ARTICLE IN PRESS

0022-5193/$ - se

doi:10.1016/j.jtb

�CorrespondE-mail addr

Journal of Theoretical Biology 246 (2007) 18–27

www.elsevier.com/locate/yjtbi

Instabilities in multiserotype disease models withantibody-dependent enhancement

Lora Billingsa,� , Ira B. Schwartzb, Leah B. Shawb, Marie McCrarya,Donald S. Burkec, Derek A.T. Cummingsc

aDepartment of Mathematical Sciences, Montclair State University, Montclair, NJ 07043, USAbUS Naval Research Laboratory, Code 6792, Nonlinear Systems Dynamics Section, Plasma Physics Division, Washington, DC 20375, USA

cJohns Hopkins Bloomberg School of Public Health, Department of International Health, Baltimore, MD 21205, USA

Received 31 March 2006; received in revised form 18 October 2006; accepted 15 December 2006

Available online 28 December 2006

Abstract

This paper investigates the complex dynamics induced by antibody-dependent enhancement (ADE) in multiserotype disease models.

ADE is the increase in viral growth rate in the presence of immunity due to a previous infection of a different serotype. The increased

viral growth rate is thought to increase the infectivity of the secondary infectious class. In our models, ADE induces the onset of

oscillations without external forcing. The oscillations in the infectious classes represent outbreaks of the disease. In this paper, we derive

approximations of the ADE parameter needed to induce oscillations and analyze the associated bifurcations that separate the types of

oscillations. We then investigate the stability of these dynamics by adding stochastic perturbations to the model. We also present a

preliminary analysis of the effect of a single serotype vaccination in the model.

r 2007 Elsevier Ltd. All rights reserved.

Keywords: Epidemics; Multiserotype; Antibody-dependent enhancement; Dengue; Vaccine

1. Introduction

As we become more sophisticated in our resources tofight disease, pathogens become more resilient in theirmeans to survive. One alarming trend is the rapid evolutionof new antigenic strains that stay one step ahead ofvaccinations. As clearly demonstrated in influenza, it isdifficult to quickly develop a vaccine that protects againstthe newest strain. Similarly, it can be difficult to developone vaccine protecting against several co-circulatingstrains. An additional challenge is posed by viruses thatexhibit antibody-dependent enhancement (ADE). ThisADE effect is the increase in the viral growth rate in asecondary infection after recovery from a primary infectionof a different disease serotype. Dengue and otherenveloped viruses have been shown to exhibit ADE invitro, as reported in Burke (1992) and Ferguson et al.

e front matter r 2007 Elsevier Ltd. All rights reserved.

i.2006.12.023

ing author. Tel.: +1973 655 7812.

ess: billingsl@mail.montclair.edu (L. Billings).

(1999b). Due to ADE, in theory, a vaccine developedagainst only one serotype could increase the infectiousnessof an individual upon infection of second serotype.The importance of understanding ADE is underscored by

the pandemic status of the multiserotype disease dengue,with no vaccine available to protect against its fourserotypes. Each year, tens of millions of dengue fever casesoccur throughout the world. Additionally, hundreds ofthousands of cases occur in a more severe form calleddengue hemorrhagic fever (DHF) (CDC website, 2006). Thecase-fatality rate of DHF is believed to be at least 2.5% butcould be reduced to 1% with proper medical treatment(WHO website, 2006). We note that DHF has been found ininfections of all serotypes, but has been associated with someserotypes more than others in certain locations, and withcertain strains within serotypes more than others. Overall,DHF is reported to occur more frequently among indivi-duals experiencing a secondary infection (Nisalak et al.,2003). Therefore, administering a vaccine against one specificserotype could possibly increase the incidence of DHF.

ARTICLE IN PRESSL. Billings et al. / Journal of Theoretical Biology 246 (2007) 18–27 19

Because a tetravalent dengue vaccine is not expected tobecome available for at least 5–10 years, the CDC urges thedevelopment of a surveillance system that provides earlywarning of an impending dengue epidemic (CDC website,2006). Understanding the dynamics of a mathematicalmodel of a multiserotype disease with ADE could help inthis effort. Foundational work on ADE factors can befound in Ferguson et al. (1999a,b), Cummings et al. (2005),and Schwartz et al. (2005). Similarly, we can followapproaches used in previous dengue models withoutADE, as in Esteva and Vargas (2000, 2003). Here, weexamine a dynamical system with a general number of co-circulating serotypes and an ADE factor that modifies thetransmissibility of secondary infections, providing deriva-tions of and extending previous work in Cummings et al.(2005) and Schwartz et al. (2005). As we begin tounderstand the complexity of the dengue model, we gaina better perspective in formulating optimal vaccinationstrategies. Dengue has become a severe and intractablepublic health problem in Thailand (Nisalak et al., 2003),and we hope that models such as this will aid in theprediction and prevention of future outbreaks. While weuse parameters for dengue, the model is general enough tobe considered for other multistrain or multiserotypediseases.

We begin with a detailed derivation of the model inSection 2. Section 3 continues with a description of thegeneral dynamics. A detailed analysis of the bifurcationsfrom steady state to oscillatory dynamics is in Section 4,and the bifurcation to chaos is in Section 5. We then studythe robustness of these dynamics by adding stochasticperturbations to the system in Section 6. In Section 7, wedescribe the effects of vaccinations against select serotypes.Finally, we discuss our results in Section 8.

