Journal of Theoretical Biology 245 (2007) 9–25

Tracking the dynamics of pathogen interactions: Modeling ecologicaland immune-mediated processes in a two-pathogen single-host system

Daniel A. Vascoa,!, Helen J. Wearinga, Pejman Rohania,b

aInstitute of Ecology, University of Georgia, Athens, GA 30602, USAbCenter for Tropical and Emerging Global Diseases, University of Georgia, Athens, GA 30602, USA

Received 16 November 2005; received in revised form 17 July 2006; accepted 21 August 2006Available online 30 August 2006

Abstract

Traditionally, epidemiological studies have focused on understanding the dynamics of a single pathogen, assuming no interactionswith other pathogens. Recently, a large body of work has begun to explore the effects of immune-mediated interactions, arising fromcross-immunity and antibody-dependent enhancement, between related pathogen strains. In addition, ecological processes such as atemporary period of convalescence and pathogen-induced mortality have led to the concept of ecological interference between unrelateddiseases. There remains, however, the need for a systematic study of both immunological and ecological processes within a singleframework. In this paper, we develop a general two-pathogen single-host model of pathogen interactions that simultaneouslyincorporates these mechanisms. We are then able to mechanistically explore how immunoecological processes mediate interactionsbetween diseases for a pool of susceptible individuals. We show that the precise nature of the interaction can induce either competitive orcooperative associations between pathogens. Understanding the dynamic implications of multi-pathogen associations has potentiallyimportant public health consequences. Such a framework may be especially helpful in disentangling the effects of partially cross-immunizing infections that affect populations with a pre-disposition towards immunosuppression such as children and the elderly.r 2006 Elsevier Ltd. All rights reserved.

Keywords: Infectious disease modeling; Ecological interference; Immune-mediated processes

1. Introduction

Outbreaks of diseases such as SARS, West Nile Virusand avian influenza have alerted us to the potentially gravepublic health threat from emerging and re-emergingpathogens (Cyranoski, 2001; Lipsitch et al., 2003; Mack-enzie et al., 2004). However, understanding the persistence,spread and evolution of extant pathogens also remains asignificant challenge. Historically, our understanding ofdisease-host systems has been built upon a population-based rather than community-based approach: epidemiol-ogists have traditionally studied infectious disease at thelevel of a single pathogen species infecting a single-hostspecies. This has led to a significant body of epidemiolo-gical research devoted to understanding the precisemechanisms underlying host–pathogen dynamics in isola-

tion. In the case of novel emerging pathogens, this isperhaps a reasonable study unit. In the case of establishedpathogens, however, it may well be an oversimplification ofthe web of ecological and immunological interactions inwhich hosts and their pathogens persist and evolve.During the past several years, single-host single-patho-

gen approaches have been extended to incorporate multiplehosts (Hudson and Greenman, 1998; Gog et al., 2002;Dobson, 2004) and multiple pathogens (Gupta et al., 1994;Ferguson et al., 2003). Much of the work on multi-hostsystems has focused on how parasites can shape hostcoexistence dynamics by mediating ‘‘apparent’’ competi-tion between host species (Holt, 1977; Tomkins et al.,2001). However, the area of community epidemiology thathas received most attention has been the dynamics ofmulti-strain pathogens, in part due to the number andimportance of microparasites with well-established anti-genic polymorphism, such as influenza, malaria, theadenoviruses, poliovirus and cholera (Andreasen et al.,

ARTICLE IN PRESS

www.elsevier.com/locate/yjtbi

0022-5193/$ - see front matter r 2006 Elsevier Ltd. All rights reserved.doi:10.1016/j.jtbi.2006.08.015

!Corresponding author. Tel.: +17065423580.E-mail address: vasco@uga.edu (D.A. Vasco).

1997; Gupta et al., 1998; Read and Taylor, 2001; Earnet al., 2002; Koelle et al., 2005). In such systems, it isempirically documented that different strains typicallyfluctuate out-of-phase with each other; epidemics of onestrain do not generally coincide with epidemics of another.Multi-strain interactions have been studied on two distinctscales: within-host immune-mediated interactions such assuperinfection and coinfection (Nowak and May, 1994;Kirschner, 1999); and between-host immune-mediatedinteractions such as cross-immunity and antibody-depen-dent enhancement (Ferguson et al., 1999; Cummings et al.,2005). In particular, the significance of partial cross-immunity in shaping the coexistence, dynamics andevolution of strains has been at the core of a large bodyof work (Elveback et al., 1968; Dietz, 1979; White et al.,1998; Gog et al., 2002; Kamo and Sasaki, 2002; Abu-Raddad and Ferguson, 2004; Pease, 1987).

Until recently, however, the possibility that epidemics ofunrelated pathogens might interact has been ignored.Rohani et al. (1998) proposed an ecological mechanism—termed interference—that may contribute to interactionamong unrelated acute infections. This ecological inter-ference arises from the (temporary or permanent) removalof potential hosts from the susceptible population for oneinfection following an infection by one of its directcompetitors. The primary mechanism for removal is theconvalescent period, during which individuals are inquarantine and hence unavailable to contract ‘‘competing’’infections. In addition, depending upon the disease andhost age and condition, individuals may suffer death as aresult of infection, in which case removal from thesusceptible pool becomes permanent and the competitiveinteraction between pathogens is predicted to becomestronger. Ecological interference has similar dynamicalconsequences to cross-immunity in strain polymorphicsystems, whereby individuals previously infected with onestrain may have partial protection to other competingstrains (Kamo and Sasaki, 2002). The significant predictionof the Rohani et al. (1998) model was that epidemics ofcompeting infections would be temporally segregated, withmajor outbreaks out-of-phase with each other (Huang andRohani, 2005, 2006; Rohani et al., 2006). Empiricalsupport for interference effects is provided in case fatalitydata for measles and whooping cough for several Europeancities in the pre- and post-WWI eras when birth rates werehigh and infection was associated with significant mortality(Rohani et al., 2003), conditions which remain pervasive inmany developing countries today.

In addition to ecological interactions, unrelated patho-gens may also interact through immune-mediated re-sponses. This is likely to take place via a host’s innate,non-specific immune response rather than the acquired,specific response that mediates interactions such as cross-immunity and antibody-dependent enhancement betweensimilar pathogen strains. There are perhaps two main waysfor this interaction to occur. First, during the initial stagesof infection, the host’s innate immune response is strongly

activated and may temporarily prevent the establishmentof other infections. Second, following infection with certainpathogens (e.g. measles and influenza), a host’s immunesystem is suppressed for a period of time during which anindividual may be more prone to colonization by other(particularly bacterial) infections (Openshaw and Tregon-ing, 2005; Openshaw, 2005; Thompson et al., 2003). In thiscase, prior infection with one pathogen may actuallyfacilitate infection by another, unrelated pathogen. Wheninfection occurs by certain enteroviruses, it is thought thattissue damage triggers a severe immunosuppressive re-sponse within patients and that this ‘‘primes’’ the patientfor other pathogens by inducing dilated cardiomyopathyand chronic inflammatory myopathy (Klingel et al., 1992).Another well-studied example is infection with humanimmunodeficiency virus (HIV) which has been shown toincrease susceptibility to infection with many otherpathogens and parasites (Corbett et al., 2002), and, inparticular, with Mycobacterium tuberculosis (TB) (Porco etal., 2001; Corbett et al., 2003). Dynamically, these kinds ofmechanisms leading to immunosuppression between un-related pathogens should exhibit strong similarities withantibody-dependent enhancement between antigenically-related pathogens, whereby individuals previously infectedwith one strain may experience more severe disease wheninfected with a second strain (e.g. dengue serotypes(Halstead, 1970)). So far, however, a dynamical explora-tion of immunosuppression as a mode of interactionbetween pathogens has been lacking.This body of past work makes clear that related and

unrelated pathogens can potentially interact through avariety of immunological and ecological mechanisms,many of which share common dynamical outcomes.However, a systematic study of the different possibleroutes of interaction between different infections is stilllacking. In this paper, we aim to fill this gap by developinga general model for examining systems with multiplepathogens. The proposed framework is flexible and may beapplied to strain polymorphic as well as to unrelateddiseases. The key ingredient of the formalism we develop isthe simultaneous inclusion of immune-mediated compo-nents (coinfection, immunosuppression/cross-enhancementand cross-immunity) and ecological factors (quarantineand infection-induced mortality). Our approach is togeneralize the transmission process within a two-pathogensingle-host community. An important feature is that theframework is built around a null model of no interaction,which allows us to demonstrate how pathogen interactionchanges single-pathogen single-host dynamics in a pre-dictable manner. We then use this framework to track thedynamics of pathogen interactions over the immunoecolo-gical ‘‘landscape’’. Specifically, we focus on four static anddynamical properties of the two-pathogen system: coex-istence and stability of equilibria, and period and phase ofperiodic attractors. We demonstrate how a suite ofimmunoecological interactions between pathogens in atwo-disease community can lead to transient or sustained

ARTICLE IN PRESSD.A. Vasco et al. / Journal of Theoretical Biology 245 (2007) 9–2510

oscillations in disease prevalence. These fluctuations maybe either asynchronous if competition is strong orsynchronous if cooperation is induced. Furthermore, weshow how the period of the two-disease attractor is affectedby key immunoecological interactions. The theory we use isbased upon numerical explorations of two key eigenmodes(eigenvalue–eigenvector pairs). Although this paper doesnot directly address the influence of seasonality, thesesignatures are crucial for gaining insight into the systemtrajectories observed under periodic forcing, and thepatterns we might expect to observe in epidemiologicaldata.

