Epidemic cycling in a multi-strain SIRS epidemic network model

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1 Zhang Theoretical Biology an Meical Moelling (2016) 13:14 DOI /s RESEARCH Open Access Epiemic cycling in a multi-strain SIRS epiemic network moel Xu-Sheng Zhang 1,2 Corresponence: xu-sheng.zhang@phe.gov.uk 1 Department of Statistics, Moelling an Economics, Centre for Infectious Disease Surveillance an Control, Public Health Englan, 61 Colinale Avenue, Lonon NW9 5EQ, UK 2 Meical Research Council Centre for Outbreak Analysis an Moelling, Department of Infectious Disease Epiemiology, Imperial College School of Public Health, Norfolk Place, Lonon W2 1PG, UK Abstract Backgroun: One common observation in infectious iseases cause by multi-strain pathogens is that both the incience of all infections an the relative fraction of infection with each strain oscillate with time (i.e., so-calle Epiemic cycling). Many ifferent mechanisms have been propose for the pervasive nature of epiemic cycling. Nevertheless, the two facts that people contact each other through a network rather than following a simple mass-action law an most infectious iseases involve multiple strains have not been consiere together for their influence on the epiemic cycling. Methos: To emonstrate how the structural contacts among people influences the ynamical patterns of multi-strain pathogens, we investigate a two strain epiemic moel in a network where every iniviual ranomly contacts with a fixe number of other iniviuals. The stanar pair approximation is applie to escribe the changing numbers of iniviuals in ifferent infection states an contact pairs. Results: We show that spatial correlation ue to contact network an interactions between strains through both ecological interference an immune response interact to generate epiemic cycling. Compare to one strain epiemic moel, the two strain moel presente here can generate epiemic cycling within a much wier parameter range that covers many infectious iseases. Conclusion: Our results suggest that co-circulation of multiple strains within a contact network provies an explanation for epiemic cycling. Keywors: Competition, Cross-immunity, Cyclical ominance of strains, Infectious iseases, Contact network, Oscillatory epiemics Backgroun Recurrent epiemics are a common behaviour of many enemic infectious iseases [1, 2]. Transmission an sprea of infectious iseases epen, in part, on the way an frequency of how people contact with each other. The mass-action law which assumes the homogeneous mixing among iniviuals has been traitionally employe in moelling contact patterns because of simplicity an mathematical tractability. Base on the mass-action law, however, simple transmission ynamics moels cannot preict sustaine oscillations in incience [3, 4]. To explain recurrent epiemics, many complicate aspects of both hosts an infectious agents have been inclue. For example, seasonal forcing ue to external riving changes in host behaviour an/or susceptibility, an the intrinsic mechanisms such as interactions between strains of the infectious agents (for a review see [5]). The moels that incorporate ifferent elaborate aspects can generate the oscillatory epiemics uner certain, usually restricte, parameter ranges Zhang. Open Access This article is istribute uner the terms of the Creative Commons Attribution 4.0 International License ( which permits unrestricte use, istribution, an reprouction in any meium, provie you give appropriate creit to the original author(s) an the source, provie a link to the Creative Commons license, an inicate if changes were mae. The Creative Commons Public Domain Deication waiver ( publicomain/zero/1.0/) applies to the ata mae available in this article, unless otherwise state.