2. Description of general n-serotype model

We begin by describing the important characteristics ofdengue epidemiology that we wish to model. In SoutheastAsia, four serotypes of dengue have circulated for at least50 years, while their reemergence in the Americas has beenmuch more recent (Gubler, 1998). Primary infection withany one serotype confers immunity to that serotype but notto the others. Since tertiary infections are rare, we assumethat individuals are immune to all four serotypes after twosequential serotype infections (Nisalak et al., 2003). It isalso hypothesized that secondary infections carry a higherviral load, causing that person to be more infectious due toADE (CDC website, 2006). We design our model to usethese properties, but for general n serotypes so that we canevaluate the advantage of developing additional serotypes.Therefore, we follow the n-serotype susceptible-infected-recovered (SIR) model formulation, similar to the onesderived by Ferguson et al. (1999a), and limit the totalnumber of infections to two.

We use a compartmental model assuming the populationis constant, so the variables represent percentages of the

total. We denote the percentage of the population that issusceptible to all serotypes at time t by sðtÞ. People enterthis group at a birth rate of m. Keeping the total populationconstant, we set the death rate md equal to the birth rate munless otherwise noted. We assume that death is equallyprobable from all compartments, since dengue infectiondoes not have a high mortality rate. Even though the raresevere form of the disease has mortality in approximately1–5% of cases, the overwhelming majority of dengueinfections do not result in death.Dengue is transmitted by mosquito, but the time scale

for transmission is sufficiently short and we assume themosquito density is sufficiently dense that we model it by amass action term, from person to person. The percentageof the population with a first (primary) infection by virusserotype i is represented by xiðtÞ. Similarly, the percentageof the population with a secondary infection by virusserotype j, but previously recovered from serotype i ðiajÞ,is represented by xijðtÞ. There are n primary and nðn� 1Þsecondary infectious groups.The rates at which susceptibles are infected by the

primary infected groups are denoted by the terms bisxi,where bi is a measure of infectivity of serotype i. Weassume that the rates at which susceptibles are infected bysecondary infected groups are higher because of ADE. Wedenote these rates in the model by the terms fjbjsxij , wherefj is the ADE factor for serotype j. The newly infectedindividuals move to the associated primary infected group.People in the primary infectious groups recover at a rate

of sxi. The newly recovered individuals move to theassociated primary recovered group riðtÞ, which representsthe percentage of the population recovered from a first(primary) infection by virus serotype i. Each primaryrecovered group is susceptible to all serotypes except itsprevious infection i. So, we model the disease transmissionin a similar way to that of the susceptibles. The newlyinfected individuals from the primary recovered groupsmove to the associated secondary infected groups.People in the secondary infectious groups recover at a

rate of sxij and move to one large secondary recoveredgroup r�ðtÞ. This group is effectively removed from thesystem, since the members are now assumed to be immuneto all serotypes. A description of how an individual wouldproceed through the model for two serotypes is shown by aflowchart in Fig. 1.To generalize the model, we can extend the total number

of serotypes to n. The number of compartments for a viruswith n serotypes and two possible infections is n2 þ nþ 2,including the fully recovered class. We write the generalmodel in this compact form for i; j ¼ 1; . . . ; n:

ds

dt¼ m� s

Xi

bixi þ fibi

Xjai

xji

!� mds,

dxi

dt¼ bisxi þ fibis

Xjai

xji � sxi � mdxi,

ARTICLE IN PRESS

β φ βsx + sx1 1 1 1 21

μs

12β φ βsx + sx

22 2 2

1x

1r

1r

x12

x12

2x

2r

2r

21x

21x

1xμ

2xμ

1xσ 2

xσ

*r

*r

μ

s

β φβr x + r x22

β φ βr x + r x1 121 1 2112 12 21

μ

μ

μ

μ

σ12

x σ21

x

μ

Fig. 1. A flow chart of how an individual would proceed through the

model for two serotypes. Notice that the ADE factors (fi) increase the

infectivity of the groups on the secondary infected level. Natural death

terms are also included for all compartments.

L. Billings et al. / Journal of Theoretical Biology 246 (2007) 18–2720

dri

dt¼ sxi � ri

Xjai

bjxj þ fjbj

Xkaj

xkj

!� mdri,

dxij

dt¼ bjrixj þ fjbjri

Xkaj

xkj � sxij � mdxij,

dr�

dt¼X

i;jðiajÞ

sxij � mdr�. ð1Þ

We can also generalize the total possible number ofinfections an individual can contract up to n. The num-ber of equations for n serotypes and n infections wouldincrease to

2þ n!þ 2Xn�1m¼1

n!

ðn�mÞ!(2)

if we continued to record the order of serotype infectionand include the fully recovered class. In this case, it wouldbe more efficient to form an index-set notation of previousinfections, similar to the approach of Andreasen et al.(1997).

Notice that the ADE factors increase the infectivity ofthe groups on the secondary infected level. They arerepresented by the parameters fi for i ¼ 1; . . . ; n, so thatfi41. This is in contrast to previous models describing theeffect of cross-immunity, or the decrease in susceptibilityfor recovered individuals. Also, cross-immunity factors arebounded between zero and one, since they decreasesusceptibility. Detailed studies of these systems can befound in Castillo-Chavez et al. (1989), Andreasen et al.(1997), Dawes and Gog (2002), and Abu-Raddad andFerguson (2005). While the steady states and reducedsystem analysis for the general cross-immunity model are

similar to the ADE model, the ADE model exhibitsinteresting oscillatory and desynchronization behavior thatis not found in cross-immunity model. For completeness,we continue with a description of the basic features of thegeneral system in Eq. (1).

