BioMed Research International

Volume 2014 (2014), Article ID 375862, 23 pages

http://dx.doi.org/10.1155/2014/375862

## The Impact of Coinfections and Their Simultaneous Transmission on Antigenic Diversity and Epidemic Cycling of Infectious Diseases

^{1}Department of Statistics, Modelling and Economics, Centre for Infectious Disease Surveillance and Control, Public Health England, 61 Colindale Avenue, London NW9 5EQ, UK^{2}Medical Research Council Centre for Outbreak Analysis and Modelling, Department of Infectious Disease Epidemiology, Imperial College Faculty of Medicine, Norfolk Place, London W2 1PG, UK^{3}Centre for Nonlinear Complex Systems, Department of Physics, Yunnan University, Kunming, Yunnan 650091, China

Received 6 February 2014; Revised 18 April 2014; Accepted 18 April 2014; Published 22 June 2014

Academic Editor: Laith Abu-Raddad

Copyright © 2014 Xu-Sheng Zhang and Ke-Fei Cao. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

Epidemic cycling in human infectious diseases is common; however, its underlying mechanisms have been poorly understood. Much effort has been made to search for external mechanisms. Multiple strains of an infectious agent were usually observed and coinfections were frequent; further, empirical evidence indicates the simultaneous transmission of coinfections. To explore intrinsic mechanisms for epidemic cycling, in this study we consider a multistrain Susceptible-Infected-Recovered-Susceptible epidemic model by including coinfections and simultaneous transmission. We show that coinfections and their simultaneous transmission widen the parameter range for coexistence and coinfections become popular when strains enhance each other and the immunity wanes quickly. However, the total prevalence is nearly independent of these characteristics and approximated by that of one-strain model. With sufficient simultaneous transmission and antigenic diversity, cyclical epidemics can be generated even when strains interfere with each other by reducing infectivity. This indicates that strain interactions within coinfections and cross-immunity during subsequent infection provide a possible intrinsic mechanism for epidemic cycling.

#### 1. Introduction

Infectious diseases cause huge morbidity and mortality in human populations and remain as a dramatic threat to public health [1, 2]. The incidence of many infectious diseases varies periodically. Cyclical infections of humans range from childhood infections such as measles, pertussis, mumps, diphtheria, varicella, and scarlet fever [3] to faecal-oral infections such as cholera and rotavirus and vector-borne diseases such as malaria and dengue fever [4, 5]. For example, seasonal flu develops as an epidemic during winter in temperate regions but it remains at very low levels during summer [6] while pertussis has 3-4-year cycles [7]. Despite the pervasive nature of oscillation, its underlying mechanisms are not well understood. From detailed analyses of empirical data, several distinct mechanisms have been proposed [4, 6, 8]: survival of disease pathogen outside host; host behaviour; seasonal changes in host susceptibility and immune defence and in vector abundance. Overall, those mechanisms have accounted only for the variation caused by external factors. Mechanically, they can be explained by a standard Susceptible-Infected-Recovered (SIR) or Susceptible-Infected-Recovered-Susceptible (SIRS) transmission model that was embedded with a seasonal transmission rate (i.e., the so-called “seasonal forcing”) [4, 5, 8–12]. However, these models cannot provide a reasonable explanation for, say, a quickly globally spreading pattern of influenza [13] and require unrealistically high reproduction rates for the observational seasonal patterns in influenza [14]. Moreover, Grassly et al. [15] show that it is the intrinsic factors (e.g., immunity following the recovery from syphilis infection), rather than the external factors (e.g., changes in human sexual behaviour), that cause the 8–11-year cycles in syphilis incidence from the 1960s to the 1980s in the United States cities. Other mechanisms were also sought [16]. A single-strain standard SIR model that was not embedded with explicit seasonal forcing cannot generate sustained oscillations in incidence unless other complications are included. For example, variants of the standard SIR model such as cyclic models of SIRS and SEIRS type with a large time delay in the removed class, models with nonlinear force of infection, models with variable population size and disease-related deaths [17], or models including an isolated class generate sustained oscillations albeit under unrealistic region of parameter space [18]. However, it has been recognised that, without consideration of the intrinsic aspects of infectious diseases such as interactions between strains of an infectious agent, the understanding of epidemic cycling cannot be complete [10, 19] or is impossible [20].