2. A generalized two-pathogen single-host model

Our generalized model builds on the traditional conceptof the SEIR (susceptible, exposed, infected, recovered)paradigm (Kermack and McKendrick, 1927; Anderson andMay, 1991). In this framework an individual is categorizedaccording to their infection status and passes sequentiallythrough the series of SEIR classes. As mentioned in theIntroduction, various models have been developed in anattempt to generalize this approach to include interactionsbetween different pathogens. In this paper we combineseveral of these approaches into a single model. Specifi-cally, ecological interactions between pathogens, such as aperiod of convalescence (Rohani et al., 1998, 2003) ordisease-induced mortality (Huang and Rohani, 2005) areincorporated as well as immune-mediated interactions suchas coinfection (Nowak and May, 1995), cross-immunity(Kamo and Sasaki, 2002), or cross-enhancement (Fergusonet al., 1999) and immunosuppression. This allows asystematic investigation of these dynamical processeswithin a unified modeling framework. We incorporate thebasic elements required to describe the natural history oftwo distinct infections of a single host:

1. All new-borns are fully susceptible to both infections.2. Upon infection, a susceptible individual enters the

exposed (infected but not yet infectious) class, and hasa relative probability of contracting the other diseasesimultaneously, modulated by the coinfection para-meter, fi !i " 1; 2#.

3. After a latent period, the individual becomes infectiousand still has the same chance of contracting the otherdisease (fi, i " 1; 2). We note that coinfection is not acompetitive process within the host in the current model:if coinfection occurs both infections are allowed to runtheir course.

4. Often when symptoms appear, the disease is diagnosedand the individual enters a convalescent phase for anaverage period given by 1=di (i " 1; 2). During con-valescence, the other infection may be contracted but thetransmission rate is modulated by the parameterxi !i " 1; 2#. If 0pxio1, convalescence may represent aperiod of quarantine (reduced contact rates) or tempor-ary cross-immunity (reduced susceptibility). If xi41,

convalescence may represent, via increased susceptibil-ity, immunosuppression (non-related pathogens) orcross-enhancement (antigenically related pathogens).

5. Depending upon the pathogen and host age andcondition, an infection may be fatal, often due tocomplications (such as pneumonia and encephalitis, inthe case of measles and pertussis). This is represented byper capita infection-induced mortality probabilities ri!i " 1; 2#.

6. Upon complete recovery, the individual is assumedimmune to the first infection but remains susceptible tothe second infection, if not previously exposed to it. Atthis stage, we introduce the term wi to explore theimplications of long-lasting cross-immunity !wio1# orcross-enhancement/immunosuppression !wi41# for thetransmission rate of disease i following infection withdisease j.

7. An alternative way of introducing cross-immunity andcross-enhancement is by assuming that infectiousness,rather than susceptibility, is altered. We incorporate thisusing the parameter Zi, so that comparisons can be madewith previous models that have used this representation.If Zio1 then those contracting infection i after the otherinfection will be less infectious (or equivalently only aproportion Zi will transmit the infection), if Zi41 theywill be more infectious.

We note that for parameters describing the interactionbetween pathogens (fi, wi, xi, Zi, i " 1; 2) the subscriptalways refers to the infecting pathogen.The mathematical representation of these assumptions is

presented as the following system of ordinary differentialequations:

dS0

dt" nN $ !l1 % l2#

S0

N$ mS0,

dE1

dt" l1

S0

N$ f2l2

E1

N$ !s1 % m#E1,

dI1dt

" s1E1 $ f2l2I1N

$ !g1 % m#I1,

dC1

dt" g1I1 $ x2l2

C1

N$ !d1 % m#C1,

dS1

dt" !1$ r1#d1C1 $ w2l2

S1

N$ mS1,

dE2

dt" l2

S0

N$ f1l1

E2

N$ !s2 % m#E2,

dI2dt

" s2E2 $ f1l1I2N

$ !g2 % m#I2,

dC2

dt" g2I2 $ x1l1

C2

N$ !d2 % m#C2,

dS2

dt" !1$ r2#d2C2 $ w1l1

S2

N$ mS2,

ARTICLE IN PRESSD.A. Vasco et al. / Journal of Theoretical Biology 245 (2007) 9–25 11

dS12

dt" !1$ r1#!1$ r2# f2l2

E1 % I1N

% x2l2C1

N

!

% f1l1E2 % I2

N% x1l1

C2

N

"

% !1$ r2#w2l2S1

N% !1$ r1#w1l1

S2

N$ mS12,

d!1dt

" l1S0

N% Z1 f1l1

E2 % I2N

% x1l1C2

N% w1l1

S2

N

! "

$ !s1 % m#!1,

d!2dt

" l2S0

N% Z2 f2l2

E1 % I1N

% x2l2C1

N% w2l2

S1

N

! "

$ !s2 % m#!2,

dl1dt

" b1s1!1 $ !g1 % m#l1,

dl2dt

" b2s2!2 $ !g2 % m#l2, (1)

where the class of all those susceptible to both infections isdenoted by S0. The variables Ei, I i and Ci (i " 1; 2)represent those currently exposed, infectious or convales-cing (respectively) after infection with disease i, with noprevious exposure to any infection. The terms Si (i " 1; 2)represent all individuals that are recovered from infection ibut still susceptible to infection j. For bookkeepingpurposes we let !i and li represent the forces of latencyand infection for pathogen i (i " 1; 2). Additionally, notethat S12 are all those no longer susceptible to eitherinfection, and may include those who are still exposedor infectious with one or both diseases (i.e. included within!1, !2, l1 or l2). The model parameters are explained inTable 1.

In this paper, N represents the total population size inthe absence of disease-induced mortality, which is definedto be constant by setting n " m. In the presence of disease-induced mortality, we do not reduce N when calculating

frequency-dependent transmission since we want to ensurethat the level of mixing/contact remains the same. With thisassumption, the S12 class plays no dynamic role in thesystem (see Appendix A for further details on how thissystem of equations is derived from first principles). It isalso worth noting that historically, disease-induced mor-tality has been incorporated into epidemiological modelsby the addition of a mortality term to equations describingthe dynamics of infected. While this formulation may beappropriate for some pathogens, a re-think is needed whenconsidering diseases such as measles and whooping cough.Here, individuals may succumb to the pathogen as a resultof complications from the infection, such as hypoxia,encephilitis or secondary lung infections, which typicallyoccur long after the effective infectious period has elapsed(Behrman and Kliegman, 1998). Hence, we have chosen tomodel this as the probability a convalescing individualsuccessfully re-enters the population.We point out that most previous models investigating

the interactions of two pathogens and a single host arenested within this generalized model as limiting cases ofcertain parameters. In particular, setting fi " xi " 0, wi "Zi " 1 for i " 1; 2 we arrive at the extension by Huang andRohani (2005) to the ecological interference model ofRohani et al. (1998). Furthermore, the lower dimensionalmodels analysed by Ferguson et al. (1999) (which is a two-strain version of a model formulated in Gupta et al.(1998) and Kamo and Sasaki (2002)) can be derived if1=si ! 0 and fi " xi " wi " 1, and 1=si ! 0 andfi " xi " wi; Zi " 1, respectively.

3. Equilibria, invasability and coexistence

We now investigate the equilibrium properties of ourtwo-disease model with respect to changes in modelparameters. When the disease transmission coefficients(bi) are constants, as we assume throughout this paper, itcan be shown that there are four possible equilibria: (i) thedisease-free equilibrium; (ii) two single disease equilibria

ARTICLE IN PRESS

Table 1Description of model parameters

Parameter Epidemiological description Typical range

n Per capita birth rate 1/40–1=75 yr$1

m Per capita death rate 1/40–1=75 yr$1

1=si Average latent period 1–15 days1=gi Average infectious period 1–20 days1=di Average convalescent period 0–9 monthsri Probability of infection-induced mortality 0–1fi Relative probability of coinfection with infection i 0–1xi Temporary quarantine/cross-immunity to infection i 0–1xi Temporary immunosuppression/cross-enhancement to infection i 41wi Permanent cross-immunity to infection i 0–1wi Permanent immunosuppression/cross-enhancement to infection i 41Zi Alternative permanent cross-immunity to infection i 0–1Zi Alternative permanent cross-enhancement to infection i 41bi Transmission rate 100–2000 yr$1

D.A. Vasco et al. / Journal of Theoretical Biology 245 (2007) 9–2512

and (iii) the two-disease coexistence equilibrium. Condi-tions for demonstrating the existence and computation ofthese equilibria are developed in the Appendix. We showthat equilibrium values for all state variables of thisdynamical system are uniquely determined by numericallysolving two coupled equations in terms of the forces ofinfection !l1; l2#.

