ + δ4( 1113857 J Elowast

( 1113857 minus λI1113868111386811138681113868

1113868111386811138681113868 0 (65)

9us the stability analysis of Nlowast is similar to that of ElowastSet θ9 ρδ31c2 and substitute δ31 for δ3 in (47) to get newconditions

12 Complexity

(h4) B4 αβxlowastσ2zlowast

θc1ylowast

+ δ31σ1ylowast

1113888 1113889


(h5) B1B2 minus B3 k1θ + k2 gt 0

(h6) B1 B2B3 minus B1B4( 1113857 minus B23 B3 k1θ + k2( 1113857 minus B

21B4

k3θ2

+ k4θ + k5 gt 0

(66)

wherek1 σ1 + δ31y

lowast( 1113857c1

k2 δ31 +σ1ylowast

+σ2zlowast


+σ2zlowast

1113874 1113875σ1ylowast

+ δ311113888 11138891113890



σ1ylowast


k3 k1 αβxlowast

+σ2zlowast

1113874 1113875c1ylowast

k4 k2c1ylowast

+ k1σ1δ31

ylowast1113888 1113889 αβx

lowast+σ2zlowast

1113874 11138751113890 1113891

+ k1 αβxlowastσ2zlowast

δ31 +σ1ylowast

minus αβxlowast

+ δ31 +σ1ylowast

+σ2zlowast

1113888 1113889

2

c1ylowast⎛⎝ ⎞⎠⎡⎢⎢⎣

+ ρ + c2( 1113857ω2xlowastylowastzlowast1113859 k5 k2 αβx

lowast+σ2zlowast

1113874 1113875σ1δ31

ylowast


k2 δ31 +σ1ylowast

1113888 1113889 minus αβxlowast

+ δ31 +σ1ylowast

+σ2zlowast

1113888 1113889

2σ1δ31ylowast

⎡⎣ ⎤⎦

+ k2 minus δ31( 1113857ρω2xlowastylowastzlowast

(67)

9erefore we can obtain the following result

Theorem 12 System (34) has a coexistence equilibrium Nlowast

when ω2gtβδ2σ1geα(αρc1 +δ1c2)c2θ9ltθltθ8 or ω2gtβδ2σ1leα(αρc1 +δ1c2)c2θ8ltθltθ9 or ω2ltβδ2 θgtθ8 holdFurther Nlowast is LAS when condition (66) is satisfied

32 Numerical Simulations

321 ACI Model (σ1 gt 0 σ3 0) We conduct numericalsimulations of the ACI model We choose ω2 00004 andthe other parameter values are given in (22) We find thestability region of Elowast (xlowast ylowast zlowast ulowast) as shown inFigure 5(a) Elowast is stable in region I and unstable in region II

Case (a) we choose a point G1 (σ1 θ) (04 05) inthe region I then system (34) has one coexistenceequilibrium

Elowast

(11560 1598 7543 6993) (68)

)e eigenvalues of characteristic equation of (47) are


(69)

so Elowast is stable as shown in Figure 6(b)Case (b) we choose a point G2 (σ1 θ) (04 055)

in the region II then system (34) has one coexistenceequilibrium

Elowast

(69131 1409 13895 11962) (70)



(71)

so Elowast is unstable as shown in Figures 6(c) and 6(d)

According to the stability condition of tumor-freeequilibrium E0 in )eorem 5 ACI curve can be ob-tained as

θ αδ3 ρσ2 minus δ1δ2( 1113857 + σ1δ2δ3

α αc1δ2 + c2σ2( 1113857 009391σ1 + 0452 (72)

We find the stability region of E0 (0 y0 z0 u0)

shown in Figure 5(b) E0 is stable in the region III andunstable in the region IV With the increase of σ1region III gradually increases while region IV grad-ually decreasesCase (c) we choose a point G3 (σ1 θ) (04 048)

in the region III then system (34) has a tumor-freeequilibrium

E0 (0 1782 6909 6596) (73)

)e eigenvalues of Jacobian matrix of E0 are


(74)

so E0 is stable as shown in Figure 6(a)Case (d) we choose a point G4 (σ1 θ) (04 053)

in the region IV then system (34) has a tumor-freeequilibrium

E0 (0 1141 6909 6212) (75)


0494 minus 0055 minus 03 + 0297i and minus 03 minus 0297i (76)

so E0 is unstableWe choose ACI parameter σ1 04 to study the rela-

tionship between the number of TCs x and the parameter θ(see Figure 5(c)) By some calculations we have θ3 ρδ3c2

Complexity 13

06 θ5 αδ3(ρσ2 minusδ1δ2) +σ1δ2δ3α(αc1 δ2 + c2σ2) 0489And Hopf bifurcation point appears at θlowastACI 05354 Wecan obtain the following result

Proposition 5 When 0lt θlt θ5 E0 exists and is LAS and Elowast

is nonexistent When θ5 lt θlt θlowastACI E0 exists and is unstable

and Elowast exists and is LAS When θlowastACI lt θlt θ3 E0 exists and isunstable and Elowast exists and is unstable When θgt θ3 E0 existsand is unstable and Elowast is nonexistent

Next we choose different ACI parametersσ1 0 01 03 05 07 09 to study the relationship be-tween the number of TCs x and the parameter θ (seeFigure 5(d))

By comparing the curves in Figure 5(d) we see that as σ1increases gradually the stable region of tumor-free equilib-rium E0 of system (34) gradually increases (the intersectionpoint of the curve and the x-coordinate gradually moves to theright) and the stability region of equilibrium Elowast in system (34)gradually increases (the blue curve gradually moves upward)9is indicates that increasing the injection volume of adoptiveT cells can not only delay the inhibitory effect of Tregs ontumor immune response but also help the immune system to

remove more TCs It also helps the immune system to controlmore TCs keeping them at a stable state

322 MAI Model (σ1 0 σ3 gt 0) We conduct numericalsimulations of the MAI model We choose ω2 00004 ξ

005 and δ4 025 and the other parameter values are givenin (22) We find the stability region of Flowast (xlowast ylowast zlowast ulowast)as shown in Figure 7(a) Flowast is stable in region I and unstablein region II

Case (a) we choose a point M1 (σ3 θ) (04 06)

in region I then system (34) has one coexistenceequilibrium

Flowast

(2413 1628 7032 5002 16) (77)

)e eigenvalues of Jacobian matrix of Flowast are

minus025 minus0146 + 0336i minus0146 minus0336i minus00499



StableI

IIUnstable

0

05

1

15

12

04 0465 055 06 07 08 09 1θ

σ1

(a)

IV

III

04

045205

06

07

08

09

θ = 00939σ1 + 0452Unstable

Stable

0 02 04 06 08 1σ1

θ

(b)

0 0489 05354 06 07 08 090

51

100

1375150

H

θ5 θ3 θ

E0

ElowastX

(c)

θ

X

03 04 05 06 07 08 09 1

H

H

H

H

H

0

50

100

1501375

H

(d)

Figure 5 (a) Stability region of Elowast and Hopf bifurcation curve in θ minus σ1 parameter plane Elowast is stable in region I and unstable in region II(b) ACI curve E0 is stable in region III and unstable in region IV (c))e bifurcation diagram of x with respect to θ for fixed σ1 04 Hopfbifurcation point appears at θlowastACI 05354 (d))e bifurcation diagram of x with respect to θ for different σ1 0 01 03 05 07 09 (theblue curves represent the stable steady states of system (34) while the red curves represent the unstable steady states)

14 Complexity

so Flowast is stable as shown in Figure 8(b)Case (b) we choose a pointM2 (σ3 θ) (04 0635) in region II then system(34) has one coexistence equilibrium

Flowast

(27345 1546 8624 593 16) (79)


minus04334 minus025 minus00644 00172

+ 0429i and 00172 minus 0429i(80)

so Flowast is unstable as shown in Figures 8(c) and 8(d)According to the stability condition of tumor-freeequilibrium F1 in )eorem 7 MAI curve can be ob-tained as

θ δ3δ4 + ξσ3( 1113857 ρσ2 minus δ1δ2( 1113857 αc1δ2 + σ2c2( 1113857

δ4 03616 σ3 + 0452(81)

We find the stability region of F1 as shown inFigure 7(b) F0 is stable in region III and unstable inregion IV With the increase of σ3 region IIIgradually increases while region IV graduallydecreasesCase (c) we choose a point M3 (σ3 θ) (04 05)

in region III then system (34) has a tumor-freeequilibrium

F1 (0 2556 6909 5349 16) (82)

)e eigenvalues of Jacobian matrix of F1 are

0

1

2

3

4

5

6

7

8

Coun

t

0 100 200 300 400 500 600

tx

y

z

u

(a)

Coun

t

100

101

102

0 100 200 300 400 500 600

tx

y

z

u

(b)

Coun

t

10ndash1

100

101

102

103

104

0 100

x

y

z

u

200 300 400 500 600t

(c)

1105

11

minus04minus02

002

1

15

2

25

3

uy

x

(d)

Figure 6 Choose σ1 04 and the other parameter values are given in (22) (a)When θ 048 E0(0 1782 6909 6596) is LAS (b)Whenθ 050 Elowast(11560 1598 7543 6993) is LAS (c) When θ 055 Elowast(69131 1409 13895 11962) is unstable (d) When θ 055 the3D phase portrait depicts TCs ECs and Tregs

Complexity 15

minus092 minus025 minus0055 minus0164 + 0428i and minus 0164 minus 0428i

(83)

so F1 is stable as shown in Figure 8(a)Case (d) we choose a pointM4 (σ3 θ) (04 06) inregion IV then system (34) has a tumor-free equilibrium

F1 (0 1575 6909 4903 16) (84)

)e eigenvalues of Jacobian matrix of F1 are0061 minus 025 minus 0055 minus 0164 + 0338i and minus 0164 minus 0338i

(85)

so F1 is unstableWe choose MAI parameter σ3 04 to study the rela-

tionship between the number of TCs x and the parameter θ(see Figure 7(c)) By some calculations we haveθ70 δ31(ρσ2 minus δ1δ2)αc1δ2 + c2σ2 0597 θ71 δ31(ρσ2minusδ1δ2)c2σ2 0703 and θ73 ρδ31c2 0792 And Hopfbifurcation point appears at θlowastMAI 0627 We can obtainthe following result

Proposition 6 When 0lt θlt θ70 F0 exists and is unstable F1exists and is LAS and Flowast exists and is LAS When

θ70 lt θlt θlowastMAI F0 exists and is unstable F1 exists and is un-

stable and Flowast exists and is unstable When θ lowastMAI lt θ lt θ71 F0exists and is unstableF1 exists and is unstable andFlowast exists andis unstable When θ71 lt θlt θ73 F0 exists and is unstable F1 isnonexistent and Flowast exists and is unstable When θgt θ3 F0exists and is unstable F1 is nonexistent and Flowast is nonexistent

Next we choose different MAI parametersσ3 0 01 03 05 07 09 to study the relationship betweenthe number of TCs x and the parameter θ (see Figure 7(d))

By comparing the curves in Figure 7(d) we can find thatwith the gradual increase of σ3 the stability region of thetumour-free equilibrium F1 of system (34) gradually increases(the intersection point of curves and x-coordinate graduallymoves to the right) 9is shows that the increase of antibodyinjection quantity can help to slow down Tregs inhibition oftumor immune responses 9e stability region of equilibriumFlowast of system (34) gradually increases (the blue curve graduallymoves upward) which means that increasing the amount ofantibody injected can help the immune system to control moreTCs By comparing Figures 5 and 7 we find that the effect ofMAI is better than that of AIC in delaying the inhibitory effectof Tregs on tumor immune response (at the same injectiondose the intersection point of curves and x-coordinate inFigure 7 moves to the right more widely than that in Figure 5)AIC is more effective than MAI in controlling TCs (the blue

002040608

112141618

2

StableUnstable


I

II

04 05 06 07 08 09 10465θ

σ3

(a)

04

045205

06

07

08

09

Stable

UnstableIV

θ

σ3

0 02 04 06 08 1

III

θ = 03616σ3 + 0452

(b)

0

22

50

100

1375150

H

0 05 0597 0703 0792 09F0

F1

X

θ70 θ71 θ73 θ

Flowast

(c)

H H H H H

H

0

50

100

1501375

X

θ03 04 05 06 07 08 09 1

(d)

Figure 7 (a) Stability region of Flowast and Hopf bifurcation curve in θ minus σ3 parameter plane Flowast is stable in region I and unstable in region II(b) MAI curve F1 is stable in region III and unstable in region IV (c))e bifurcation diagram of x with respect to θ for fixed σ3 04 Hopfbifurcation point appears at θlowastMAI 0627 (d))e bifurcation diagrams of x with respect to θ for different σ3 0 01 03 05 07 09 (theblue curves represent the stable steady states of system (34) while the red curves represent the unstable steady states)

16 Complexity

curve in Figure 7 moves up more than the blue curves inFigure 5 at the same injection dose)

323 Combined Immunotherapy Model (σ1 gt 0 σ3 gt 0)We conduct numerical simulations of the combined immu-notherapy model We choose ω2 00004

ξ 005 and δ4 025 and the other parameter values aregiven in (22) We find the stability region of Nlowast (xlowast ylowast

zlowast ulowast vlowast) as shown in Figure 9(a) When σ3 increasesgradually the stability region of Nlowast also increases gradually

Case (a) we choose a point L1 (θ σ1) (07 04) onthe left of the curve σ3 05 then system (34) has onecoexistence equilibrium

Nlowast

(1156 159 754 499 2) (86)


minus006 minus 0176 minus 025 minus 022 minus 037i and minus 022 + 037i

(87)

so Nlowast is stable as shown in Figure 10(b)Case (b) we choose a point L2 (θ σ1) (078 04)

on the right of the curve σ3 05 then system (34) hasone coexistence equilibrium

Nlowast

(77887 1381 15936 9698 2) (88)

ndash10

2

4

6

8

10

Coun

t

0 100 200 300 400 500 600

tx

y

z

v

u

(a)

Coun

t

0 100 200 300 400 500 600

tx

y

z

v

u

10ndash4

100

10ndash2

104

102

(b)

Coun

t

0 100 200 300 400 500 600

tx

y

z

v

u

10ndash1

101

100

102

(c)

076078

08minus01

001

02

05

1

15

2

25

uy

x

(d)

Figure 8 Choose σ3 04 and the other parameter values are given in (22) (a) When θ 05 F1(0 2556 6909 5349 16) is LAS(b) When θ 06 Flowast(2413 1628 7032 5002 16) is LAS (c) When θ 0635 Flowast(27345 1546 8624 593 16) is unstable (d) Whenθ 0635 the 3D phase portrait depicts TCs ECs and Treg

Complexity 17


minus 0906238 minus 00248 minus 025 000641minus 066i and 000641 + 066i

(89)

so Nlowast is unstable as shown in Figures 10(c) and 10(d)According to the stability condition of tumor-freeequilibrium N0 in )eorem 11 combined immuno-therapy surface can be obtained as


α αc1δ2 + σ2c2( 1113857

0055σ1 + 02647( 1113857 1366σ3 + 1707( 1113857

(90)

)erefore the stability region of tumor-free equilib-rium N0 of system (34) is obtained (see Figure 9(b))Case (c) we choose a point L3 (σ1 σ2 θ) (05 05

04) below surface (90) then system (34) has a tumor-free equilibrium

N0 (0 83 6909 7505 2) (91)

)e eigenvalues of Jacobian matrix of (64) areminus6664 minus 0055 minus 025 minus 0204 + 069i and minus 0204 minus 069i

(92)

so N0 is stable as shown in Figure 10(a)Case (d) we choose a point L4 (σ1 σ2 θ)

(05 05 08) above surface (90) then system (34) hasa tumor-free equilibrium

N0 (0 0886 6909 432 2) (93)


075 minus 0055 minus 025 minus 0453 + 0311i and

minus 0453 minus 0311i(94)

so N0 is unstableWe choose combined immunotherapy parameter σ1

04 and σ3 04 to study the relationship between thenumber of TCs x and the parameter θ (see Figure 9(c))By some calculations we have θ8 αδ31(ρσ2 minus δ1δ2) +

04 05 06 07 08 09 10

02040608

112141618

2


σ3 = 01 σ3 = 05 σ3 = 08

θ

σ1

(a)

005

1

002040608104

05

06

07

08

09

1

11

Unstable

Stable

θ

σ3

σ1

(b)

090727 0791064600

50

709

100

1375150

H

N0

X

θθ8 θ9

Nlowast

(c)

X

0 01 02 03 04 05 06 07 08 09 1

H

H

H

H

(σ1 σ3) = (07 07)

(σ1 σ3) = (04 04)

(σ1 σ3) = (0 04)

(σ1 σ3) = (04 0)

0

50

100

1501375

θ

(d)

Figure 9 (a) Choose different σ3 01 05 08 Stability region of Nlowast and Hopf bifurcation curve in θ minus σ1 parameter plane Nlowast is stable tothe left of the curve and unstable to the right of the curve (b) Combined immunotherapy surface N0 is stable below the surface and unstableabove the surface (c) When σ1 04 and σ3 04 the bifurcation diagram of x with respect to θ for system (34) Hopf bifurcation pointappears at θlowastCI 0727 (d) Choose different (σ1 σ3) (04 0) (0 04) (04 04) (07 07) )e bifurcation diagram of x with respect toθ (the stable steady state is represented by the blue curve and the unstable one corresponds to the red curve)

18 Complexity

σ1δ2δ31α(αc1δ2 + σ2c2) 0646 and θ9 ρδ31 c2 0791And Hopf bifurcation point appears at θlowastCI 0727 We canobtain the following result

Proposition 7 When 0lt θlt θ8 N0 exists and is LAS andNlowast is nonexistent When θ8 lt θlt θ

lowastCI N0 exists and is un-

stable and Nlowast exists and is LAS When θlowastCI lt θlt θ9 N0 existsand is unstable and Nlowast exists and is unstable When θ gt θ9N0 exists and is unstable and Nlowast is nonexistent

For combined immunotherapy by comparing the curvesin Figure 9(d) we know that with the increase of σ1 and σ3the stability region of the tumour-free equilibrium of N0 ofsystem (34) gradually increases (the intersection point ofcurves and x-coordinate gradually moves to the right) andthe stability region of equilibrium Nlowast of system (34)gradually increases (the blue curve gradually moves up-ward) By comparing Figure 9 with Figures 5 and 7 it can be

seen that combined immunotherapy has better effects ondelaying the inhibitory effect of Tregs on tumor immuneresponse and helps the immune system to control more TCsthan ACI or MAI

4 Discussion and Conclusion

Tregs-mediated tumor immune escape is one of the coremechanisms of tumor immune regulation And Tregs havebeen found to mediate tumor evasion and immune escape inmany different solid tumors [46] )e study on Tregs has avery high research value and application prospect in theimmunotherapy of tumors If the activity of Tregs is con-trolled or blocked during the tumor immune response or abarrier is set to prevent Tregs from migrating into the tumormicroenvironment then the effect of tumor immunotherapycan be improved [47]

xyz

vu

0

5

10

15

20

Coun

t

20 40 60 80 1000t

(a)

0 100 200 300 400 500 600t

xyz

vu

Coun

t

100

102

101

(b)

xyz

vu

Coun

t

100

103

102

101

100 200 300 400 500 6000t

(c)

0975098

0985099

0

0204

17

18

19

2

21

22

uy

x

(d)

Figure 10 (a) When σ1 σ3 05 θ 04 N0(0 83 6909 7505 2) is LAS (b) When σ1 04 σ3 05 θ 07Nlowast(1156 159 754 499 2) is LAS (c) When σ1 04 σ3 05 θ 078 Nlowast(77887 1381 15936 9698 2) is unstable (d) Whenσ1 04 σ3 05 θ 078 the 3D phase portrait depicts TCs ECs and Tregs

Complexity 19

First we developed a mathematical model to study theinhibitory role of Tregs in the tumor immune system For thelower recognition of tumor antigens by the immune systemthe stronger the inhibition effect of Tregs on ECs TCs caneasily escape the control of the immune system (seeFigure 4(a)) When the immune system is highly sensitive totumor antigens the immune system activates ECs thestronger the inhibition effect of Tregs on ECs the morecomplicated interactions between TCs and immune cells(see Figure 4(b))

Second we incorporated the previousmathematicalmodelwith three types of immunotherapy to obtain ACImodel MAImodel and combined both ACI andMAI model)rough thetheoretical analysis and numerical simulations we found thatACI can control more TCs but have no obvious effects onreducing the inhibitory effect of Tregs on ECs (see Figure 5)MAI can effectively reduce the inhibitory effect of Tregs onECs but cannot control more TCs (see Figure 7) Howevercombination immunotherapy with ACI and MAI is moreeffective than single immunotherapy It can not only signif-icantly reduce the inhibitory effect of Tregs on ECs but alsohelp the immune system to kill TCs to the maximum extent(see Figure 9) )erefore we recommend the use of combinedimmunotherapy in the treatment of tumors Besides clinicaltrials are needed to further evaluate the safety and efficacy ofcombined immunotherapy

)is paper focused on the general process of tumorimmune response with negative feedback Using themathematical model it is possible to simulate the state oftumors in the immune system at different inhibition states)e results of the study can contribute to the understandingof tumor immunity at the same time it also provides newideas for the treatment of tumors However due to thecomplexity and heterogeneity of tumor microenvironmentthere is still a certain gap between the mathematical modeland the description of the real interactions between thetumor and immune system )erefore specific tumormicroenvironment and heterogeneous tumoral pop-ulations should be considered in practical application tomake the model more realistic [34] In addition to thebilinear incidence model considered in this paper non-bilinear model with saturation incidence should beemployed in the further study [35] Besides other im-portant factors such as immune activation delays[36ndash38 48] stochastic effects [39 40 49] and impulsiveperturbations [41 42 50] can be considered in the mod-elling of the tumor immune system )e tumor immuneresponse dynamics in vivo is very complex and not wellunderstood primarily because the measurements of thenecessary parameters are difficult in vivo [43]

Data Availability

No data were used to support this study

Conflicts of Interest

)e authors declare that there are no conflicts of interestregarding the publication of this paper

Acknowledgments

)is research was supported by the National Natural ScienceFoundation of China (nos 11871235 11871238 and11901225) Natural Science Foundation of Hubei Province(no 2019CFB189) Fundamental Research Funds for theCentral Universities (no CCNU18XJ041) Aoyama GakuinUniversity research grant ldquoOngoing Research Supportrdquo andJapan Society for the Promotion of Science ldquoGrand-in-Aid20K03755rdquo

References

[1] P Boyle ldquoMeasuring progress against cancer in Europe hasthe 15 decline targeted for 2000 come aboutrdquo Annals ofOncology vol 14 pp 1312ndash1325 2003

[2] T Boon and P Van der Bruggen ldquoHuman tumor antigensrecognized by T lymphocytesrdquo 9e Journal of ExperimentalMedicine vol 183 no 3 pp 725ndash729 1996

[3] K E De Visser A Eichten and L M Coussens ldquoParadoxicalroles of the immune system during cancer developmentrdquoNature Reviews Cancer vol 6 no 1 pp 24ndash37 2006

[4] Y Dong R Miyazaki R Miyazaki and Y TakeuchildquoMathematical modeling on helper Tcells in a tumor immunesystemrdquo Discrete amp Continuous Dynamical Systems-B vol 19no 1 pp 55ndash72 2014

[5] A Talkington C Dantoin and R Durrett ldquoOrdinary dif-ferential equation models for adoptive immunotherapyrdquoBulletin of Mathematical Biology vol 9 pp 1ndash25 2017