Due to the high mutation rate and other genetic processes (such as recombination and reassortment), pathogens can quickly generate a lot of different genotypes. The striking fact of infectious agents is their antigenically diverse strains that are cocirculating within host populations [21–26] and concurrent infections with multiple strains [27–31]. Polymorphic pathogens such as dengue virus [22], malaria parasite* Plasmodium falciparum* [21], influenza virus [23, 24], and* Streptococcus pneumoniae* [26] are responsible for enormous health burdens of human populations [26, 32, 33]. Understanding the underlying mechanisms of antigenic diversity and cyclical epidemics will provide theoretical bases for their evolution under selective pressure such as the replacement of strains and the emergence of resistance strains under pharmacological control measures [25]. In practice, this will also be supplied with the basis for the design of effective control strategies such as vaccination [34].

In general, once two different pathogen strains infect the same host individuals, they will interact with each other [32]. The interaction can be classified into direct interactions of pathogen genes or gene products (such as recombination in bacterium* Streptococcus* and reassortment in influenza A viruses) and indirect interactions that result from alteration in the host environment such as immunological responses [35]. The indirect interactions may be synergistic, neutral, or antagonistic in view of changes in transmissibility and virulence [2, 33, 35–37]. For example, epidemiological evidence suggests that primary infection with one strain of dengue virus can enhance transmission of subsequent infection with another strain [38–40]. Further, the strains present in coinfections can transmit simultaneously and separately [29, 41, 42]. The direct observation of transmission process from one to another person is hard even for single transmission, let alone simultaneous transmission of multiple strains. To our knowledge, the one direct evidence of simultaneous transmission (or cotransmission) came from the field study during 2009 pandemic influenza by Liu et al. [29]. Cross-immunity between different subtypes of influenza has been observed [43–46]. Strong cross-immunity has been observed among antigenic drift variants of the same influenza subtype; for example, the first wave of the 1918 (H1N1) pandemic provided up to 94% protection against clinical illness during the second wave [47]. Cross-immunity was also found between different strains of other infectious agents [48, 49]. These characteristics surely affect pathogen diversity and dynamical behaviour of polymorphic pathogens. Though a large body of epidemiological researches are devoted to understanding the mechanisms of host single-strain pathogen dynamics, the increased evidence has gathered that the characteristics of polymorphic pathogen causing infections prove deviant from the predictions from one-strain models. For example, estimation of characteristics of transmission dynamics of polymorphic malaria* Plasmodium falciparum* that is based on one-strain epidemic model is misleading [50]; further, coinfections may imply that the survivorship of strains is not necessarily maximized by maximizing the basic reproductive number [51].

Transmission consequences of coinfection and superinfection have received a great deal of interest over the last 30 years [37, 52–58]. Theoretical modelling and empirical analyses show that pathogen interactions could have profound implications for the pathogen communities [51, 59–61]. For example, it has been shown that cocirculation of different diseases and cross-immunity [59] and antibody-dependent enhancement in transmission [38, 39, 62] can induce cyclical and chaotic epidemics. Cross-immunity alone within a two-strain SIR or SIRS model cannot generate sustained oscillation [12, 63–65] unless other factors such as age-dependent survival [53, 57] or isolation [66] are included. Following the traditional assumption that individuals, while being infected with one strain, are temporarily not susceptible to the other [52, 53, 62, 67], most of all these models consider only subsequent infection (i.e., individuals that recovered from primary infection with one strain were further infected by others). Even in those that included coinfections, different strains are not assumed to simultaneously transmit [12, 58, 61, 67, 68].

Recent empirical evidence has challenged the traditional assumption and indicated the concurrent infections during the infectious period of primary infection [29, 32, 41, 69, 70]. Due to technical difficulty and complexity involved in distinguishing multiple strains, multiple-strain infections are surely underestimated as argued by Balmer and Tanner [32]. The available empirical studies suggest that the occurrence of multiple strains was significantly more common than expected by chance. For example, Vaccarella et al. [70] found that the observed-to-expected ratio for concurrent infections with two human papillomavirus (HPV) types was 1.16 and for 3–7 types was 1.04. This implies that prior infection by one strain may facilitate infection by the other and strain interactions within coinfections can be synergism [42, 61]. For influenza that spreads from person-to-person through aerosols of droplets nuclei, for example, infection with one strain may increase the aerosolisation and hence the transmissibility of coinfecting other strains [71]. It is hence necessary to explore the possible role that strain interactions within concurrent infections might play in maintaining pathogen diversity and generating oscillatory diseases.