2 Zhang Theoretical Biology an Meical Moelling (2016) 13:14 Page 2 of 11 The actual contact patterns among people surely eviate from the mass-action law [6, 7]. For example, contact patterns between people may isplay the characteristics of scale-free networks [8] or small-worl networks [9]. A recent stuy shows that it is the contact heterogeneity, rather than transmission efficiency, that limits the emergence an sprea of canine influenza virus [10]. This inicates the crucial role of contact structure in infection transmission an sprea. Applying the network frameworks into infectious isease moelling has attracte much theoretical attention an shown some novel features (e.g. [11 15]). Letting infection sprea on a homogeneous population with a fixe ranom network structure, Rozhnova an Nunes [16] illustrate sustaine cyclical epiemics within a one strain susceptible-infective-recovere-susceptible (SIRS) moel. They show that a combination of intrinsic stochasticity ue to a finite population size an spatial correlation ue to limite contacts may be enough to prouce realistic oscillatory patterns observe in recurrent epiemics. As they observe, however, the phase of sustaine oscillations for parameter values that correspon to iseases gets thinner with the number of contacts each iniviual has (which is efine as the egree in network theory) so quickly that the oscillatory phase isappears once the egree excees six. Another striking characteristic of enemic infectious iseases is the fact that they are mostly cause by multi-strain pathogens an the ominant strain alters between epiemics [5]. For chilhoo iseases, for example, it might have been traitionally thought that only one strain is involve in each isease. With avance techniques such as polymerase chain reaction an phylogenetic analysis, it has been now recognise that more than one genotypes (or strains in general sense) are cocirculate in, say, measles [17, 18], chicken pox [19 24], rubella [25, 26], pertussis [27 30], mycoplasma pneumoniae [31], an han-foot-mouth isease [32]. Further, the accumulative evience that reinfection oes occur in, for instance, measles [33 36], chicken pox [37 40], rubella [41 44] an pertussis [45, 46], mycoplasma pneumoniae [47, 48], an han-foot-mouth isease [49], inicates that immunity against these chilhoo iseases that were built through nature infection or vaccination wanes. Many other infectious iseases are also cause by multi-strain pathogens such as cholera, engue, influenza, malaria, Neisseria meningities an respiratory syncytial virus infection. Although the structure network plays an important role in infection transmission an polymorphic infectious iseases are quite common, these two characteristics have not yet been collectively investigate on their potential role in generating sustaine epiemic cycling. In this stuy, we consier a two strain SIRS epiemic moel (e.g., [50]) an assume that immunity wanes either because of immune loss within the human boy or immune escapement ue to changes in the circulating strains. Further, following Rozhnova an Nunes [16], two strains are assume to co-circulate within a ranom network of a fixe egree. We investigate how cross-immunity between strains an spatial correlation ue to contact structure interplay to prouce the epiemic cycling, i.e., the concomitant occurrence of sustaine oscillations in the total incience an the alternation of ominant strains. When ealing with polymorphic pathogens, it is worth pointing out the meanings of strains. In empirical stuies, strains of a pathogen are usually efine serologically or phenotypically. In theoretical moelling, however, strains have been efine immunologically or genetically [51, 52]. In this stuy we assume this theoretical traition to allow the moel framework to be wiely applicable.