Given the number of parameters in our model, onemight expect a range of different effects on these equilibria.Intuitively, one possible consequence of pathogen interac-tions among infections is reduced abundance. Surprisingly,however, detailed analyses have demonstrated that ecolo-gical interference does not manifest itself by significantlyaltering infection prevalence; changes in model parameterssuch as the convalescent period translate into negligiblechanges in the number of infectives of either infection(Huang and Rohani, 2005). Perhaps more surprisingly,ecological factors have been shown to exert little influenceon the coexistence likelihood of pathogens. To demonstratein detail how the different immunological and ecologicalparameters of the generalized model affect the possibilityof pathogen coexistence, we now state the invasabilitycriterion. Given that the basic reproductive ratio of eachdisease is determined by

Ri0 "

bisi!si % m#!gi % m#

; i " 1; 2,

it is then straightforward to show that pathogen j can onlyinvade the single-disease equilibrium of pathogen i if

Rj04

Ri0

1% ai!Ri0 $ 1#

where Ri041 (2)

and

ai "Zj

si % mfjm%

sigi % m

fjm%gi

di % m!xjm% wj!1$ ri#di#

! "! ",

(3)

for i; j " 1; 2 and iaj. Of greater insight is that ai &Zjwj!1$ ri# under a fairly non-restrictive set of assump-tions: namely, that host lifespan is significantly greater thanthe length of the latent, infectious and convalescent periods(m5si; gi; di) and that xj and Zj are not too large(Zjfjm5si; gi and Zjxjm5di). For a limiting case of thismodel !fi " xi " 0; wi " Zi " 1; i; j " 1; 2#, Gumel et al.(2003) showed that the two disease coexistence equilibriummust be stable when both single-disease equilibria lose theirstability, and this appears to hold for the general modelpresented here. Fig. 1 illustrates how condition (2) affectsthe coexistence of the two diseases as a function of the keyparameters r (" r1 " r2 assuming symmetry) and w(" w1 " w2 assuming symmetry, which is equivalent tovarying Z " Z1 " Z2). Disease-induced mortality !r40#decreases the region of coexistence in exactly the sameway as permanent cross-immunity after infection !wo1#: atthe population level, these two mechanisms have the samedynamical consequences. However, immune-mediated in-teractions may also act in the opposite direction: if

infection with one pathogen in some way primes the hostfor infection with another !w41#, the coexistence regioncan significantly expand, so that one disease can invadeanother even if its R0 is below 1.

4. Stability and qualitative dynamics of the model

In simple host–pathogen models, for R041, the systemexhibits damped oscillations towards a globally stableendemic equilibrium. However, when competitive orcooperative interactions between pathogens are permitted,ecological theory would suggest that a range of dynamicaloutcomes is possible. In this section our emphasis is onusing linear stability analysis of the pathogen coexistenceequilibrium, guided by numerical integration of the fullnonlinear system, to describe pathogen dynamics for anumber of different epidemiological scenarios. For eachscenario, a two-parameter bifurcation diagram illustratesthe stability of the coexistence equilibrium, and, when it isunstable, the qualitative oscillatory dynamics we expect toobserve. These explorations of parameter space are dividedinto those that are possible for multi-strain (related)pathogens and those that are possible for unrelatedpathogens. In addition, for the analyses of dynamicspresented in this paper, we will focus on interactionsinvolving host susceptibility rather than infectiousness andassume that Zi " 1 !i " 1; 2#. For purposes of illustration,we adopt three baseline sets of parameter values from well-studied childhood diseases (Anderson and May, 1991)

ARTICLE IN PRESS

Fig. 1. Invasability diagram determining two-pathogen coexistence. Thisfigure demonstrates that coexistence of the two infections can be affectedby immunosuppression and disease-induced mortality. In the absence ofimmune-mediated interactions !fi " xi " wi " 1; i " 1; 2#, large levels ofdisease-induced mortality (a 50% level is shown on the dot-dashed line),can cause the region of two-disease coexistence to shrink. In contrast,strong levels of permanent immunosuppression (fi " xi " 1; wi " 2;ri " 0; i " 1; 2) can expand the coexistence domain (dashed line).Epidemiological parameters used to construct this diagram were formeasles (labeled disease 1 in figure) and pertussis (labeled disease 2 infigure) and are given in Table 2.

D.A. Vasco et al. / Journal of Theoretical Biology 245 (2007) 9–25 13

(see Table 2). However, in some analyses we also allowepidemiological parameters to vary in addition to thosemodulating pathogen interactions. In this section we donot systematically vary the birth rate (m) or disease-inducedmortality (ri, i " 1; 2), but the effects of these variations onpathogen interactions will be discussed later (we set m "0:02 and ri " 0, i " 1; 2, except where otherwise noted).

4.1. Interactions between two related pathogens (strains)

For the case of two strains, we assume that theepidemiological characteristics of each strain are the same,so that one set of our baseline parameters applies to bothstrains (for illustration, we use measles). In Fig. 2, wecompare the following three case studies motivated bypossible antigenic strain interactions:

(a) temporary symmetric cross-immunity/enhancement be-tween strains;

(b) temporary complete cross-immunity followed by con-tinued symmetric cross-immunity/enhancement be-tween strains;

(c) permanent symmetric cross-immunity/enhancementbetween strains.

For (a) and (b), we simultaneously vary the average lengthof the period of temporary cross-immunity. For (c), we co-vary the basic reproductive ratio of the strains. We findthat a temporary period of cross-enhancement candestabilize the equilibrium, leading to synchronous cyclesin strain dynamics (Fig. 2a) in a similar manner topermanent enhancement (Ferguson et al., 1999) (Fig. 2c),even for average periods of a few days. As the strength andduration of cross-enhancement increases the dynamicsbecome chaotic (not shown). However, if there is a short-lived period of cross-immunity before enhancement occurs(of at least 3 weeks), as is thought to be the case for dengueserotypes (Wearing and Rohani, 2006), then synchrony islost and the pathogens cycle out-of-phase with each other(Fig. 2b). Permanent symmetric (partial) cross-immunityhas previously been shown to produce stable dynamicsindependent of R0 (Kamo and Sasaki, 2002) and this isreproduced in Fig. 2c. However, if partial cross-immunityis assumed to be temporary or follows a period of complete

cross-immunity, then sustained asynchronous oscillationscan arise (Figs. 2a and b).

4.2. Interactions between two unrelated pathogens

For the case of two unrelated pathogens, we assume thatthe epidemiological characteristics of each infection aredifferent. We also consider two different sets of diseaseparameters: one where the diseases share the same R0 buthave different infectious periods (measles–pertussis) andone where the diseases have different R0’s and differentinfectious periods (measles–rubella). In this manner, and intandem with the results of the previous section, we candetect the changes in pathogen interaction due todifferences between the lengths of the infectious period,

ARTICLE IN PRESS

Table 2Baseline parameter sets used in the epidemic modeling

Parameter Measles Pertussis Rubella

1=s 8 days 8 days 9 days1=g 5 days 10 days 11 days1=d 7 days 14 days 14 daysR0 17 17 7

b "R0!g% m#!s% m#

s1242 yr$1 621 yr$1 233 yr$1

TI'2p

#######################################################1

m!R0 $ 1#1

m% s%

1

m% g

! "s2.1 yr 2.5 yr 4.2 yr

Fig. 2. Stability diagrams of different epidemiological scenarios forrelated strains. The gray region corresponds to a stable coexistenceequilibrium. White regions are unstable and correspond to either stableantiphase or synchronous limit cycles: (a) temporary symmetric immunity/enhancement between strains; (b) temporary cross-immunity followed bycontinued symmetric immunity/enhancement between strains; (c) perma-nent symmetric cross-immunity/enhancement between strains. The base-line epidemic parameter set is that given for measles in Table 2. Note thatin all figures, the units of d are yr$1.

D.A. Vasco et al. / Journal of Theoretical Biology 245 (2007) 9–2514

ARTICLE IN PRESS

Fig. 3. Stability diagrams of different epidemiological scenarios for unrelated pathogens. The gray region corresponds to a stable coexistence equilibrium.White regions are unstable and correspond to either stable out-of-phase or synchronous attractors: (a) and (f) quarantine/temporary immunosuppressionbetween diseases; (b) and (g) quarantine followed by permanent immunosuppression; (c) and (h) quarantine/temporary immunosuppression followed bypermanent immunosuppression between diseases; (d) and (i) quarantine followed by infection-induced mortality; (e) and (j) quarantine followed bypermanent asymmetric immunosuppression. The baseline epidemic parameter sets are given in Table 2.