3. General dynamics of the system

The known steady states of Eq. (1) can be described asdisease free and endemic equilibria. The disease freeequilibrium is the trivial case where no disease is present.The entire population is susceptible, and all othercompartments are empty. Its stability is determined bythe basic reproduction number, or the maximum of thespectrum of the associated next generation matrix. We candefine the basic reproduction number for each serotype as

Ri ¼bi

mþ s, (3)

resulting in a basic reproduction number for the system ofR0 ¼ maxi¼1;...;n Ri. When R041, the disease free equili-brium is unstable. Note that this value does not depend onthe ADE factor.There are other boundary equilibria, defined by the die

out of one or more serotypes. For example, we analyze theother cases for n ¼ 2. The equilibrium that represents thesurvival of serotype 1 and the die out of serotype 2 is

ðs;x1;x2; r1; r2;x21;x12Þ

¼sþ mb1

;mðb1 � s� mÞb1ðsþ mÞ

; 0;sðb1 � s� mÞb1ðsþ mÞ

; 0; 0; 0

� �. ð4Þ

Note that this boundary equilibrium does not depend onthe ADE factor. For the stability analysis, we evaluate theJacobian of the system at the boundary equilibrium to findthe following set of eigenvalues:

�m;�s� m;�b1mmþ s

;�b1m�

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffib21m2 � 4mðsþ mÞ2o

q2ðsþ mÞ

;

8<:ðsþ mÞ2ðb2 � b1Þ þ b2sf2o

b1ðsþ mÞ

9=;, ð5Þ

with o ¼ b1 � m� s. The equilibrium is stable for para-meters that keep all eigenvalues negative or have negativereal part. This is true when R141 and

R2oR1

1þ ðsf2=ðsþ mÞÞðR1 � 1Þ, (6)

as defined in Eq. (3). A similar analysis can be performedfor the remaining boundary equilibrium, when onlyserotype 2 survives. In the symmetric case with R1 ¼ R2

(or b1 ¼ b2), neither boundary equilibrium will ever bestable.

ARTICLE IN PRESS

1.85 1.86 1.87 1.88 1.89 1.9 1.91 1.9240

50

60

70

80

90

100

φL 2 N

orm

steady state

1.883 1.884 1.885 1.88642

46

50

54

φ

L2 N

orm

Subcritical Hopf

L. Billings et al. / Journal of Theoretical Biology 246 (2007) 18–27 21

If the disease free equilibrium and both boundary equili-bria are unstable, then there exists a nonzero equilibrium,which we call the endemic equilibrium. In the cases of n42,we can consider the event of one or more serotypes dyingout as the reduced nonzero endemic cases for a system withthe surviving serotypes. We continue the next section withan analysis of the stability of the endemic equilibrium for ageneral n serotypes.

4. The endemic equilibrium

We analyze the symmetric case, defined by setting thecontact rates and ADE factors of all the serotypes equal(bi ¼ b and fi ¼ f, for i ¼ 1; . . . ; n) and the death rateequal to the birth rate (md ¼ m). Using the parameters listedin Table 1 and n ¼ 4, Fig. 2 plots the extrema of a timeseries of length t ¼ 100 years over a range of f. Here, thedisease free and boundary equilibria are not stable. As fincreases, trajectories change from converging to stableendemic equilibrium to complicated oscillatory dynamics.We numerically approximate that a Hopf bifurcationoccurs at f � 1:88333, where the endemic equilibriumloses stability.

Table 1

Model parameters

Parameter Value Reference

m, 1/host life span, years�1 0.02 Ferguson et al. (1999a)

b, transmission coefficient, years�1 400 Ferguson et al. (1999b)

s, recovery rate, years�1 100 Ferguson et al. (1999a)

fi , ADE factor of serotype i 1–5 –

1 1.5 2 2.5 3

0.04

0.06

0.08

0.1

0.12

0.14

0.16

0.18

φ

Susceptible

s (

Extr

em

a)

MinMax

Fig. 2. A bifurcation diagram for the symmetric case of Eq. (1) with

n ¼ 4. Shown is a Poincare section constructed from the extrema of a time

series for susceptibles of length t ¼ 100 years after a transient of t ¼ 5000

years. The minima are denoted by grey points and the maxima are overlaid

in black points.

Fig. 3. A stability diagram of the periodic orbits of Eq. (1) with n ¼ 4 as a

function of f near the Hopf bifurcation. We plot the L2 norm of the orbits

and denote the stable parameter regions by the thick solid curves. See text

for details.

When the endemic equilibrium is stable, the modeldynamics are symmetric. All primary infectious, primaryrecovered, and secondary infectious variables are equalwithin their own classes. For a small window of f beyondthe Hopf bifurcation, the symmetry continues in a perioddoubling bifurcation pattern leading to chaos. We show thetransition from steady state to oscillations by numericallyapproximating the L2 norm of the orbits1 in Fig. 3. Thesteady state is represented by the black curve along thebottom of the graph. Notice that the Hopf bifurcation issubcritical, since the unstable periodic orbit collides withthe steady state as the steady state loses stability. See theinset for a close-up of the bifurcation. The unstableperiodic orbit originates from a saddle-node bifurcation,shown by its connection to another thick black stablebranch. There is a co-existing periodic orbit overlaid ingrey. As the primary branch of the Hopf bifurcation losesstability at f � 1:88557, trajectories approach the greystable periodic orbit through a hysteresis-type bifurcation.As f increases, the co-existing grey periodic orbit under-goes a period doubling route to chaos. The periods of theorbits are plotted in Fig. 4.Using the symmetry between serotypes, we can reduce

the symmetric n-serotype system in Eq. (1) to analyzethe endemic equilibrium and Hopf bifurcation. First, set

1The L2 norm for an orbit zðtÞ over period p is defined as ð1=pR p

0 zðtÞT

zðtÞdtÞ1=2.