[6] Y-P Lai C-J Jeng and S-C Chen ldquo)e roles of CD4+Tcellsin tumor immunityrdquo ISRN Immunology vol 2011 pp 1ndash62011

[7] T Maj W Wang J Crespo et al ldquoOxidative stress controlsregulatory Tcell apoptosis and suppressor activity and PD-L1-blockade resistance in tumorrdquo Nature Immunology vol 18no 12 pp 1332ndash1341 2017

[8] S Wilson and D Levy ldquoA mathematical model of the en-hancement of tumor vaccine efficacy by immunotherapyrdquo Bul-letin of Mathematical Biology vol 74 no 7 pp 1485ndash1500 2012

[9] A Albert M Freedman and A S Perelson ldquoTumors and theimmune system the effects of a tumor growth modulatorrdquoMathematical Biosciences vol 50 no 1-2 pp 25ndash58 1980

[10] A drsquoOnofrio ldquoA general framework for modeling tumor-immune system competition and immunotherapy mathe-matical analysis and biomedical inferencesrdquo Physica Dvol 208 pp 220ndash235 2005

[11] D Kirschner and J C Panetta ldquoModeling immunotherapy ofthe tumor - immune interactionrdquo Journal of MathematicalBiology vol 37 no 3 pp 235ndash252 1998

[12] S Amarnath C W Mangus J C M Wang et al ldquo)e PDL1-PD1 Axis converts human TH1 cells into regulatory T cellsrdquoScience Translational Medicine vol 3 no 111 pp 111ndash120 2011

[13] L Spain S Diem and J Larkin ldquoManagement of toxicities ofimmune checkpoint inhibitorsrdquo Cancer Treatment Reviewsvol 44 pp 51ndash60 2016

[14] K S Peggs S A Quezada A J Korman and J P AllisonldquoPrinciples and use of anti-CTLA4 antibody in human cancerimmunotherapyrdquo Current Opinion in Immunology vol 18no 2 pp 206ndash213 2006

[15] A A Hurwitz B A Foster E D Kwon et al ldquoCombinationimmunotherapy of primary prostate cancer in a transgenicmouse model using CTLA-4 blockaderdquo Cancer Researchvol 60 no 9 pp 2444ndash2448 2000

20 Complexity

[16] J G EgenM S Kuhns and J P Allison ldquoCTLA-4 new insightsinto its biological function and use in tumor immunotherapyrdquoNature Immunology vol 3 no 7 pp 611ndash618 2002

[17] T Takahashi T Tagami S Yamazaki et al ldquoImmunologicself-tolerance maintained by Cd25+Cd4+Regulatory T cellsconstitutively expressing cytotoxic T lymphocyte-associatedantigen 4rdquo 9e Journal of Experimental Medicine vol 192no 2 pp 303ndash310 2000

[18] Y-F Ma C Chen D Li et al ldquoTargeting of interleukin (IL)-17A inhibits PDL1 expression in tumor cells and inducesanticancer immunity in an estrogen receptor-negative murinemodel of breast cancerrdquo Oncotarget vol 8 no 5 pp 7614ndash7624 2016

[19] K R Jordan V Borges and M D Mccarter ldquoAbstract 1671immunosuppressive myeloid-derived suppressor cellsexpressing PDL1 are increased in human melanoma tumortissuerdquo Cancer Research vol 74 p 1671 2014

[20] E Baldelli V Calvert and K A Hodge ldquoAbstract 5656quantitative measurement of PDL1 expression across tumortypes using laser capture microdissection and reverse phaseprotein microarrayrdquo Cancer Research vol 77 p 5656 2017

[21] E E West H-T Jin A-U Rasheed et al ldquoPD-L1 blockadesynergizes with IL-2 therapy in reinvigorating exhaustedT cellsrdquo Journal of Clinical Investigation vol 123 no 6pp 2604ndash2615 2013

[22] L M Francisco P T Sage and A H Sharpe ldquo)e PD-1pathway in tolerance and autoimmunityrdquo ImmunologicalReviews vol 236 no 1 pp 219ndash242 2010

[23] B Brady ldquoDramatic survival benefit with nivolumab inmelanomardquo Cancer Research vol 6 p OF7 2016

[24] D Kazandjian D L Suzman G Blumenthal et al ldquoFDAapproval summary nivolumab for the treatment of metastaticnon-small cell lung cancer with progression on or afterplatinum-based chemotherapyrdquo9e Oncologist vol 21 no 5pp 634ndash642 2016

[25] C Voena and R Chiarle ldquoAdvances in cancer immunologyand caner immuntherapyrdquo Cancer Research vol 21pp 125ndash133 2016

[26] N V Stepanova ldquoCourse of the immune reaction during thedevelopment of a malignant tumourrdquo Cancer Researchvol 24 pp 917ndash923 1979

[27] V A Kuznetsov I A Makalkin M A Taylor andA S Perelson ldquoNonlinear dynamics of immunogenic tumorsparameter estimation and global bifurcation analysisrdquo Bul-letin ofMathematical Biology vol 56 no 2 pp 295ndash321 1994

[28] O Sotolongo-Costa L Morales Molina D Rodrıguez PerezJ C Antoranz and M Chacon Reyes ldquoBehavior of tumorsunder nonstationary therapyrdquo Physica D Nonlinear Phe-nomena vol 178 no 3-4 pp 242ndash253 2003

[29] M Galach ldquoDynamics of the tumor-immune system com-petition-the effect of time delayrdquo International Journal ofApplied Mathematics and Computer Science vol 13pp 395ndash406 2003

[30] L G de Pillis W Gu and A E Radunskaya ldquoMixed im-munotherapy and chemotherapy of tumors modeling ap-plications and biological interpretationsrdquo Journal of9eoretical Biology vol 238 no 4 pp 841ndash862 2006

[31] S Bunimovich-Mendrazitsky H Byrne and L StoneldquoMathematical model of pulsed immunotherapy for super-ficial bladder cancerrdquo Bulletin of Mathematical Biologyvol 70 no 7 pp 2055ndash2076 2008

[32] A Radunskaya R Kim and T Woods ldquoMathematicalmodeling of tumor immune interactions a closer look at the

role of a PD-L1 inhibitor in cancer immunotherapyrdquo Spora AJournal of Biomathematics vol 4 pp 25ndash41 2018

[33] Y Shu J Huang Y Dong and Y Takeuchi ldquoMathematicalmodeling and bifurcation analysis of pro- and anti-tumormacrophagesrdquo Applied Mathematical Modelling vol 88pp 758ndash773 2020

[34] E Piretto M Delitala and M Ferraro ldquoHow combinationtherapies shape drug resistance in heterogeneous tumoralpopulationsrdquo Letters in Biomathematics vol 5 no 2pp S160ndashS177 2018

[35] H Dritschel S Waters A Roller and H Byrne ldquoA math-ematical model of cytotoxic and helper T cell interactions in atumourmicroenvironmentrdquo Letters in Biomathematics vol 5no 2 pp S36ndashS68 2018

[36] Y Dong G Huang R Miyazaki and Y Takeuchi ldquoDynamicsin a tumor immune system with time delaysrdquo AppliedMathematics and Computation vol 252 pp 99ndash113 2015

[37] M Yu Y Dong and Y Takeuchi ldquoDual role of delay effects ina tumour-immune systemrdquo Journal of Biological Dynamicsvol 11 no 2 pp 334ndash347 2017

[38] M Yu G Huang Y Dong and Y Takeuchi ldquoComplicateddynamics of tumor-immune system interaction model withdistributed time delayrdquo Discrete amp Continuous DynamicalSystems-B vol 25 no 7 pp 2391ndash2406 2020

[39] G Caravagna A drsquoOnofrio P Milazzo and R BarbutildquoTumour suppression by immune system through stochasticoscillationsrdquo Journal of 9eoretical Biology vol 265 no 3pp 336ndash345 2010

[40] J T George and H Levine ldquoStochastic modeling of tumorprogression and immune evasionrdquo Journal of 9eoreticalBiology vol 458 pp 148ndash155 2018

[41] H-P Ren Y Yang M S Baptista and C Grebogi ldquoTumourchemotherapy strategy based on impulse control theoryrdquoPhilosophical Transactions of the Royal Society A Mathe-matical Physical and Engineering Sciences vol 375 no 2088p 20160221 2017

[42] Y Deng and M Liu ldquoAnalysis of a stochastic tumor-immunemodel with regime switching and impulsive perturbationsrdquoApplied Mathematical Modelling vol 78 pp 482ndash504 2020

[43] G E Mahlbacher K C Reihmer and H B FrieboesldquoMathematical modeling of tumor-immune cell interactionsrdquoJournal of 9eoretical Biology vol 469 pp 47ndash60 2019

[44] J K Hale 9eory of Functional Differential EquationsSpringer New York NY USA 1977

[45] A Dhooge W Govaerts and Y A Kuznetsov ldquoMatcontrdquoACM SIGSAM Bulletin vol 38 no 1 pp 21-22 2004

[46] A Facciabene G T Motz and G Coukos ldquoT-regulatory cellskey players in tumor immune escape and angiogenesisrdquoCancer Research vol 72 no 9 pp 2162ndash2171 2012

[47] K Shitara and H Nishikawa ldquoRegulatory T cells a potentialtarget in cancer immunotherapyrdquo Annals of the New YorkAcademy of Sciences vol 1417 no 1 pp 104ndash115 2018

[48] T Zhang J Wang Y Li Z Jiang and X Han ldquoDynamicsanalysis of a delayed virus model with two different trans-mission methods and treatmentsrdquo Advances in DifferenceEquations vol 1 p 2020 2020

[49] H Zhang and T Zhang ldquo)e stationary distribution of a mi-croorganism flocculation model with stochastic perturbationrdquoApplied Mathematics Letters vol 103 p 106217 2020

[50] T Zhang N Gao N Gao T Wang H Liu and Z JiangldquoGlobal dynamics of a model for treating microorganisms insewage by periodically adding microbial flocculantsrdquo Math-ematical Biosciences and Engineering vol 17 no 1 pp 179ndash201 2020

Complexity 21

Tregs have become an important target in tumor im-munotherapy because of their contribution to tumor im-mune escape Cytotoxic T lymphocyte antigen 4 (CTLA-4) isa marker that is expressed on the surface of activated T cellsand transmits inhibitory signals in the immune response[14ndash16] Blocking CTLA-4 can reduce the inhibitory activityof Tregs and the anti-CTLA-4 humanized monoclonalantibody Ipilimumab and Tremelimumab are used to treatadvanced melanoma and malignant mesothelioma re-spectively [17] Similar to CTLA-4 programmed death re-ceptor 1 (PD-1) can also promote the activation anddevelopment of Tregs [18ndash21] Blocking PD-1 can preventthe development of Tregs and prevent the conversion ofHTCs into Tregs [22] Currently OPDIVO (Nivolumab) ananti-PD-1 monoclonal antibody has been approved by theUS FDA for the treatment of melanoma renal cell carci-noma and non-small cell lung cancer [23ndash25] Establish-ment of a mathematical model to study the immunotherapyon the reduction of Tregs inhibition has both theoretical andpractical significance

In order to describe the mechanisms of hostrsquos ownimmune response to against TCs various types of mathe-matical models have been proposed [26ndash43] )e modellingof the tumor immune system described by ordinary dif-ferential equations (ODEs) has a long history which can betraced back to the classic research of Stepanova in 1980 [26]In 1994 Kuznetsov et al established the famous two-di-mensional ODEs model postulating that tumor growthfollows the Logistic growth pattern )ey evaluated theparameters of the model by fitting experimental data frommice [27] In 2003 Stolongo-Costa et al assumed that TCsfollows the exponential growth pattern and constructed atwo-dimensional ODEs model )ey analyzed the basicproperties of the model and provided conditions for stabilityof the tumor-free equilibrium explaining its epidemiologicalsignificance [28] In 2004 Galach simplified Kuznetsovrsquossystem to account for the effect of immune delay on thetumor immune system [29] In 2014 Dong et al constructeda three-dimensional ODEs model focusing on the effects ofHTCs on the tumor immune system [4]

In 1998 Kirschner and Panetta generalized Kuznet-sovndashTaylor model and illustrated the dynamics betweenTCs ECs and IL-2 )ey firstly introduced ACI into theirmodel which can explain both short-term tumor oscillationsin tumor sizes as well as long-term tumor relapse [11] In2003 in order to study the role of cytokine therapy in theactivation of the immune system Stolongo-Costa et alintroduced cycle therapy term Fcos2ωt and established acycle immunotherapy model )ey obtained some thresh-olds of the frequency and intensity of immunotherapy [28]In 2006 de Pillis et al constructed the six-dimensionalODEs model to investigate the effects of combined che-motherapy and immunotherapy on tumor control )eybriefly analyzed the nature of the model and discussed theoptimal treatment using optimal control theory [30] In2008 Bunimovich-Mendrazitsky et al established a pulseddifferential equation model with Bacillus Calmette-Guerintumor immunotherapy )ey obtained the critical thresholdand pulse frequency of BCG injection dose that could

successfully treat superficial bladder cancer [31] In 2012Wilson and Levy established a mathematical model con-taining Tregs)ey studied the absence of treatment vaccinetreatment anti-TGF treatment and combination vaccineand anti-TGF treatment as well as sensitivity analysis ofsome important parameters [8] In 2018 Radunskaya et alestablished a mathematical model with blood spleen andtumor compartments to study PD-L1 inhibitors in the roleof tumor immunotherapy )e model was used to fit pa-rameters with the experimental data)e results showed thatincreasing the resistance of PD-L1 doses can greatly improvethe clearance rate of tumor [32]

)is paper investigates the role of Tregs in the tumorimmune system )erefore we incorporate the fourthpopulation of Tregs into the previous system in [8] For themathematical simplicity a bilinear term also has been usedto describe the interactions between immune response andtumor To our knowledge HTCs can recognize TCs andpromote the growth of ECs And ECs can provide directprotective immunity by attacking TCs When there are moreHTCs and ECs in order to maintain immune homeostasisthe body will produce corresponding Tregs to suppress ECsand Tregs originating from both HTCs and ECs )en weestablish a four-dimensional ODEs model described asbelow

dT(t)


dE(t)

dt pE(t)H(t) minus qR(t)E(t) minus d1E(t)

dH(t)


dR(t)

dt r1E(t) + r2H(t) minus d3R(t)

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(1)

where T(t) E(t) H(t) and R(t) represent the populationsof TCs ECs HTCs and Tregs respectively )e firstequation describes the rate change of TCs population )etumor follows logistic growth dynamics with growth rate aand the maximum capacity is 1b n represents the loss rateof TCs by ECs interaction)e second equation describes therate change of the ECs population d1 is the mortality rate ofECs p is the activation rate of ECs by HTCs and q is theinhibition rate of Tregs on ECs )e third equation describesthe rate change of the HTCs population s2 is birth rate ofHTCs produced in the bone marrow and HTCs have anatural lifespan of an average 1d2 days k2 is HTCs stim-ulation rate by the presence of identified tumor antigens)efourth equation gives the rate change of the Tregs pop-ulation r1 and r2 are the activation rates of Tregs by ECs andHTCs respectively d3 represents per capita decay rate ofTregs A diagram of the various interactions between thesecell populations is shown in Figure 1

We nondimensionalize model (1) by taking the followingscaling

2 Complexity

t τ



nT0 β bT0

ρ p

n ω2

k2

n θ

qR0

nT0 δ1

d1

nT0 δ2

d2

nT0 δ3

d3

nT0 σ2

s2

nT0H0 c1

r1

nR0 c2

r2

nR0

(2)


dx(t)


dy(t)


dz(t)


du(t)


⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(3)


x(0) x0 ge 0

y(0) y0 ge 0

z(0) z0 ge 0

u(0) u0 ge 0

(4)


2 Model Analysis





x(t) x0e1113938

t


y(t) y0e1113938

t


z(t) z0e1113938

t


+ σ2 1113946t

0e1113938

t


ds ge 0

u(t) u0eminusδ3t

+ 1113946t

0c1y(s) + c2z(s)( 1113857e


(5)







⎧⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎩

(6)


P0 x0 y0 z0 u0( 1113857 0 0σ2δ2

c2σ2δ2δ3

1113888 1113889 (7)


rium namely


θc1δ2σ2δ2

ρσ2 minus δ1δ2

δ2θ1113888 1113889

(8)

TCs

HTCsECs

Tregs

nET

aT (1 ndash bT)

k2TH

pEH

r2Hr1E

qRE

d2Hd1E

d3R

S2

Promotion

Inhibition


Complexity 3



P2 x2 y2 z2 u2( 1113857 1β

0βσ2

βδ2 minus ω2


1113888 1113889

(9)


(6) we have

F(x) Ax2

+ Bx + C 0 (10)



Plowast



lowast( 1113857

σ2δ2 minus ω2x

lowast1113888



lowast( 11138571113889

(11)






















J(P)




0 c1 c2 minusδ3

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠

(12)


det(J(P) minus λI)





1113868111386811138681113868111386811138681113868111386811138681113868111386811138681113868111386811138681113868111386811138681113868

1113868111386811138681113868111386811138681113868111386811138681113868111386811138681113868111386811138681113868111386811138681113868

0 (13)




δ2δ31113888 1113889 0

(14)





θc1δ21113888 1113889

λ + δ2( 1113857 λ2 + δ3λ + θc1y11113872 1113873 0

(15)

4 Complexity







(16)




λ4 + A1λ3

+ A2λ2

+ A3λ + A4 0 (17)

where

A1 αβxlowast


A2 θc1ylowast


( 1113857xlowast

+ αβxlowast


( 1113857δ3 gt 0

A3 θc1 αβxlowast


( 1113857ylowast


( 1113857xlowast





(18)



A1 gt 0 A2 gt 0 A3 gt 0 A4 gt 0 A1A2


(19)





( 1113857ylowast


( 1113857xlowast



(20)

where

B1 αβxlowast


( 11138572δ3 + αβx





lowast( 1113857c1y

lowast

minus αβxlowast


+ δ3( 11138572 αβc1 δ2 minus ω2x

lowast( 1113857x

lowastylowast

(


B2 αβxlowast


( 11138572δ3 + αβx


lowast1113872 1113873δ231113960


lowast( 1113857x

lowast+ ρω2x


( 11138571113859

minus αβxlowast


+ δ3( 11138572ρω2δ3x


(21)




α 1636 β 0002 δ1 03743 σ2 038

δ2 0055 ρ 048 c1 015 c2 02 δ3 025

(22)





1113896 (20)

Complexity 5

θ0 δ3 ρσ2 minus δ1δ2( 1113857

αc1δ2 + c2σ2 0452

θ1 δ3 ρσ2 minus δ1δ2( 1113857

c2σ2 05323

θ3 ρδ3c2

06


βc2σ2 615625ω2 + 05323

(23)





Plowast1 (776449 138195 80448 7265) (24)



minus 001991 + 0378i(25)


Plowast2 (284485 162669 705506 662006) (26)





Plowast3 (72569 139855 8094 7314) (28)



(29)


Plowast4 (26365 154973 854816 776837) (30)



(31)










6 Complexity

0452 05 05323 06 065

Stable

I

IIIIIUnstable

Stable


times10ndash3

ω2

0

02

04

06

08

1

ωlowast2θ2

θ1 θ3 θ


10ndash1

100

101

102

103

Coun

t

x

y

z

u

0 100 200 300 400 500 600

t

(a)

x

y

z

u

0

2

4

6

8

Coun

t

0 100 200 300 400 500 600

t

(b)

10ndash1

100

101

102

103

x

y

z

u

0 100 200 300 400 500 600t

Coun

t

(c)

10ndash2

10ndash1

100

101

102

103

Coun

t

0 100 200 300 400 500 600

tx

y

z

u

(d)

084 086 08809 092 094

minus10

1minus2

minus1

0

1

2

3

uy

x

(e)


Complexity 7

3 Treatment Model


dT(t)


dE(t)


dH(t)


dR(t)


dW(t)

dt s3 minus d4W(t)

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(32)



W(t) W0v(τ) σ1 s1

nE0T0 ξ

mW0

nT0 σ3

s3

nW0T0 δ4

d4

nT0

(33)


dx(t)


dy(t)


dz(t)


du(t)


dv(t)


⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(34)


x(0) x0 ge 0 y(0) y0 ge 0 z(0) z0 ge 0 u(0)

u0 ge 0 v(0) v0 ge 0(35)




0 02 0452 05323 05938 0650

100

200

300

400

500

600

P0

P1

P2

PlowastX

θ0 θ2θ1 θ

(a)

P0 06506053230200

135

50

100

1375

150

H

0452 04657

X

θ0 θ3θ1θ4 θ

Plowast

P1

(b)


8 Complexity








σ3 minus δ4v 0

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎩

(36)






⎧⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎩

(37)


E0 x0 y0 z0 u0( 1113857 0 y0σ2δ2

c1δ2y0 + c2σ2

δ2δ31113888 1113889

(38)




have

F(x) C3x3

+ C2x2

+ C1x + C0



C1 θα 2αβc1δ2 + αc1ω2 + βc2σ2( 1113857



(39)


Elowast



lowast( 1113857

σ2δ2 minus ω2x

lowast1113888



1113889

(40)




Elowast1 x



lowast1( 1113857

σ2δ2 minus ω2x

lowast1

1113888



1113889

(41)

Elowast2 x



lowast2( 1113857

σ2δ2 minus ω2x

lowast2

1113888



1113889

(42)


Complexity 9


Fprime(x) 3C3x2

+ 2C2x + C1 0 (43)



Δ

radic3C3 and

x+2 minusC2 +

Δ



Δ

radic3C3












λ minus α + y0( 1113857 λ + δ2( 1113857 λ2 +σ1y0

+ δ31113888 1113889λ + θc1y0 +σ1δ3y0

1113888 1113889 0

(44)


B2 minus 4AC



λ4 + B1λ3

+ B2λ2

+ B3λ + B4 0 (45)

where

B1 αβxlowast

+ δ3 +σ1ylowast

+σ2zlowastgt 0

B2 αβxlowast

+σ2zlowast

1113874 1113875σ1ylowast


+ δ3σ1ylowast

+ θc1ylowast gt 0

B3 θc1ylowast

+ δ3σ1ylowast

1113888 1113889 αβxlowast

+σ2zlowast


δ3 +σ1ylowast

1113888 1113889



θc1ylowast

+ δ3σ1ylowast


(46)





θc1ylowast

+ δ3σ1ylowast

1113888 1113889




21B4

(47)

whereK1 σ1 + δ3y

lowast( 1113857c1

K2 δ3 +σ1ylowast

+σ2zlowast


+σ2zlowast

1113874 1113875σ1ylowast

+ δ31113888 11138891113890



σ1ylowast


K3 K1 αβxlowast

+σ2zlowast

1113874 1113875c1ylowast

K4 K2c1ylowast

+ K1σ1δ3ylowast

1113888 1113889 αβxlowast

+σ2zlowast

1113874 11138751113890 1113891


δ3 +σ1ylowast

minus αβxlowast

+ δ3 +σ1ylowast

+σ2zlowast

1113888 1113889

2



K5 K2 αβxlowast

+σ2zlowast

1113874 1113875σ1δ3ylowast


K2 δ3 +σ1ylowast

1113888 11138891113890

minus αβxlowast

+ δ3 +σ1ylowast

+σ2zlowast

1113888 1113889

2σ1δ3ylowast


(48)

10 Complexity







σ3 minus δ4v 0

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎩

(49)





⎧⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎩

(50)