D.A. Vasco et al. / Journal of Theoretical Biology 245 (2007) 9–25 15

and those due solely to differences between values of R0. InFig. 3, we compare the following five case studies:

(a) quarantine/temporary immunosuppression betweendiseases;

(b) quarantine followed by permanent immunosuppres-sion;

(c) quarantine/temporary immunosuppression followed bypermanent immunosuppression between diseases;

(d) quarantine followed by infection-induced mortality;(e) quarantine followed by permanent asymmetric immu-

nosuppression.

In each investigation we assume that the coinfectionparameter !f# is equal to the quarantine/temporaryimmunosuppression parameter !x# to allow for directcomparisons with results from the previous section.However, if f is set equal to 1, representing the nullposition of non-interacting coinfection, then similarqualitative results are obtained although the regions ofinstability tend to be smaller.

Our analysis of the effects of quarantine and temporaryimmunosuppression on the dynamics of unrelated diseasesshows similar results to those of cross-immunity andenhancement on strain dynamics. Figs. 2a, 3a and f, allshare similar qualitative features: quarantine/cross-immu-nity periods longer than a certain amount of time lead tosustained out-of-phase cycles, whereas temporary immu-nosuppression/enhancement leads to sustained synchro-nous cycles. The major difference is that the unrelatedpathogens with asymmetric infection parameters generatesmaller regions of instabilities, in particular the pathogenswith very different R0’s (measles–rubella). Moreover, thelength of quarantine required to achieve sustained cycles(with x " fo1) in the unrelated pathogens is much longerthan is realistic for many diseases.

If we assume short periods of quarantine and examinehow immunosuppression following these periods affectsthe dynamics, then we find a substantial difference betweenthe diagram for measles–pertussis (Fig. 3b) compared withthat for measles–rubella (Fig. 3g). The interaction betweenunrelated pathogens with similar R0’s but differentinfectious periods (measles–pertussis) closely resemblesthat for related pathogens with identical parameters(measles–measles, Fig. 2b). The interaction betweenunrelated pathogens with different R0’s (measles–rubella)shows that for relatively short periods of quarantine,permanent immunosuppression can give rise to sustainedsynchronous cycles; asynchronous cycles only occur formuch longer periods of quarantine. This difference ispartially explained by considering Figs. 3c and h, in whichwe simultaneously vary the quarantine/temporary immu-nosuppression parameter with the permanent immunosup-pression parameter. The unrelated pathogens with verydifferent R0’s (measles–rubella) interact to produce syn-chronous cycles over a much larger area of parameterspace. We might infer from this that pathogens with similar

R0’s are more susceptible to competition induced by aperiod of quarantine. This key difference is also demon-strated in Figs. 3d and i, in which we vary the probabilityof infection-induced mortality against the period ofquarantine. We can see that mortality only has a significanteffect on stability if the convalescent period is also long,but this region is much larger for measles–pertussis thanmeasles–rubella. Again, asynchronous cycles are easier toinduce in the unrelated pathogens with similar R0’s.Until this point, we have assumed that the interaction

parameters have been symmetric, i.e. the same for bothunrelated pathogens. Figs. 3e and j illustrate the effect ofassuming that immunosuppression following quarantineoccurs for only one of the two pathogens. Interestingly, inboth cases, almost all the unstable regions observed inFigs. 3b and g become stable.In summary, although we observe a range of behavior

varying a suite of immunological and ecological para-meters, there are two opposing mechanisms that drive theunderlying dynamics: competition and cooperation. Cross-immunity, quarantine or disease-induced mortality createcompetition for the pool of susceptibles, of which thedynamical consequence is a temporal separation of out-breaks of each pathogen. Conversely, cross-enhancementand immunosuppression lead to facilitation betweenpathogens: infection with one strain or disease increasesan individual’s chance of contracting the second and thusepidemics are more likely to coincide. These observationsgive us a very intuitive signature of interaction: competitivemechanisms lead to out-of-phase dynamics, cooperativemechanisms lead to synchronous dynamics. However, themechanisms do not always generate sustained cyclicdynamics and when they are acting together, for examplewhen there is a period of quarantine/cross-immunity priorto enchancement/immunosuppression, then one (competi-tion) seems to dominate the other.In this section, we have focused on the stability of the

coexistence equilibrium and the phase of the cyclicdynamics, but cycle period may also provide a keyinteraction signature. In the next section, we pursue amore rigorous examination of a subset of the analysespresented here.

5. Period and phase of attractors in two-pathogen epidemics

We now investigate the properties of period and phase,which characterize oscillations of the coexistence attractor.This allows us to explore how ecological and immune-mediated processes determine disease-specific dynamicaloutcomes. We show that the interaction of two principaleigenmodes (eigenvalue–eigenvector pairs) determines thedynamical outcome of our model. Instability of thecoexistence equilibrium is generated when the real part ofthe eigenvalue from one of these pairs moves from beingnegative to positive across the imaginary axis, leading to aHopf bifurcation. Depending upon how the eigenmodeinteractions generate the instability, the periodic attractors

ARTICLE IN PRESSD.A. Vasco et al. / Journal of Theoretical Biology 245 (2007) 9–2516

may be either in-phase or out-of-phase. Thus, diseaseinteractions may be tracked by continuously following themagnitude and direction of the two eigenmodes inparameter space. Our computational results demonstratehow ecological and immune-mediated processes maycouple the amplitude and frequency components of twocomplex eigenmodes of our model. Although it is clear thatin a 14-dimensional model many other eigenmode interac-tions are possible, we focus on the two complex pairs thatappear to dominate the dynamic stability of the system.These two pairs were determined by performing acombination of extensive numerical exploration of theparameter space underlying the model and model simpli-fication and mathematical analysis. The model simplifica-tion involves lumping several transient classes into one sothat the state-space dimension reduces from 14 to 7.Approximate analytic results can then be obtained thatallow better understanding the observed dynamics of thehigher dimensional model.

Due to the complexity of our model we chose twoanalyses from the previous section to demonstrate thismechanism:

(a) permanent symmetric cross-immunity/enhancementbetween strains;

(b) quarantine followed by permanent immunosuppressionbetween diseases.

5.1. Determinants of attractor properties

Of the 14 eigenvalues determined by the Jacobian ofEqs. (1) five principally determine the stability pro-perties of the attractors studied in this paper. One of theseis always real and is determined by the host lifetime m$1.The others are two pairs of complex eigenvalues. These arecoupled by immune-mediated parameters (wi, fi, xi, Zi,i " 1; 2) and their interactions with the epidemiologicalparameters (m, gi, si, di, i " 1; 2). Since our modeldecouples into two independent epidemiological modelswhen wi " fi " xi " Zi " 1 the eigenmode interactiondisappears at this point. We designate these two eigenmode(eigenvalue, eigenvector) pairs as (y1, v1) and (y2, v2).Throughout this section the eigenmodes will be labeled inthe graphs as red and blue colors, respectively, unlessotherwise noted.

We now reduce the general model to the special case oftwo interacting strains, in which the classes Ei; I i;Ci;Si arecollapsed into a single class, Xi (see Appendix C). The stateof the resulting model takes the form !S0;X 1;X 2;!1; !2; l1; l2#, where each of the remaining five statevariables represents the same quantity corresponding toits value in Eqs. (1). As stated in Section 2 this reduces ourmodel to one similar to that studied previously by Kamoand Sasaki (2002). Symmetrizing this reduced class ofmodels it can be shown that the periods and phases of astable periodic attractor are principally determined by two

dominant coupled eigenmodes, with eigenvalues,

y1 & $!2$ f#m

R0!g% m#!s% m#s

2g% i

R0!g% m#!s% m#s

m! "1=2

,

(4)

y2 & $fm

R0!g% m#!s% m#s

2g% i f

R0!g% m#!s% m#s

m! "1=2

(5)

if fo1 and

y1 & $fm

R0!g% m#!s% m#s

2g% i f

R0!g% m#!s% m#s

m! "1=2

,

(6)

y2 & $!2$ f#m

R0!g% m#!s% m#s

2g% i

R0!g% m#!s% m#s

m! "1=2

(7)

if f41. In other words, y1 ! y2 and y2 ! y1 if f41 andthe reverse occurs if fo1. Eqs. (4) and (5) were derivedpreviously by Kamo and Sasaki (2002) for the case0ofo1. Their analysis of model equilibria was restrictedto the cross-immunity case and the equilibria theyexamined are not defined when f41 or the special casesf " 0 and 1. Nonetheless, their two-epidemic modeldecouples into two independent epidemics in the sameway that ours does when f " 1 and all their results for thecase 0ofo1 can be used as benchmarks for the reducedsymmetric case we analyse in this section. Future work,that is beyond the scope of the present paper, should allowshowing that several of the results we state in the rest ofthis section also apply in some form, to the modelsdeveloped by Gupta et al. (1998) and Ferguson et al.(1999). We showed earlier (see Fig. 2c) that for this class ofmodels, when f42 the system spontaneously erupts into astable periodic synchronous attractor. One can approxi-mately compute the dominant interepidemic period for theattractor from the imaginary parts (eigenfrequencies) ofone of the two dominant eigenmodes:

T(I "

2pIm!yi#

, (8)

where ( represents the dominant eigenmode propagatingthrough the attractor. Which mode dominates the periodi-city properties of the attractor depends in a complex wayupon the types of biological interactions and forcingmechanisms acting upon the system. We examine severalexamples in Sections 5.2 and 5.3.Phase of a periodic attractor for our system can be

studied by using the associated eigenvectors (vi, i " 1; 2)corresponding to Eqs. (4)–(7). Each vi has components thatrepresent the seven state variables !S0;X 1;X 2; !1; !2; l1; l2#.Since we are studying a symmetrized two-strain model wecan write pairs of components (for example, X 1;X 2) of

ARTICLE IN PRESSD.A. Vasco et al. / Journal of Theoretical Biology 245 (2007) 9–25 17

each eigenvector as a ratio,

X 1

X 2"

r1eiO1

r2eiO2"

r1r2ei!O1$O2# "

r1r2ei!DO#, (9)

where ri " !a2i % b2i # is the magnitude of the complexeigenvalue yi with ai " Re!yi#, bi " Im!yi#; Oi is the phaseangle in the complex plane and is equal to tan$1!bi=ai# andDO " O1 $ O2, the phase difference between infectedtrajectories of the two strains. For these definitions thetwo strains exhibit an in-phase attractor if DO " 0 and aout-of-phase (or antiphase) periodic attractor if DO " p.Thus, each of the eigenmodes represents a in-phase and anout-of-phase signature of a periodic attractor. Accordingto Eqs. (4)–(7) the two eigenmode signatures will exhibit aswitch, depending on whether fo1 or f41. Thus, whetheran eigenmode exhibits an in-phase or out-of-phasesignature is determined by the immune-mediation para-meter f. Analysis of the entire eigensystem shows thatthere exists an eigenphase that synchronizes and desyn-chronizes the dynamical system as its eigensystem rotatesby p radians.

5.2. Permanent symmetric immune-interaction betweenstrains

We now explore how the effects of permanent cross-immunity (fo1) and facilitation (f41) determine uniquephase signatures for the reduced symmetric two-strain

model. The results are shown in Fig. 4. Fig. 4a shows howthe predicted period changes as a function of the R0’s andthe cross-immunity/enhancement parameter f " x " w.Our reduced model predicts that with cross-immunity f "x " w " 0:8 we would obtain a pair of attractors, onedetermined by a synchronous eigenvalue (the blue surface)and one determined by an out-of-phase eigenvalue (the redsurface) both with period 2. As the system increases theintensity of cross-immunity to f " x " w " 0:2 our modelpredicts a period 8 out-of-phase attractor.For the case f " x " w41 we examine the onset of a

synchronous limit cycle as f " x " w42. Fig. 4b showsthat this limit cycle occurs as a result of a Hopf bifurcationarising from the same eigenvalue that resulted in the out-of-phase attractor in the case of cross-immunity. Thiseigenvalue switching was predicted earlier by Eqs. (4)–(7).Fig. 4c shows this phase switching as a function of R0 bynumerically computing the components of the eigenvectors(Eq. (9)) corresponding to the dominant eigenvalues.

5.3. Quarantine followed by permanent immunosuppression

As shown previously in Fig. 1 and discussed in Section 3,immunosuppression can lead to facilitation betweenunrelated pathogens, whereby one pathogen can increasethe likelihood of persistence with another. We alsodemonstrated in Section 4.2 that reducing the convalescent

ARTICLE IN PRESS

Fig. 4. Scenario for temporary symmetric immune-mediated measles–measles interactions: (a) interepidemic periods computed from imaginary parts oftwo eigenvalues (period " 2p=Im!yi#), versus (R0, f " x " w) parameter space; (b) real parts of two eigenvalues (Re!yi#) versus (R0, f " x " w) parameterspace; (c) phase differences of !X 1;X 2# components of two complex eigenvectors !vi# versus (R0, f " x " w) parameter space.

D.A. Vasco et al. / Journal of Theoretical Biology 245 (2007) 9–2518

period can modulate the re-infection effect by furtherreducing competition for susceptibles. Conversely, byincreasing the convalescent period, and making convales-cent individuals less likely to be available for re-infection,this parameter can enhance competition. These results wereobtained assuming a period of complete quarantine(f " x " 0), but Fig. 5 shows that each of these effectshas a unique disease interaction signature which is stillpresent in the null case (f " x " 1). While both themeasles–pertussis and measles–rubella scenarios showsignificant regions of synchronous and asynchronousattractors in parameter space, the relative size of theseregions varies. In general, however, decreasing the con-valescent period with increasing immunosuppression leadsto synchronous cycling, whereas increasing the convales-cent period leads to out-of-phase attractors. The major roleof the convalescent class is to delay the onset of thepermanent immunosuppressive effect: thus, the degree offacilitation between pathogens is significantly modulatedby this delay.

Intuitively, we would expect that decreasing the birthrate should increase the intensity of competition betweendiseases for the pool of susceptibles, since it is replenishedat a slower rate. Fig. 5b shows that increasing the birth rate(from m " 0:01 to 0.02) causes the synchronous region todisappear for the measles–pertussis scenario. Conversely,for the measles–rubella scenario, increasing the birth ratecauses the out-of-phase region to disappear (Fig. 5d). Thus,increasing the birth rate for diseases with very differentR0’s (measles–rubella) appears to increase the effects offacilitation but decrease those of interference: the oppositeof what occurs for diseases with similar R0’s (measles–per-tussis). Such diseases may experience higher competitionfor the pool of susceptibles. Fig. 5f shows that the realparts of two complex pairs of eigenvalues exhibit twodistinct phase signatures depending on whether diseases areshowing interference or facilitation. In Fig. 5f the real partof the eigenvalue y1 (represented as the red surface) losesstability and gives rise to stable synchronous attractors.The real part of the eigenvalue y2 (represented as the bluesurface) loses stability to give rise to regions of both stablein-phase and out-of-phase attractors.

Fig. 5e shows that the eigenfrequency (imaginary part)of the eigenvalue y2 (represented as the blue surface) issensitive to permanent immunosuppression while theeigenfrequency of the eigenvalue y1 is relatively insensitiveunless wo1 (which would represent some form ofpermanent quarantine between the two diseases). Inspec-tion of Fig. 5e demonstrates that the sensitive eigenfre-quency of this complex pair determines the expecteddominant period of the time series. Unless the convalescentperiod becomes very long (i.e. d becomes very small) botheigenfrequencies remain almost stationary over the part ofparameter space we consider. Table 2 gives the intrinsicepidemiological periods (TI ) of measles, pertussis andrubella, which are computed using the baseline parameterslisted in the table. The intrinsic period for uncoupled

epidemics is approximately two years for both measles andpertussis. Using our computational results it is nowpossible to explore the effects of immunoecologicalinteractions in modulating the intrinsic period of thecoupled epidemic time series via interaction of the twoprincipal eigenmodes, as shown in Fig. 5g.The boldface letters A, B and C in Fig. 5a designate the

regions in !d; w# parameter space in which eigenvalueinteractions between y1 and y2 modulate the dynamicaloutcomes. Thus, at point A our theory predicts an intrinsicperiod of the coupled time series with period 3.3 yr in-phasecycles. The observed period of 3.3 yr in the time seriesgenerated by numerical integration of the full system (Fig.6c) bears out this prediction. At point B of Fig. 5a ourtheory predicts an intrinsic period of the coupled timeseries with period 1.5 yr out-of-phase cycles. Again, Fig. 6bshows that the period 1:6 is close to what we observe in thecorresponding solution of the full system. At point C ofFig. 5a our theory predicts an intrinsic period of thecoupled time series with 3.3 yr in-phase cycles. Time serieswith a period 3.3 yr in-phase cycling are presented in Fig.6a. Fig. 6d shows that the eigenvalue value interactionsurfaces also determine the maximum amplitude of thecycles. As the real part of the dominant eigenvalue passesthrough zero and continues to increase through !d; w#parameter space, the amplitude associated with thedominant eigenvalue also increases.