ARTICLE IN PRESS

1.85 1.86 1.87 1.88 1.89 1.9 1.91 1.920

5

10

15

20

φ

period

Fig. 4. A graph of the periods (in years) of Eq. (1) with n ¼ 4 as a function

of f near the Hopf bifurcation. The thick parts of the curves denote the

ranges where those periodic orbits are stable.

L. Billings et al. / Journal of Theoretical Biology 246 (2007) 18–2722

the variables in each of the classes equal to define thevariables of a new reduced system: susceptible y1 ¼ s,primary infectious y2 ¼ xi, primary recovered y3 ¼ ri, andsecondary infectious y4 ¼ xij , for i; j ¼ 1; . . . ; n. For n

serotypes, the associated reduced model is

dy1

dt¼ m� nb y1y2 � n ðn� 1Þbf y1y4 � mdy1,

dy2

dt¼ b y1y2 þ ðn� 1Þbfy1y4 � s y2 � mdy2,

dy3

dt¼ sy2 � ðn� 1Þby3y2 � ðn� 1Þ2bfy3y4 � mdy3,

dy4

dt¼ by3y2 þ ðn� 1Þbfy3y4 � sy4 � mdy4. ð7Þ

There is a nice agreement between the full and reducedsystem for values of f near one. In both systems, the steadystate loses stability through a Hopf bifurcation. The limitcycles in the full system quickly increase in period andbecome unstable. The limit cycles in the reduced system donot lose stability, and in comparison, their extrema boundthe full system. Therefore, we will use the reduced systemto analyze only the Hopf bifurcation, and not the chaoticregime for larger values of f.

To analyze the destabilization of the endemic equili-brium, one would ideally linearize about the fixed pointand derive the spectra as a function of f. Due to the lengthand complexity of a direct calculation of the eigenvaluesfor this system, we approach the analysis using perturba-tion theory.2 A small parameter can be introduced bynoticing the following. The parameter m ¼ 0:02 year�1 issmall compared to the parameters s and b; since s and b

2We note that Abu-Raddad and Ferguson (2005) recently derived

recursion formulas for an endemic equilibrium in a similar cross-immunity

system, and that approach might be possible for this ADE model.

are both Oð1=mÞ: Therefore, we rescale the parameters inrelation to m by letting b ¼ b0=m; and s ¼ s0=m, where weassume b0 and s0 are on the order of unity. We also removethe mortality terms in each of the differential equations ofthe reduced system by setting md ¼ 0. Since they areoriginally of OðmÞ; they will be small and have a negligibleeffect on the steady states.The rescaled reduced system has the form

dy1

dt¼ m�

1

mnb0y1y2 �

1

mnðn� 1Þb0fy1y4,

dy2

dt¼

1

mb0y1y2 þ

1

mðn� 1Þb0fy1y4 �

1

ms0y2,

dy3

dt¼

1

ms0y2 �

1

mðn� 1Þb0y3y2 �

1

mðn� 1Þ2b0fy3y4,

dy4

dt¼

1

mb0y3y2 þ

1

mðn� 1Þb0fy3y4 �

1

ms0y4 ð8Þ

and yields the following steady state endemic equilibrium:

ðy1; y2; y3; y4Þ

¼s0

b0ðfþ 1Þ;m2

ns0;

s0b0ðn� 1Þðfþ 1Þ

;m2

nðn� 1Þs0

� �. ð9Þ

We would like to find where the Hopf bifurcation occurs asa function of f and n. We do this by evaluating the stabilityof the steady state solution through linearization. There-fore, we take the Jacobian of the vector field of the reducedmodel about the endemic steady state, and examine thecharacteristic polynomial for the eigenvalues. The char-acteristic polynomial is given by

F ðzÞ ¼ c4z4 þ c3z3 þ c2z

2 þ c1zþ c0, (10)

where the polynomial coefficients are given by c0 ¼ b20ðn� 1Þðfþ 1Þ2=n, c1 ¼ 2b20mðn� 1Þðfþ 1Þ2=s0nþ b0s0=m, c2 ¼

b20m2ðn� 1Þðfþ 1Þ2=s20nþ b0ðð3n� 1Þðfþ 1Þ � fÞ=n, c3 ¼

b0m ð2n� 1Þðfþ 1Þ=s0nþ s0=m, c4 ¼ 1.Since the coefficients c1 and c3 contain terms of order

1=m, we define a rescaled function as

F ðzÞ ¼ mF ðzÞ, (11)

in which the coefficients are now given by mci, i ¼ 0; . . . ; 4.This, in turn, allows us to apply regular perturbationtheory, and we assume the solutions to Eq. (11) are of theform

li ¼ zðmÞ ¼ z0 þ z1mþOðm2Þ. (12)

The second order approximation of the eigenvalues are

l1 ¼ �bðn� 1Þðfþ 1Þ2m

ns,

ARTICLE IN PRESS

l2;3 ¼bmððn� 1Þf2

� nf� nÞ

2ns

� 1þbððn� 1Þð3ðn� 1Þf4

þ 2ð3 n� 4Þf3� 2ðnþ 2Þf2

Þ � n2ðfþ 1Þ2Þm8n2s2

� � ffiffiffiffiffiffibm

pi. ð13Þ

L. Billings et al. / Journal of Theoretical Biology 246 (2007) 18–27 23

To determine the magnitude of the fourth eigenvalue, wenotice from the characteristic polynomial coefficients thatthe solution must have one root which is of order 1=m:Therefore, we define a rescaled function

GðzÞ ¼ m3F ðz=mÞ, (14)

and again consider perturbation theory. The analysis yieldsa solution to GðzÞ ¼ 0 of z0 ¼ �s0 for zeroth order, and thefirst order term is computed to be zero. Changing back tothe original variables, the fourth eigenvalue can beapproximated by a large negative value l4 ¼ �s.