In this study we focus on the intrinsic factors of cyclical epidemics within a multistrain SIRS model by assuming that coinfections can directly transmit from people to people but at a low rate in relation to single transmission. We consider the indirect strain interactions that do not produce novel strains (cf. [59, 72]). In view of evidence of strong cross-immunity among strains (e.g., [47]), we assume a special situation where the patients recovered from infection will become completely cross-immune against any subsequent infections during their protective immunity periods. We thus exclude subsequent infection (cf. [38, 62, 67]). For concurrent infection of strains of a polymorphic pathogen, we have not found any informative data about the impact of order of infection and therefore ignore the impact of infection passage and superinfection [37, 42, 54]. We investigate how the interplay of complete cross-immunity during subsequent infection and strain interactions within coinfection affects antigenic diversity structure and generates cyclical epidemics.

#### 2. Models and Methods

We consider a human population in which there are cocirculating strains and every individual is capable of being simultaneously infected by at most strains. For convenience, we denote the coinfection of multiplicity as , here and representing the strains present in the coinfection, which is independent of order of such infections. Let denote the fraction of individuals that carry the coinfection . In this study we define the multiplicity of strains in a coinfection as the within-in host antigenic diversity and the number () of cocirculating strains in the host population as the antigenic diversity of the population. We consider a SIRS epidemic model by assuming the coinfection can simultaneously transmit among individuals at rate ; the coinfection simultaneously progresses to the recovered at rate (cf. [34, 73]); strain interactions within coinfection occur via the change in infectivity (cf. [34]). Infections are acquired through a mass action term with the force of infection for the coinfection given by Here represents the factor that strains transmit simultaneously from a patient coinfected with strains, which reflects the strain interaction within coinfection. If strains interfere with each other, ; otherwise, implies enhancement in infectivity of strains. The factor here is distinct from that proposed in dengue virus models (e.g., [38, 62]) where it reflects the change in transmissibility during secondary infection.

Further we assume that strain interaction also occurs via the change in susceptibility: once recovered from primary infection, patients become completely immune against any strain during the immunity period (1/), which is assumed to be the same for people recovered from different infections (cf. [51, 61]). This assumption neglects the complicated history of infection and thus largely simplifies the system (cf. [51, 63, 67, 74]). Further we assume that the birth to the susceptible and the loss of population by death are introduced through the same constant rate , therefore a constant population size. These assumptions lead to a system of coupled differential equations Here is used to stand for empty set. In the above equations, , and represents the fraction of the susceptible. For coinfection of multiplicity , there are different combinations. A very large population is assumed to ignore the stochasticity.

The system at an arbitrary number of strains is intractable. We simplify the problem to investigate only the symmetric multistrain system where all strains share the same epidemiological properties: and , (cf. [51, 67]). Further, strain interactions are identical and depend only on the multiplicity of coinfections: , for and . At the equilibrium of such symmetric system the fraction of individuals that were coinfected with strains is the same among different combinations; for example, . The model system reduces to that of equations with the force of infection, The prevalence of coinfections of multiplicity is and the total prevalence is . For further simplicity we assume that all the efficiencies of partial transmission from coinfections are the same (i.e., ) and the infectious periods of all infections are the same (i.e., ).

#### 3. Results

Though analytical results are hard to obtain for the general model system, they can be obtained for a two-strain model which was given in Appendix. Otherwise, our results are mainly based on the numerical integrations of (3) and (4). We first illustrate the invasion conditions for a pathogen strain to the host population in which other strains were established and then demonstrate how coinfections affect the antigenic diversity structure and facilitate the emergence of cyclical epidemics.

##### 3.1. Dynamics of Invasion and Coexistence of Multiple Strains

Consider a situation where a host population was already at equilibrium with endemic strains, each of which has a basic reproductive number . What is the condition for a new indistinguishable strain to invade into the population? In the case of , as shown in Appendix, it is that the efficiency of partial transmission from coinfections and the simultaneous transmission rate of double infections satisfy the inequality This approximated relationship holds for pathogens that have a much shorter infectious period than the duration of immunity such as influenza virus.