F0 0 0σ2δ2

c2σ2δ2δ31

σ3δ4

1113888 1113889


θc1δ2σ2δ2

ρσ2 minus δ1δ2

θδ2σ3δ4

1113888 1113889

F2 1β

0βσ2

βδ2 minus ω2


σ3δ4

1113888 1113889

Flowast


lowast( 1113857

σ2δ2 minus ω2x

lowast1113888



σ3δ4

1113889

(51)






J(F)





0 0 0 0 minusδ4

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠





0 0 0 0 minusδ4

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠

(52)







lowast( 1113857x

lowastylowast




( 1113857ylowast


( 1113857xlowast



(55)

where

B1 αβxlowast


( 11138572δ31 + αβx


lowast( 1113857δ2311113872



lowast( 1113857c1y

lowast

minus αβxlowast


+ δ31( 11138572 αβc1 δ2 minus ω2x

lowast( 1113857x

lowastylowast

(


B2 αβxlowast


( 11138572δ31 + αβx


lowast( 1113857δ2311113872


lowast( 1113857x

lowast+ ρω2x


( 1113857

minus αβxlowast


+ δ31( 11138572ρω2δ31x


(56)

We set

Complexity 11

θ70 δ31 ρσ2 minus δ1δ2( 1113857

αc1δ2 + c2σ2

θ71 δ31 ρσ2 minus δ1δ2( 1113857

c2σ2


βc2σ2

θ73 ρδ31c2

(57)











⎧⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎩

(58)



N0 x0 y0 z0 u0 v0( 1113857 0 y0σ2δ2

c1δ2y0 + c2σ2

δ2δ31σ3δ4

1113888 1113889

(59)




Nlowast



lowast( 1113857

σ2δ2 minus ω2x

lowast1113888



σ3δ4

1113889

(60)


2 +







(61)


J N0( 1113857

α minus y0 0 0 0 0

0 minusσ1y0

ρy0 minusθy0 0



0 0 0 0 minusδ4

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠

(62)


J N0( 1113857 minus λI1113868111386811138681113868

1113868111386811138681113868 λ + δ4( 1113857 J E0( 1113857 minus λI1113868111386811138681113868

1113868111386811138681113868 0 (63)




J Nlowast

( 1113857


0 minusσ1ylowast


ω2zlowast 0 ω2x



0 0 0 0 minusδ4

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠

(64)


J Nlowast

( 1113857 minus λI1113868111386811138681113868

1113868111386811138681113868 λ + δ4( 1113857 J Elowast

( 1113857 minus λI1113868111386811138681113868

1113868111386811138681113868 0 (65)


12 Complexity


θc1ylowast

+ δ31σ1ylowast

1113888 1113889




21B4

k3θ2

+ k4θ + k5 gt 0

(66)

wherek1 σ1 + δ31y

lowast( 1113857c1

k2 δ31 +σ1ylowast

+σ2zlowast


+σ2zlowast

1113874 1113875σ1ylowast

+ δ311113888 11138891113890



σ1ylowast


k3 k1 αβxlowast

+σ2zlowast

1113874 1113875c1ylowast

k4 k2c1ylowast

+ k1σ1δ31

ylowast1113888 1113889 αβx

lowast+σ2zlowast

1113874 11138751113890 1113891


δ31 +σ1ylowast

minus αβxlowast

+ δ31 +σ1ylowast

+σ2zlowast

1113888 1113889

2



lowast+σ2zlowast

1113874 1113875σ1δ31

ylowast


k2 δ31 +σ1ylowast


+ δ31 +σ1ylowast

+σ2zlowast

1113888 1113889

2σ1δ31ylowast

⎡⎣ ⎤⎦


(67)







Elowast

(11560 1598 7543 6993) (68)



(69)



Elowast

(69131 1409 13895 11962) (70)



(71)




α αc1δ2 + c2σ2( 1113857 009391σ1 + 0452 (72)




E0 (0 1782 6909 6596) (73)



(74)



E0 (0 1141 6909 6212) (75)





Complexity 13












Flowast

(2413 1628 7032 5002 16) (77)





StableI

IIUnstable

0

05

1

15

12

04 0465 055 06 07 08 09 1θ

σ1

(a)

IV

III

04

045205

06

07

08

09

θ = 00939σ1 + 0452Unstable

Stable

0 02 04 06 08 1σ1

θ

(b)

0 0489 05354 06 07 08 090

51

100

1375150

H

θ5 θ3 θ

E0

ElowastX

(c)

θ

X

03 04 05 06 07 08 09 1

H

H

H

H

H

0

50

100

1501375

H

(d)


14 Complexity


Flowast

(27345 1546 8624 593 16) (79)



+ 0429i and 00172 minus 0429i(80)



δ4 03616 σ3 + 0452(81)



F1 (0 2556 6909 5349 16) (82)


0

1

2

3

4

5

6

7

8

Coun

t

0 100 200 300 400 500 600

tx

y

z

u

(a)

Coun

t

100

101

102

0 100 200 300 400 500 600

tx

y

z

u

(b)

Coun

t

10ndash1

100

101

102

103

104

0 100

x

y

z

u

200 300 400 500 600t

(c)

1105

11

minus04minus02

002

1

15

2

25

3

uy

x

(d)


Complexity 15


(83)


F1 (0 1575 6909 4903 16) (84)


(85)








002040608

112141618

2

StableUnstable


I

II

04 05 06 07 08 09 10465θ

σ3

(a)

04

045205

06

07

08

09

Stable

UnstableIV

θ

σ3

0 02 04 06 08 1

III

θ = 03616σ3 + 0452

(b)

0

22

50

100

1375150

H

0 05 0597 0703 0792 09F0

F1

X

θ70 θ71 θ73 θ

Flowast

(c)

H H H H H

H

0

50

100

1501375

X

θ03 04 05 06 07 08 09 1

(d)


16 Complexity






Nlowast

(1156 159 754 499 2) (86)



(87)



Nlowast

(77887 1381 15936 9698 2) (88)

ndash10

2

4

6

8

10

Coun

t

0 100 200 300 400 500 600

tx

y

z

v

u

(a)

Coun

t

0 100 200 300 400 500 600

tx

y

z

v

u

10ndash4

100

10ndash2

104

102

(b)

Coun

t

0 100 200 300 400 500 600

tx

y

z

v

u

10ndash1

101

100

102

(c)

076078

08minus01

001

02

05

1

15

2

25

uy

x

(d)


Complexity 17



(89)



α αc1δ2 + σ2c2( 1113857

0055σ1 + 02647( 1113857 1366σ3 + 1707( 1113857

(90)



N0 (0 83 6909 7505 2) (91)


(92)



N0 (0 0886 6909 432 2) (93)






04 05 06 07 08 09 10

02040608

112141618

2


σ3 = 01 σ3 = 05 σ3 = 08

θ

σ1

(a)

005

1

002040608104

05

06

07

08

09

1

11

Unstable

Stable

θ

σ3

σ1

(b)

090727 0791064600

50

709

100

1375150

H

N0

X

θθ8 θ9

Nlowast

(c)

X

0 01 02 03 04 05 06 07 08 09 1

H

H

H

H

(σ1 σ3) = (07 07)

(σ1 σ3) = (04 04)

(σ1 σ3) = (0 04)

(σ1 σ3) = (04 0)

0

50

100

1501375

θ

(d)


18 Complexity









xyz

vu

0

5

10

15

20

Coun

t

20 40 60 80 1000t

(a)

0 100 200 300 400 500 600t

xyz

vu

Coun

t

100

102

101

(b)

xyz

vu

Coun

t

100

103

102

101

100 200 300 400 500 6000t

(c)

0975098

0985099

0

0204

17

18

19

2

21

22

uy

x

(d)


Complexity 19




Data Availability




Acknowledgments


References
















20 Complexity





































Complexity 21

t τ



nT0 β bT0

ρ p

n ω2

k2

n θ

qR0

nT0 δ1

d1

nT0 δ2

d2

nT0 δ3

d3

nT0 σ2

s2

nT0H0 c1

r1

nR0 c2

r2

nR0

(2)


dx(t)


dy(t)


dz(t)


du(t)


⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(3)


x(0) x0 ge 0

y(0) y0 ge 0

z(0) z0 ge 0

u(0) u0 ge 0

(4)


2 Model Analysis





x(t) x0e1113938

t


y(t) y0e1113938

t


z(t) z0e1113938

t


+ σ2 1113946t

0e1113938

t


ds ge 0

u(t) u0eminusδ3t

+ 1113946t

0c1y(s) + c2z(s)( 1113857e


(5)







⎧⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎩

(6)


P0 x0 y0 z0 u0( 1113857 0 0σ2δ2

c2σ2δ2δ3

1113888 1113889 (7)


rium namely


θc1δ2σ2δ2

ρσ2 minus δ1δ2

δ2θ1113888 1113889

(8)

TCs

HTCsECs

Tregs

nET

aT (1 ndash bT)

k2TH

pEH

r2Hr1E

qRE

d2Hd1E

d3R

S2

Promotion

Inhibition


Complexity 3



P2 x2 y2 z2 u2( 1113857 1β

0βσ2

βδ2 minus ω2


1113888 1113889

(9)


(6) we have

F(x) Ax2

+ Bx + C 0 (10)



Plowast



lowast( 1113857

σ2δ2 minus ω2x

lowast1113888



lowast( 11138571113889

(11)






















J(P)




0 c1 c2 minusδ3

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠

(12)


det(J(P) minus λI)





1113868111386811138681113868111386811138681113868111386811138681113868111386811138681113868111386811138681113868111386811138681113868

1113868111386811138681113868111386811138681113868111386811138681113868111386811138681113868111386811138681113868111386811138681113868

0 (13)




δ2δ31113888 1113889 0

(14)





θc1δ21113888 1113889

λ + δ2( 1113857 λ2 + δ3λ + θc1y11113872 1113873 0

(15)

4 Complexity







(16)




λ4 + A1λ3

+ A2λ2

+ A3λ + A4 0 (17)

where

A1 αβxlowast


A2 θc1ylowast


( 1113857xlowast

+ αβxlowast


( 1113857δ3 gt 0

A3 θc1 αβxlowast


( 1113857ylowast


( 1113857xlowast





(18)



A1 gt 0 A2 gt 0 A3 gt 0 A4 gt 0 A1A2


(19)





( 1113857ylowast


( 1113857xlowast



(20)

where

B1 αβxlowast


( 11138572δ3 + αβx





lowast( 1113857c1y

lowast

minus αβxlowast


+ δ3( 11138572 αβc1 δ2 minus ω2x

lowast( 1113857x

lowastylowast

(


B2 αβxlowast


( 11138572δ3 + αβx


lowast1113872 1113873δ231113960


lowast( 1113857x

lowast+ ρω2x


( 11138571113859

minus αβxlowast


+ δ3( 11138572ρω2δ3x


(21)




α 1636 β 0002 δ1 03743 σ2 038

δ2 0055 ρ 048 c1 015 c2 02 δ3 025

(22)





1113896 (20)

Complexity 5

θ0 δ3 ρσ2 minus δ1δ2( 1113857

αc1δ2 + c2σ2 0452

θ1 δ3 ρσ2 minus δ1δ2( 1113857

c2σ2 05323

θ3 ρδ3c2

06


βc2σ2 615625ω2 + 05323

(23)





Plowast1 (776449 138195 80448 7265) (24)



minus 001991 + 0378i(25)


Plowast2 (284485 162669 705506 662006) (26)





Plowast3 (72569 139855 8094 7314) (28)



(29)


Plowast4 (26365 154973 854816 776837) (30)



(31)










6 Complexity

0452 05 05323 06 065

Stable

I

IIIIIUnstable

Stable


times10ndash3

ω2

0

02

04

06

08

1

ωlowast2θ2

θ1 θ3 θ


10ndash1

100

101

102

103

Coun

t

x

y

z

u

0 100 200 300 400 500 600

t

(a)

x

y

z

u

0

2

4

6

8

Coun

t

0 100 200 300 400 500 600

t

(b)

10ndash1

100

101

102

103

x

y

z

u

0 100 200 300 400 500 600t

Coun

t

(c)

10ndash2

10ndash1

100

101

102

103

Coun

t

0 100 200 300 400 500 600

tx

y

z

u

(d)

084 086 08809 092 094

minus10

1minus2

minus1

0

1

2

3

uy

x

(e)


Complexity 7

3 Treatment Model


dT(t)


dE(t)


dH(t)


dR(t)


dW(t)

dt s3 minus d4W(t)

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(32)



W(t) W0v(τ) σ1 s1

nE0T0 ξ

mW0

nT0 σ3

s3

nW0T0 δ4

d4

nT0

(33)


dx(t)


dy(t)


dz(t)


du(t)


dv(t)


⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(34)


x(0) x0 ge 0 y(0) y0 ge 0 z(0) z0 ge 0 u(0)

u0 ge 0 v(0) v0 ge 0(35)




0 02 0452 05323 05938 0650

100

200

300

400

500

600

P0

P1

P2

PlowastX

θ0 θ2θ1 θ

(a)

P0 06506053230200

135

50

100

1375

150

H

0452 04657

X

θ0 θ3θ1θ4 θ

Plowast

P1

(b)


8 Complexity








σ3 minus δ4v 0

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎩

(36)






⎧⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎩

(37)


E0 x0 y0 z0 u0( 1113857 0 y0σ2δ2

c1δ2y0 + c2σ2

δ2δ31113888 1113889

(38)




have

F(x) C3x3

+ C2x2

+ C1x + C0



C1 θα 2αβc1δ2 + αc1ω2 + βc2σ2( 1113857



(39)


Elowast



lowast( 1113857

σ2δ2 minus ω2x

lowast1113888



1113889

(40)




Elowast1 x



lowast1( 1113857

σ2δ2 minus ω2x

lowast1

1113888



1113889

(41)

Elowast2 x



lowast2( 1113857

σ2δ2 minus ω2x

lowast2

1113888



1113889

(42)


Complexity 9


Fprime(x) 3C3x2

+ 2C2x + C1 0 (43)



Δ

radic3C3 and

x+2 minusC2 +

Δ



Δ

radic3C3












λ minus α + y0( 1113857 λ + δ2( 1113857 λ2 +σ1y0

+ δ31113888 1113889λ + θc1y0 +σ1δ3y0

1113888 1113889 0

(44)


B2 minus 4AC



λ4 + B1λ3

+ B2λ2

+ B3λ + B4 0 (45)

where

B1 αβxlowast

+ δ3 +σ1ylowast

+σ2zlowastgt 0

B2 αβxlowast

+σ2zlowast

1113874 1113875σ1ylowast


+ δ3σ1ylowast

+ θc1ylowast gt 0

B3 θc1ylowast

+ δ3σ1ylowast

1113888 1113889 αβxlowast

+σ2zlowast


δ3 +σ1ylowast

1113888 1113889



θc1ylowast

+ δ3σ1ylowast


(46)





θc1ylowast

+ δ3σ1ylowast

1113888 1113889




21B4

(47)

whereK1 σ1 + δ3y

lowast( 1113857c1

K2 δ3 +σ1ylowast

+σ2zlowast


+σ2zlowast

1113874 1113875σ1ylowast

+ δ31113888 11138891113890



σ1ylowast


K3 K1 αβxlowast

+σ2zlowast

1113874 1113875c1ylowast

K4 K2c1ylowast

+ K1σ1δ3ylowast

1113888 1113889 αβxlowast

+σ2zlowast

1113874 11138751113890 1113891


δ3 +σ1ylowast

minus αβxlowast

+ δ3 +σ1ylowast

+σ2zlowast

1113888 1113889

2



K5 K2 αβxlowast

+σ2zlowast

1113874 1113875σ1δ3ylowast


K2 δ3 +σ1ylowast

1113888 11138891113890

minus αβxlowast

+ δ3 +σ1ylowast

+σ2zlowast

1113888 1113889

2σ1δ3ylowast


(48)

10 Complexity







σ3 minus δ4v 0

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎩

(49)





⎧⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎩

(50)



F0 0 0σ2δ2

c2σ2δ2δ31

σ3δ4

1113888 1113889


θc1δ2σ2δ2

ρσ2 minus δ1δ2

θδ2σ3δ4

1113888 1113889

F2 1β

0βσ2

βδ2 minus ω2


σ3δ4

1113888 1113889

Flowast


lowast( 1113857

σ2δ2 minus ω2x

lowast1113888



σ3δ4

1113889

(51)






J(F)





0 0 0 0 minusδ4

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠





0 0 0 0 minusδ4

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠

(52)







lowast( 1113857x

lowastylowast




( 1113857ylowast


( 1113857xlowast



(55)

where

B1 αβxlowast


( 11138572δ31 + αβx


lowast( 1113857δ2311113872



lowast( 1113857c1y

lowast

minus αβxlowast


+ δ31( 11138572 αβc1 δ2 minus ω2x

lowast( 1113857x

lowastylowast

(


B2 αβxlowast


( 11138572δ31 + αβx


lowast( 1113857δ2311113872


lowast( 1113857x

lowast+ ρω2x


( 1113857

minus αβxlowast


+ δ31( 11138572ρω2δ31x


(56)

We set

Complexity 11

θ70 δ31 ρσ2 minus δ1δ2( 1113857

αc1δ2 + c2σ2

θ71 δ31 ρσ2 minus δ1δ2( 1113857

c2σ2


βc2σ2

θ73 ρδ31c2

(57)











⎧⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎩

(58)



N0 x0 y0 z0 u0 v0( 1113857 0 y0σ2δ2

c1δ2y0 + c2σ2

δ2δ31σ3δ4

1113888 1113889

(59)




Nlowast



lowast( 1113857

σ2δ2 minus ω2x

lowast1113888



σ3δ4

1113889

(60)


2 +







(61)


J N0( 1113857

α minus y0 0 0 0 0

0 minusσ1y0

ρy0 minusθy0 0



0 0 0 0 minusδ4

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠

(62)


J N0( 1113857 minus λI1113868111386811138681113868

1113868111386811138681113868 λ + δ4( 1113857 J E0( 1113857 minus λI1113868111386811138681113868

1113868111386811138681113868 0 (63)




J Nlowast

( 1113857


0 minusσ1ylowast


ω2zlowast 0 ω2x



0 0 0 0 minusδ4

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠

(64)


J Nlowast

( 1113857 minus λI1113868111386811138681113868

1113868111386811138681113868 λ + δ4( 1113857 J Elowast

( 1113857 minus λI1113868111386811138681113868

1113868111386811138681113868 0 (65)


12 Complexity


θc1ylowast

+ δ31σ1ylowast

1113888 1113889




21B4

k3θ2

+ k4θ + k5 gt 0

(66)

wherek1 σ1 + δ31y

lowast( 1113857c1

k2 δ31 +σ1ylowast

+σ2zlowast


+σ2zlowast

1113874 1113875σ1ylowast

+ δ311113888 11138891113890



σ1ylowast


k3 k1 αβxlowast

+σ2zlowast

1113874 1113875c1ylowast

k4 k2c1ylowast

+ k1σ1δ31

ylowast1113888 1113889 αβx

lowast+σ2zlowast

1113874 11138751113890 1113891


δ31 +σ1ylowast

minus αβxlowast

+ δ31 +σ1ylowast

+σ2zlowast

1113888 1113889

2



lowast+σ2zlowast

1113874 1113875σ1δ31

ylowast


k2 δ31 +σ1ylowast


+ δ31 +σ1ylowast

+σ2zlowast

1113888 1113889

2σ1δ31ylowast

⎡⎣ ⎤⎦


(67)







Elowast

(11560 1598 7543 6993) (68)



(69)



Elowast

(69131 1409 13895 11962) (70)



(71)




α αc1δ2 + c2σ2( 1113857 009391σ1 + 0452 (72)




E0 (0 1782 6909 6596) (73)



(74)



E0 (0 1141 6909 6212) (75)





Complexity 13












Flowast

(2413 1628 7032 5002 16) (77)





StableI

IIUnstable

0

05

1

15

12

04 0465 055 06 07 08 09 1θ

σ1

(a)

IV

III

04

045205

06

07

08

09

θ = 00939σ1 + 0452Unstable

Stable

0 02 04 06 08 1σ1

θ

(b)

0 0489 05354 06 07 08 090

51

100

1375150

H

θ5 θ3 θ

E0

ElowastX

(c)

θ

X

03 04 05 06 07 08 09 1

H

H

H

H

H

0

50

100

1501375

H

(d)


14 Complexity


Flowast

(27345 1546 8624 593 16) (79)



+ 0429i and 00172 minus 0429i(80)



δ4 03616 σ3 + 0452(81)



F1 (0 2556 6909 5349 16) (82)


0

1

2

3

4

5

6

7

8

Coun

t

0 100 200 300 400 500 600

tx

y

z

u

(a)

Coun

t

100

101

102

0 100 200 300 400 500 600

tx

y

z

u

(b)

Coun

t

10ndash1

100

101

102

103

104

0 100

x

y

z

u

200 300 400 500 600t

(c)

1105

11

minus04minus02

002

1

15

2

25

3

uy

x

(d)


Complexity 15


(83)


F1 (0 1575 6909 4903 16) (84)


(85)








002040608

112141618

2

StableUnstable


I

II

04 05 06 07 08 09 10465θ

σ3

(a)

04

045205

06

07

08

09

Stable

UnstableIV

θ

σ3

0 02 04 06 08 1

III

θ = 03616σ3 + 0452

(b)

0

22

50

100

1375150

H

0 05 0597 0703 0792 09F0

F1

X

θ70 θ71 θ73 θ

Flowast

(c)

H H H H H

H

0

50

100

1501375

X

θ03 04 05 06 07 08 09 1

(d)


16 Complexity






Nlowast

(1156 159 754 499 2) (86)



(87)



Nlowast

(77887 1381 15936 9698 2) (88)

ndash10

2

4

6

8

10

Coun

t

0 100 200 300 400 500 600

tx

y

z

v

u

(a)

Coun

t

0 100 200 300 400 500 600

tx

y

z

v

u

10ndash4

100

10ndash2

104

102

(b)

Coun

t

0 100 200 300 400 500 600

tx

y

z

v

u

10ndash1

101

100

102

(c)

076078

08minus01

001

02

05

1

15

2

25

uy

x

(d)


Complexity 17



(89)



α αc1δ2 + σ2c2( 1113857

0055σ1 + 02647( 1113857 1366σ3 + 1707( 1113857

(90)



N0 (0 83 6909 7505 2) (91)


(92)



N0 (0 0886 6909 432 2) (93)






04 05 06 07 08 09 10

02040608

112141618

2


σ3 = 01 σ3 = 05 σ3 = 08

θ

σ1

(a)

005

1

002040608104

05

06

07

08

09

1

11

Unstable

Stable

θ

σ3

σ1

(b)

090727 0791064600

50

709

100

1375150

H

N0

X

θθ8 θ9

Nlowast

(c)

X

0 01 02 03 04 05 06 07 08 09 1

H

H

H

H

(σ1 σ3) = (07 07)

(σ1 σ3) = (04 04)

(σ1 σ3) = (0 04)

(σ1 σ3) = (04 0)

0

50

100

1501375

θ

(d)


18 Complexity









xyz

vu

0

5

10

15

20

Coun

t

20 40 60 80 1000t

(a)

0 100 200 300 400 500 600t

xyz

vu

Coun

t

100

102

101

(b)

xyz

vu

Coun

t

100

103

102

101

100 200 300 400 500 6000t

(c)

0975098

0985099

0

0204

17

18

19

2

21

22

uy

x

(d)