3 Zhang Theoretical Biology an Meical Moelling (2016) 13:14 Page 3 of 11 Methos Within the two strain SIRS moel, the population is classifie into eight ifferent compartments an moelle as a ranom network of a fixe egree κ. Iniviuals are enote by noes an contacts between iniviuals by eges. The epiemic ynamics is etermine by the following transmission an transition processes. Susceptible noes (S) become infecte with strain i, i ={1,2},atrateλ through an ege with a noe of primary infection I i or a noe of seconary infection J i. Primarily infecte noes (I i ) recover at rate γ to become fully immune (R i ) to the infecting strain i an partially so to the other strain. The recovere iniviuals (R i ) lose immunity at rate σ to become susceptible again, or become seconarily infecte at rate (1-ψ)λ through an ege with a noe of infection (I 3-i or J 3-i ) to become seconarily infecte J 3-i, i ={1,2}. Here ψ reflects the reuction in susceptibility ue to the previous exposure to other strain (i.e., cross-immunity). Noes of seconary infection J i, i = {1,2} recover at rate γ to become fully immune against all strains (i.e., R). Noes of fully immune (R) lose immunity at rate σ to become susceptible again. These transitions an transmissions are efine accoring to the pairs or triplets involve in the process [16, 53]. For simplicity we ignore the clustering in the network (c.f., [15, 53]). Following Eames an Keeling [53], the numbers of people in eight ifferent statuses are represente by [S], [I 1 ], [I 2 ], [J 1 ], [J 2 ], [R 1 ], [R 2 ], an [R]. The aitional mortality cause by the virulence of infections is ignore, an birth an eath occur at the same rate μ to maintain a constant population size, N =[S]+[I 1 ]+[I 2 ]+[J 1 ]+[J 2 ]+[R 1 ] +[R 2 ] + [R]. There are (8 7)/2 = 28 heterogeneous pairs within the network in which the two noes of a pair are of ifferent states. The number of homogenous pairs can be foun from these of heterogeneous pairs: e.g., [RR]= κ N X! Y X XR an Y R X R [SS]= κs X SX. The state of the system is efine by seven integers of noes an X S 28 integers of heterogeneous pairs. To focus on the impact of spatial correlation (i.e., competition among the limite number of partners) an cross-immunity between strains, two strains are assume to be antigenically inistinguishable. Transmission of infection among noes occurs through pair-link an the change of pairs is etermine by the triples. The stanar pair approximation SIRS moel of two strains is escribe by a set of = 35 ifferential equations, Equations escribing the changing numbers of noes t S ¼ μ ð N S Þ λ ð SI 1 þsj 1 þsi 2 þsj 2 ÞþσðR þr 1 þr 2 Þ t I 1 ¼ ðμ þ γþi 1 þλðsi 1 þsj 1 Þ t I 2 ¼ ðμ þ γþi 2 þλðsi 2 þsj 2 Þ t R 1 ¼ γi 1 ðμ þ σþr 1 λð1 ψþðr 1 I 2 þr 1 J 2 Þ t R 2 ¼ γi 2 ðμ þ σþr 2 λð1 ψþðr 2 I 1 þr 2 J 1 Þ t J 1 ¼ λð1 ψþðr 2 I 1 þr 2 J 1 Þ ðμ þ γþj 1 t J 2 ¼ λð1 ψþðr 1 I 2 þr 1 J 2 Þ ðμ þ γþj 2

4 Zhang Theoretical Biology an Meical Moelling (2016) 13:14 Page 4 of 11 Equations escribing the changing numbers of pairs t SI 1 ¼ λðssi 1 þssj 1 I 1 SI 1 J 1 SI 1 I 2 SI 1 J 2 SI 1 Þ ðλ þ γþsi 1 þσðri 1 þr 1 I 1 þr 2 I 1 ÞþμκI ð 1 2SI 1 Þ t SI 2 ¼ λðssi 2 þssj 2 I 1 SI 2 J 1 SI 2 I 2 SI 2 J 2 SI 2 Þ ðλ þ γþsi 2 þσðri 2 þr 1 I 2 þr 2 I 2 ÞþμκI ð 2 2SI 2 Þ t SR 1 ¼ λði 1 SR 1 þj 1 SR 1 þi 2 SR 1 þj 2 SR 1 Þ λð1 ψþðsr 1 I 2 þsr 1 J 2 ÞþγSI 1 þσðr 1 R 1 þr 1 R 2 þr 1 R SR 1 ÞþμκR ð 1 2SR 1 Þ t SR 2 ¼ λði 1 SR 2 þj 1 SR 2 þi 2 SR 2 þj 2 SR 2 λð1 ψþðsr 2 I 1 þsr 2 J 1 ÞþγSI 2 þσðr 1 R 2 þr 2 R 2 þr 2 R SR 2 ÞþμκR ð 2 2SR 2 Þ t SR ¼ λ ð I 1SR þj 1 SR þi 2 SR þj 2 SR ÞþγðSJ 1 þsj 2 Þ þσðr 1 R þr 2 R þrr SR ÞþμκR ð 2SR Þ t SJ 1 ¼ λð1 ψþðsr 2 I 1 þsr 2 J 1 Þ λði 1 SJ 1 þj 1 SJ 1 þi 2 SJ 1 þj 2 SJ 1 Þ ðλ þ γþsj 1 þσðrj 1 þr 1 J 1 þr 2 J 1 ÞþμκJ ð 1 2SJ 1 Þ t SJ 2 ¼ λð1 ψþðsr 1 I 2 þsr 1 J 2 Þ λði 1 SJ 2 þj 1 SJ 2 þi 2 SJ 2 þj 2 SJ 2 Þ ðλ þ γþsj 2 þσðrj 2 þr 1 J 2 þr 2 J 2 ÞþμκJ ð 2 2SJ 2 Þ t I 1I 2 ¼ λð2i 1 SI 2 þj 1 SI 2 þi 1 SJ 2 Þ 2ðγ þ μþi 1 I 2 t I 