Considering all four eigenvalues of the system, we noticethat l1 and l4 are always negative for n41 and ourparameters. Therefore, the real part of the complexconjugate pair of eigenvalues determines the stability ofthe endemic equilibrium. We can approximate the locationof the Hopf bifurcation by setting the real part of l2;3 ¼ 0.Solving for f, we find the boundary curve

f ¼nþ

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffinð5n� 4Þ

p2ðn� 1Þ

, (15)

which is graphed in Fig. 5. Below the curve, the real part ofl2;3o0 and the endemic equilibrium is stable. Above thecurve, the endemic equilibrium is unstable and solutionsoscillate. We can see that as the number of serotypesincreases, the amount of ADE needed for oscillatingsolutions (outbreaks) decreases. The change in the Hopfpoint decreases as the number of serotypes increases, andthe value of f limits to ð1þ

ffiffiffi5pÞ=2 as n approaches infinity.

Therefore, we can conclude that in this class of multi-serotype systems, the effects of ADE can cause a Hopfbifurcation for any number of serotypes. However, there isnot much advantage in generating a large number of

1 2 3 4 5 6 7 8

0

1

2

3

4

number of serotypes

φ

oscillations

steady state

Fig. 5. The graph of Eq. (15) describing the Hopf bifurcation as a

function of number of serotypes n and ADE factor f. Notice that as the

number of serotypes increases, smaller ADE factors are needed for

oscillatory dynamics.

serotypes if the goal is to lower the ADE factor necessaryto induce oscillatory dynamics. Also, there is a balancebetween the number of co-circulating serotypes and themagnitude of ADE factors that do not induce oscillationsthat may threaten the persistence of each serotype (asdiscussed in Cummings et al., 2005).

5. Chaotic dynamics

Beyond the Hopf bifurcation, a common perioddoubling cascade leads to chaotic dynamics and serotypedesynchronization. We turn to a Lyapunov exponentcalculation to approximate the bifurcation value to chaos.We compute the maximum Lyapunov exponent byintegrating the linear variational equations along solutionsto the full model and using a time series of length t ¼

100; 000 years for a random initial condition, aftertransients are removed. Fig. 6 graphs these numericalapproximations. The maximum exponent is negative forfo1:883, which agrees with our stability analysis of theendemic equilibrium. The maximum exponent is approxi-mately zero for 1:883ofo1:91, which agrees with theperiodic behavior after the Hopf bifurcation shown inFig. 3. Chaotic behavior is indicated by a positivemaximum Lyapunov exponent, which can be found forparameter values in the region f41:91.For values of f just beyond the transition to chaos, the

orbits gain and lose synchronization in a predictablepattern. A typical trajectory will oscillate with the variablessynchronized by class within a slowly increasing envelope.Then, the variables unlock to separate phases for eachserotype before possibly resynchronizing at a later time.We show a time series of all the secondary variablesin Fig. 7 with a window of synchronous behavior.

1.75 1.8 1.85 1.9 1.95 2 2.05-0.02

0

0.02

0.04

0.06

0.08

0.1

0.12

Ma

xim

um

Lya

pu

no

v E

xp

on

en

t

φ

Fig. 6. A graph of the maximum Lyapunov exponent of Eq. (1) with

n ¼ 4 as a function of f. We use a time series of length t ¼ 100; 000 years

for a random initial condition, after transients are removed.

ARTICLE IN PRESS

0 20 40 60 80 100 120-16

-14

-12

-10

-8

-6

-4

time (years)

0 20 40 60 80 100 120

-14

-12

-10

-8

-6

-4

time (years)

log 1

0(x 1

,x2,

x 2,x

4)

log 1

0(x 1

,x2,

x 2,x

4)

Standard deviation = 0.02

Fig. 7. On the left, a graph of a window of synchronous behavior for Eq. (1) with n ¼ 4 and f ¼ 2:12. We plot log10 of the primary infections as a function

of time (years). On the right, the same plot with noise added at a standard deviation of 0.02. Notice the lack of long synchronous behavior before the

lowest amplitude oscillations compared to the deterministic case on the left.

L. Billings et al. / Journal of Theoretical Biology 246 (2007) 18–2724

A numerical approximation of the Lyapunov exponentsover the desynchronized regions indicates that trajectory isexhibiting chaotic behavior. As f increases, the trajectoriesspend more time exhibiting this unlocked behavior and wenote the associated increase in the maximum Lyapunovexponent for the whole time series.

It is interesting to note that within each serotype, themaximum values of each primary and secondary infectedtime series remain synchronized for all f, and we only see avariation in amplitude. Across serotypes, we see phasedifferences that can reorder outbreaks in each burstgroup. In this parameter region, other behaviors are alsopossible. Synchronization can occur between two, three,or all four serotypes. As discussed in Cummings et al. (2005),synchronization can act in a boom-bust fashion, driving thenumbers of infected individuals to greater extremes. If thepercentage of infected individuals is sufficiently low, it couldrepresent the fadeout of one or more serotypes.

6. Adding noise

We continue with an analysis of stochastic perturbationsto the system in Eq. (1) to explore the robustness ofsolutions in this model to the effect of randomness. We addcontinuous multiplicative white noise to each of our diseasestate compartments, so that the randomness is propor-tional to the variable. Equivalently, we include additivenoise in our numerical integration in the natural log space.The derivation of this equivalence follows.