Complexity 19




Data Availability




Acknowledgments


References
















20 Complexity





































Complexity 21



P2 x2 y2 z2 u2( 1113857 1β

0βσ2

βδ2 minus ω2


1113888 1113889

(9)


(6) we have

F(x) Ax2

+ Bx + C 0 (10)



Plowast



lowast( 1113857

σ2δ2 minus ω2x

lowast1113888



lowast( 11138571113889

(11)






















J(P)




0 c1 c2 minusδ3

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠

(12)


det(J(P) minus λI)





1113868111386811138681113868111386811138681113868111386811138681113868111386811138681113868111386811138681113868111386811138681113868

1113868111386811138681113868111386811138681113868111386811138681113868111386811138681113868111386811138681113868111386811138681113868

0 (13)




δ2δ31113888 1113889 0

(14)





θc1δ21113888 1113889

λ + δ2( 1113857 λ2 + δ3λ + θc1y11113872 1113873 0

(15)

4 Complexity







(16)




λ4 + A1λ3

+ A2λ2

+ A3λ + A4 0 (17)

where

A1 αβxlowast


A2 θc1ylowast


( 1113857xlowast

+ αβxlowast


( 1113857δ3 gt 0

A3 θc1 αβxlowast


( 1113857ylowast


( 1113857xlowast





(18)



A1 gt 0 A2 gt 0 A3 gt 0 A4 gt 0 A1A2


(19)





( 1113857ylowast


( 1113857xlowast



(20)

where

B1 αβxlowast


( 11138572δ3 + αβx





lowast( 1113857c1y

lowast

minus αβxlowast


+ δ3( 11138572 αβc1 δ2 minus ω2x

lowast( 1113857x

lowastylowast

(


B2 αβxlowast


( 11138572δ3 + αβx


lowast1113872 1113873δ231113960


lowast( 1113857x

lowast+ ρω2x


( 11138571113859

minus αβxlowast


+ δ3( 11138572ρω2δ3x


(21)




α 1636 β 0002 δ1 03743 σ2 038

δ2 0055 ρ 048 c1 015 c2 02 δ3 025

(22)





1113896 (20)

Complexity 5

θ0 δ3 ρσ2 minus δ1δ2( 1113857

αc1δ2 + c2σ2 0452

θ1 δ3 ρσ2 minus δ1δ2( 1113857

c2σ2 05323

θ3 ρδ3c2

06


βc2σ2 615625ω2 + 05323

(23)





Plowast1 (776449 138195 80448 7265) (24)



minus 001991 + 0378i(25)


Plowast2 (284485 162669 705506 662006) (26)





Plowast3 (72569 139855 8094 7314) (28)



(29)


Plowast4 (26365 154973 854816 776837) (30)



(31)










6 Complexity

0452 05 05323 06 065

Stable

I

IIIIIUnstable

Stable


times10ndash3

ω2

0

02

04

06

08

1

ωlowast2θ2

θ1 θ3 θ


10ndash1

100

101

102

103

Coun

t

x

y

z

u

0 100 200 300 400 500 600

t

(a)

x

y

z

u

0

2

4

6

8

Coun

t

0 100 200 300 400 500 600

t

(b)

10ndash1

100

101

102

103

x

y

z

u

0 100 200 300 400 500 600t

Coun

t

(c)

10ndash2

10ndash1

100

101

102

103

Coun

t

0 100 200 300 400 500 600

tx

y

z

u

(d)

084 086 08809 092 094

minus10

1minus2

minus1

0

1

2

3

uy

x

(e)


Complexity 7

3 Treatment Model


dT(t)


dE(t)


dH(t)


dR(t)


dW(t)

dt s3 minus d4W(t)

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(32)



W(t) W0v(τ) σ1 s1

nE0T0 ξ

mW0

nT0 σ3

s3

nW0T0 δ4

d4

nT0

(33)


dx(t)


dy(t)


dz(t)


du(t)


dv(t)


⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(34)


x(0) x0 ge 0 y(0) y0 ge 0 z(0) z0 ge 0 u(0)

u0 ge 0 v(0) v0 ge 0(35)




0 02 0452 05323 05938 0650

100

200

300

400

500

600

P0

P1

P2

PlowastX

θ0 θ2θ1 θ

(a)

P0 06506053230200

135

50

100

1375

150

H

0452 04657

X

θ0 θ3θ1θ4 θ

Plowast

P1

(b)


8 Complexity








σ3 minus δ4v 0

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎩

(36)






⎧⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎩

(37)


E0 x0 y0 z0 u0( 1113857 0 y0σ2δ2

c1δ2y0 + c2σ2

δ2δ31113888 1113889

(38)




have

F(x) C3x3

+ C2x2

+ C1x + C0



C1 θα 2αβc1δ2 + αc1ω2 + βc2σ2( 1113857



(39)


Elowast



lowast( 1113857

σ2δ2 minus ω2x

lowast1113888



1113889

(40)




Elowast1 x



lowast1( 1113857

σ2δ2 minus ω2x

lowast1

1113888



1113889

(41)

Elowast2 x



lowast2( 1113857

σ2δ2 minus ω2x

lowast2

1113888



1113889

(42)


Complexity 9


Fprime(x) 3C3x2

+ 2C2x + C1 0 (43)



Δ

radic3C3 and

x+2 minusC2 +

Δ



Δ

radic3C3












λ minus α + y0( 1113857 λ + δ2( 1113857 λ2 +σ1y0

+ δ31113888 1113889λ + θc1y0 +σ1δ3y0

1113888 1113889 0

(44)


B2 minus 4AC



λ4 + B1λ3

+ B2λ2

+ B3λ + B4 0 (45)

where

B1 αβxlowast

+ δ3 +σ1ylowast

+σ2zlowastgt 0

B2 αβxlowast

+σ2zlowast

1113874 1113875σ1ylowast


+ δ3σ1ylowast

+ θc1ylowast gt 0

B3 θc1ylowast

+ δ3σ1ylowast

1113888 1113889 αβxlowast

+σ2zlowast


δ3 +σ1ylowast

1113888 1113889



θc1ylowast

+ δ3σ1ylowast


(46)





θc1ylowast

+ δ3σ1ylowast

1113888 1113889




21B4

(47)

whereK1 σ1 + δ3y

lowast( 1113857c1

K2 δ3 +σ1ylowast

+σ2zlowast


+σ2zlowast

1113874 1113875σ1ylowast

+ δ31113888 11138891113890



σ1ylowast


K3 K1 αβxlowast

+σ2zlowast

1113874 1113875c1ylowast

K4 K2c1ylowast

+ K1σ1δ3ylowast

1113888 1113889 αβxlowast

+σ2zlowast

1113874 11138751113890 1113891


δ3 +σ1ylowast

minus αβxlowast

+ δ3 +σ1ylowast

+σ2zlowast

1113888 1113889

2



K5 K2 αβxlowast

+σ2zlowast

1113874 1113875σ1δ3ylowast


K2 δ3 +σ1ylowast

1113888 11138891113890

minus αβxlowast

+ δ3 +σ1ylowast

+σ2zlowast

1113888 1113889

2σ1δ3ylowast


(48)

10 Complexity







σ3 minus δ4v 0

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎩

(49)





⎧⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎩

(50)



F0 0 0σ2δ2

c2σ2δ2δ31

σ3δ4

1113888 1113889


θc1δ2σ2δ2

ρσ2 minus δ1δ2

θδ2σ3δ4

1113888 1113889

F2 1β

0βσ2

βδ2 minus ω2


σ3δ4

1113888 1113889

Flowast


lowast( 1113857

σ2δ2 minus ω2x

lowast1113888



σ3δ4

1113889

(51)






J(F)





0 0 0 0 minusδ4

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠





0 0 0 0 minusδ4

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠

(52)







lowast( 1113857x

lowastylowast




( 1113857ylowast


( 1113857xlowast



(55)

where

B1 αβxlowast


( 11138572δ31 + αβx


lowast( 1113857δ2311113872



lowast( 1113857c1y

lowast

minus αβxlowast


+ δ31( 11138572 αβc1 δ2 minus ω2x

lowast( 1113857x

lowastylowast

(


B2 αβxlowast


( 11138572δ31 + αβx


lowast( 1113857δ2311113872


lowast( 1113857x

lowast+ ρω2x


( 1113857

minus αβxlowast


+ δ31( 11138572ρω2δ31x


(56)

We set

Complexity 11

θ70 δ31 ρσ2 minus δ1δ2( 1113857

αc1δ2 + c2σ2

θ71 δ31 ρσ2 minus δ1δ2( 1113857

c2σ2


βc2σ2

θ73 ρδ31c2

(57)











⎧⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎩

(58)



N0 x0 y0 z0 u0 v0( 1113857 0 y0σ2δ2

c1δ2y0 + c2σ2

δ2δ31σ3δ4

1113888 1113889

(59)




Nlowast



lowast( 1113857

σ2δ2 minus ω2x

lowast1113888



σ3δ4

1113889

(60)


2 +







(61)


J N0( 1113857

α minus y0 0 0 0 0

0 minusσ1y0

ρy0 minusθy0 0



0 0 0 0 minusδ4

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠

(62)


J N0( 1113857 minus λI1113868111386811138681113868

1113868111386811138681113868 λ + δ4( 1113857 J E0( 1113857 minus λI1113868111386811138681113868

1113868111386811138681113868 0 (63)




J Nlowast

( 1113857


0 minusσ1ylowast


ω2zlowast 0 ω2x



0 0 0 0 minusδ4

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠

(64)


J Nlowast

( 1113857 minus λI1113868111386811138681113868

1113868111386811138681113868 λ + δ4( 1113857 J Elowast

( 1113857 minus λI1113868111386811138681113868

1113868111386811138681113868 0 (65)


12 Complexity


θc1ylowast

+ δ31σ1ylowast

1113888 1113889




21B4

k3θ2

+ k4θ + k5 gt 0

(66)

wherek1 σ1 + δ31y

lowast( 1113857c1

k2 δ31 +σ1ylowast

+σ2zlowast


+σ2zlowast

1113874 1113875σ1ylowast

+ δ311113888 11138891113890



σ1ylowast


k3 k1 αβxlowast

+σ2zlowast

1113874 1113875c1ylowast

k4 k2c1ylowast

+ k1σ1δ31

ylowast1113888 1113889 αβx

lowast+σ2zlowast

1113874 11138751113890 1113891


δ31 +σ1ylowast

minus αβxlowast

+ δ31 +σ1ylowast

+σ2zlowast

1113888 1113889

2



lowast+σ2zlowast

1113874 1113875σ1δ31

ylowast


k2 δ31 +σ1ylowast


+ δ31 +σ1ylowast

+σ2zlowast

1113888 1113889

2σ1δ31ylowast

⎡⎣ ⎤⎦


(67)







Elowast

(11560 1598 7543 6993) (68)



(69)



Elowast

(69131 1409 13895 11962) (70)



(71)




α αc1δ2 + c2σ2( 1113857 009391σ1 + 0452 (72)




E0 (0 1782 6909 6596) (73)



(74)



E0 (0 1141 6909 6212) (75)





Complexity 13












Flowast

(2413 1628 7032 5002 16) (77)





StableI

IIUnstable

0

05

1

15

12

04 0465 055 06 07 08 09 1θ

σ1

(a)

IV

III

04

045205

06

07

08

09

θ = 00939σ1 + 0452Unstable

Stable

0 02 04 06 08 1σ1

θ

(b)

0 0489 05354 06 07 08 090

51

100

1375150

H

θ5 θ3 θ

E0

ElowastX

(c)

θ

X

03 04 05 06 07 08 09 1

H

H

H

H

H

0

50

100

1501375

H

(d)


14 Complexity


Flowast

(27345 1546 8624 593 16) (79)



+ 0429i and 00172 minus 0429i(80)



δ4 03616 σ3 + 0452(81)



F1 (0 2556 6909 5349 16) (82)


0

1

2

3

4

5

6

7

8

Coun

t

0 100 200 300 400 500 600

tx

y

z

u

(a)

Coun

t

100

101

102

0 100 200 300 400 500 600

tx

y

z

u

(b)

Coun

t

10ndash1

100

101

102

103

104

0 100

x

y

z

u

200 300 400 500 600t

(c)

1105

11

minus04minus02

002

1

15

2

25

3

uy

x

(d)


Complexity 15


(83)


F1 (0 1575 6909 4903 16) (84)


(85)








002040608

112141618

2

StableUnstable


I

II

04 05 06 07 08 09 10465θ

σ3

(a)

04

045205

06

07

08

09

Stable

UnstableIV

θ

σ3

0 02 04 06 08 1

III

θ = 03616σ3 + 0452

(b)

0

22

50

100

1375150

H

0 05 0597 0703 0792 09F0

F1

X

θ70 θ71 θ73 θ

Flowast

(c)

H H H H H

H

0

50

100

1501375

X

θ03 04 05 06 07 08 09 1

(d)


16 Complexity






Nlowast

(1156 159 754 499 2) (86)



(87)



Nlowast

(77887 1381 15936 9698 2) (88)

ndash10

2

4

6

8

10

Coun

t

0 100 200 300 400 500 600

tx

y

z

v

u

(a)

Coun

t

0 100 200 300 400 500 600

tx

y

z

v

u

10ndash4

100

10ndash2

104

102

(b)

Coun

t

0 100 200 300 400 500 600

tx

y

z

v

u

10ndash1

101

100

102

(c)

076078

08minus01

001

02

05

1

15

2

25

uy

x

(d)


Complexity 17



(89)



α αc1δ2 + σ2c2( 1113857

0055σ1 + 02647( 1113857 1366σ3 + 1707( 1113857

(90)



N0 (0 83 6909 7505 2) (91)


(92)



N0 (0 0886 6909 432 2) (93)






04 05 06 07 08 09 10

02040608

112141618

2


σ3 = 01 σ3 = 05 σ3 = 08

θ

σ1

(a)

005

1

002040608104

05

06

07

08

09

1

11

Unstable

Stable

θ

σ3

σ1

(b)

090727 0791064600

50

709

100

1375150

H

N0

X

θθ8 θ9

Nlowast

(c)

X

0 01 02 03 04 05 06 07 08 09 1

H

H

H

H

(σ1 σ3) = (07 07)

(σ1 σ3) = (04 04)

(σ1 σ3) = (0 04)

(σ1 σ3) = (04 0)

0

50

100

1501375

θ

(d)


18 Complexity









xyz

vu

0

5

10

15

20

Coun

t

20 40 60 80 1000t

(a)

0 100 200 300 400 500 600t

xyz

vu

Coun

t

100

102

101

(b)

xyz

vu

Coun

t

100

103

102

101

100 200 300 400 500 6000t

(c)

0975098

0985099

0

0204

17

18

19

2

21

22

uy

x

(d)


Complexity 19




Data Availability




Acknowledgments


References
















20 Complexity





































Complexity 21







(16)




λ4 + A1λ3

+ A2λ2

+ A3λ + A4 0 (17)

where

A1 αβxlowast


A2 θc1ylowast


( 1113857xlowast

+ αβxlowast


( 1113857δ3 gt 0

A3 θc1 αβxlowast


( 1113857ylowast


( 1113857xlowast





(18)



A1 gt 0 A2 gt 0 A3 gt 0 A4 gt 0 A1A2


(19)





( 1113857ylowast


( 1113857xlowast



(20)

where

B1 αβxlowast


( 11138572δ3 + αβx





lowast( 1113857c1y

lowast

minus αβxlowast


+ δ3( 11138572 αβc1 δ2 minus ω2x

lowast( 1113857x

lowastylowast

(


B2 αβxlowast


( 11138572δ3 + αβx


lowast1113872 1113873δ231113960


lowast( 1113857x

lowast+ ρω2x


( 11138571113859

minus αβxlowast


+ δ3( 11138572ρω2δ3x


(21)




α 1636 β 0002 δ1 03743 σ2 038

δ2 0055 ρ 048 c1 015 c2 02 δ3 025

(22)





1113896 (20)

Complexity 5

θ0 δ3 ρσ2 minus δ1δ2( 1113857

αc1δ2 + c2σ2 0452

θ1 δ3 ρσ2 minus δ1δ2( 1113857

c2σ2 05323

θ3 ρδ3c2

06


βc2σ2 615625ω2 + 05323

(23)





Plowast1 (776449 138195 80448 7265) (24)



minus 001991 + 0378i(25)


Plowast2 (284485 162669 705506 662006) (26)





Plowast3 (72569 139855 8094 7314) (28)



(29)


Plowast4 (26365 154973 854816 776837) (30)



(31)










6 Complexity

0452 05 05323 06 065

Stable

I

IIIIIUnstable

Stable


times10ndash3

ω2

0

02

04

06

08

1

ωlowast2θ2

θ1 θ3 θ


10ndash1

100

101

102

103

Coun

t

x

y

z

u

0 100 200 300 400 500 600

t

(a)

x

y

z

u

0

2

4

6

8

Coun

t

0 100 200 300 400 500 600

t

(b)

10ndash1

100

101

102

103

x

y

z

u

0 100 200 300 400 500 600t

Coun

t

(c)

10ndash2

10ndash1

100

101

102

103

Coun

t

0 100 200 300 400 500 600

tx

y

z

u

(d)

084 086 08809 092 094

minus10

1minus2

minus1

0

1

2

3

uy

x

(e)


Complexity 7

3 Treatment Model


dT(t)


dE(t)


dH(t)


dR(t)


dW(t)

dt s3 minus d4W(t)

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(32)



W(t) W0v(τ) σ1 s1

nE0T0 ξ

mW0

nT0 σ3

s3

nW0T0 δ4

d4

nT0

(33)


dx(t)


dy(t)


dz(t)


du(t)


dv(t)


⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(34)


x(0) x0 ge 0 y(0) y0 ge 0 z(0) z0 ge 0 u(0)

u0 ge 0 v(0) v0 ge 0(35)




0 02 0452 05323 05938 0650

100

200

300

400

500

600

P0

P1

P2

PlowastX

θ0 θ2θ1 θ

(a)

P0 06506053230200

135

50

100

1375

150

H

0452 04657

X

θ0 θ3θ1θ4 θ

Plowast

P1

(b)


8 Complexity








σ3 minus δ4v 0

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎩

(36)






⎧⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎩

(37)


E0 x0 y0 z0 u0( 1113857 0 y0σ2δ2

c1δ2y0 + c2σ2

δ2δ31113888 1113889

(38)




have

F(x) C3x3

+ C2x2

+ C1x + C0



C1 θα 2αβc1δ2 + αc1ω2 + βc2σ2( 1113857



(39)


Elowast



lowast( 1113857

σ2δ2 minus ω2x

lowast1113888



1113889

(40)




Elowast1 x



lowast1( 1113857

σ2δ2 minus ω2x

lowast1

1113888



1113889

(41)

Elowast2 x



lowast2( 1113857

σ2δ2 minus ω2x

lowast2

1113888



1113889

(42)


Complexity 9


Fprime(x) 3C3x2

+ 2C2x + C1 0 (43)



Δ

radic3C3 and

x+2 minusC2 +

Δ



Δ

radic3C3












λ minus α + y0( 1113857 λ + δ2( 1113857 λ2 +σ1y0

+ δ31113888 1113889λ + θc1y0 +σ1δ3y0

1113888 1113889 0

(44)


B2 minus 4AC



λ4 + B1λ3

+ B2λ2

+ B3λ + B4 0 (45)

where

B1 αβxlowast

+ δ3 +σ1ylowast

+σ2zlowastgt 0

B2 αβxlowast

+σ2zlowast

1113874 1113875σ1ylowast


+ δ3σ1ylowast

+ θc1ylowast gt 0

B3 θc1ylowast

+ δ3σ1ylowast

1113888 1113889 αβxlowast

+σ2zlowast


δ3 +σ1ylowast

1113888 1113889



θc1ylowast

+ δ3σ1ylowast


(46)





θc1ylowast

+ δ3σ1ylowast

1113888 1113889




21B4

(47)

whereK1 σ1 + δ3y

lowast( 1113857c1

K2 δ3 +σ1ylowast

+σ2zlowast


+σ2zlowast

1113874 1113875σ1ylowast

+ δ31113888 11138891113890



σ1ylowast


K3 K1 αβxlowast

+σ2zlowast

1113874 1113875c1ylowast

K4 K2c1ylowast

+ K1σ1δ3ylowast

1113888 1113889 αβxlowast

+σ2zlowast

1113874 11138751113890 1113891


δ3 +σ1ylowast

minus αβxlowast

+ δ3 +σ1ylowast

+σ2zlowast

1113888 1113889

2



K5 K2 αβxlowast

+σ2zlowast

1113874 1113875σ1δ3ylowast


K2 δ3 +σ1ylowast

1113888 11138891113890

minus αβxlowast

+ δ3 +σ1ylowast

+σ2zlowast

1113888 1113889

2σ1δ3ylowast


(48)

10 Complexity







σ3 minus δ4v 0

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎩

(49)





⎧⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎩

(50)



F0 0 0σ2δ2

c2σ2δ2δ31

σ3δ4

1113888 1113889


θc1δ2σ2δ2

ρσ2 minus δ1δ2

θδ2σ3δ4

1113888 1113889

F2 1β

0βσ2

βδ2 minus ω2


σ3δ4

1113888 1113889

Flowast


lowast( 1113857

σ2δ2 minus ω2x

lowast1113888



σ3δ4

1113889

(51)






J(F)





0 0 0 0 minusδ4

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠





0 0 0 0 minusδ4

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠

(52)







lowast( 1113857x

lowastylowast




( 1113857ylowast


( 1113857xlowast



(55)

where

B1 αβxlowast


( 11138572δ31 + αβx


lowast( 1113857δ2311113872



lowast( 1113857c1y

lowast

minus αβxlowast


+ δ31( 11138572 αβc1 δ2 minus ω2x

lowast( 1113857x

lowastylowast

(


B2 αβxlowast


( 11138572δ31 + αβx


lowast( 1113857δ2311113872


lowast( 1113857x

lowast+ ρω2x


( 1113857

minus αβxlowast


+ δ31( 11138572ρω2δ31x


(56)

We set

Complexity 11

θ70 δ31 ρσ2 minus δ1δ2( 1113857

αc1δ2 + c2σ2

θ71 δ31 ρσ2 minus δ1δ2( 1113857

c2σ2


βc2σ2

θ73 ρδ31c2

(57)











⎧⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎩

(58)



N0 x0 y0 z0 u0 v0( 1113857 0 y0σ2δ2

c1δ2y0 + c2σ2

δ2δ31σ3δ4

1113888 1113889

(59)




Nlowast



lowast( 1113857

σ2δ2 minus ω2x

lowast1113888



σ3δ4

1113889

(60)


2 +







(61)


J N0( 1113857

α minus y0 0 0 0 0

0 minusσ1y0

ρy0 minusθy0 0



0 0 0 0 minusδ4

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠

(62)


J N0( 1113857 minus λI1113868111386811138681113868

1113868111386811138681113868 λ + δ4( 1113857 J E0( 1113857 minus λI1113868111386811138681113868

1113868111386811138681113868 0 (63)




J Nlowast

( 1113857


0 minusσ1ylowast


ω2zlowast 0 ω2x



0 0 0 0 minusδ4

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠

(64)


J Nlowast

( 1113857 minus λI1113868111386811138681113868

1113868111386811138681113868 λ + δ4( 1113857 J Elowast

( 1113857 minus λI1113868111386811138681113868

1113868111386811138681113868 0 (65)