Though condition (5) was analytically obtained for the system of two strains, numerical simulations confirm it as a good approximation for multistrain systems (see Table 1). Results show that when the efficiency of partial transmission from coinfections is less than the critical value , only a single strain persists stably. Otherwise, a new indistinguishable strain can successfully invade and coexist with the strains in the previous endemic. That is, once condition (5) is satisfied, arbitrary number of strains can be maintained. Table 1 also indicates that simultaneous transmissions of triple and quadruple infections have a very weak impact on the invasion and pathogen diversity. This implies that, with simultaneous transmission of double infections, pathogen diversity can increase even under fairly strong interference between strains (i.e., is much smaller than the unit) (cf. [73]). The critical value is 50% in the absence of simultaneous transmission of double infections; with simultaneous transmission , the critical value further reduces as predicted in (5) (Table 1). Therefore, given that the basic reproductive number of each strain is larger than unit, equilibrium dynamics fall into two classes depending on the efficiency of partial transmission from coinfections and simultaneous transmission: stable single-strain persistence and multistrain persistence with stable diversity (cf. [67]). Hence it is the strain interaction within coinfection (i.e., and ) that changes the traditional competitive exclusion [67, 73] and facilitates the increase in antigenic diversity.

##### 3.2. Structure of Antigenic Diversity

Under the traditional circumstance where only one strain can transmit (i.e., but for ), the antigenic diversity structure is shown in Figure 1. It shows that as the antigenic diversity () of coinfections increases, the proportion of coinfections of the same strains (i.e., ) decrease as 1/; however, their prevalence (i.e., ) remains roughly the same. The reduction in with multiplicity is so remarkable that single infections are the most common infections (Figures 1(a) and 1(b)). When transmissibility is relatively low, triple and quadruple infections are prohibited (Figures 1(a) and 1(c)). However, with a high transmissibility, high diverse infections can safely survive (Figures 1(b) and 1(d)). Nevertheless, the total prevalence of all different forms of infections is nearly independent of the number of cocirculating strains and approximated as in the one-strain model system (see Figure 1(e)). This approximation can be simply confirmed from (3) and (4) under the situation where coinfections are rare. Ignoring coinfections, (3) and (4) reduce to that of one-strain SIR model with the prevalence of one strain replaced by total prevalence of strains (cf. [67]). The derivation of approximation (6) for a special situation of two strains is given in Appendix.

The influences of simultaneous transmission of coinfections are shown in Figures 2 and 3 under two different immunity periods. Increasing increases the prevalence of coinfections but reduces single infections when strains interfere with each other (Figures 2(a) and 3(a)). However, if strains interact to enhance the infectivity within coinfections , cyclical epidemics may emerge when immunity lasts longer (Figure 2(b)) or the prevalence of highly diverse coinfections increases substantially while single infections decrease greatly (Figure 3(b)). However, the impact of simultaneous transmission of more complicated coinfections is much weaker such that hardly have any effect (Figures 2(e), 2(f), 3(e), and 3(f)). Under the circumstance of , the total prevalence is dominated by single infections while the triple infections or higher complex coinfections increase but still remain at very low levels if the immunity period is long (cf. Figure 1(d)) and (Figures 2(a), 2(c), 2(e), and 3(a)). However, when strains facilitate each other to enhance their infectivity , single infections cannot dominate (Figures 2(b), 2(d), and 2(f)). Furthermore, in the situation of short immunity period, complicated infections such as double, triple, and quadruple infections become more popular than single infections (Figures 3(b), 3(d), and 3(f)). This indicates that decreasing the duration of immunity can increase coinfection and stabilize the coexistence (cf. Figures 2(b) and 3(b); see also Figure 5(a)). When the coexistence is steady, the total prevalence slightly increases with these characteristics. Given all others being the same, increasing from 0 to only increases less than 3% while increasing from 0 to 0.75 and increasing from 0 to 0.6 only increase by 0.9% and 0.3%, respectively.

The interaction between strains is also reflected in the efficiency of partial transmission from coinfection . When is not strong (≤1), single infections dominate (Figure 1). When is strong, given all others being the same, coinfections such as double and triple coinfections become common and predominate (Figure 3) and may also induce cyclical epidemics (Figure 2(b)). Nevertheless, as increases from 0 to 2, coinfections predominate but the total prevalence only increases about 5%. Overall, the results cannot exceed 106% of (6) for all the steady scenarios examined in Figures 2 and 3. These observations are also supported by the results listed in Table 2. This indicates that (6) is a good approximation for the total prevalence of steady coexistence. Thus strain interactions (i.e., and ) determine the structure of antigenic diversity but cannot much alter the total prevalence.