1R 1 ¼ λði 1 SR 1 þj 1 SR 1 Þ λð1 ψþði 1 R 1 I 2 þi 1 R 1 J 2 ÞþγI 1 I 1 ðσ þ γ þ 2μÞI 1 R 1 t I 1R 2 ¼ λði 1 SR 2 þj 1 SR 2 Þ λð1 ψþði 1 R 2 I 1 þi 1 R 2 J 1 þi 1 R 2 ÞþγI 2 I 1 ðσ þ γ þ 2μÞI 1 R 2 t I 1R ¼ λði 1 SR þj 1 SR ÞþγðI 1 J 1 þi 1 J 2 Þ ðσ þ γ þ 2μÞI 1 R t I 1J 1 ¼ λði 1 SJ 1 þj 1 SJ 1 þsj 1 Þþλð1 ψþði 1 R 2 I 1 þi 1 R 2 J 1 þi 1 R 2 Þ 2ðγ þ μþi 1 J 1 t I 1J 2 ¼ λði 1 SJ 2 þj 1 SJ 2 Þþλð1 ψþði 1 R 1 I 2 þi 1 R 1 J 2 Þ 2ðγ þ μþi 1 J 2 ð1þ t I 2R 1 ¼ λði 2 SR 1 þj 2 SR 1 Þ λð1 ψþði 2 R 1 I 2 þi 2 R 1 J 2 þi 2 R 1 ÞþγI 2 I 1 ðσ þ γ þ 2μÞI 2 R 1 t I 2R 2 ¼ λði 2 SR 2 þj 2 SR 2 Þ λð1 ψþði 2 R 2 I 1 þi 2 R 2 J 1 ÞþγI 2 I 2 ðσ þ γ þ 2μÞI 2 R 2 t I 2R ¼ λði 2 SR þj 2 SR ÞþγðJ 1 I 2 þj 2 I 2 Þ ðσ þ γ þ 2μÞI 2 R t I 2J 1 ¼ λði 2 SJ 1 þj 2 SJ 1 Þþλð1 ψþði 2 R 2 I 1 þi 2 R 2 J 1 Þ 2ðγ þ μþi 2 J 1 t I 2J 2 ¼ λði 2 SJ 2 þj 2 SJ 2 þsj 2 Þþλð1 ψþði 2 R 1 I 2 þi 2 R 1 J 2 þi 2 R 1 Þ 2ðγ þ μþi 2 J 2 t R 1R 2 ¼ λð1 ψþðr 1 R 2 I 1 þr 1 R 2 J 1 þi 2 R 1 R 2 þj 2 R 1 R 2 ÞþγðI 1 R 2 þr 1 I 2 Þ 2ðσ þ μþr 1 R 2 t R 1R ¼ λð1 ψþði 2 R 1 R þj 2 R 1 R ÞþγðI 1 R þr 1 J 1 þr 1 J 2 Þ 2ðσ þ μþr 1 R t R 1J 1 ¼ λð1 ψþðr 1 R 2 I 1 þr 1 R 2 J 1 I 2 R 1 J 1 J 2 R 1 J 1 ÞþγI 1 J 1 ðγ þ σ þ 2μÞR 1 J 1 t R 1J 2 ¼ λð1 ψþðr 1 R 1 I 2 þr 1 R 1 J 2 I 2 R 1 J 2 J 2 R 1 J 2 R 1 J 2 ÞþγI 1 J 2 ðγ þ σ þ 2μÞR 1 J 2 t R 2R ¼ λð1 ψþði 1 R 2 R þj 1 R 2 R ÞþγðI 2 R þr 2 J 2 þr 2 J 1 Þ 2ðσ þ μþr 2 R t R 2J 1 ¼ λð1 ψþðr 2 R 2 I 1 þr 2 R 2 J 1 I 1 R 2 J 1 J 1 R 2 J 1 R 2 J 1 ÞþγI 2 J 1 ðγ þ σ þ 2μÞR 2 J 1 t R 2J 2 ¼ λð1 ψþðr 2 R 1 I 2 þr 2 R 1 J 2 I 1 R 2 J 2 J 1 R 2 J 2 ÞþγI 2 J 2 ðγ þ σ þ 2μÞR 2 J 2 t RJ 1 ¼ λð1 ψþðrr 2 I 1 þrr 2 J 1 ÞþγðJ 1 J 2 þj 1 J 1 Þ ðγ þ σ þ 2μÞRJ 1 t RJ 2 ¼ λð1 ψþðrr 1 I 2 þrr 1 J 2 ÞþγðJ 1 J 2 þj 2 J 2 Þ ðγ þ σ þ 2μÞRJ 2 t J 1J 2 ¼ λð1 ψþði 1 R 2 J 2 þj 1 R 2 J 2 þj 1 R 1 I 2 þj 1 R 1 J 2 Þ 2ðγ þ μþj 1 J 2 (1)

5 Zhang Theoretical Biology an Meical Moelling (2016) 13:14 Page 5 of 11 Here [XYZ] represents the number of triple XYZ with noe Y having contacts with both X an Z. To close the system, the number of triples is approximate in terms of the number of pairs as in [53], XYZ k 1 XY YZ k Y ð2þ Noting that there are 15 8 = 120 transmission an transition processes in the two strain SIRS moel, the above eqs. (1, 2) can also be obtaine by a coarse escription where the effect of the change in state of a given noe on the κ pairs that it forms is average over each pair type [16]. The complexity of the two strain ynamics allows us to investigate the combine effects of cross-immunity an competition between two strains on ynamic patterns of enemic infectious iseases, along with spatial correlation embee within the ranom network. To ignore the stochasticity ue to the limite size of population, here we consier a human population of a very large size N. The fourth orer Runge-Kutta algorithm is use to solve the eqs. (1, 2) an the programme is coe in R3.2.0 [54] to simulate the ynamical process. Results an iscussion For simplicity, the time scale is set so that γ =1 (i.e. the average infectious uration is taken as the time unit). The numerical calculations show that the final ynamic patterns of epiemic time series are inepenent of the initial conitions. The phase iagram of the two strain SIRS moel is shown in Fig. 1, which is ivie into three parts as that for one strain moel (see Fig. 1 of [16]). When infection rate λ is less than a critical value λ c (σ + 1)/[(κ 2) + σ(κ 1)] from [16], isease cannot survive (iseasefree phase). For a given infection rate λ that is larger than λ c, only a steay enemic Fig. 1 Phase iagram in the (λ,σ) plane for pair approximation moel of two strain SIRS moel. Other parameters: κ =4,μ = , an ψ = 0.0. The bounary of the one strain moel of [16] is inclue for comparison. Region I is the constant enemics phase where the number of new infections is balance by the number of recoveries an region II the oscillatory epiemics