12 Complexity


θc1ylowast

+ δ31σ1ylowast

1113888 1113889




21B4

k3θ2

+ k4θ + k5 gt 0

(66)

wherek1 σ1 + δ31y

lowast( 1113857c1

k2 δ31 +σ1ylowast

+σ2zlowast


+σ2zlowast

1113874 1113875σ1ylowast

+ δ311113888 11138891113890



σ1ylowast


k3 k1 αβxlowast

+σ2zlowast

1113874 1113875c1ylowast

k4 k2c1ylowast

+ k1σ1δ31

ylowast1113888 1113889 αβx

lowast+σ2zlowast

1113874 11138751113890 1113891


δ31 +σ1ylowast

minus αβxlowast

+ δ31 +σ1ylowast

+σ2zlowast

1113888 1113889

2



lowast+σ2zlowast

1113874 1113875σ1δ31

ylowast


k2 δ31 +σ1ylowast


+ δ31 +σ1ylowast

+σ2zlowast

1113888 1113889

2σ1δ31ylowast

⎡⎣ ⎤⎦


(67)







Elowast

(11560 1598 7543 6993) (68)



(69)



Elowast

(69131 1409 13895 11962) (70)



(71)




α αc1δ2 + c2σ2( 1113857 009391σ1 + 0452 (72)




E0 (0 1782 6909 6596) (73)



(74)



E0 (0 1141 6909 6212) (75)





Complexity 13












Flowast

(2413 1628 7032 5002 16) (77)





StableI

IIUnstable

0

05

1

15

12

04 0465 055 06 07 08 09 1θ

σ1

(a)

IV

III

04

045205

06

07

08

09

θ = 00939σ1 + 0452Unstable

Stable

0 02 04 06 08 1σ1

θ

(b)

0 0489 05354 06 07 08 090

51

100

1375150

H

θ5 θ3 θ

E0

ElowastX

(c)

θ

X

03 04 05 06 07 08 09 1

H

H

H

H

H

0

50

100

1501375

H

(d)


14 Complexity


Flowast

(27345 1546 8624 593 16) (79)



+ 0429i and 00172 minus 0429i(80)



δ4 03616 σ3 + 0452(81)



F1 (0 2556 6909 5349 16) (82)


0

1

2

3

4

5

6

7

8

Coun

t

0 100 200 300 400 500 600

tx

y

z

u

(a)

Coun

t

100

101

102

0 100 200 300 400 500 600

tx

y

z

u

(b)

Coun

t

10ndash1

100

101

102

103

104

0 100

x

y

z

u

200 300 400 500 600t

(c)

1105

11

minus04minus02

002

1

15

2

25

3

uy

x

(d)


Complexity 15


(83)


F1 (0 1575 6909 4903 16) (84)


(85)








002040608

112141618

2

StableUnstable


I

II

04 05 06 07 08 09 10465θ

σ3

(a)

04

045205

06

07

08

09

Stable

UnstableIV

θ

σ3

0 02 04 06 08 1

III

θ = 03616σ3 + 0452

(b)

0

22

50

100

1375150

H

0 05 0597 0703 0792 09F0

F1

X

θ70 θ71 θ73 θ

Flowast

(c)

H H H H H

H

0

50

100

1501375

X

θ03 04 05 06 07 08 09 1

(d)


16 Complexity






Nlowast

(1156 159 754 499 2) (86)



(87)



Nlowast

(77887 1381 15936 9698 2) (88)

ndash10

2

4

6

8

10

Coun

t

0 100 200 300 400 500 600

tx

y

z

v

u

(a)

Coun

t

0 100 200 300 400 500 600

tx

y

z

v

u

10ndash4

100

10ndash2

104

102

(b)

Coun

t

0 100 200 300 400 500 600

tx

y

z

v

u

10ndash1

101

100

102

(c)

076078

08minus01

001

02

05

1

15

2

25

uy

x

(d)


Complexity 17



(89)



α αc1δ2 + σ2c2( 1113857

0055σ1 + 02647( 1113857 1366σ3 + 1707( 1113857

(90)



N0 (0 83 6909 7505 2) (91)


(92)



N0 (0 0886 6909 432 2) (93)






04 05 06 07 08 09 10

02040608

112141618

2


σ3 = 01 σ3 = 05 σ3 = 08

θ

σ1

(a)

005

1

002040608104

05

06

07

08

09

1

11

Unstable

Stable

θ

σ3

σ1

(b)

090727 0791064600

50

709

100

1375150

H

N0

X

θθ8 θ9

Nlowast

(c)

X

0 01 02 03 04 05 06 07 08 09 1

H

H

H

H

(σ1 σ3) = (07 07)

(σ1 σ3) = (04 04)

(σ1 σ3) = (0 04)

(σ1 σ3) = (04 0)

0

50

100

1501375

θ

(d)


18 Complexity









xyz

vu

0

5

10

15

20

Coun

t

20 40 60 80 1000t

(a)

0 100 200 300 400 500 600t

xyz

vu

Coun

t

100

102

101

(b)

xyz

vu

Coun

t

100

103

102

101

100 200 300 400 500 6000t

(c)

0975098

0985099

0

0204

17

18

19

2

21

22

uy

x

(d)


Complexity 19




Data Availability




Acknowledgments


References
















20 Complexity





































Complexity 21

θ0 δ3 ρσ2 minus δ1δ2( 1113857

αc1δ2 + c2σ2 0452

θ1 δ3 ρσ2 minus δ1δ2( 1113857

c2σ2 05323

θ3 ρδ3c2

06


βc2σ2 615625ω2 + 05323

(23)





Plowast1 (776449 138195 80448 7265) (24)



minus 001991 + 0378i(25)


Plowast2 (284485 162669 705506 662006) (26)





Plowast3 (72569 139855 8094 7314) (28)



(29)


Plowast4 (26365 154973 854816 776837) (30)



(31)










6 Complexity

0452 05 05323 06 065

Stable

I

IIIIIUnstable

Stable


times10ndash3

ω2

0

02

04

06

08

1

ωlowast2θ2

θ1 θ3 θ


10ndash1

100

101

102

103

Coun

t

x

y

z

u

0 100 200 300 400 500 600

t

(a)

x

y

z

u

0

2

4

6

8

Coun

t

0 100 200 300 400 500 600

t

(b)

10ndash1

100

101

102

103

x

y

z

u

0 100 200 300 400 500 600t

Coun

t

(c)

10ndash2

10ndash1

100

101

102

103

Coun

t

0 100 200 300 400 500 600

tx

y

z

u

(d)

084 086 08809 092 094

minus10

1minus2

minus1

0

1

2

3

uy

x

(e)


Complexity 7

3 Treatment Model


dT(t)


dE(t)


dH(t)


dR(t)


dW(t)

dt s3 minus d4W(t)

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(32)



W(t) W0v(τ) σ1 s1

nE0T0 ξ

mW0

nT0 σ3

s3

nW0T0 δ4

d4

nT0

(33)


dx(t)


dy(t)


dz(t)


du(t)


dv(t)


⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(34)


x(0) x0 ge 0 y(0) y0 ge 0 z(0) z0 ge 0 u(0)

u0 ge 0 v(0) v0 ge 0(35)




0 02 0452 05323 05938 0650

100

200

300

400

500

600

P0

P1

P2

PlowastX

θ0 θ2θ1 θ

(a)

P0 06506053230200

135

50

100

1375

150

H

0452 04657

X

θ0 θ3θ1θ4 θ

Plowast

P1

(b)


8 Complexity








σ3 minus δ4v 0

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎩

(36)






⎧⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎩

(37)


E0 x0 y0 z0 u0( 1113857 0 y0σ2δ2

c1δ2y0 + c2σ2

δ2δ31113888 1113889

(38)




have

F(x) C3x3

+ C2x2

+ C1x + C0



C1 θα 2αβc1δ2 + αc1ω2 + βc2σ2( 1113857



(39)


Elowast



lowast( 1113857

σ2δ2 minus ω2x

lowast1113888



1113889

(40)




Elowast1 x



lowast1( 1113857

σ2δ2 minus ω2x

lowast1

1113888



1113889

(41)

Elowast2 x



lowast2( 1113857

σ2δ2 minus ω2x

lowast2

1113888



1113889

(42)


Complexity 9


Fprime(x) 3C3x2

+ 2C2x + C1 0 (43)



Δ

radic3C3 and

x+2 minusC2 +

Δ



Δ

radic3C3












λ minus α + y0( 1113857 λ + δ2( 1113857 λ2 +σ1y0

+ δ31113888 1113889λ + θc1y0 +σ1δ3y0

1113888 1113889 0

(44)


B2 minus 4AC



λ4 + B1λ3

+ B2λ2

+ B3λ + B4 0 (45)

where

B1 αβxlowast

+ δ3 +σ1ylowast

+σ2zlowastgt 0

B2 αβxlowast

+σ2zlowast

1113874 1113875σ1ylowast


+ δ3σ1ylowast

+ θc1ylowast gt 0

B3 θc1ylowast

+ δ3σ1ylowast

1113888 1113889 αβxlowast

+σ2zlowast


δ3 +σ1ylowast

1113888 1113889



θc1ylowast

+ δ3σ1ylowast


(46)





θc1ylowast

+ δ3σ1ylowast

1113888 1113889




21B4

(47)

whereK1 σ1 + δ3y

lowast( 1113857c1

K2 δ3 +σ1ylowast

+σ2zlowast


+σ2zlowast

1113874 1113875σ1ylowast

+ δ31113888 11138891113890



σ1ylowast


K3 K1 αβxlowast

+σ2zlowast

1113874 1113875c1ylowast

K4 K2c1ylowast

+ K1σ1δ3ylowast

1113888 1113889 αβxlowast

+σ2zlowast

1113874 11138751113890 1113891


δ3 +σ1ylowast

minus αβxlowast

+ δ3 +σ1ylowast

+σ2zlowast

1113888 1113889

2



K5 K2 αβxlowast

+σ2zlowast

1113874 1113875σ1δ3ylowast


K2 δ3 +σ1ylowast

1113888 11138891113890

minus αβxlowast

+ δ3 +σ1ylowast

+σ2zlowast

1113888 1113889

2σ1δ3ylowast


(48)

10 Complexity







σ3 minus δ4v 0

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎩

(49)





⎧⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎩

(50)



F0 0 0σ2δ2

c2σ2δ2δ31

σ3δ4

1113888 1113889


θc1δ2σ2δ2

ρσ2 minus δ1δ2

θδ2σ3δ4

1113888 1113889

F2 1β

0βσ2

βδ2 minus ω2


σ3δ4

1113888 1113889

Flowast


lowast( 1113857

σ2δ2 minus ω2x

lowast1113888



σ3δ4

1113889

(51)






J(F)





0 0 0 0 minusδ4

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠





0 0 0 0 minusδ4

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠

(52)







lowast( 1113857x

lowastylowast




( 1113857ylowast


( 1113857xlowast



(55)

where

B1 αβxlowast


( 11138572δ31 + αβx


lowast( 1113857δ2311113872



lowast( 1113857c1y

lowast

minus αβxlowast


+ δ31( 11138572 αβc1 δ2 minus ω2x

lowast( 1113857x

lowastylowast

(


B2 αβxlowast


( 11138572δ31 + αβx


lowast( 1113857δ2311113872


lowast( 1113857x

lowast+ ρω2x


( 1113857

minus αβxlowast


+ δ31( 11138572ρω2δ31x


(56)

We set

Complexity 11

θ70 δ31 ρσ2 minus δ1δ2( 1113857

αc1δ2 + c2σ2

θ71 δ31 ρσ2 minus δ1δ2( 1113857

c2σ2


βc2σ2

θ73 ρδ31c2

(57)











⎧⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎩

(58)



N0 x0 y0 z0 u0 v0( 1113857 0 y0σ2δ2

c1δ2y0 + c2σ2

δ2δ31σ3δ4

1113888 1113889

(59)




Nlowast



lowast( 1113857

σ2δ2 minus ω2x

lowast1113888



σ3δ4

1113889

(60)


2 +







(61)


J N0( 1113857

α minus y0 0 0 0 0

0 minusσ1y0

ρy0 minusθy0 0



0 0 0 0 minusδ4

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠

(62)


J N0( 1113857 minus λI1113868111386811138681113868

1113868111386811138681113868 λ + δ4( 1113857 J E0( 1113857 minus λI1113868111386811138681113868

1113868111386811138681113868 0 (63)




J Nlowast

( 1113857


0 minusσ1ylowast


ω2zlowast 0 ω2x



0 0 0 0 minusδ4

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠

(64)


J Nlowast

( 1113857 minus λI1113868111386811138681113868

1113868111386811138681113868 λ + δ4( 1113857 J Elowast

( 1113857 minus λI1113868111386811138681113868

1113868111386811138681113868 0 (65)


12 Complexity


θc1ylowast

+ δ31σ1ylowast

1113888 1113889




21B4

k3θ2

+ k4θ + k5 gt 0

(66)

wherek1 σ1 + δ31y

lowast( 1113857c1

k2 δ31 +σ1ylowast

+σ2zlowast


+σ2zlowast

1113874 1113875σ1ylowast

+ δ311113888 11138891113890



σ1ylowast


k3 k1 αβxlowast

+σ2zlowast

1113874 1113875c1ylowast

k4 k2c1ylowast

+ k1σ1δ31

ylowast1113888 1113889 αβx

lowast+σ2zlowast

1113874 11138751113890 1113891


δ31 +σ1ylowast

minus αβxlowast

+ δ31 +σ1ylowast

+σ2zlowast

1113888 1113889

2



lowast+σ2zlowast

1113874 1113875σ1δ31

ylowast


k2 δ31 +σ1ylowast


+ δ31 +σ1ylowast

+σ2zlowast

1113888 1113889

2σ1δ31ylowast

⎡⎣ ⎤⎦


(67)







Elowast

(11560 1598 7543 6993) (68)



(69)



Elowast

(69131 1409 13895 11962) (70)



(71)




α αc1δ2 + c2σ2( 1113857 009391σ1 + 0452 (72)




E0 (0 1782 6909 6596) (73)



(74)



E0 (0 1141 6909 6212) (75)





Complexity 13












Flowast

(2413 1628 7032 5002 16) (77)





StableI

IIUnstable

0

05

1

15

12

04 0465 055 06 07 08 09 1θ

σ1

(a)

IV

III

04

045205

06

07

08

09

θ = 00939σ1 + 0452Unstable

Stable

0 02 04 06 08 1σ1

θ

(b)

0 0489 05354 06 07 08 090

51

100

1375150

H

θ5 θ3 θ

E0

ElowastX

(c)

θ

X

03 04 05 06 07 08 09 1

H

H

H

H

H

0

50

100

1501375

H

(d)


14 Complexity


Flowast

(27345 1546 8624 593 16) (79)



+ 0429i and 00172 minus 0429i(80)



δ4 03616 σ3 + 0452(81)



F1 (0 2556 6909 5349 16) (82)


0

1

2

3

4

5

6

7

8

Coun

t

0 100 200 300 400 500 600

tx

y

z

u

(a)

Coun

t

100

101

102

0 100 200 300 400 500 600

tx

y

z

u

(b)

Coun

t

10ndash1

100

101

102

103

104

0 100

x

y

z

u

200 300 400 500 600t

(c)

1105

11

minus04minus02

002

1

15

2

25

3

uy

x

(d)


Complexity 15


(83)


F1 (0 1575 6909 4903 16) (84)


(85)








002040608

112141618

2

StableUnstable


I

II

04 05 06 07 08 09 10465θ

σ3

(a)

04

045205

06

07

08

09

Stable

UnstableIV

θ

σ3

0 02 04 06 08 1

III

θ = 03616σ3 + 0452

(b)

0

22

50

100

1375150

H

0 05 0597 0703 0792 09F0

F1

X

θ70 θ71 θ73 θ

Flowast

(c)

H H H H H

H

0

50

100

1501375

X

θ03 04 05 06 07 08 09 1

(d)


16 Complexity






Nlowast

(1156 159 754 499 2) (86)



(87)



Nlowast

(77887 1381 15936 9698 2) (88)

ndash10

2

4

6

8

10

Coun

t

0 100 200 300 400 500 600

tx

y

z

v

u

(a)

Coun

t

0 100 200 300 400 500 600

tx

y

z

v

u

10ndash4

100

10ndash2

104

102

(b)

Coun

t

0 100 200 300 400 500 600

tx

y

z

v

u

10ndash1

101

100

102

(c)

076078

08minus01

001

02

05

1

15

2

25

uy

x

(d)


Complexity 17



(89)



α αc1δ2 + σ2c2( 1113857

0055σ1 + 02647( 1113857 1366σ3 + 1707( 1113857

(90)



N0 (0 83 6909 7505 2) (91)


(92)



N0 (0 0886 6909 432 2) (93)






04 05 06 07 08 09 10

02040608

112141618

2


σ3 = 01 σ3 = 05 σ3 = 08

θ

σ1

(a)

005

1

002040608104

05

06

07

08

09

1

11

Unstable

Stable

θ

σ3

σ1

(b)

090727 0791064600

50

709

100

1375150

H

N0

X

θθ8 θ9

Nlowast

(c)

X

0 01 02 03 04 05 06 07 08 09 1

H

H

H

H

(σ1 σ3) = (07 07)

(σ1 σ3) = (04 04)

(σ1 σ3) = (0 04)

(σ1 σ3) = (04 0)

0

50

100

1501375

θ

(d)


18 Complexity









xyz

vu

0

5

10

15

20

Coun

t

20 40 60 80 1000t

(a)

0 100 200 300 400 500 600t

xyz

vu

Coun

t

100

102

101

(b)

xyz

vu

Coun

t

100

103

102

101

100 200 300 400 500 6000t

(c)

0975098

0985099

0

0204

17

18

19

2

21

22

uy

x

(d)


Complexity 19




Data Availability




Acknowledgments


References
















20 Complexity





































Complexity 21

0452 05 05323 06 065

Stable

I

IIIIIUnstable

Stable


times10ndash3

ω2

0

02

04

06

08

1

ωlowast2θ2

θ1 θ3 θ


10ndash1

100

101

102

103

Coun

t

x

y

z

u

0 100 200 300 400 500 600

t

(a)

x

y

z

u

0

2

4

6

8

Coun

t

0 100 200 300 400 500 600

t

(b)

10ndash1

100

101

102

103

x

y

z

u

0 100 200 300 400 500 600t

Coun

t

(c)

10ndash2

10ndash1

100

101

102

103

Coun

t

0 100 200 300 400 500 600

tx

y

z

u

(d)

084 086 08809 092 094

minus10

1minus2

minus1

0

1

2

3

uy

x

(e)


Complexity 7

3 Treatment Model


dT(t)


dE(t)


dH(t)


dR(t)


dW(t)

dt s3 minus d4W(t)

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(32)



W(t) W0v(τ) σ1 s1

nE0T0 ξ

mW0

nT0 σ3

s3

nW0T0 δ4

d4

nT0

(33)


dx(t)


dy(t)


dz(t)


du(t)


dv(t)


⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(34)


x(0) x0 ge 0 y(0) y0 ge 0 z(0) z0 ge 0 u(0)

u0 ge 0 v(0) v0 ge 0(35)




0 02 0452 05323 05938 0650

100

200

300

400

500

600

P0

P1

P2

PlowastX

θ0 θ2θ1 θ

(a)

P0 06506053230200

135

50

100

1375

150

H

0452 04657

X

θ0 θ3θ1θ4 θ

Plowast

P1

(b)


8 Complexity








σ3 minus δ4v 0

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎩

(36)






⎧⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎩

(37)


E0 x0 y0 z0 u0( 1113857 0 y0σ2δ2

c1δ2y0 + c2σ2

δ2δ31113888 1113889

(38)




have

F(x) C3x3

+ C2x2

+ C1x + C0



C1 θα 2αβc1δ2 + αc1ω2 + βc2σ2( 1113857



(39)


Elowast



lowast( 1113857

σ2δ2 minus ω2x

lowast1113888



1113889

(40)




Elowast1 x



lowast1( 1113857

σ2δ2 minus ω2x

lowast1

1113888



1113889

(41)

Elowast2 x



lowast2( 1113857

σ2δ2 minus ω2x

lowast2

1113888



1113889

(42)


Complexity 9


Fprime(x) 3C3x2

+ 2C2x + C1 0 (43)



Δ

radic3C3 and

x+2 minusC2 +

Δ



Δ

radic3C3












λ minus α + y0( 1113857 λ + δ2( 1113857 λ2 +σ1y0

+ δ31113888 1113889λ + θc1y0 +σ1δ3y0

1113888 1113889 0

(44)


B2 minus 4AC



λ4 + B1λ3

+ B2λ2

+ B3λ + B4 0 (45)

where

B1 αβxlowast

+ δ3 +σ1ylowast

+σ2zlowastgt 0

B2 αβxlowast

+σ2zlowast

1113874 1113875σ1ylowast


+ δ3σ1ylowast

+ θc1ylowast gt 0

B3 θc1ylowast

+ δ3σ1ylowast

1113888 1113889 αβxlowast

+σ2zlowast


δ3 +σ1ylowast

1113888 1113889



θc1ylowast

+ δ3σ1ylowast


(46)





θc1ylowast

+ δ3σ1ylowast

1113888 1113889




21B4

(47)

whereK1 σ1 + δ3y

lowast( 1113857c1

K2 δ3 +σ1ylowast

+σ2zlowast


+σ2zlowast

1113874 1113875σ1ylowast

+ δ31113888 11138891113890



σ1ylowast


K3 K1 αβxlowast

+σ2zlowast

1113874 1113875c1ylowast

K4 K2c1ylowast

+ K1σ1δ3ylowast

1113888 1113889 αβxlowast

+σ2zlowast

1113874 11138751113890 1113891


δ3 +σ1ylowast

minus αβxlowast

+ δ3 +σ1ylowast

+σ2zlowast

1113888 1113889

2



K5 K2 αβxlowast

+σ2zlowast

1113874 1113875σ1δ3ylowast


K2 δ3 +σ1ylowast

1113888 11138891113890

minus αβxlowast

+ δ3 +σ1ylowast

+σ2zlowast

1113888 1113889

2σ1δ3ylowast


(48)

10 Complexity







σ3 minus δ4v 0

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎩

(49)





⎧⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎩

(50)



F0 0 0σ2δ2

c2σ2δ2δ31

σ3δ4

1113888 1113889


θc1δ2σ2δ2

ρσ2 minus δ1δ2

θδ2σ3δ4

1113888 1113889

F2 1β

0βσ2

βδ2 minus ω2


σ3δ4

1113888 1113889

Flowast


lowast( 1113857

σ2δ2 minus ω2x

lowast1113888



σ3δ4

1113889

(51)






J(F)





0 0 0 0 minusδ4

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠





0 0 0 0 minusδ4

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠

(52)







lowast( 1113857x

lowastylowast




( 1113857ylowast


( 1113857xlowast



(55)

where

B1 αβxlowast


( 11138572δ31 + αβx


lowast( 1113857δ2311113872



lowast( 1113857c1y

lowast

minus αβxlowast


+ δ31( 11138572 αβc1 δ2 minus ω2x

lowast( 1113857x

lowastylowast

(


B2 αβxlowast


( 11138572δ31 + αβx


lowast( 1113857δ2311113872


lowast( 1113857x

lowast+ ρω2x


( 1113857

minus αβxlowast


+ δ31( 11138572ρω2δ31x


(56)

We set

Complexity 11

θ70 δ31 ρσ2 minus δ1δ2( 1113857

αc1δ2 + c2σ2

θ71 δ31 ρσ2 minus δ1δ2( 1113857

c2σ2


βc2σ2

θ73 ρδ31c2

(57)











⎧⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎩

(58)