6. Conclusions and discussion

The mathematical model we develop in this paper willallow investigators to gain a better understanding ofcurrently available extensive data sets that have accumu-lated from case report data with respect to multi-pathogenepidemics. Thus, we are now in a better position to addressthe potential benefits and pitfalls that may arise from usingmechanistic models of multi-pathogen epidemics to devel-op robust statistical methodologies. Our framework alsoallows the systematic exploration of public health implica-tions arising from interactions between pathogens. Hence,it may present a novel way of understanding prediction andforecasting of multi-pathogen interacting epidemics indisease communities.There are several advantages of mechanistic modeling of

multi-pathogen epidemics. One significant advantage isthat specified mechanisms of disease transmission andspread can be precisely stated and analysed. Our modeladds to previous work on mechanistic modeling diseasestrain interactions (Elveback et al., 1968; Dietz, 1979;White et al., 1998; Gog et al., 2002; Kamo and Sasaki,2002; Abu-Raddad and Ferguson, 2004). With respect toincorporation of immune-mediated processes in twodisease models, we considered the additional problem ofmechanistically distinguishing between related and unre-lated pathogens. In our model we have started as simply aspossible. We model the immune-mediated processes in ourmodel in terms of three parameters w, f, x (for simplicity of

ARTICLE IN PRESSD.A. Vasco et al. / Journal of Theoretical Biology 245 (2007) 9–25 19

ARTICLE IN PRESS

Fig. 5. Stability diagrams of different epidemiological scenarios for unrelated pathogens. The gray region corresponds to a stable coexistence equilibrium.White regions are unstable coexistence equilibrium points and correspond to either stable out-of-phase or synchronous attractors. (a) measles–pertussisinteraction with low birth rate !m " 0:01#; (b) measles–pertussis interaction with standard birth rate !m " 0:02#; (c) measles–rubella interaction with lowbirth rate !m " 0:01#; (d) measles–rubella interaction with standard birth rate !m " 0:02#; (e) plots of the interepidemic periods computed from imaginaryparts of two eigenvalues (period " 2p=Im!yi#), for (d, w) parameter space; (f) plots of the real parts of two eigenvalues (Re!yi#) in (d, w), formeasles–pertussis interaction; (g) contour plot showing how the period varies for the in-phase and out-of-phase attractors A, B and C regions. The baselineepidemic parameter sets are given in Table 2.

D.A. Vasco et al. / Journal of Theoretical Biology 245 (2007) 9–2520

exposition, we assume here symmetry between the immune-mediated parameters). Each of these are distinguished bythe permanency or temporary action of the immuneresponse in mediating the transmission and spread ofdisease. The parameter w represents the effect of permanentcross-immunity (or removal/quarantine) when it is lessthan one and immunosuppression when it is greater thatone. The parameters f and x represent temporary cross-immunity (or removal/quarantine) or immunosuppression.With this parameterization we were able to distinguishimmune-mediated effects between related strains versusunrelated pathogens. Although simple, this parameteriza-tion allowed us to distinguish model mechanisms and infertheir dynamical effects. In our approach, dependingupon how these three parameters are specified in themodel, we were able to add complexity one step at a time sothat the dynamical effects of key mechanisms could beidentified. This proved to be particularly important indistinguishing between the interaction signatures arisingfrom oscillatory instabilities in related strains versusunrelated pathogens.

A second advantage is the introduction of key ecologicalinteractions into multi-pathogen epidemics. Previous work

has demonstrated the importance of introducing ecologicalmechanisms for interaction, such as convalescence anddisease-induced mortality (Rohani et al., 1998, 2003;Huang and Rohani, 2005, 2006). Our model simulta-neously incorporates both ecological and immune-mediated processes so that a comprehensive comparisonbetween these processes could be performed. Thus, ourmodel may permit the exploration of how covariation ofecological and immune-mediated effects may produceunique dynamical interaction signatures. One last advan-tage of our mechanistic approach is that the interactionparameters are hierarchically incorporated or nested withinthe dynamical equations. For example, if we want toexamine the null hypothesis that diseases do not interactthen we may assert that w " f " x " 1. This is equivalentto assuming that the dynamical system decouples into twoindependent epidemics. If we want to test the hypothesisthat permanent or temporary cross-immunity does notexist between two unrelated pathogens then we may assertthat (wX1, fX1, xX1) must hold in the parameter spacefor this case. For each of these cases we have shown thatthere are clear dynamical consequences associated witheach assertion.

ARTICLE IN PRESS

Fig. 6. Each panel illustrates summary properties characteristic of the two-disease time series: amplitude, phase and period of coupled oscillations. In thispaper we use these summary properties to mine disease-specific interference or facilitation signatures out of immunoecological interactions. To generatethese time series a measles–pertussis interaction scenario was assumed. The baseline epidemic parameter sets are given in Table 2: (a) synchronous (in-phase) infected time series with low birth rate (m " 0:01), short convalescence (1=d " 5:2 days) and large permanent immunosuppression effect (w " 5); (b)out-of-phase infected time series with low birth rate (m " 0:01), long convalescence (1=d " 36:25 days) and large permanent immunosuppression (w " 5);(c) synchronous (in-phase) infected time series with low birth rate (m " 0:01), long convalescence (1=d " 36:25 days) and permanent quarantine effect(w " 0:1); (d) amplitude-response as measured by the maximum amplitude of square-root of force of infection for measles plotted versus convalescent rate.Out-of-phase attractors show increasing amplitude as convalescent rate decreases. Synchronous attractors show increasing amplitude as convalescent rateincreases. The parameters determining the time series (a)–(c) of this figure are points of !w; d# parameter spaces shown in Fig. 5 from the regions labeledattractors C, B and A, respectively. Also see Fig. 5 with corresponding exposition in the text describing how eigenmode interactions generate oscillatoryinstabilities and attractor properties.

D.A. Vasco et al. / Journal of Theoretical Biology 245 (2007) 9–25 21

While work is just beginning on the construction ofmodels general enough to incorporate realistic andimmune-mediated processes into multi-pathogen interac-tions, the prognosis for progress is a good one. Combinedwith disease-specific scenario modeling and a strongdatabase, a predictive theory of multi-pathogen epidemicsis now on the horizon.

Acknowledgments

We thank an anonymous reviewer for their carefulcomments on this paper. This work was funded by grantsfrom the National Science Foundation (DEB-0343176), theNational Institutes of Health (1R01GM069111-01A1) andthe Ellison Medical Foundation to PR.

Appendix A

In this appendix, we provide the full equations trackingthe entire immune history of a general two-pathogensingle-host system. The equations we present in thisappendix would be used when developing a stochasticanalogue of the system, while the reduced system ofequations (1), are more conducive to an investigation ofdeterministic dynamics, as has been carried out in thispaper.

First, we define the state variables Xij by their subscripts,which track status with regards to each pathogen: i denotesinfection status (susceptible, exposed, infectious, convales-cent or recovered) with respect to pathogen 1 and j denotesinfection status with respect to pathogen 2.

dXSS

dt" nN $ !l1 % l2#

XSS

N$ mXSS, (A.1)

dXES

dt" l1

XSS

N$ f2l2

XES

N$ !s1 % m#XES, (A.2)

dXIS

dt" s1XES $ f2l2

XIS

N$ !g1 % m#XIS, (A.3)

dXCS

dt" g1XIS $ x2l2

XCS

N$ !d1 % m#XCS, (A.4)

dXRS

dt" !1$ r1#d1XCS $ w2l2

XRS

N$ mXRS, (A.5)

dXSE

dt" l2

XSS

N$ f1l1

XSE

N$ !s2 % m#XSE , (A.6)

dXSI

dt" s2XSE $ f1l1

XSI

N$ !g2 % m#XSI , (A.7)

dXSC

dt" g2XSI $ x1l1

XSC

N$ !d2 % m#XSC , (A.8)

dXSR

dt" !1$ r2#d2XSC $ w1l1

XSR

N$ mXSR, (A.9)

dXEE

dt" f2l2

XES

N% f1l1

XSE

N$ !s1 % s2 % m#XEE ,

(A.10)

dXIE

dt" s1XEE % f2l2

XIS

N$ !s2 % g1 % m#XIE , (A.11)

dXCE

dt" g1XIE % x2l2

XCS

N$ !s2 % d1 % m#XCE , (A.12)

dXRE

dt" !1$ r1#d1XCE % w2l2

XRS

N$ !s2 % m#XRE ,

(A.13)

dXEI

dt" s2XEE % f1l1

XSI

N$ !s1 % g2 % m#XEI , (A.14)

dXEC

dt" g2XEI % x1l1

XSC

N$ !s1 % d2 % m#XEC , (A.15)

dXER

dt" !1$ r2#d2XEC % w1l1

XSR

N$ !s1 % m#XER,

(A.16)

dXII

dt" s1XEI % s2XIE $ !g1 % g2 % m#XII , (A.17)

dXCI

dt" s2XCE % g1XII $ !g2 % d1 % m#XCI , (A.18)

dXRI

dt" s2XRE % !1$ r1#d1XCI $ !g2 % m#XRI , (A.19)

dXIC

dt" s1XEC % g2XII $ !g1 % d2 % m#XIC , (A.20)

dXIR

dt" s1XER % !1$ r2#d2XIC $ !g1 % m#XIR, (A.21)

dXCC

dt" g1XIC % g2XCI $ !d1 % d2 % m#XCC , (A.22)

dXRC

dt" g2XRI % !1$ r1#d1XCC $ !d2 % m#XRC , (A.23)

dXCR

dt" g1XIR % !1$ r2#d2XCC $ !d1 % m#XCR, (A.24)

dXRR

dt" !1$ r2#d2XRC % !1$ r1#d1XCR $ mXRR, (A.25)

where l1 " b1!XIS % XIE % XII % XIC % XIR# and l2 "b2!XSI % XEI % XII % XCI % XRI # are the pathogen-speci-fic forces of infection. Note that to incorporate theparameter Z—which modulates infectiousness—into thisframework requires additional compartments to keep trackof the order in which individuals become infected. For easeof exposition, we do not present these here, however, we dodescribe how Z is incorporated within the reduced frame-work below.