phase. The critical infection rate that separates isease-free phase an region I is λ c 0.5. Data for the four chilhoo infectious iseases are from [16]. As the infectious perio is use as the unit of time, the birth an eath rate of μ = is equivalent to a life span of about 50 years in the moel system for infectious iseases which have infectious perios of 1-2 weeks (such as Measles, chickenpox, rubella). It is worth mentioning that our preicte threshol waning rate of immunity in one strain SIRS moel with μ = (i.e., the ashe line) are only slightly smaller than that presente in [16] who o not consier the birth rate

6 Zhang Theoretical Biology an Meical Moelling (2016) 13:14 Page 6 of 11 with constant incience is possible if immunity waning rate σ is larger than a critical value σ c (region I: constant incience phase); otherwise, the sustaine oscillatory epiemic emerges (region II: oscillatory incience phase). Comparison of our moel with another two strain moel that assumes homogeneous mixing [50] suggests that the spatial correlation ue to network structure inuces the sustaine epiemic cycling as in the one strain moel [16]. From Fig. 1, it is obvious that the critical value σ c for the two strain moel is much higher than that for the one strain moel. (Note that resonant amplification of stochastic fluctuations ue to a finite population size can only slightly increase the values of σ c in the one strain eterministic moel [16]). This inicates that uner the circumstance of the same epiemic characteristics, the two strain moel allows for oscillatory epiemics in infectious iseases that have much shorter immunity perios, an thus expans moel parameter range for oscillatory epiemics. The publishe ata for chilhoo infectious iseases that occur recurrently fall into the oscillatory phase of the two strain moel (see Fig. 1); comparably, only some infection ata are within the oscillatory phase of the one strain moel [16]. This ifference results from the competition between strains. The competition comes from two ifferent aspects. One is ecological interference [55] that infectiousness with one strain avois further being infecte by another strain as in multi-strain moels (e.g. [56]). This acts equivalently as a kin of convalesce with respect to another strain an enhances the emergence of sustaine oscillations in incience [57, 58]. The other is spatial correlation ue to contact network structure. The limite number of noes each noe links in the contact network leas to the competition, which increases as the egree κ ecreases an then inuces cyclical epiemics [16]. These two aspects work together to expan greatly the oscillatory phase in the two strain moel. Introuction of cross-immunity between strains further enlarges the oscillatory phase in the two strain moel (Figs. 2 an 3). In contrast to the one strain moel where oscillatory phase isappears on networks of egree κ > 6 [16], the oscillatory epiemics in the two strain moel persist on contact networks of a very high egree κ (Fig. 2). This implies that the ecological interference an cross-immunity in some ways compensate weakene spatial correlation at highly contact networks. Therefore, the two strain SIRS Fig. 2 The effect of the egree on the oscillatory phase uner three levels of cross-immunity. Region I is for constant enemics where the number of new infections is balance by the number of recoveries an region II for the oscillatory epiemics. The transmission rate is λ = 10 an the birth rate is μ =