Let Y ¼ fs; xi; ri; xijgi;j¼1;...;n and _Y ¼ F ðYÞ. Define natur-al log coordinates y ¼ fyig, where lnðY iÞ ¼ yi (or Y i ¼ eyi )and _y ¼ f ðyÞ. Let Z be a vector of random variables withmean zero and standard deviation s. We add multiplicativenoise to the original system in the following way:

_Y i ¼ F iðYÞ þ ZiY i. (16)

Standard change of coordinates relates the two systems by_Y i=Y i ¼ _yi, so

_Y i ¼ Y if iðlnYÞ þ ZiY i (17)

and

_y ¼ f ðyÞ þ Z. (18)

Eq. (18) was integrated numerically with additive noiseusing the Milstein method, which is a second orderRunge–Kutta method for adding white noise to ordinarydifferential equations.In Cummings et al. (2005), we reported that for

stochastic perturbations with small standard deviations,fadeouts occur less frequently for values of f ranging from2.0 to 2.5. We hypothesize that in this parameter range, thedecrease is because noise breaks the synchrony of thesolutions. Note that the typical behavior in the time seriesof the infected classes for the deterministic system with f inthe range from 2.0 to 2.5 is a period of synchronousbehavior between serotypes, with regular oscillations ofspreading maximum/minimum amplitudes, followed by aperiod of asynchronous behavior, where one or moreserotypes can experience fadeout dynamics. See the leftgraph in Fig. 7 as an example. The addition of noise in thesystem at these parameter values appears to keep theclasses of components from becoming synchronized inphase and amplitude, so the fadeouts occur less frequentlyand without the precursor of synchronized oscillations, asshown in the right graph of Fig. 7.We also note that in the larger parameter regions of f,

the deterministic system does not typically undergosynchronization before fadeouts. In Cummings et al.(2005) we reported that for stochastic perturbations withsmall standard deviations, fadeouts occur more frequentlyfor values of f between 2.5 and 3.0. Recall from the Hopfbifurcation analysis that the underlying unstable periodicorbits in the system have increasingly larger periods as weincrease f. We hypothesize that the addition of noise couldinduce solutions to follow these paths for periods of time,with their extreme maximum/minimum amplitudes, andincrease the occurrence of fadeouts.

7. Vaccinations

While a few candidate vaccines are under development,an effective licensed tetravalent vaccine against dengue willprobably not be available for at least 5–10 years (CDCwebsite, 2006). Though it is unclear whether successfulvaccination against one serotype would place individuals atrisk of severe disease in the same way that a natural

ARTICLE IN PRESSL. Billings et al. / Journal of Theoretical Biology 246 (2007) 18–27 25

infection does, there is significant concern that this is thecase. Therefore, vaccines that do not provide solidimmunity to all four serotypes are not considered seriouscandidates for licensure (CDC website, 2006). However, itis clear that even in more advanced vaccines, there isconcern that some vaccine failure might occur, potentiallyputting recipients at risk of severe disease upon exposure toserotypes from which vaccine-derived immunity not onlydoes not protect, but enhances their infection.

Here, we consider the impact of vaccine failure in theextreme by considering the impact of a monotypic vaccinethat grants full immunity to one serotype but does notgrant immunity to the other three serotypes. As an exampleof this sort of vaccine, Steven Whitehead and colleagues atthe Laboratory for Infectious Diseases, National Institutesof Health, Bethesda, Maryland have developed separatevaccine candidates for each of the four serotypes (Halsteadet al., 2005). We assume that the immunity granted by thisvaccine is similar to natural infection in its potential toprovide antibody-dependent enhancement. However, weassume that one natural infection following vaccination isnot sufficient to create immunity to all serotypes. Instead,individuals may become infected with two serotypesnaturally following vaccination. Using our modelingframework, we compare the benefit of reducing oneserotype in the population through vaccination to thepotential increase in secondary infections due to theimmunity granted to vaccine recipients.

To investigate the effects of implementing a single-serotype vaccination campaign, we assume that the entirepopulation has been vaccinated against serotype four(arbitrarily chosen) and that it has 100% efficacy. Becausethere will be no infections of that serotype, we can removethe compartments associated with that serotype from thegeneral model. The major difference in this model is thatthe force of infection by primary infected groups nowincludes the ADE factor. We write the general model inthis compact form for i; j ¼ 1; . . . ; n� 1:

ds

dt¼ m� s

Xi

fibi xi þXjai

xji

!� mds,

dxi

dt¼ fibis xi þ

Xjai

xji

!� sxi � mdxi,

dri

dt¼ sxi � ri

Xjai

fjbj xj þXkaj

xkj

!� mdri,

dxij

dt¼ fjbjri xj þ

Xkaj

xkj

!� sxij � mdxij. ð19Þ

For md ¼ m, the disease free equilibrium of Eq. (19) issimilar to the non-vaccination model. The entire popula-tion is susceptible and all other compartments are empty.Again, its stability is determined by a basic reproductionnumber, or the maximum of the spectrum of the associatednext generation matrix. We can define the basic reproduc-

tion number for each serotype as

Ri ¼bifi

mþ s, (20)

resulting in a basic reproduction number for the system ofR0 ¼ maxi¼1;...;n Ri. When R0o1, the disease free equili-brium is stable and the disease dies out.As we increase fi, the disease free equilibrium undergoes

a transcritical bifurcation and exchanges stability with anendemic equilibrium. For the symmetric case of all bi ¼ band fi ¼ f, the steady state is similar to the non-vaccination model in that when the endemic equilibriumis stable, the model dynamics are synchronized, orsymmetric, with respect to serotype. In other words, allprimary infectious, primary recovered, and secondaryinfectious variables equal within their own classes. Dueto the vaccination, the reduced system for n serotypes nowhas n� 1 serotypes, and the system can be generalized as

dz1

dt¼ m� ðn� 1Þbfz1z2 � ðn� 1Þðn� 2Þbfz1z4 � mdz1,

dz2

dt¼ bfz1z2 þ ðn� 2Þbfz1z4 � sz2 � mdz2,

dz3

dt¼ sz2 � ðn� 2Þbfz3z2 � ðn� 2Þ2bfz3z4 � mdz3,

dz4

dt¼ bfz3z2 þ ðn� 2Þbfz3z4 � sz4 � mdz4. ð21Þ

We can approximate the endemic equilibrium for thevaccinated system using the reduced non-mortality modelðmd ¼ 0Þ:

ðz1; z2; z3; z4Þ

¼s

2bf;

mðn� 1Þs

;s

2ðn� 2Þbf;

mðn� 1Þðn� 2Þs

� �. ð22Þ

For the parameters we have been addressing, we can shownumerically by linearizing the system about this equili-brium that it is locally stable for 1pfp5.We would like to compare this full-vaccination model

equilibrium to the equilibrium of the non-vaccinationmodel (with no-mortality) from Eq. (9). In originalparameters it has the form

ðy1; y2; y3; y4Þ

¼s

bðfþ 1Þ;m

ns;

sbðn� 1Þðfþ 1Þ

;m

nðn� 1Þs

� �. ð23Þ

While one could argue that the total number of infected aredecreased in the vaccination model, since z2 þ z4oy2 þ y4,we are interested in the total number of secondaryinfections, which has the higher incidence of the moresevere form of dengue, DHF. Both infectious classes ofthe vaccination model are considered secondary, becausethe vaccination was equivalent to a primary infection.Therefore, we find y4oz2 þ z4 because 0on2 � 2nþ 2 for

ARTICLE IN PRESSL. Billings et al. / Journal of Theoretical Biology 246 (2007) 18–2726

all n 2 R, and conclude that single-serotype vaccinationscould potentially increase the number of DHF cases forADE values associated with an endemic steady-state in thenon-vaccination model (before the Hopf bifurcation).

We note that single serotype vaccines have been tested inpopulations with no exposure to dengue and found to have53–100% seroconversion rates (Sun et al., 2003). No single-serotype vaccines are being evaluated among persons atrisk of exposure to dengue nor being designed for routineuse because it is assumed that this would have a negativeimpact on health in any setting. Our quantitative modelfinds supporting evidence for this assumption.

8. Conclusions

In this paper, we have analyzed the dynamics of amultiserotype disease model with antibody-dependentenhancement. The model considered consists of compart-ments describing the fractions of susceptible, primary andsecondary infectious, and recovered individuals. Althoughconsidered primarily for four serotypes and two sequentialinfections, the model is easily generalized to an arbitrarynumber of serotypes and infections.

We focus on the impact of the ADE parameter in thedynamics. Since ADE is modeled as an enhancement ofmass action contacts in the secondary infectious rate, it isexpected to play an important role in oscillatory onset ofoutbreaks. Indeed, we analyze quantitatively and numeri-cally the sub-critical Hopf bifurcation from an endemicsteady state to oscillatory dynamics. From the analysis, it isshown that the underlying period is quite sensitive tochanges in ADE, and the window of parameter values forwhich stable periodic behavior occurs is quite small. Atransition to chaotic, desynchronized behavior occurs forlarger values of ADE and is robust over a very large rangeof parameter values, with occasional small windows ofperiodic behavior.

One interesting numerical finding is that there exist twodistinct branches of periodic orbits which have slightlyoverlapping parameter values and which exhibit bi-stablesolutions, as shown in Fig. 4. One branch occurs as a Hopfbifurcation, and the other branch appears as a saddle-nodebifurcation. The stable branch of the saddle-node bifurca-tion has roughly 2.5 times the period of the Hopf branch.Since it is bi-stable over a small region, as ADE is changed,the dynamics may exhibit hysteresis, in which it switchesfrom one type of behavior to another.

Beyond the range of ADE values exhibiting synchro-nized periodic dynamics, there exists a complicated set ofchaotic oscillations, which is comprised of both synchro-nous and asynchronous outbreaks of the multiple sero-types. In the case where the dynamics is deterministic,synchronized behavior is in the form of an unstable focus,where the primary infectives grow in amplitude until somecritical value, at which the infectives desynchronize andexhibit long period, large amplitude oscillations. It ispossible that such synchronized behavior might be used to

predict a probability of fadeout of one or more serotypes.However, at present, there is no evidence in the data whichpoints to such an indicator.In addition to the deterministic case, we made a

preliminary study of the effects of stochastic perturbationson the population. For small noise in the chaotic region,the dynamics exhibits asynchronous behavior most of thetime, with similar increasing infectives as in the determi-nistic case. However, it is possible that stochastic perturba-tions can either increase or decrease the probability offadeout, depending on the deterministic behavior of thesystem, as well as the ADE value.Vaccination strategies need to be considered carefully to

avoid an unintended effect. Since ADE is a function ofsecondary contact rates, the effect of vaccination on oneserotype is expected to have a significant role in theasymptotic dynamics. If one vaccinates against a singleserotype, the other infectious serotype classes all havecontacts which are enhanced due to the ADE secondaryinfection hypothesis. In the endemic equilibrium, weshowed that the total number of secondary cases is greaterduring the use of our modeled monotypic vaccination andconjecture that it is possible to potentially increase thenumber of secondary infections at steady state.We hypothesize that the dynamics for some ADE