ARTICLE IN PRESSD.A. Vasco et al. / Journal of Theoretical Biology 245 (2007) 9–2522

These equations reduce to the system of equations (1) inthe following way:

(i) Define Xss " S0, XiS " i1 and XSj " j2 so that Eqs.(A.1)–(A.9) are exactly the first nine equations of (1).

(ii) Also define !1 " XES % XEE % XEI % XEC % XER and!2 " XSE % XEE % XIE % XCE % XRE , so that

d!1dt

" l1S0

N% f1l1

E2 % I2N

% x1l1C2

N% w1l1

S2

N$ r2d2XEC $ !s1 % m#!1,

d!2dt

" l2S0

N% f2l2

E1 % I1N

% x2l2C1

N% w2l2

S1

N$ r1d1XCE $ !s2 % m#!2,

dl1dt

" b1s1!1 $ r2d2XIC $ !g1 % m#l1,

dl2dt

" b2s2!2 $ r1d1XCI $ !g2 % m#l2.

When r1d1XEC , r2d2XCE , r1d1XCI and r2d2XCI arerelatively small then these terms can be neglected andwe obtain the equations given in (1) (with Z1 " Z2 " 1).Numerical investigation can be used to show that forthe acute infections (those with ‘‘short’’ latent andinfectious periods) examined in this paper, thisapproximation is reasonable. For chronic infections,disease-induced mortality is likely to occur whilst anindividual is still infected and therefore an alternativerepresentation is necessary. Of course, when there is nodisease-induced mortality then this approximation isexact, but we note that it is also exact whenfi " xi " 0; i " 1; 2.

(iii) With this approximation, and since we assume that Nremains constant even in the presence of disease-induced mortality, once individuals have been exposedto both pathogens it is no longer dynamicallyimportant to keep track of the final 16 variables (i.e.solve Eqs. (A.10)–(A.25) explicitly). We therefore lumpall these states—XEE through XRR—into a single class,S12. For short latent, infectious and convalescentclasses (relative to the average life expectancy), thefollowing equation is thus a reasonable approximationtracking those individuals who have been exposed toboth pathogens and do not suffer disease-inducedmortality:

dS12

dt" !1$ r1#!1$ r2# f2l2

E1 % I1N

% x2l2C1

N

!

% f1l1E2 % I2

N% x1l1

C2

N

"

% !1$ r2#w2l2S1

N% !1$ r1#w1l1

S2

N$ mS12.

This approximation discounts mortality earlier than itoccurs and so slightly underestimates survival.

A.1. Incorporating Z: the forces of latency (!) andinfection (l)

We briefly describe how we can modify the reducedsystem to incorporate the parameter Z. Let EiP (I iP) denoteindividuals exposed to (infected with) pathogen i as aprimary infection and EiS (I iS) denote individuals exposedto (infected with) pathogen i as a secondary infection. Then

dE1P

dt" l1

S0

N$ !s1 % m#E1P, (A.26)

dE2P

dt" l2

S0

N$ !s2 % m#E2P, (A.27)

dE1S

dt" f1l1

E2 % I2N

% x1l1C2

N% w1l1

S2

N$ !s1 % m#E1S, !A:28#

dE2S

dt" f2l2

E1 % I1N

% x2l2C1

N% w2l2

S1

N$ !s2 % m#E2S, !A:29#

dI1Pdt

" s1E1P $ !g1 % m#I1P, (A.30)

dI2Pdt

" s2E2P $ !g2 % m#I2P, (A.31)

dI1Sdt

" s1E1S $ !g1 % m#I1S, (A.32)

dI2Sdt

" s2E2S $ !g2 % m#I2S (A.33)

and defining !i " EiP % ZiEiS and li " bi!I iP % ZiI iS#,i " 1; 2 we obtain the equations presented in (1).

Appendix B

In this appendix we state the equations used tonumerically compute the equilibrium states, the Jacobian,and eigenvalues of the model specified by Eqs. (1). Asmentioned in the main text, we note that N remainsconstant even in the presence of disease-induced mortalityso that the state equation for dS12=dt is uncoupled fromthe dynamical system. As it plays no role in determiningthe equilibrium of the other state variables it will thereforebe ignored in this appendix.First, determine each of the state variables as a function

of the forces of infection !l1; l2#,

S0 "m

l1 % l2 % m,

Ei "li

fj lj % si % m

!m

l1 % l2 % m

!

,

ARTICLE IN PRESSD.A. Vasco et al. / Journal of Theoretical Biology 245 (2007) 9–25 23

I i "lisi

fj lj % gi % m

!1

fj lj % si % m

!m

l1 % l2 % m

!

,

Ci "ligi

xjlj % di % m

!si

fj lj % gi % m

!1

fj lj % si % m

!

)m

l1 % l2 % m

!

,

Si "li!1$ ri#diwj lj % m

!gi

xj lj % di % m

!si

fj lj % gi % m

!

)1

fj lj % si % m

!m

l1 % l2 % m

!

,

!i " !liS0 % Zi!filiEi % fili I i % xiliCi % wiliSi##!si % m#$1,

(B.1)

where the symbol ^ represents the equilibrium value of thestate variable scaled by N, and the indices i and j representthe equilibrium states for the disease i interacting withdisease j (where i and j can take on the values 1; 2, withiaj). Equilibria of Eqs. (1) can now be computed asnonnegative solutions of the implicit equations:

li " bisi !i gi % m$ %$1

. (B.2)

Appendix C

In this section we demonstrate the reduction of the 14-dimensional system to a seven-dimensional system andshow how these two systems decouple into two single-disease models.

Assume that there is no disease-induced mortality foreither disease, r1 " r2 " 0. Assume that w1 " x1 " f1,w2 " x2 " f2 and Z1 " Z2 " 1. Let Xi " Ei % I i % Ci % Si,i " 1; 2. Then

dS0

dt" nN $ !l1 % l2#

S0

N$ mS0, (C.1)

dX 1

dt" l1

S0

N$ f2l2

X 1

N$ mX 1, (C.2)

dX 2

dt" l2

S0

N$ f1l1

Z2

N$ mX 2, (C.3)

dS12

dt" f2l2

X 1

N% f1l1

X 2

N$ mS12, (C.4)

d!1dt

" l1S0

N% f1l1

X 2

N$ !s1 % m#!1, (C.5)

d!2dt

" l2S0

N% f2l2

X 1

N$ !s2 % m#!2, (C.6)

dl1dt

" b1s1!1 $ !g1 % m#l1, (C.7)

dl2dt

" b2s2!2 $ !g2 % m#l2. (C.8)

Now assume f1 " f2 " 1 and let Zi " S0 % Xi, i " 1; 2

dS0

dt" nN $ !l1 % l2#

S0

N$ mS0, (C.9)

dZ1

dt" nN $ l2

Z1

N$ mZ1, (C.10)

dZ2

dt" nN $ l1

Z2

N$ mZ2, (C.11)

dS12

dt" l2

Z1 $ S0

N% l1

Z2 $ S0

N$ mS12, (C.12)

d!1dt

" l1Z2

N$ !s1 % m#!1, (C.13)

d!2dt

" l2Z1

N$ !s2 % m#!2, (C.14)

dl1dt

" b1s1!1 $ !g1 % m#l1, (C.15)

dl2dt

" b2s2!2 $ !g2 % m#l2. (C.16)

These equations decouple into two dynamically distinctsystems (Z1; l2; !2) and (Z2; l1; !1):

dZ1

dt" nN $ l2

Z1

N$ mZ1, (C.17)

d!2dt

" l2Z1

N$ !s2 % m#!2, (C.18)

dl2dt

" b2s2!2 $ !g2 % m#l2, (C.19)

dZ2

dt" nN $ l1

Z2

N$ mZ2, (C.20)

d!1dt

" l1Z2

N$ !s1 % m#!1, (C.21)

dl1dt

" b1s1!1 $ !g1 % m#l1. (C.22)

References

Abu-Raddad, L.J., Ferguson, N.M., 2004. The impact of cross-immunity,mutation and stochastic extinction on pathogen diversity. Proc. R.Soc. London Ser. B—Biol. Sci. 271, 2431–2438.