N0 x0 y0 z0 u0 v0( 1113857 0 y0σ2δ2

c1δ2y0 + c2σ2

δ2δ31σ3δ4

1113888 1113889

(59)




Nlowast



lowast( 1113857

σ2δ2 minus ω2x

lowast1113888



σ3δ4

1113889

(60)


2 +







(61)


J N0( 1113857

α minus y0 0 0 0 0

0 minusσ1y0

ρy0 minusθy0 0



0 0 0 0 minusδ4

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠

(62)


J N0( 1113857 minus λI1113868111386811138681113868

1113868111386811138681113868 λ + δ4( 1113857 J E0( 1113857 minus λI1113868111386811138681113868

1113868111386811138681113868 0 (63)




J Nlowast

( 1113857


0 minusσ1ylowast


ω2zlowast 0 ω2x



0 0 0 0 minusδ4

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠

(64)


J Nlowast

( 1113857 minus λI1113868111386811138681113868

1113868111386811138681113868 λ + δ4( 1113857 J Elowast

( 1113857 minus λI1113868111386811138681113868

1113868111386811138681113868 0 (65)


12 Complexity


θc1ylowast

+ δ31σ1ylowast

1113888 1113889




21B4

k3θ2

+ k4θ + k5 gt 0

(66)

wherek1 σ1 + δ31y

lowast( 1113857c1

k2 δ31 +σ1ylowast

+σ2zlowast


+σ2zlowast

1113874 1113875σ1ylowast

+ δ311113888 11138891113890



σ1ylowast


k3 k1 αβxlowast

+σ2zlowast

1113874 1113875c1ylowast

k4 k2c1ylowast

+ k1σ1δ31

ylowast1113888 1113889 αβx

lowast+σ2zlowast

1113874 11138751113890 1113891


δ31 +σ1ylowast

minus αβxlowast

+ δ31 +σ1ylowast

+σ2zlowast

1113888 1113889

2



lowast+σ2zlowast

1113874 1113875σ1δ31

ylowast


k2 δ31 +σ1ylowast


+ δ31 +σ1ylowast

+σ2zlowast

1113888 1113889

2σ1δ31ylowast

⎡⎣ ⎤⎦


(67)







Elowast

(11560 1598 7543 6993) (68)



(69)



Elowast

(69131 1409 13895 11962) (70)



(71)




α αc1δ2 + c2σ2( 1113857 009391σ1 + 0452 (72)




E0 (0 1782 6909 6596) (73)



(74)



E0 (0 1141 6909 6212) (75)





Complexity 13












Flowast

(2413 1628 7032 5002 16) (77)





StableI

IIUnstable

0

05

1

15

12

04 0465 055 06 07 08 09 1θ

σ1

(a)

IV

III

04

045205

06

07

08

09

θ = 00939σ1 + 0452Unstable

Stable

0 02 04 06 08 1σ1

θ

(b)

0 0489 05354 06 07 08 090

51

100

1375150

H

θ5 θ3 θ

E0

ElowastX

(c)

θ

X

03 04 05 06 07 08 09 1

H

H

H

H

H

0

50

100

1501375

H

(d)


14 Complexity


Flowast

(27345 1546 8624 593 16) (79)



+ 0429i and 00172 minus 0429i(80)



δ4 03616 σ3 + 0452(81)



F1 (0 2556 6909 5349 16) (82)


0

1

2

3

4

5

6

7

8

Coun

t

0 100 200 300 400 500 600

tx

y

z

u

(a)

Coun

t

100

101

102

0 100 200 300 400 500 600

tx

y

z

u

(b)

Coun

t

10ndash1

100

101

102

103

104

0 100

x

y

z

u

200 300 400 500 600t

(c)

1105

11

minus04minus02

002

1

15

2

25

3

uy

x

(d)


Complexity 15


(83)


F1 (0 1575 6909 4903 16) (84)


(85)








002040608

112141618

2

StableUnstable


I

II

04 05 06 07 08 09 10465θ

σ3

(a)

04

045205

06

07

08

09

Stable

UnstableIV

θ

σ3

0 02 04 06 08 1

III

θ = 03616σ3 + 0452

(b)

0

22

50

100

1375150

H

0 05 0597 0703 0792 09F0

F1

X

θ70 θ71 θ73 θ

Flowast

(c)

H H H H H

H

0

50

100

1501375

X

θ03 04 05 06 07 08 09 1

(d)


16 Complexity






Nlowast

(1156 159 754 499 2) (86)



(87)



Nlowast

(77887 1381 15936 9698 2) (88)

ndash10

2

4

6

8

10

Coun

t

0 100 200 300 400 500 600

tx

y

z

v

u

(a)

Coun

t

0 100 200 300 400 500 600

tx

y

z

v

u

10ndash4

100

10ndash2

104

102

(b)

Coun

t

0 100 200 300 400 500 600

tx

y

z

v

u

10ndash1

101

100

102

(c)

076078

08minus01

001

02

05

1

15

2

25

uy

x

(d)


Complexity 17



(89)



α αc1δ2 + σ2c2( 1113857

0055σ1 + 02647( 1113857 1366σ3 + 1707( 1113857

(90)



N0 (0 83 6909 7505 2) (91)


(92)



N0 (0 0886 6909 432 2) (93)






04 05 06 07 08 09 10

02040608

112141618

2


σ3 = 01 σ3 = 05 σ3 = 08

θ

σ1

(a)

005

1

002040608104

05

06

07

08

09

1

11

Unstable

Stable

θ

σ3

σ1

(b)

090727 0791064600

50

709

100

1375150

H

N0

X

θθ8 θ9

Nlowast

(c)

X

0 01 02 03 04 05 06 07 08 09 1

H

H

H

H

(σ1 σ3) = (07 07)

(σ1 σ3) = (04 04)

(σ1 σ3) = (0 04)

(σ1 σ3) = (04 0)

0

50

100

1501375

θ

(d)


18 Complexity









xyz

vu

0

5

10

15

20

Coun

t

20 40 60 80 1000t

(a)

0 100 200 300 400 500 600t

xyz

vu

Coun

t

100

102

101

(b)

xyz

vu

Coun

t

100

103

102

101

100 200 300 400 500 6000t

(c)

0975098

0985099

0

0204

17

18

19

2

21

22

uy

x

(d)


Complexity 19




Data Availability




Acknowledgments


References
















20 Complexity





































Complexity 21

3 Treatment Model


dT(t)


dE(t)


dH(t)


dR(t)


dW(t)

dt s3 minus d4W(t)

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(32)



W(t) W0v(τ) σ1 s1

nE0T0 ξ

mW0

nT0 σ3

s3

nW0T0 δ4

d4

nT0

(33)


dx(t)


dy(t)


dz(t)


du(t)


dv(t)


⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(34)


x(0) x0 ge 0 y(0) y0 ge 0 z(0) z0 ge 0 u(0)

u0 ge 0 v(0) v0 ge 0(35)




0 02 0452 05323 05938 0650

100

200

300

400

500

600

P0

P1

P2

PlowastX

θ0 θ2θ1 θ

(a)

P0 06506053230200

135

50

100

1375

150

H

0452 04657

X

θ0 θ3θ1θ4 θ

Plowast

P1

(b)


8 Complexity








σ3 minus δ4v 0

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎩

(36)






⎧⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎩

(37)


E0 x0 y0 z0 u0( 1113857 0 y0σ2δ2

c1δ2y0 + c2σ2

δ2δ31113888 1113889

(38)




have

F(x) C3x3

+ C2x2

+ C1x + C0



C1 θα 2αβc1δ2 + αc1ω2 + βc2σ2( 1113857



(39)


Elowast



lowast( 1113857

σ2δ2 minus ω2x

lowast1113888



1113889

(40)




Elowast1 x



lowast1( 1113857

σ2δ2 minus ω2x

lowast1

1113888



1113889

(41)

Elowast2 x



lowast2( 1113857

σ2δ2 minus ω2x

lowast2

1113888



1113889

(42)


Complexity 9


Fprime(x) 3C3x2

+ 2C2x + C1 0 (43)



Δ

radic3C3 and

x+2 minusC2 +

Δ



Δ

radic3C3












λ minus α + y0( 1113857 λ + δ2( 1113857 λ2 +σ1y0

+ δ31113888 1113889λ + θc1y0 +σ1δ3y0

1113888 1113889 0

(44)


B2 minus 4AC



λ4 + B1λ3

+ B2λ2

+ B3λ + B4 0 (45)

where

B1 αβxlowast

+ δ3 +σ1ylowast

+σ2zlowastgt 0

B2 αβxlowast

+σ2zlowast

1113874 1113875σ1ylowast


+ δ3σ1ylowast

+ θc1ylowast gt 0

B3 θc1ylowast

+ δ3σ1ylowast

1113888 1113889 αβxlowast

+σ2zlowast


δ3 +σ1ylowast

1113888 1113889



θc1ylowast

+ δ3σ1ylowast


(46)





θc1ylowast

+ δ3σ1ylowast

1113888 1113889




21B4

(47)

whereK1 σ1 + δ3y

lowast( 1113857c1

K2 δ3 +σ1ylowast

+σ2zlowast


+σ2zlowast

1113874 1113875σ1ylowast

+ δ31113888 11138891113890



σ1ylowast


K3 K1 αβxlowast

+σ2zlowast

1113874 1113875c1ylowast

K4 K2c1ylowast

+ K1σ1δ3ylowast

1113888 1113889 αβxlowast

+σ2zlowast

1113874 11138751113890 1113891


δ3 +σ1ylowast

minus αβxlowast

+ δ3 +σ1ylowast

+σ2zlowast

1113888 1113889

2



K5 K2 αβxlowast

+σ2zlowast

1113874 1113875σ1δ3ylowast


K2 δ3 +σ1ylowast

1113888 11138891113890

minus αβxlowast

+ δ3 +σ1ylowast

+σ2zlowast

1113888 1113889

2σ1δ3ylowast


(48)

10 Complexity







σ3 minus δ4v 0

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎩

(49)





⎧⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎩

(50)



F0 0 0σ2δ2

c2σ2δ2δ31

σ3δ4

1113888 1113889


θc1δ2σ2δ2

ρσ2 minus δ1δ2

θδ2σ3δ4

1113888 1113889

F2 1β

0βσ2

βδ2 minus ω2


σ3δ4

1113888 1113889

Flowast


lowast( 1113857

σ2δ2 minus ω2x

lowast1113888



σ3δ4

1113889

(51)






J(F)





0 0 0 0 minusδ4

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠





0 0 0 0 minusδ4

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠

(52)







lowast( 1113857x

lowastylowast




( 1113857ylowast


( 1113857xlowast



(55)

where

B1 αβxlowast


( 11138572δ31 + αβx


lowast( 1113857δ2311113872



lowast( 1113857c1y

lowast

minus αβxlowast


+ δ31( 11138572 αβc1 δ2 minus ω2x

lowast( 1113857x

lowastylowast

(


B2 αβxlowast


( 11138572δ31 + αβx


lowast( 1113857δ2311113872


lowast( 1113857x

lowast+ ρω2x


( 1113857

minus αβxlowast


+ δ31( 11138572ρω2δ31x


(56)

We set

Complexity 11

θ70 δ31 ρσ2 minus δ1δ2( 1113857

αc1δ2 + c2σ2

θ71 δ31 ρσ2 minus δ1δ2( 1113857

c2σ2


βc2σ2

θ73 ρδ31c2

(57)











⎧⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎩

(58)



N0 x0 y0 z0 u0 v0( 1113857 0 y0σ2δ2

c1δ2y0 + c2σ2

δ2δ31σ3δ4

1113888 1113889

(59)




Nlowast



lowast( 1113857

σ2δ2 minus ω2x

lowast1113888



σ3δ4

1113889

(60)


2 +







(61)


J N0( 1113857

α minus y0 0 0 0 0

0 minusσ1y0

ρy0 minusθy0 0



0 0 0 0 minusδ4

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠

(62)


J N0( 1113857 minus λI1113868111386811138681113868

1113868111386811138681113868 λ + δ4( 1113857 J E0( 1113857 minus λI1113868111386811138681113868

1113868111386811138681113868 0 (63)




J Nlowast

( 1113857


0 minusσ1ylowast


ω2zlowast 0 ω2x



0 0 0 0 minusδ4

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠

(64)


J Nlowast

( 1113857 minus λI1113868111386811138681113868

1113868111386811138681113868 λ + δ4( 1113857 J Elowast

( 1113857 minus λI1113868111386811138681113868

1113868111386811138681113868 0 (65)


12 Complexity


θc1ylowast

+ δ31σ1ylowast

1113888 1113889




21B4

k3θ2

+ k4θ + k5 gt 0

(66)

wherek1 σ1 + δ31y

lowast( 1113857c1

k2 δ31 +σ1ylowast

+σ2zlowast


+σ2zlowast

1113874 1113875σ1ylowast

+ δ311113888 11138891113890



σ1ylowast


k3 k1 αβxlowast

+σ2zlowast

1113874 1113875c1ylowast

k4 k2c1ylowast

+ k1σ1δ31

ylowast1113888 1113889 αβx

lowast+σ2zlowast

1113874 11138751113890 1113891


δ31 +σ1ylowast

minus αβxlowast

+ δ31 +σ1ylowast

+σ2zlowast

1113888 1113889

2



lowast+σ2zlowast

1113874 1113875σ1δ31

ylowast


k2 δ31 +σ1ylowast


+ δ31 +σ1ylowast

+σ2zlowast

1113888 1113889

2σ1δ31ylowast

⎡⎣ ⎤⎦


(67)







Elowast

(11560 1598 7543 6993) (68)



(69)



Elowast

(69131 1409 13895 11962) (70)



(71)




α αc1δ2 + c2σ2( 1113857 009391σ1 + 0452 (72)




E0 (0 1782 6909 6596) (73)



(74)



E0 (0 1141 6909 6212) (75)





Complexity 13












Flowast

(2413 1628 7032 5002 16) (77)





StableI

IIUnstable

0

05

1

15

12

04 0465 055 06 07 08 09 1θ

σ1

(a)

IV

III

04

045205

06

07

08

09

θ = 00939σ1 + 0452Unstable

Stable

0 02 04 06 08 1σ1

θ

(b)

0 0489 05354 06 07 08 090

51

100

1375150

H

θ5 θ3 θ

E0

ElowastX

(c)

θ

X

03 04 05 06 07 08 09 1

H

H

H

H

H

0

50

100

1501375

H

(d)


14 Complexity


Flowast

(27345 1546 8624 593 16) (79)



+ 0429i and 00172 minus 0429i(80)



δ4 03616 σ3 + 0452(81)



F1 (0 2556 6909 5349 16) (82)


0

1

2

3

4

5

6

7

8

Coun

t

0 100 200 300 400 500 600

tx

y

z

u

(a)

Coun

t

100

101

102

0 100 200 300 400 500 600

tx

y

z

u

(b)

Coun

t

10ndash1

100

101

102

103

104

0 100

x

y

z

u

200 300 400 500 600t

(c)

1105

11

minus04minus02

002

1

15

2

25

3

uy

x

(d)


Complexity 15


(83)


F1 (0 1575 6909 4903 16) (84)


(85)








002040608

112141618

2

StableUnstable


I

II

04 05 06 07 08 09 10465θ

σ3

(a)

04

045205

06

07

08

09

Stable

UnstableIV

θ

σ3

0 02 04 06 08 1

III

θ = 03616σ3 + 0452

(b)

0

22

50

100

1375150

H

0 05 0597 0703 0792 09F0

F1

X

θ70 θ71 θ73 θ

Flowast

(c)

H H H H H

H

0

50

100

1501375

X

θ03 04 05 06 07 08 09 1

(d)


16 Complexity






Nlowast

(1156 159 754 499 2) (86)



(87)



Nlowast

(77887 1381 15936 9698 2) (88)

ndash10

2

4

6

8

10

Coun

t

0 100 200 300 400 500 600

tx

y

z

v

u

(a)

Coun

t

0 100 200 300 400 500 600

tx

y

z

v

u

10ndash4

100

10ndash2

104

102

(b)

Coun

t

0 100 200 300 400 500 600

tx

y

z

v

u

10ndash1

101

100

102

(c)

076078

08minus01

001

02

05

1

15

2

25

uy

x

(d)


Complexity 17



(89)



α αc1δ2 + σ2c2( 1113857

0055σ1 + 02647( 1113857 1366σ3 + 1707( 1113857

(90)



N0 (0 83 6909 7505 2) (91)


(92)



N0 (0 0886 6909 432 2) (93)






04 05 06 07 08 09 10

02040608

112141618

2


σ3 = 01 σ3 = 05 σ3 = 08

θ

σ1

(a)

005

1

002040608104

05

06

07

08

09

1

11

Unstable

Stable

θ

σ3

σ1

(b)

090727 0791064600

50

709

100

1375150

H

N0

X

θθ8 θ9

Nlowast

(c)

X

0 01 02 03 04 05 06 07 08 09 1

H

H

H

H

(σ1 σ3) = (07 07)

(σ1 σ3) = (04 04)

(σ1 σ3) = (0 04)

(σ1 σ3) = (04 0)

0

50

100

1501375

θ

(d)


18 Complexity









xyz

vu

0

5

10

15

20

Coun

t

20 40 60 80 1000t

(a)

0 100 200 300 400 500 600t

xyz

vu

Coun

t

100

102

101

(b)

xyz

vu

Coun

t

100

103

102

101

100 200 300 400 500 6000t

(c)

0975098

0985099

0

0204

17

18

19

2

21

22

uy

x

(d)


Complexity 19




Data Availability




Acknowledgments


References
















20 Complexity





































Complexity 21








σ3 minus δ4v 0

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎩

(36)






⎧⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎩

(37)


E0 x0 y0 z0 u0( 1113857 0 y0σ2δ2

c1δ2y0 + c2σ2

δ2δ31113888 1113889

(38)




have

F(x) C3x3

+ C2x2

+ C1x + C0



C1 θα 2αβc1δ2 + αc1ω2 + βc2σ2( 1113857



(39)


Elowast



lowast( 1113857

σ2δ2 minus ω2x

lowast1113888



1113889

(40)




Elowast1 x



lowast1( 1113857

σ2δ2 minus ω2x

lowast1

1113888



1113889

(41)

Elowast2 x



lowast2( 1113857

σ2δ2 minus ω2x

lowast2

1113888



1113889

(42)


Complexity 9


Fprime(x) 3C3x2

+ 2C2x + C1 0 (43)



Δ

radic3C3 and

x+2 minusC2 +

Δ



Δ

radic3C3












λ minus α + y0( 1113857 λ + δ2( 1113857 λ2 +σ1y0

+ δ31113888 1113889λ + θc1y0 +σ1δ3y0

1113888 1113889 0

(44)


B2 minus 4AC



λ4 + B1λ3

+ B2λ2

+ B3λ + B4 0 (45)

where

B1 αβxlowast

+ δ3 +σ1ylowast

+σ2zlowastgt 0

B2 αβxlowast

+σ2zlowast

1113874 1113875σ1ylowast


+ δ3σ1ylowast

+ θc1ylowast gt 0

B3 θc1ylowast

+ δ3σ1ylowast

1113888 1113889 αβxlowast

+σ2zlowast


δ3 +σ1ylowast

1113888 1113889



θc1ylowast

+ δ3σ1ylowast


(46)





θc1ylowast

+ δ3σ1ylowast

1113888 1113889




21B4

(47)

whereK1 σ1 + δ3y

lowast( 1113857c1

K2 δ3 +σ1ylowast

+σ2zlowast


+σ2zlowast

1113874 1113875σ1ylowast

+ δ31113888 11138891113890



σ1ylowast


K3 K1 αβxlowast

+σ2zlowast

1113874 1113875c1ylowast

K4 K2c1ylowast

+ K1σ1δ3ylowast

1113888 1113889 αβxlowast

+σ2zlowast

1113874 11138751113890 1113891


δ3 +σ1ylowast

minus αβxlowast

+ δ3 +σ1ylowast

+σ2zlowast

1113888 1113889

2



K5 K2 αβxlowast

+σ2zlowast

1113874 1113875σ1δ3ylowast


K2 δ3 +σ1ylowast

1113888 11138891113890

minus αβxlowast

+ δ3 +σ1ylowast

+σ2zlowast

1113888 1113889

2σ1δ3ylowast


(48)

10 Complexity







σ3 minus δ4v 0

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎩

(49)





⎧⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎩

(50)



F0 0 0σ2δ2

c2σ2δ2δ31

σ3δ4

1113888 1113889


θc1δ2σ2δ2

ρσ2 minus δ1δ2

θδ2σ3δ4

1113888 1113889

F2 1β

0βσ2

βδ2 minus ω2


σ3δ4

1113888 1113889

Flowast


lowast( 1113857

σ2δ2 minus ω2x

lowast1113888



σ3δ4

1113889

(51)






J(F)





0 0 0 0 minusδ4

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠





0 0 0 0 minusδ4

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠

(52)







lowast( 1113857x

lowastylowast




( 1113857ylowast


( 1113857xlowast



(55)

where

B1 αβxlowast


( 11138572δ31 + αβx


lowast( 1113857δ2311113872



lowast( 1113857c1y

lowast

minus αβxlowast


+ δ31( 11138572 αβc1 δ2 minus ω2x

lowast( 1113857x

lowastylowast

(


B2 αβxlowast


( 11138572δ31 + αβx


lowast( 1113857δ2311113872


lowast( 1113857x

lowast+ ρω2x


( 1113857

minus αβxlowast


+ δ31( 11138572ρω2δ31x


(56)

We set

Complexity 11

θ70 δ31 ρσ2 minus δ1δ2( 1113857

αc1δ2 + c2σ2

θ71 δ31 ρσ2 minus δ1δ2( 1113857

c2σ2


βc2σ2

θ73 ρδ31c2

(57)











⎧⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎩

(58)



N0 x0 y0 z0 u0 v0( 1113857 0 y0σ2δ2

c1δ2y0 + c2σ2

δ2δ31σ3δ4

1113888 1113889

(59)




Nlowast



lowast( 1113857

σ2δ2 minus ω2x

lowast1113888



σ3δ4

1113889

(60)


2 +







(61)


J N0( 1113857

α minus y0 0 0 0 0

0 minusσ1y0

ρy0 minusθy0 0



0 0 0 0 minusδ4

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠

(62)


J N0( 1113857 minus λI1113868111386811138681113868

1113868111386811138681113868 λ + δ4( 1113857 J E0( 1113857 minus λI1113868111386811138681113868

1113868111386811138681113868 0 (63)




J Nlowast

( 1113857


0 minusσ1ylowast


ω2zlowast 0 ω2x



0 0 0 0 minusδ4

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠

(64)


J Nlowast

( 1113857 minus λI1113868111386811138681113868

1113868111386811138681113868 λ + δ4( 1113857 J Elowast

( 1113857 minus λI1113868111386811138681113868

1113868111386811138681113868 0 (65)


12 Complexity


θc1ylowast

+ δ31σ1ylowast

1113888 1113889




21B4

k3θ2

+ k4θ + k5 gt 0

(66)

wherek1 σ1 + δ31y

lowast( 1113857c1

k2 δ31 +σ1ylowast

+σ2zlowast


+σ2zlowast

1113874 1113875σ1ylowast

+ δ311113888 11138891113890



σ1ylowast


k3 k1 αβxlowast

+σ2zlowast

1113874 1113875c1ylowast

k4 k2c1ylowast

+ k1σ1δ31

ylowast1113888 1113889 αβx

lowast+σ2zlowast

1113874 11138751113890 1113891


δ31 +σ1ylowast

minus αβxlowast

+ δ31 +σ1ylowast

+σ2zlowast

1113888 1113889

2



lowast+σ2zlowast

1113874 1113875σ1δ31

ylowast


k2 δ31 +σ1ylowast


+ δ31 +σ1ylowast

+σ2zlowast

1113888 1113889

2σ1δ31ylowast

⎡⎣ ⎤⎦


(67)