##### 3.3. Cyclic Epidemics

As shown above, coinfections facilitate coexistence; further they also aid the generation of sustained epidemic oscillations. Stability analysis and simulations show that with sufficiently enhanced transmissibility from coinfections the dynamic system will experience a Hopf bifurcation at which steady coexistences develop into limit cycles (Appendix). Figure 4 shows the impact of simultaneous transmission of coinfections and antigenic diversity on the critical value of efficiency of partial transmission from coinfections for emergence of cyclical epidemics. In the absence of simultaneous transmission of coinfections, . The observation under this special situation is similar to the current view that multistrain oscillations can occur only in the case of enhancement if cross-immunity during subsequent infection is very strong [38, 62, 75]. Two previous multistrain models [59, 63] which ignored current infection concluded that as cross-immunity becomes very strong, cyclical and chaotic epidemics vanish. In the presence of simultaneous transmission of double infections and high antigenic diversity, reduces to below unit. For example, when , for the situation of ten cocirculating strains. Comparatively, the simultaneous transmissions of triple and quadruple infections only have a much weak effect (Figures 4(b) and 4(c)). These indicate that has a predominant effect on the critical value . This implies that even with interference among strains, cyclic epidemics can still be generated provided there is enough simultaneous transmission of double infections and sufficient antigenic diversity.

Figure 5 shows that cyclic epidemics are also controlled by other model parameters. For a given immunity period, Figure 5(a) shows that cyclic epidemics become impossible for a too small or too large reproductive number . While for a given basic reproductive number, it also shows that the oscillation is prohibited under short immunity. Once emerged, the period of cyclic epidemics increases with immunity period and efficiency of partial transmission from coinfections but decreases with the basic reproductive number (Figures 5(b) and 5(c) and Figure 6). However, infectious period can only slightly increase the interepidemic period (data not shown). Figure 6 shows some example time series of cyclical epidemics under different strain interactions. When strains interfere with each other , single infections dominate the total prevalence; however, when they enhance each other to increase infectivity , the fluctuations in prevalence become large (cf. Figure 10) and the fraction of complicated coinfections can become popular as seen in steady coexistence shown in Figure 1(d).

#### 4. Discussion

In this study we investigate a multistrain SIRS epidemic model that includes coinfections and interactions between strains within coinfections. Though remaining at very low prevalence relative to single infections, coinfections and their simultaneous transmission substantially change the behaviour of infectious dynamics. They enable multiple strains to readily coexist and therefore maintain antigenic diversity. Further, interplay of strain interactions within coinfections and complete cross-immunity during subsequent infection induces sustained cyclical epidemics over a wide range of biologically realistic parameter space. In contrast to the heavily sought external seasonal forcing, this simple model provides a possible intrinsic mechanism for epidemic cycling of infectious diseases.

##### 4.1. Invasion of New Strains and Antigenic Diversity

In contrast to other coinfection models [51], we include simultaneous transmission (or cotransmission) of multiple strains within concurrent infection. Even under total cross-immunity, coinfections increase the chance of invasion by a new strain into a population at endemic with other strains (Figures 7 and 8; [40]) and simultaneous transmission enables a less transmissible strain to successfully invade (Table 4). This allows multiple strains to coexist and increases the pathogen diversity though, in its absence, only one strain would persist, as suggested in traditional competitive exclusion (Table 1; [67, 73]). This is comparable with multistrain models for dengue virus which include cross-reactive interaction during subsequent infection by other strains [76, 77]. Kooi et al. [76] show that though heterogeneity in transmission among strains reduces coexistence, the cross-reaction in second infection increases it. Interestingly, Mier-y-T-Teran-Romero et al. [77] show that combination of antibody-dependent enhancement and slight heterogeneity in transmission among strains increases persistence of four strains, suggesting evolutionary advantage of antibody-dependent enhancement. In contrast to Abu-Raddad et al. [51], our results show that, during the establishment process of a new strain, the total prevalence remains roughly unchanged. This observation suggests invisible invasion of genotypically different but antigenic indistinguishable new strains. When strains within coinfections can be transmitted simultaneously, coinfections can become prevalent if pathogen strains are highly transmissible (Figures 1, 2, and 3). However, the long immunity period and interference between strains within coinfection will reduce the opportunities of high complicated coinfection and the antigenic diversity. This is comparable with predicted patterns of antigenic diversity by other models [25].