Anderson, R.M., May, R.M., 1991. Infectious Diseases of Humans:Dynamics and Control. Oxford University Press, Oxford.

Andreasen, V., Lin, J., Levin, S.A., 1997. The dynamics of co-circulatinginfluenza strains conferring partial cross-immunity. J. Math. Biol. 35,825–842.

Behrman, R.E., Kliegman, R.M., 1998. Nelson Essentials of Pediatrics.Saunders, Philadelphia.

ARTICLE IN PRESSD.A. Vasco et al. / Journal of Theoretical Biology 245 (2007) 9–2524

Corbett, E.L., Steketee, R.W., ter Kuile, F.O., Latif, A.S., Kamali, A.,Hayes, R.J., 2002. HIV-1/AIDS and the control of other infectiousdiseases in Africa. Lancet 359, 2177–2187.

Corbett, E.L., Watt, C.J., Walker, N., Maher, D., Williams, B.G.,Raviglione, M.C., Dye, C., 2003. The growing burden of tuberculosis:global trends and interactions with the HIV epidemic. Arch. Intern.Med. 163, 1009–1021.

Cummings, D.A.T., Schwartz, I.B., Billings, L., Shaw, L.B., Burke, D.S.,2005. Dynamics effects of antibody-dependent enchancement on thefitness of viruses. Proc. Natl Acad. Sci. USA 102, 15259–15264.

Cyranoski, D., 2001. Outbreak of chicken flu rattles Hong Kong. Nature412, 261.

Dobson, A.P., 2004. Population dynamics of pathogens with multiple hostspecies. Am. Nat. 164, S64–S78.

Dietz, K., 1979. Epidemiologic interference of virus populations. J. Math.Biol. 8, 291–300.

Earn, D.J.D., Dushoff, J., Levin, S.A., 2002. Ecology and evolution of theflu. Trends Ecol. Evol. 17, 334–340.

Elveback, L., Ackerman, E., Young, G., Fox, J.P., 1968. A stochasticmodel for competition between viral agents in the presence ofinterference. Am. J. Epidemiol. 87, 373–384.

Ferguson, N.M., Anderson, R., Gupta, S., 1999. The effect of antibody-dependent enhancement on the transmission dynamics and persis-tence of multiple-strain pathogens. Proc. Natl Acad. Sci. USA 96,790–794.

Ferguson, N.M., Galvani, A.P., Bush, R.M., 2003. Ecological andimmunological determinants of influenza evolution. Nature 422,428–433.

Gog, J., Woodroffe, R., Swinton, J., 2002. Disease in endangeredmetapopulations: the importance of alternative hosts. Proc. R. Soc.London Ser. B—Biol. Sci. 269, 671–676.

Gumel, A.B., Moghadas, S.M., Yuan, Y., Yu, P., 2003. Bifurcation andstability analysis of a 13-d SEIC model using normal form reductionand numerical analysis. Dynamics Continuous Discrete ImpulsiveSystems Ser. B. Appl. Algorithms 10, 317–330.

Gupta, S., Swinton, J., Anderson, R.M., 1994. Theoretical-studies of theeffects of heterogeneity in the parasite population on the transmissiondynamics of malaria. Proc. Roy. Soc. London Ser. B—Biol. Sci. 256,231–238.

Gupta, S., Ferguson, N., Anderson, R., 1998. Chaos, persistence, andevolution of strain structure in antigenically diverse infectious agents.Science 280, 912–915.

Halstead, S.B., 1970. Observations related to pathogenesis of DHF. VI.Hypotheses and discussion. Yale J. Biol. Med. 42, 350–361.

Holt, R.D., 1977. Predation, apparent competition, and structure of preycommunities. Theor. Popul. Biol. 12, 197–229.

Huang, Y., Rohani, P., 2005. The dynamical implications of diseaseinterference: correlations and coexistence. Theor. Popul. Biol. 68,205–215.

Huang, Y., Rohani, P., 2006. Age-structured effects and diseaseinterference in childhood infections. Proc. R. Soc. London Ser. B—Biol. Sci. 273, 1229–1237.

Hudson, P., Greenman, J., 1998. Competition mediated by para-sites: biological and theoretical progress. Trends Ecol. Evol. 13,387–390.

Kamo, M., Sasaki, A., 2002. The effect of cross-immunity and seasonalforcing in a multi-strain epidemic model. Physica D 165, 228–241.

Kermack, W.O., McKendrick, A.G., 1927. A contribution to themathematical theory of epidemics. Proc. R. Soc. London Ser. A 115,700–721.

Kirschner, D., 1999. Dynamics of co-infection with M. tuberculosis andHIV-1. Theor. Popul. Biol. 55, 94–109.

Klingel, K., Hohenadl, C., Canu, A., Albrecht, M., Seemann, M., Mall,G., Kandolf, R., 1992. Ongoing enterovirus-induced myocarditis isassociated with persistent heart muscle infection: quantitative analysisof virus replication, tissue damage, and inflammation. Proc. NatlAcad. Sci. USA 89, 314–318.

Koelle, K., Rodo, X., Pascual, M., Yunus, M., Mostafa, G., 2005.Refractory periods and climate forcing in cholera dynamics. Nature436, 696–700.

Lipsitch, M., et al., 2003. Transmission dynamics and control of severeacute respiratory syndrome. Science 300, 1966–1970.

Mackenzie, J.S., Gubler, D.J., Petersen, L.R., 2004. Emerging flaviviruses:the spread and resurgence of Japanese encephalitis, West Nile anddengue viruses. Nature Med. 10, S98–S109.

Nowak, M.A., May, R.M., 1994. Superinfection and the evolution ofparasite virulence. Proc. Natl Acad. Sci. USA 91, 4877–4881.

Nowak, M.A., May, R.M., 1995. Coinfection and the evolution ofparasite virulence. Proc. R. Soc. London Ser. B—Biol. Sci. 261,209–215.

Openshaw, P.J., 2005. Antiviral immune responses and lung inflammationafter respiratory synctial virus infection. Proc. Am. Thorac. Soc. 2,121–125.

Openshaw, P.J., Tregoning, J.S., 2005. Immune responses and diseaseenhancement during respiratory synctial virus infection. Clin. Micro-biol. Rev. 18, 541–555.

Pease, C.M., 1987. An evolutionary epidemiological mechanism, withapplications to type A influenza. Theor. Popul. Biol. 31, 422–452.

Porco, T.C., Small, P.M., Blower, S.M., 2001. Amplification dynamics:predicting the effect of HIV on tuberculosis outbreaks. J. AcquiredImmune Deficiency Syndr. 28, 437–444.

Read, A.F., Taylor, L.H., 2001. The ecology of genetically diverseinfections. Science 292, 1099–1102.

Rohani, P., Earn, D.J.D., Finkenstadt, B.F., Grenfell, B.T., 1998.Population dynamic interference among childhood diseases. Proc. R.Soc. London Ser. B—Biol. Sci. 265, 2033–2041.

Rohani, P., Green, C.J., Mantilla-Beniers, N.B., Grenfell, B.T., 2003.Ecological interference among fatal infections. Nature 422, 885–888.

Rohani, P., Wearing, H.J., Vasco, D.A., Huang, Y., 2006. Understandinghost-multi-pathogen systems: the interaction between ecology andimmunology. In: Osfeld, R., Keesing, F., Eviner, V. (Eds.), Ecology ofInfectious Diseases. Princeton University Press, Princeton.

Thompson, W.W., Shay, D., Weintraup, E., Brammer, L., Cox, N.,Anderson, L., Fukuda, K., 2003. Mortality associated with influenzaand respiratory synctial virus in the united states. J. Am. Med. Assoc.289, 179–186.

Tomkins, D.M., Greenman, J.V., Hudson, P.J., 2001. Differential impactof a shared nematode parasite on two gamebird hosts: implications forapparent competition. Parasitology 122, 187–193.

Wearing, H.J., Rohani, P., 2006. Ecological and immunological determi-nants of dengue epidemics. Proc. Natl Acad. Sci. 103, 11802–11807.

White, L.J., Cox, M.J., Medley, G.F., 1998. Cross immunity andvaccination against multiple microparasite strains. IMA J. Math.Appl. Med. Biol. 15, 211–233.

ARTICLE IN PRESSD.A. Vasco et al. / Journal of Theoretical Biology 245 (2007) 9–25 25