parameter value greater than 1.91 (the onset of chaoticoscillations) could qualitatively agree with the Bangkokdata reported in Nisalak et al. (2003). Vaccination schemesagainst all four serotypes can be tested with this model, andoptimal strategies could be determined. Because of thesensitivity to the ADE parameter, we conclude that onewould have to prove the strategy robust over a range ofparameters, with and without stochastic perturbations.More detailed studies of vaccination strategies in thismodel are the subject of a future paper.One interesting aspect of the multiserotype dynamics is

the existence of very large fluctuations which lead to one ormore serotypes possibly becoming extinct. Since thedynamics exhibits these large fluctuations in chaotic cases,there must be a large amplitude, long-period unstable orbitwhich attracts the dynamics by way of its stable manifold.For the unstable periodic orbits having long periods, it wasseen that they also possess large amplitudes which, in thelog variables, correspond to small fractions in the original‘‘fraction’’ variables. Since the unstable orbits of longperiods exist over almost all parameters of interest, it isquite possible they are the orbits which have a highprobability of leading the dynamics to fadeout. Furtherstudies involving the underlying unstable orbits will bestudied in a future paper.

Acknowledgments

L.B. and M.M. were supported by the National ScienceFoundation under Grants DMS-0414087 and CTS-0319555. L.B.S. and I.B.S. are supported by the Office ofNaval Research, Center for Army Analysis, and the Armed

ARTICLE IN PRESSL. Billings et al. / Journal of Theoretical Biology 246 (2007) 18–27 27

Forces Medical Intelligence Center. L.B.S. is currently aNational Research Council post doctoral fellow. D.C. andD.B. were supported by the National Institute of GeneralMedical Sciences of the National Institutes of HealthGrant U01-GM070749-01 and a consortium of theNational Oceanic and Atmospheric Administration, theEnvironmental Protection Agency, the National Aeronau-tics and Space Administration, and the National ScienceFoundation from Grant NA04OAR4310138.

References

Abu-Raddad, L.J., Ferguson, N.M., 2005. Characterizing the symmetric

equilibrium of multi-strain host-pathogen systems in the presence of

cross immunity. J. Math. Biol. 50, 531–558.

Andreasen, V., Lin, J., Levin, S.A., 1997. The dynamics of cocirculating

influenza strains conferring partial cross-immunity. J. Math. Biol. 35,

825–842.

Burke, D.S., 1992. Human HIV vaccine trials: does antibody-dependent

enhancement pose a genuine risk? Perspect. Biol. Med. 35, 511–530.

Castillo-Chavez, C., Hethcote, H.W., Andreasen, V., Levin, S.A., Liu,

W.M., 1989. Epidemiological models with age structure, proportionate

mixing, and cross-immunity. J. Math. Biol. 27, 233–258.

CDC website, 2006. hhttp://www.cdc.gov/ncidod/dvbid/dengue/i.

Cummings, D.A.T., Schwartz, I.B., Billings, L., Shaw, L.B., Burke, D.S.,

2005. Dynamic effects of antibody dependent enhancement on the

fitness of viruses. Proc. Natl Acad. Sci. USA 102, 15259–15264.

Dawes, J.H.P., Gog, J.R., 2002. The onset of oscillatory dynamics in

models of multistrain diseases. J. Math. Biol. 45, 471–510.

Esteva, L., Vargas, C., 2000. Influence of vertical and mechanical

transmission on the dynamics of dengue disease. Math. Biosci. 167,

51–64.

Esteva, L., Vargas, C., 2003. Coexistence of different serotypes of dengue

virus. J. Math. Biol. 46, 31–47.

Ferguson, N.M., Anderson, R., Gupta, S., 1999a. The effect of antibody-

dependent enhancement on the transmission dynamics and persistence

of multiple-strain pathogens. Proc. Natl Acad. Sci. USA 96, 790–794.

Ferguson, N.M., Donnelly, C.A., Anderson, R.M., 1999b. Transmission

dynamics and epidemiology of dengue: insights from age-stratified

sero-prevalence surveys. Phil. Trans. R. Soc. London, Ser. B 354,

757–768.

Gubler, D.J., 1998. Dengue and dengue hemorrhagic fever. Clin.

Microbiol. Rev. 11, 480–496.

Halstead, S.B., Heinz, F.X., Barrett, A.D.T., Roehrig, J.T., 2005. Dengue

virus: molecular basis of cell entry and pathogenesis, 25–27 June 2003,

Vienna, Austria. Vaccine 23, 849–856.

Nisalak, A., Endy, T.P., Nimmannitya, S., Kalayanarooj, S., Thisaya-

korn, U., Scott, R.M., Burke, D.S., Hoke, C.H., Innis, B.L., Vaughn,

D.W., 2003. Serotype-specific dengue virus circulation and dengue

disease in Bangkok, Thailand from 1973 to 1999. Am. J. Trop. Med.

Hyg. 68, 191–202.

Schwartz, I.B., Shaw, L.B., Cummings, D.A.T., Billings, L., McCrary, M.,

Burke, D., 2005. Chaotic desynchronization of multi-strain diseases.

Phys. Rev. E 72, art. no. –066201.

Sun, W., Edelman, R., Kanesa-Thasan, N., Eckels, K.H., Putnak, J.R.,

King, A.D., Houng, H.-S., Tang, D., Scherer, J.M., Hoke Jr., C.H.,

Innis, B.L., 2003. Vaccination of human volunteers with monovalent

and tetravalent live-attenuated dengue vaccine candidates. Am. J.

Trop. Med. Hyg. 69, 24–31.

WHO website, 2006. hhttp://www.who.int/mediacentre/factsheets/fs117/en/i.