7 Zhang Theoretical Biology an Meical Moelling (2016) 13:14 Page 7 of 11 Fig. 3 The effect of cross-immunity on the oscillatory phase. The parameter area for cycling epiemics (II) increases with cross-immunity but reuces rapily when cross-immunity becomes complete. Other parameters: κ =8,λ = 10 an two birth rates are assume: μ = (soli line) an μ = σ (ashe line). In the later situation the threshol waning rate of immunity nearly halves, which inicates that most of iniviuals stay in the recovery an immune compartments epiemic moel can easily explain the oscillatory behaviours observe in chilhoo infectious iseases. Uner the extreme circumstance of complete cross-immunity, the oscillatory phase ecreases consierably (Fig. 3); for the situation shown in Fig. 2 recurrent epiemics emerge only on contact networks of a egree κ <5. Gupta et al. [59] investigate multi-strain SIR moels within a ranomly mixing population in which strains can exchange through mutation an recombination. They illustrate that host immunity ictates the structure an ynamics of the pathogen population, with cyclical ominance of non-overlapping sets of strains occurring only at intermeiate levels of cross-immunity. Though Buckee et al. [12] show a significant role of host contact network structure in meiating pathogen strain structure an ynamics, qualitatively similar patterns ictate by cross-immunity remains: cyclical epiemics emerges only at the intermeiate levels of cross-immunity. Spreaing through a ranom network structure with a fixe egree, however, our two strain SIRS epiemics show cyclical epiemics even uner circumstances of no cross-immunity (Figs. 1, 2 an 3) an of complete crossimmunity (Figs. 2 an 3). The ifference comes from that, comparing to the SIR moels of [12, 59], here we consier waning immunity which takes place through both immunity loss an immunity escapement. Although we o not explicitly moel the continuous changes in two strains, these changes have been taken into an reflecte in the waning immunity of our moel. The moels of Gupta et al. [59] an Buckee et al. [12] o not assume immunity loss in human boy; however, immunity escapement occurs ue to the continuous exchanges in strains through mutation an recombination. Uner the two extreme circumstances: complete cross-immunity wherein there is no immunity escapement, an no cross-immunity wherein all strains become inepenently an act as a single strain, it is a quite straightforwar result that no recurrent epiemics will be generate within their moels. This is compatible with the pattern shown in Fig. 2: When each iniviual can contact many others, our moel will also prohibit recurrent epiemics uner the two extreme circumstances. Without cross-immunity, our moel emonstrates that the total incience oscillates an two strains anti-synchronize as shown in Fig. 4a, which can be unerstoo as a

8 Zhang Theoretical Biology an Meical Moelling (2016) 13:14 Page 8 of 11 Fig. 4 Examples of incience time series uner eight ifferent levels of cross-immunity. Other parameters: κ =4,λ = 10, σ = 0.005, an μ = Two strains are represente by ifferent lines. In graph (f) ψ =1, two strains completely synchronize so that their inciences overlap consequence of competition between strains an spatial correlation of noes within the ranom network as argue above (cf., [58, 59]). On another extreme situation of full cross-immunity (ψ = 1.0), the total incience oscillates but two strains synchronizes as shown in Fig. 4f. This is in agreement with the conclusions from [5] who consier complete cross-immune strains within a well-mixing population. However, the unerlying mechanisms for oscillatory epiemics are ifferent. In this stuy it is ue to the interplay of competition an spatial correlation in contact structure while in [5], it is ue to the enhance infectivity within concurrent infection. With intermeiate levels of cross-immunity, recurrent epiemics oscillate irregularly an ominant strains alternate between epiemics (Figs. 4b e; cf., [12, 59]). Three possible interactions have been examine for sustaine oscillatory epiemics: ecological interference which arises from the exclusion of strain uring the infectious perio [56 58], an immune-meiate competition which takes into effect uring the immunity perio [12, 59], network-meiate spatial correlation which makes the whole population in some way act as many spatial subpopulations [16]. It is worth pointing out the ifference between the first two interactions. The ecological interference occurs uring the infectious perio an because of the removal of iniviuals from the susceptible pool uring the perio of being infecte with one strain (see [55, 60]). The