Elowast

(11560 1598 7543 6993) (68)



(69)



Elowast

(69131 1409 13895 11962) (70)



(71)




α αc1δ2 + c2σ2( 1113857 009391σ1 + 0452 (72)




E0 (0 1782 6909 6596) (73)



(74)



E0 (0 1141 6909 6212) (75)





Complexity 13












Flowast

(2413 1628 7032 5002 16) (77)





StableI

IIUnstable

0

05

1

15

12

04 0465 055 06 07 08 09 1θ

σ1

(a)

IV

III

04

045205

06

07

08

09

θ = 00939σ1 + 0452Unstable

Stable

0 02 04 06 08 1σ1

θ

(b)

0 0489 05354 06 07 08 090

51

100

1375150

H

θ5 θ3 θ

E0

ElowastX

(c)

θ

X

03 04 05 06 07 08 09 1

H

H

H

H

H

0

50

100

1501375

H

(d)


14 Complexity


Flowast

(27345 1546 8624 593 16) (79)



+ 0429i and 00172 minus 0429i(80)



δ4 03616 σ3 + 0452(81)



F1 (0 2556 6909 5349 16) (82)


0

1

2

3

4

5

6

7

8

Coun

t

0 100 200 300 400 500 600

tx

y

z

u

(a)

Coun

t

100

101

102

0 100 200 300 400 500 600

tx

y

z

u

(b)

Coun

t

10ndash1

100

101

102

103

104

0 100

x

y

z

u

200 300 400 500 600t

(c)

1105

11

minus04minus02

002

1

15

2

25

3

uy

x

(d)


Complexity 15


(83)


F1 (0 1575 6909 4903 16) (84)


(85)








002040608

112141618

2

StableUnstable


I

II

04 05 06 07 08 09 10465θ

σ3

(a)

04

045205

06

07

08

09

Stable

UnstableIV

θ

σ3

0 02 04 06 08 1

III

θ = 03616σ3 + 0452

(b)

0

22

50

100

1375150

H

0 05 0597 0703 0792 09F0

F1

X

θ70 θ71 θ73 θ

Flowast

(c)

H H H H H

H

0

50

100

1501375

X

θ03 04 05 06 07 08 09 1

(d)


16 Complexity






Nlowast

(1156 159 754 499 2) (86)



(87)



Nlowast

(77887 1381 15936 9698 2) (88)

ndash10

2

4

6

8

10

Coun

t

0 100 200 300 400 500 600

tx

y

z

v

u

(a)

Coun

t

0 100 200 300 400 500 600

tx

y

z

v

u

10ndash4

100

10ndash2

104

102

(b)

Coun

t

0 100 200 300 400 500 600

tx

y

z

v

u

10ndash1

101

100

102

(c)

076078

08minus01

001

02

05

1

15

2

25

uy

x

(d)


Complexity 17



(89)



α αc1δ2 + σ2c2( 1113857

0055σ1 + 02647( 1113857 1366σ3 + 1707( 1113857

(90)



N0 (0 83 6909 7505 2) (91)


(92)



N0 (0 0886 6909 432 2) (93)






04 05 06 07 08 09 10

02040608

112141618

2


σ3 = 01 σ3 = 05 σ3 = 08

θ

σ1

(a)

005

1

002040608104

05

06

07

08

09

1

11

Unstable

Stable

θ

σ3

σ1

(b)

090727 0791064600

50

709

100

1375150

H

N0

X

θθ8 θ9

Nlowast

(c)

X

0 01 02 03 04 05 06 07 08 09 1

H

H

H

H

(σ1 σ3) = (07 07)

(σ1 σ3) = (04 04)

(σ1 σ3) = (0 04)

(σ1 σ3) = (04 0)

0

50

100

1501375

θ

(d)


18 Complexity









xyz

vu

0

5

10

15

20

Coun

t

20 40 60 80 1000t

(a)

0 100 200 300 400 500 600t

xyz

vu

Coun

t

100

102

101

(b)

xyz

vu

Coun

t

100

103

102

101

100 200 300 400 500 6000t

(c)

0975098

0985099

0

0204

17

18

19

2

21

22

uy

x

(d)


Complexity 19




Data Availability




Acknowledgments


References
















20 Complexity





































Complexity 21


Fprime(x) 3C3x2

+ 2C2x + C1 0 (43)



Δ

radic3C3 and

x+2 minusC2 +

Δ



Δ

radic3C3












λ minus α + y0( 1113857 λ + δ2( 1113857 λ2 +σ1y0

+ δ31113888 1113889λ + θc1y0 +σ1δ3y0

1113888 1113889 0

(44)


B2 minus 4AC



λ4 + B1λ3

+ B2λ2

+ B3λ + B4 0 (45)

where

B1 αβxlowast

+ δ3 +σ1ylowast

+σ2zlowastgt 0

B2 αβxlowast

+σ2zlowast

1113874 1113875σ1ylowast


+ δ3σ1ylowast

+ θc1ylowast gt 0

B3 θc1ylowast

+ δ3σ1ylowast

1113888 1113889 αβxlowast

+σ2zlowast


δ3 +σ1ylowast

1113888 1113889



θc1ylowast

+ δ3σ1ylowast


(46)





θc1ylowast

+ δ3σ1ylowast

1113888 1113889




21B4

(47)

whereK1 σ1 + δ3y

lowast( 1113857c1

K2 δ3 +σ1ylowast

+σ2zlowast


+σ2zlowast

1113874 1113875σ1ylowast

+ δ31113888 11138891113890



σ1ylowast


K3 K1 αβxlowast

+σ2zlowast

1113874 1113875c1ylowast

K4 K2c1ylowast

+ K1σ1δ3ylowast

1113888 1113889 αβxlowast

+σ2zlowast

1113874 11138751113890 1113891


δ3 +σ1ylowast

minus αβxlowast

+ δ3 +σ1ylowast

+σ2zlowast

1113888 1113889

2



K5 K2 αβxlowast

+σ2zlowast

1113874 1113875σ1δ3ylowast


K2 δ3 +σ1ylowast

1113888 11138891113890

minus αβxlowast

+ δ3 +σ1ylowast

+σ2zlowast

1113888 1113889

2σ1δ3ylowast


(48)

10 Complexity







σ3 minus δ4v 0

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎩

(49)





⎧⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎩

(50)



F0 0 0σ2δ2

c2σ2δ2δ31

σ3δ4

1113888 1113889


θc1δ2σ2δ2

ρσ2 minus δ1δ2

θδ2σ3δ4

1113888 1113889

F2 1β

0βσ2

βδ2 minus ω2


σ3δ4

1113888 1113889

Flowast


lowast( 1113857

σ2δ2 minus ω2x

lowast1113888



σ3δ4

1113889

(51)






J(F)





0 0 0 0 minusδ4

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠





0 0 0 0 minusδ4

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠

(52)







lowast( 1113857x

lowastylowast




( 1113857ylowast


( 1113857xlowast



(55)

where

B1 αβxlowast


( 11138572δ31 + αβx


lowast( 1113857δ2311113872



lowast( 1113857c1y

lowast

minus αβxlowast


+ δ31( 11138572 αβc1 δ2 minus ω2x

lowast( 1113857x

lowastylowast

(


B2 αβxlowast


( 11138572δ31 + αβx


lowast( 1113857δ2311113872


lowast( 1113857x

lowast+ ρω2x


( 1113857

minus αβxlowast


+ δ31( 11138572ρω2δ31x


(56)

We set

Complexity 11

θ70 δ31 ρσ2 minus δ1δ2( 1113857

αc1δ2 + c2σ2

θ71 δ31 ρσ2 minus δ1δ2( 1113857

c2σ2


βc2σ2

θ73 ρδ31c2

(57)











⎧⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎩

(58)



N0 x0 y0 z0 u0 v0( 1113857 0 y0σ2δ2

c1δ2y0 + c2σ2

δ2δ31σ3δ4

1113888 1113889

(59)




Nlowast



lowast( 1113857

σ2δ2 minus ω2x

lowast1113888



σ3δ4

1113889

(60)


2 +







(61)


J N0( 1113857

α minus y0 0 0 0 0

0 minusσ1y0

ρy0 minusθy0 0



0 0 0 0 minusδ4

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠

(62)


J N0( 1113857 minus λI1113868111386811138681113868

1113868111386811138681113868 λ + δ4( 1113857 J E0( 1113857 minus λI1113868111386811138681113868

1113868111386811138681113868 0 (63)




J Nlowast

( 1113857


0 minusσ1ylowast


ω2zlowast 0 ω2x



0 0 0 0 minusδ4

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠

(64)


J Nlowast

( 1113857 minus λI1113868111386811138681113868

1113868111386811138681113868 λ + δ4( 1113857 J Elowast

( 1113857 minus λI1113868111386811138681113868

1113868111386811138681113868 0 (65)


12 Complexity


θc1ylowast

+ δ31σ1ylowast

1113888 1113889




21B4

k3θ2

+ k4θ + k5 gt 0

(66)

wherek1 σ1 + δ31y

lowast( 1113857c1

k2 δ31 +σ1ylowast

+σ2zlowast


+σ2zlowast

1113874 1113875σ1ylowast

+ δ311113888 11138891113890



σ1ylowast


k3 k1 αβxlowast

+σ2zlowast

1113874 1113875c1ylowast

k4 k2c1ylowast

+ k1σ1δ31

ylowast1113888 1113889 αβx

lowast+σ2zlowast

1113874 11138751113890 1113891


δ31 +σ1ylowast

minus αβxlowast

+ δ31 +σ1ylowast

+σ2zlowast

1113888 1113889

2



lowast+σ2zlowast

1113874 1113875σ1δ31

ylowast


k2 δ31 +σ1ylowast


+ δ31 +σ1ylowast

+σ2zlowast

1113888 1113889

2σ1δ31ylowast

⎡⎣ ⎤⎦


(67)







Elowast

(11560 1598 7543 6993) (68)



(69)



Elowast

(69131 1409 13895 11962) (70)



(71)




α αc1δ2 + c2σ2( 1113857 009391σ1 + 0452 (72)




E0 (0 1782 6909 6596) (73)



(74)



E0 (0 1141 6909 6212) (75)





Complexity 13












Flowast

(2413 1628 7032 5002 16) (77)





StableI

IIUnstable

0

05

1

15

12

04 0465 055 06 07 08 09 1θ

σ1

(a)

IV

III

04

045205

06

07

08

09

θ = 00939σ1 + 0452Unstable

Stable

0 02 04 06 08 1σ1

θ

(b)

0 0489 05354 06 07 08 090

51

100

1375150

H

θ5 θ3 θ

E0

ElowastX

(c)

θ

X

03 04 05 06 07 08 09 1

H

H

H

H

H

0

50

100

1501375

H

(d)


14 Complexity


Flowast

(27345 1546 8624 593 16) (79)



+ 0429i and 00172 minus 0429i(80)



δ4 03616 σ3 + 0452(81)



F1 (0 2556 6909 5349 16) (82)


0

1

2

3

4

5

6

7

8

Coun

t

0 100 200 300 400 500 600

tx

y

z

u

(a)

Coun

t

100

101

102

0 100 200 300 400 500 600

tx

y

z

u

(b)

Coun

t

10ndash1

100

101

102

103

104

0 100

x

y

z

u

200 300 400 500 600t

(c)

1105

11

minus04minus02

002

1

15

2

25

3

uy

x

(d)


Complexity 15


(83)


F1 (0 1575 6909 4903 16) (84)


(85)








002040608

112141618

2

StableUnstable


I

II

04 05 06 07 08 09 10465θ

σ3

(a)

04

045205

06

07

08

09

Stable

UnstableIV

θ

σ3

0 02 04 06 08 1

III

θ = 03616σ3 + 0452

(b)

0

22

50

100

1375150

H

0 05 0597 0703 0792 09F0

F1

X

θ70 θ71 θ73 θ

Flowast

(c)

H H H H H

H

0

50

100

1501375

X

θ03 04 05 06 07 08 09 1

(d)


16 Complexity






Nlowast

(1156 159 754 499 2) (86)



(87)



Nlowast

(77887 1381 15936 9698 2) (88)

ndash10

2

4

6

8

10

Coun

t

0 100 200 300 400 500 600

tx

y

z

v

u

(a)

Coun

t

0 100 200 300 400 500 600

tx

y

z

v

u

10ndash4

100

10ndash2

104

102

(b)

Coun

t

0 100 200 300 400 500 600

tx

y

z

v

u

10ndash1

101

100

102

(c)

076078

08minus01

001

02

05

1

15

2

25

uy

x

(d)


Complexity 17



(89)



α αc1δ2 + σ2c2( 1113857

0055σ1 + 02647( 1113857 1366σ3 + 1707( 1113857

(90)



N0 (0 83 6909 7505 2) (91)


(92)



N0 (0 0886 6909 432 2) (93)






04 05 06 07 08 09 10

02040608

112141618

2


σ3 = 01 σ3 = 05 σ3 = 08

θ

σ1

(a)

005

1

002040608104

05

06

07

08

09

1

11

Unstable

Stable

θ

σ3

σ1

(b)

090727 0791064600

50

709

100

1375150

H

N0

X

θθ8 θ9

Nlowast

(c)

X

0 01 02 03 04 05 06 07 08 09 1

H

H

H

H

(σ1 σ3) = (07 07)

(σ1 σ3) = (04 04)

(σ1 σ3) = (0 04)

(σ1 σ3) = (04 0)

0

50

100

1501375

θ

(d)


18 Complexity









xyz

vu

0

5

10

15

20

Coun

t

20 40 60 80 1000t

(a)

0 100 200 300 400 500 600t

xyz

vu

Coun

t

100

102

101

(b)

xyz

vu

Coun

t

100

103

102

101

100 200 300 400 500 6000t

(c)

0975098

0985099

0

0204

17

18

19

2

21

22

uy

x

(d)


Complexity 19




Data Availability




Acknowledgments


References
















20 Complexity





































Complexity 21







σ3 minus δ4v 0

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎩

(49)





⎧⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎩

(50)



F0 0 0σ2δ2

c2σ2δ2δ31

σ3δ4

1113888 1113889


θc1δ2σ2δ2

ρσ2 minus δ1δ2

θδ2σ3δ4

1113888 1113889

F2 1β

0βσ2

βδ2 minus ω2


σ3δ4

1113888 1113889

Flowast


lowast( 1113857

σ2δ2 minus ω2x

lowast1113888



σ3δ4

1113889

(51)






J(F)





0 0 0 0 minusδ4

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠





0 0 0 0 minusδ4

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠

(52)







lowast( 1113857x

lowastylowast




( 1113857ylowast


( 1113857xlowast



(55)

where

B1 αβxlowast


( 11138572δ31 + αβx


lowast( 1113857δ2311113872



lowast( 1113857c1y

lowast

minus αβxlowast


+ δ31( 11138572 αβc1 δ2 minus ω2x

lowast( 1113857x

lowastylowast

(


B2 αβxlowast


( 11138572δ31 + αβx


lowast( 1113857δ2311113872


lowast( 1113857x

lowast+ ρω2x


( 1113857

minus αβxlowast


+ δ31( 11138572ρω2δ31x


(56)

We set

Complexity 11

θ70 δ31 ρσ2 minus δ1δ2( 1113857

αc1δ2 + c2σ2

θ71 δ31 ρσ2 minus δ1δ2( 1113857

c2σ2


βc2σ2

θ73 ρδ31c2

(57)











⎧⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎩

(58)



N0 x0 y0 z0 u0 v0( 1113857 0 y0σ2δ2

c1δ2y0 + c2σ2

δ2δ31σ3δ4

1113888 1113889

(59)




Nlowast



lowast( 1113857

σ2δ2 minus ω2x

lowast1113888



σ3δ4

1113889

(60)


2 +







(61)


J N0( 1113857

α minus y0 0 0 0 0

0 minusσ1y0

ρy0 minusθy0 0



0 0 0 0 minusδ4

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠

(62)


J N0( 1113857 minus λI1113868111386811138681113868

1113868111386811138681113868 λ + δ4( 1113857 J E0( 1113857 minus λI1113868111386811138681113868

1113868111386811138681113868 0 (63)




J Nlowast

( 1113857


0 minusσ1ylowast


ω2zlowast 0 ω2x



0 0 0 0 minusδ4

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠

(64)


J Nlowast

( 1113857 minus λI1113868111386811138681113868

1113868111386811138681113868 λ + δ4( 1113857 J Elowast

( 1113857 minus λI1113868111386811138681113868

1113868111386811138681113868 0 (65)


12 Complexity


θc1ylowast

+ δ31σ1ylowast

1113888 1113889




21B4

k3θ2

+ k4θ + k5 gt 0

(66)

wherek1 σ1 + δ31y

lowast( 1113857c1

k2 δ31 +σ1ylowast

+σ2zlowast


+σ2zlowast

1113874 1113875σ1ylowast

+ δ311113888 11138891113890



σ1ylowast


k3 k1 αβxlowast

+σ2zlowast

1113874 1113875c1ylowast

k4 k2c1ylowast

+ k1σ1δ31

ylowast1113888 1113889 αβx

lowast+σ2zlowast

1113874 11138751113890 1113891


δ31 +σ1ylowast

minus αβxlowast

+ δ31 +σ1ylowast

+σ2zlowast

1113888 1113889

2



lowast+σ2zlowast

1113874 1113875σ1δ31

ylowast


k2 δ31 +σ1ylowast


+ δ31 +σ1ylowast

+σ2zlowast

1113888 1113889

2σ1δ31ylowast

⎡⎣ ⎤⎦


(67)







Elowast

(11560 1598 7543 6993) (68)



(69)



Elowast

(69131 1409 13895 11962) (70)



(71)




α αc1δ2 + c2σ2( 1113857 009391σ1 + 0452 (72)




E0 (0 1782 6909 6596) (73)



(74)



E0 (0 1141 6909 6212) (75)





Complexity 13












Flowast

(2413 1628 7032 5002 16) (77)





StableI

IIUnstable

0

05

1

15

12

04 0465 055 06 07 08 09 1θ

σ1

(a)

IV

III

04

045205

06

07

08

09

θ = 00939σ1 + 0452Unstable

Stable

0 02 04 06 08 1σ1

θ

(b)

0 0489 05354 06 07 08 090

51

100

1375150

H

θ5 θ3 θ

E0

ElowastX

(c)

θ

X

03 04 05 06 07 08 09 1

H

H

H

H

H

0

50

100

1501375

H

(d)


14 Complexity


Flowast

(27345 1546 8624 593 16) (79)



+ 0429i and 00172 minus 0429i(80)



δ4 03616 σ3 + 0452(81)



F1 (0 2556 6909 5349 16) (82)


0

1

2

3

4

5

6

7

8

Coun

t

0 100 200 300 400 500 600

tx

y

z

u

(a)

Coun

t

100

101

102

0 100 200 300 400 500 600

tx

y

z

u

(b)

Coun

t

10ndash1

100

101

102

103

104

0 100

x

y

z

u

200 300 400 500 600t

(c)

1105

11

minus04minus02

002

1

15

2

25

3

uy

x

(d)


Complexity 15


(83)


F1 (0 1575 6909 4903 16) (84)


(85)








002040608

112141618

2

StableUnstable


I

II

04 05 06 07 08 09 10465θ

σ3

(a)

04

045205

06

07

08

09

Stable

UnstableIV

θ

σ3

0 02 04 06 08 1

III

θ = 03616σ3 + 0452

(b)

0

22

50

100

1375150

H

0 05 0597 0703 0792 09F0

F1

X

θ70 θ71 θ73 θ

Flowast

(c)

H H H H H

H

0

50

100

1501375

X

θ03 04 05 06 07 08 09 1

(d)


16 Complexity






Nlowast

(1156 159 754 499 2) (86)



(87)



Nlowast

(77887 1381 15936 9698 2) (88)

ndash10

2

4

6

8

10

Coun

t

0 100 200 300 400 500 600

tx

y

z

v

u

(a)

Coun

t

0 100 200 300 400 500 600

tx

y

z

v

u

10ndash4

100

10ndash2

104

102

(b)

Coun

t

0 100 200 300 400 500 600

tx

y

z

v

u

10ndash1

101

100

102

(c)

076078

08minus01

001

02

05

1

15

2

25

uy

x

(d)


Complexity 17



(89)



α αc1δ2 + σ2c2( 1113857

0055σ1 + 02647( 1113857 1366σ3 + 1707( 1113857

(90)



N0 (0 83 6909 7505 2) (91)


(92)



N0 (0 0886 6909 432 2) (93)






04 05 06 07 08 09 10

02040608

112141618

2


σ3 = 01 σ3 = 05 σ3 = 08

θ

σ1

(a)

005

1

002040608104

05

06

07

08

09

1

11

Unstable

Stable

θ

σ3

σ1

(b)

090727 0791064600

50

709

100

1375150

H

N0

X

θθ8 θ9

Nlowast

(c)

X

0 01 02 03 04 05 06 07 08 09 1

H

H

H

H

(σ1 σ3) = (07 07)

(σ1 σ3) = (04 04)

(σ1 σ3) = (0 04)

(σ1 σ3) = (04 0)

0

50

100

1501375

θ

(d)


18 Complexity









xyz

vu

0

5

10

15

20

Coun

t

20 40 60 80 1000t

(a)

0 100 200 300 400 500 600t

xyz

vu

Coun

t

100

102

101

(b)

xyz

vu

Coun

t

100

103

102

101

100 200 300 400 500 6000t

(c)

0975098

0985099

0

0204

17

18

19

2

21

22

uy

x

(d)


Complexity 19




Data Availability




Acknowledgments


References
















20 Complexity





































Complexity 21

θ70 δ31 ρσ2 minus δ1δ2( 1113857

αc1δ2 + c2σ2

θ71 δ31 ρσ2 minus δ1δ2( 1113857

c2σ2


βc2σ2

θ73 ρδ31c2

(57)











⎧⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎩

(58)



N0 x0 y0 z0 u0 v0( 1113857 0 y0σ2δ2

c1δ2y0 + c2σ2

δ2δ31σ3δ4

1113888 1113889

(59)




Nlowast



lowast( 1113857

σ2δ2 minus ω2x

lowast1113888



σ3δ4

1113889

(60)


2 +







(61)


J N0( 1113857

α minus y0 0 0 0 0

0 minusσ1y0

ρy0 minusθy0 0



0 0 0 0 minusδ4

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠

(62)


J N0( 1113857 minus λI1113868111386811138681113868

1113868111386811138681113868 λ + δ4( 1113857 J E0( 1113857 minus λI1113868111386811138681113868

1113868111386811138681113868 0 (63)




J Nlowast

( 1113857


0 minusσ1ylowast


ω2zlowast 0 ω2x



0 0 0 0 minusδ4

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠

(64)


J Nlowast

( 1113857 minus λI1113868111386811138681113868

1113868111386811138681113868 λ + δ4( 1113857 J Elowast

( 1113857 minus λI1113868111386811138681113868

1113868111386811138681113868 0 (65)