9 Zhang Theoretical Biology an Meical Moelling (2016) 13:14 Page 9 of 11 immune-meiate competition (i.e., cross-immunity) takes place when iniviuals recover from infection with one strain an then become immune against the infecting strain an also (partially) immune against another strain uring the immunity perio. As emonstrate, each of these together with other factors can inuce oscillatory epiemics. However, the conitions for oscillatory epiemics appear to be quite restricte. The three interactions act collectively in our two strain SIRS epiemic moel an epiemic cycling is shown to emerge in a much less restrictive parameter space. Although we focus on a system of two strains, the results of this stuy are generally applicable to the multi-strain setting. Technically, it is not trivial to write own the ifferential equations even for the epiemic system cause by three strains; however, the logical reason for this is simply that with more strains co-circulating within the same network, the three interactions just iscusse will increase, which consequently facilitates the emergence of recurrent epiemics. In this stuy, we show the potential of SIRS epiemic moels that allow multiple strains to co-circulate within a network structural population to generate recurrent epiemics. Although chilhoo iseases were use in Fig. 1 as examples to emonstrate how our two strain moel makes recurrent epiemics more likely than the one strain moel of Rozhnova an Nunes [16], the moel framework presente here shoul be also applicable to other infectious iseases cause by multi-strain pathogens. To explore the specific mechanism for a particular infectious isease cause by multiple strains, however, we nee to estimate the exact values of moel parameters by fitting the moel outputs to the empirical ynamical patterns observe in human populations such as the uration an size of epiemics an the gap between epiemics (as in [61]). This is the issue we will aim at in future. Conclusion We consier the pair approximation of a two strain SIRS epiemic moel in a host population with every iniviual in contact with a fixe number of other iniviuals. We show that interactions between strains ue to ecological interference an limite contacts within a network structure population can inuce sustaine oscillatory patterns in both total incience an ominant strains. Though our moel is simplifie in that clustering an heterogeneity in contact network an stochasticity have been neglecte, inclusion of these more realistic aspects will facilitate the iversity [12, 15] an fluctuation [16, 62] an therefore cannot change our conclusion. Our results suggest another possible mechanism for the observe epiemic cycling: interplay of spatial correlation ue to contact network an interactions between strains through exclusion of other infection uring the infectious perio an immune protection uring the immunity perio (cf., [5]). Competing interests The author eclares that he has no competing interests. Authors contribution Conceive an esign: XSZ. Performe: XSZ. Analyze ata: XSZ. Wrote the paper: XSZ. Acknowlegments This research was fune by the Public Health Englan. The author woul like to thank two anonymous reviewers for their helpful comments an constructive suggestions on the paper an Richar Peboy an Emilia Vynnycky for their valuable iscussions about chilhoo iseases an polymorphic pathogens. Receive: 11 February 2016 Accepte: 11 April 2016

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