12 Complexity


θc1ylowast

+ δ31σ1ylowast

1113888 1113889




21B4

k3θ2

+ k4θ + k5 gt 0

(66)

wherek1 σ1 + δ31y

lowast( 1113857c1

k2 δ31 +σ1ylowast

+σ2zlowast


+σ2zlowast

1113874 1113875σ1ylowast

+ δ311113888 11138891113890



σ1ylowast


k3 k1 αβxlowast

+σ2zlowast

1113874 1113875c1ylowast

k4 k2c1ylowast

+ k1σ1δ31

ylowast1113888 1113889 αβx

lowast+σ2zlowast

1113874 11138751113890 1113891


δ31 +σ1ylowast

minus αβxlowast

+ δ31 +σ1ylowast

+σ2zlowast

1113888 1113889

2



lowast+σ2zlowast

1113874 1113875σ1δ31

ylowast


k2 δ31 +σ1ylowast


+ δ31 +σ1ylowast

+σ2zlowast

1113888 1113889

2σ1δ31ylowast

⎡⎣ ⎤⎦


(67)







Elowast

(11560 1598 7543 6993) (68)



(69)



Elowast

(69131 1409 13895 11962) (70)



(71)




α αc1δ2 + c2σ2( 1113857 009391σ1 + 0452 (72)




E0 (0 1782 6909 6596) (73)



(74)



E0 (0 1141 6909 6212) (75)





Complexity 13












Flowast

(2413 1628 7032 5002 16) (77)





StableI

IIUnstable

0

05

1

15

12

04 0465 055 06 07 08 09 1θ

σ1

(a)

IV

III

04

045205

06

07

08

09

θ = 00939σ1 + 0452Unstable

Stable

0 02 04 06 08 1σ1

θ

(b)

0 0489 05354 06 07 08 090

51

100

1375150

H

θ5 θ3 θ

E0

ElowastX

(c)

θ

X

03 04 05 06 07 08 09 1

H

H

H

H

H

0

50

100

1501375

H

(d)


14 Complexity


Flowast

(27345 1546 8624 593 16) (79)



+ 0429i and 00172 minus 0429i(80)



δ4 03616 σ3 + 0452(81)



F1 (0 2556 6909 5349 16) (82)


0

1

2

3

4

5

6

7

8

Coun

t

0 100 200 300 400 500 600

tx

y

z

u

(a)

Coun

t

100

101

102

0 100 200 300 400 500 600

tx

y

z

u

(b)

Coun

t

10ndash1

100

101

102

103

104

0 100

x

y

z

u

200 300 400 500 600t

(c)

1105

11

minus04minus02

002

1

15

2

25

3

uy

x

(d)


Complexity 15


(83)


F1 (0 1575 6909 4903 16) (84)


(85)








002040608

112141618

2

StableUnstable


I

II

04 05 06 07 08 09 10465θ

σ3

(a)

04

045205

06

07

08

09

Stable

UnstableIV

θ

σ3

0 02 04 06 08 1

III

θ = 03616σ3 + 0452

(b)

0

22

50

100

1375150

H

0 05 0597 0703 0792 09F0

F1

X

θ70 θ71 θ73 θ

Flowast

(c)

H H H H H

H

0

50

100

1501375

X

θ03 04 05 06 07 08 09 1

(d)


16 Complexity






Nlowast

(1156 159 754 499 2) (86)



(87)



Nlowast

(77887 1381 15936 9698 2) (88)

ndash10

2

4

6

8

10

Coun

t

0 100 200 300 400 500 600

tx

y

z

v

u

(a)

Coun

t

0 100 200 300 400 500 600

tx

y

z

v

u

10ndash4

100

10ndash2

104

102

(b)

Coun

t

0 100 200 300 400 500 600

tx

y

z

v

u

10ndash1

101

100

102

(c)

076078

08minus01

001

02

05

1

15

2

25

uy

x

(d)


Complexity 17



(89)



α αc1δ2 + σ2c2( 1113857

0055σ1 + 02647( 1113857 1366σ3 + 1707( 1113857

(90)



N0 (0 83 6909 7505 2) (91)


(92)



N0 (0 0886 6909 432 2) (93)






04 05 06 07 08 09 10

02040608

112141618

2


σ3 = 01 σ3 = 05 σ3 = 08

θ

σ1

(a)

005

1

002040608104

05

06

07

08

09

1

11

Unstable

Stable

θ

σ3

σ1

(b)

090727 0791064600

50

709

100

1375150

H

N0

X

θθ8 θ9

Nlowast

(c)

X

0 01 02 03 04 05 06 07 08 09 1

H

H

H

H

(σ1 σ3) = (07 07)

(σ1 σ3) = (04 04)

(σ1 σ3) = (0 04)

(σ1 σ3) = (04 0)

0

50

100

1501375

θ

(d)


18 Complexity









xyz

vu

0

5

10

15

20

Coun

t

20 40 60 80 1000t

(a)

0 100 200 300 400 500 600t

xyz

vu

Coun

t

100

102

101

(b)

xyz

vu

Coun

t

100

103

102

101

100 200 300 400 500 6000t

(c)

0975098

0985099

0

0204

17

18

19

2

21

22

uy

x

(d)


Complexity 19




Data Availability




Acknowledgments


References
















20 Complexity





































Complexity 21


θc1ylowast

+ δ31σ1ylowast

1113888 1113889




21B4

k3θ2

+ k4θ + k5 gt 0

(66)

wherek1 σ1 + δ31y

lowast( 1113857c1

k2 δ31 +σ1ylowast

+σ2zlowast


+σ2zlowast

1113874 1113875σ1ylowast

+ δ311113888 11138891113890



σ1ylowast


k3 k1 αβxlowast

+σ2zlowast

1113874 1113875c1ylowast

k4 k2c1ylowast

+ k1σ1δ31

ylowast1113888 1113889 αβx

lowast+σ2zlowast

1113874 11138751113890 1113891


δ31 +σ1ylowast

minus αβxlowast

+ δ31 +σ1ylowast

+σ2zlowast

1113888 1113889

2



lowast+σ2zlowast

1113874 1113875σ1δ31

ylowast


k2 δ31 +σ1ylowast


+ δ31 +σ1ylowast

+σ2zlowast

1113888 1113889

2σ1δ31ylowast

⎡⎣ ⎤⎦


(67)







Elowast

(11560 1598 7543 6993) (68)



(69)



Elowast

(69131 1409 13895 11962) (70)



(71)




α αc1δ2 + c2σ2( 1113857 009391σ1 + 0452 (72)




E0 (0 1782 6909 6596) (73)



(74)



E0 (0 1141 6909 6212) (75)





Complexity 13












Flowast

(2413 1628 7032 5002 16) (77)





StableI

IIUnstable

0

05

1

15

12

04 0465 055 06 07 08 09 1θ

σ1

(a)

IV

III

04

045205

06

07

08

09

θ = 00939σ1 + 0452Unstable

Stable

0 02 04 06 08 1σ1

θ

(b)

0 0489 05354 06 07 08 090

51

100

1375150

H

θ5 θ3 θ

E0

ElowastX

(c)

θ

X

03 04 05 06 07 08 09 1

H

H

H

H

H

0

50

100

1501375

H

(d)


14 Complexity


Flowast

(27345 1546 8624 593 16) (79)



+ 0429i and 00172 minus 0429i(80)



δ4 03616 σ3 + 0452(81)



F1 (0 2556 6909 5349 16) (82)


0

1

2

3

4

5

6

7

8

Coun

t

0 100 200 300 400 500 600

tx

y

z

u

(a)

Coun

t

100

101

102

0 100 200 300 400 500 600

tx

y

z

u

(b)

Coun

t

10ndash1

100

101

102

103

104

0 100

x

y

z

u

200 300 400 500 600t

(c)

1105

11

minus04minus02

002

1

15

2

25

3

uy

x

(d)


Complexity 15


(83)


F1 (0 1575 6909 4903 16) (84)


(85)








002040608

112141618

2

StableUnstable


I

II

04 05 06 07 08 09 10465θ

σ3

(a)

04

045205

06

07

08

09

Stable

UnstableIV

θ

σ3

0 02 04 06 08 1

III

θ = 03616σ3 + 0452

(b)

0

22

50

100

1375150

H

0 05 0597 0703 0792 09F0

F1

X

θ70 θ71 θ73 θ

Flowast

(c)

H H H H H

H

0

50

100

1501375

X

θ03 04 05 06 07 08 09 1

(d)


16 Complexity






Nlowast

(1156 159 754 499 2) (86)



(87)



Nlowast

(77887 1381 15936 9698 2) (88)

ndash10

2

4

6

8

10

Coun

t

0 100 200 300 400 500 600

tx

y

z

v

u

(a)

Coun

t

0 100 200 300 400 500 600

tx

y

z

v

u

10ndash4

100

10ndash2

104

102

(b)

Coun

t

0 100 200 300 400 500 600

tx

y

z

v

u

10ndash1

101

100

102

(c)

076078

08minus01

001

02

05

1

15

2

25

uy

x

(d)


Complexity 17



(89)



α αc1δ2 + σ2c2( 1113857

0055σ1 + 02647( 1113857 1366σ3 + 1707( 1113857

(90)



N0 (0 83 6909 7505 2) (91)


(92)



N0 (0 0886 6909 432 2) (93)






04 05 06 07 08 09 10

02040608

112141618

2


σ3 = 01 σ3 = 05 σ3 = 08

θ

σ1

(a)

005

1

002040608104

05

06

07

08

09

1

11

Unstable

Stable

θ

σ3

σ1

(b)

090727 0791064600

50

709

100

1375150

H

N0

X

θθ8 θ9

Nlowast

(c)

X

0 01 02 03 04 05 06 07 08 09 1

H

H

H

H

(σ1 σ3) = (07 07)

(σ1 σ3) = (04 04)

(σ1 σ3) = (0 04)

(σ1 σ3) = (04 0)

0

50

100

1501375

θ

(d)


18 Complexity









xyz

vu

0

5

10

15

20

Coun

t

20 40 60 80 1000t

(a)

0 100 200 300 400 500 600t

xyz

vu

Coun

t

100

102

101

(b)

xyz

vu

Coun

t

100

103

102

101

100 200 300 400 500 6000t

(c)

0975098

0985099

0

0204

17

18

19

2

21

22

uy

x

(d)


Complexity 19




Data Availability




Acknowledgments


References
















20 Complexity





































Complexity 21












Flowast

(2413 1628 7032 5002 16) (77)





StableI

IIUnstable

0

05

1

15

12

04 0465 055 06 07 08 09 1θ

σ1

(a)

IV

III

04

045205

06

07

08

09

θ = 00939σ1 + 0452Unstable

Stable

0 02 04 06 08 1σ1

θ

(b)

0 0489 05354 06 07 08 090

51

100

1375150

H

θ5 θ3 θ

E0

ElowastX

(c)

θ

X

03 04 05 06 07 08 09 1

H

H

H

H

H

0

50

100

1501375

H

(d)


14 Complexity


Flowast

(27345 1546 8624 593 16) (79)



+ 0429i and 00172 minus 0429i(80)



δ4 03616 σ3 + 0452(81)



F1 (0 2556 6909 5349 16) (82)


0

1

2

3

4

5

6

7

8

Coun

t

0 100 200 300 400 500 600

tx

y

z

u

(a)

Coun

t

100

101

102

0 100 200 300 400 500 600

tx

y

z

u

(b)

Coun

t

10ndash1

100

101

102

103

104

0 100

x

y

z

u

200 300 400 500 600t

(c)

1105

11

minus04minus02

002

1

15

2

25

3

uy

x

(d)


Complexity 15


(83)


F1 (0 1575 6909 4903 16) (84)


(85)








002040608

112141618

2

StableUnstable


I

II

04 05 06 07 08 09 10465θ

σ3

(a)

04

045205

06

07

08

09

Stable

UnstableIV

θ

σ3

0 02 04 06 08 1

III

θ = 03616σ3 + 0452

(b)

0

22

50

100

1375150

H

0 05 0597 0703 0792 09F0

F1

X

θ70 θ71 θ73 θ

Flowast

(c)

H H H H H

H

0

50

100

1501375

X

θ03 04 05 06 07 08 09 1

(d)


16 Complexity






Nlowast

(1156 159 754 499 2) (86)



(87)



Nlowast

(77887 1381 15936 9698 2) (88)

ndash10

2

4

6

8

10

Coun

t

0 100 200 300 400 500 600

tx

y

z

v

u

(a)

Coun

t

0 100 200 300 400 500 600

tx

y

z

v

u

10ndash4

100

10ndash2

104

102

(b)

Coun

t

0 100 200 300 400 500 600

tx

y

z

v

u

10ndash1

101

100

102

(c)

076078

08minus01

001

02

05

1

15

2

25

uy

x

(d)


Complexity 17



(89)



α αc1δ2 + σ2c2( 1113857

0055σ1 + 02647( 1113857 1366σ3 + 1707( 1113857

(90)



N0 (0 83 6909 7505 2) (91)


(92)



N0 (0 0886 6909 432 2) (93)






04 05 06 07 08 09 10

02040608

112141618

2


σ3 = 01 σ3 = 05 σ3 = 08

θ

σ1

(a)

005

1

002040608104

05

06

07

08

09

1

11

Unstable

Stable

θ

σ3

σ1

(b)

090727 0791064600

50

709

100

1375150

H

N0

X

θθ8 θ9

Nlowast

(c)

X

0 01 02 03 04 05 06 07 08 09 1

H

H

H

H

(σ1 σ3) = (07 07)

(σ1 σ3) = (04 04)

(σ1 σ3) = (0 04)

(σ1 σ3) = (04 0)

0

50

100

1501375

θ

(d)


18 Complexity









xyz

vu

0

5

10

15

20

Coun

t

20 40 60 80 1000t

(a)

0 100 200 300 400 500 600t

xyz

vu

Coun

t

100

102

101

(b)

xyz

vu

Coun

t

100

103

102

101

100 200 300 400 500 6000t

(c)

0975098

0985099

0

0204

17

18

19

2

21

22

uy

x

(d)


Complexity 19




Data Availability




Acknowledgments


References
















20 Complexity





































Complexity 21


Flowast

(27345 1546 8624 593 16) (79)



+ 0429i and 00172 minus 0429i(80)



δ4 03616 σ3 + 0452(81)



F1 (0 2556 6909 5349 16) (82)


0

1

2

3

4

5

6

7

8

Coun

t

0 100 200 300 400 500 600

tx

y

z

u

(a)

Coun

t

100

101

102

0 100 200 300 400 500 600

tx

y

z

u

(b)

Coun

t

10ndash1

100

101

102

103

104

0 100

x

y

z

u

200 300 400 500 600t

(c)

1105

11

minus04minus02

002

1

15

2

25

3

uy

x

(d)


Complexity 15


(83)


F1 (0 1575 6909 4903 16) (84)


(85)








002040608

112141618

2

StableUnstable


I

II

04 05 06 07 08 09 10465θ

σ3

(a)

04

045205

06

07

08

09

Stable

UnstableIV

θ

σ3

0 02 04 06 08 1

III

θ = 03616σ3 + 0452

(b)

0

22

50

100

1375150

H

0 05 0597 0703 0792 09F0

F1

X

θ70 θ71 θ73 θ

Flowast

(c)

H H H H H

H

0

50

100

1501375

X

θ03 04 05 06 07 08 09 1

(d)


16 Complexity






Nlowast

(1156 159 754 499 2) (86)



(87)



Nlowast

(77887 1381 15936 9698 2) (88)

ndash10

2

4

6

8

10

Coun

t

0 100 200 300 400 500 600

tx

y

z

v

u

(a)

Coun

t

0 100 200 300 400 500 600

tx

y

z

v

u

10ndash4

100

10ndash2

104

102

(b)

Coun

t

0 100 200 300 400 500 600

tx

y

z

v

u

10ndash1

101

100

102

(c)

076078

08minus01

001

02

05

1

15

2

25

uy

x

(d)


Complexity 17



(89)



α αc1δ2 + σ2c2( 1113857

0055σ1 + 02647( 1113857 1366σ3 + 1707( 1113857

(90)



N0 (0 83 6909 7505 2) (91)


(92)



N0 (0 0886 6909 432 2) (93)






04 05 06 07 08 09 10

02040608

112141618

2


σ3 = 01 σ3 = 05 σ3 = 08

θ

σ1

(a)

005

1

002040608104

05

06

07

08

09

1

11

Unstable

Stable

θ

σ3

σ1

(b)

090727 0791064600

50

709

100

1375150

H

N0

X

θθ8 θ9

Nlowast

(c)

X

0 01 02 03 04 05 06 07 08 09 1

H

H

H

H

(σ1 σ3) = (07 07)

(σ1 σ3) = (04 04)

(σ1 σ3) = (0 04)

(σ1 σ3) = (04 0)

0

50

100

1501375

θ

(d)


18 Complexity









xyz

vu

0

5

10

15

20

Coun

t

20 40 60 80 1000t

(a)

0 100 200 300 400 500 600t

xyz

vu

Coun

t

100

102

101

(b)

xyz

vu

Coun

t

100

103

102

101

100 200 300 400 500 6000t

(c)

0975098

0985099

0

0204

17

18

19

2

21

22

uy

x

(d)


Complexity 19




Data Availability




Acknowledgments


References
















20 Complexity





































Complexity 21


(83)


F1 (0 1575 6909 4903 16) (84)


(85)








002040608

112141618

2

StableUnstable


I

II

04 05 06 07 08 09 10465θ

σ3

(a)

04

045205

06

07

08

09

Stable

UnstableIV

θ

σ3

0 02 04 06 08 1

III

θ = 03616σ3 + 0452

(b)

0

22

50

100

1375150

H

0 05 0597 0703 0792 09F0

F1

X

θ70 θ71 θ73 θ

Flowast

(c)

H H H H H

H

0

50

100

1501375

X

θ03 04 05 06 07 08 09 1

(d)


16 Complexity






Nlowast

(1156 159 754 499 2) (86)



(87)



Nlowast

(77887 1381 15936 9698 2) (88)

ndash10

2

4

6

8

10

Coun

t

0 100 200 300 400 500 600

tx

y

z

v

u

(a)

Coun

t

0 100 200 300 400 500 600

tx

y

z

v

u

10ndash4

100

10ndash2

104

102

(b)

Coun

t

0 100 200 300 400 500 600

tx

y

z

v

u

10ndash1

101

100

102

(c)

076078

08minus01

001

02

05

1

15

2

25

uy

x

(d)


Complexity 17



(89)



α αc1δ2 + σ2c2( 1113857

0055σ1 + 02647( 1113857 1366σ3 + 1707( 1113857

(90)



N0 (0 83 6909 7505 2) (91)


(92)



N0 (0 0886 6909 432 2) (93)






04 05 06 07 08 09 10

02040608

112141618

2


σ3 = 01 σ3 = 05 σ3 = 08

θ

σ1

(a)

005

1

002040608104

05

06

07

08

09

1

11

Unstable

Stable

θ

σ3

σ1

(b)

090727 0791064600

50

709

100

1375150

H

N0

X

θθ8 θ9

Nlowast

(c)

X

0 01 02 03 04 05 06 07 08 09 1

H

H

H

H

(σ1 σ3) = (07 07)

(σ1 σ3) = (04 04)

(σ1 σ3) = (0 04)

(σ1 σ3) = (04 0)

0

50

100

1501375

θ

(d)


18 Complexity









xyz

vu

0

5

10

15

20

Coun

t

20 40 60 80 1000t

(a)

0 100 200 300 400 500 600t

xyz

vu

Coun

t

100

102

101

(b)

xyz

vu

Coun

t

100

103

102

101

100 200 300 400 500 6000t

(c)

0975098

0985099

0

0204

17

18

19

2

21

22

uy

x

(d)


Complexity 19




Data Availability




Acknowledgments


References
















20 Complexity





































Complexity 21






Nlowast

(1156 159 754 499 2) (86)



(87)



Nlowast

(77887 1381 15936 9698 2) (88)

ndash10

2

4

6

8

10

Coun

t

0 100 200 300 400 500 600

tx

y

z

v

u

(a)

Coun

t

0 100 200 300 400 500 600

tx

y

z

v

u

10ndash4

100

10ndash2

104

102

(b)

Coun

t

0 100 200 300 400 500 600

tx

y

z

v

u

10ndash1

101

100

102

(c)

076078

08minus01

001

02

05

1

15

2

25

uy

x

(d)


Complexity 17



(89)



α αc1δ2 + σ2c2( 1113857

0055σ1 + 02647( 1113857 1366σ3 + 1707( 1113857

(90)



N0 (0 83 6909 7505 2) (91)


(92)



N0 (0 0886 6909 432 2) (93)






04 05 06 07 08 09 10

02040608

112141618

2


σ3 = 01 σ3 = 05 σ3 = 08

θ

σ1

(a)

005

1

002040608104

05

06

07

08

09

1

11

Unstable

Stable

θ

σ3

σ1

(b)

090727 0791064600

50

709

100

1375150

H

N0

X

θθ8 θ9

Nlowast

(c)

X

0 01 02 03 04 05 06 07 08 09 1

H

H

H

H

(σ1 σ3) = (07 07)

(σ1 σ3) = (04 04)

(σ1 σ3) = (0 04)

(σ1 σ3) = (04 0)

0

50

100

1501375

θ

(d)


18 Complexity









xyz

vu

0

5

10

15

20

Coun

t

20 40 60 80 1000t

(a)

0 100 200 300 400 500 600t

xyz

vu

Coun

t

100

102

101

(b)

xyz

vu

Coun

t

100

103

102

101

100 200 300 400 500 6000t

(c)

0975098

0985099

0

0204

17

18

19

2

21

22

uy

x

(d)


Complexity 19




Data Availability




Acknowledgments


References
















20 Complexity





































Complexity 21



(89)



α αc1δ2 + σ2c2( 1113857

0055σ1 + 02647( 1113857 1366σ3 + 1707( 1113857

(90)



N0 (0 83 6909 7505 2) (91)


(92)



N0 (0 0886 6909 432 2) (93)






04 05 06 07 08 09 10

02040608

112141618

2


σ3 = 01 σ3 = 05 σ3 = 08

θ

σ1

(a)

005

1

002040608104

05

06

07

08

09

1

11

Unstable

Stable

θ

σ3

σ1

(b)

090727 0791064600

50

709

100

1375150

H

N0

X

θθ8 θ9

Nlowast

(c)

X

0 01 02 03 04 05 06 07 08 09 1

H

H

H

H

(σ1 σ3) = (07 07)

(σ1 σ3) = (04 04)

(σ1 σ3) = (0 04)

(σ1 σ3) = (04 0)

0

50

100

1501375

θ

(d)


18 Complexity









xyz

vu

0

5

10

15

20

Coun

t

20 40 60 80 1000t

(a)

0 100 200 300 400 500 600t

xyz

vu

Coun

t

100

102

101

(b)

xyz

vu

Coun

t

100

103

102

101

100 200 300 400 500 6000t

(c)

0975098

0985099

0

0204

17

18

19

2

21

22

uy

x

(d)


Complexity 19




Data Availability




Acknowledgments


References
















20 Complexity





































Complexity 21









xyz

vu

0

5

10

15

20

Coun

t

20 40 60 80 1000t

(a)

0 100 200 300 400 500 600t

xyz

vu

Coun

t

100

102

101

(b)

xyz

vu

Coun

t

100

103

102

101

100 200 300 400 500 6000t

(c)

0975098

0985099

0

0204

17

18

19

2

21

22

uy

x

(d)


Complexity 19




Data Availability




Acknowledgments


References
















20 Complexity





































Complexity 21




Data Availability




Acknowledgments


References
















20 Complexity





































Complexity 21





































Complexity 21

Research Article Mathematical Modelling of the Inhibitory Role ...

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Transcript of Research Article Mathematical Modelling of the Inhibitory Role ...