The observed levels of antigenic diversity may be large but surely are not infinite, say, influenza A [78]. So what limits the antigenic diversity? As the total prevalence is roughly constrained by (6) when multiple strains are in steady coexistence, the fraction of each single infection decreases proportionally with the antigenic diversity. In view of the actual limited size of host populations, stochastic extinction may be one of the forces that can limit the antigenic diversity [25, 67, 74]. Further, when infections experience oscillations, trough extinction (Figure 6 and Figure 12; [65]) may be another mechanism. One important issue is that many newcomers, such as mutants, recombinants, reassortants, and immigrants, may have different epidemiological properties. Simulations show that any less transmissible newcomer will vanish no matter what the values of and are. The newcomers will replace endemic strains once they are more transmissible, which may also reduce pathogen diversity. Because of quick introduction of newcomers, the infections that were observed may never reach a static status. These factors may work together to restrain antigenic diversity.

##### 4.2. Epidemic Cycling

Though coinfection enhances coexistence of pathogen strains it alone cannot induce cyclical epidemics (Figure 5(a)). For example, Zhang et al. [36] considered a two-plant virus model in which the coinfection was included. No cyclical epidemic emerges because the infected plants die and there is no recovered compartment. The two-disease SIR models investigated by Blyuss and Kyrychko [79] and Martcheva and Pilyugin [57] included coinfections but neglected simultaneous transmission and cross-immunity. The sustained cycles were not generated in the former though induced in the latter, which is due to enhanced transmissibility of the strain that is weaker on its own.

Immunity is a common course of infection [2, 3] and is an important factor that leads to sustained oscillation [11, 15, 20]. Empirical analyses show that, for example, vaccination increases the degree of seasonal variation in the incidence of rotavirus [5]. However, theoretical investigations show that cross-immunity alone within a two-strain SIR model cannot support cyclical epidemics [63, 66, 80]. To simplify the study, we assume complete immunity and complete cross-immunity but the immunity decays. Therefore coinfection in this study is only referred to as concurrent infection (cf. [2, 38]). This is comparable with the assumption of transient strain-transcending immunity [74, 78] which states that patients who recovered from infection can be temporally protected against other strains and permanently against the previous infection strain. The decay of immunity in our model is assumed to reflect genotype changes in strains due to antigenic drift.

In some sense the immunity period can be regarded as the “strength” of the immunity: a short immunity period implying a weak immunity. It was demonstrated here that cyclical epidemics are prohibited under weak immunity (Figure 5(a); [59]). Figure 11 shows that pathogens that induce very short immunity can readily coexist and they remain steady if there is no immunity. Whilst those that induce lifelong immunity in host more likely lead to competitive exclusion, which suggests that extremely strong cross-immunity avoids cyclical epidemics (cf. [59, 63]). In the presence of simultaneous transmission, strain interactions within coinfections interplay with cross-immunity to generate cyclical epidemics (Figures 4 and 5). The prevalent view of intrinsic mechanisms for cyclical epidemics in the situation of strong cross-immunity is the enhancement in sequential infections between strains [38, 59, 62, 63, 75]. In contrast with this, our results reveal that, with simultaneous transmissibility of strains and antigenic diversity, cyclical epidemics could be induced even if strains within coinfections interfere with each other. Furthermore, our results show that it is the simultaneous transmission of double infections that enables this, while the simultaneous transmissibility of more highly diverse coinfections only has minor effects (Figures 4 and 5).

In our model, the interepidemic period is collectively determined by model parameters. Our analyses reveal that the combination of a short immunity period and a high gives rise to cyclical epidemics of short periods (Figure 5). This prediction is consistent with the results of Ferguson et al. [38] and is qualitatively in agreement with the empirically periodic patterns of childhood infection dynamics: seasonal flu that has a short immunity duration of 3–8 years [12, 81] and about 2 [82] was bound to annual cycles [24] while pertussis of a longer immunity period of about 30 years [83] leads to 3-4-year cycles [7]. Moreover, the interepidemic period initially increases with efficiencies of partial transmission from coinfections but quickly saturates (Figure 5(c) and Figure 6). It first increases with simultaneous transmission rate of double infections and then decreases (Figure 12) while it is insensitive to simultaneous transmission of higher diverse coinfections when epidemic cycle first emerges (Figure 4(d)). Unfortunately, we currently do not have empirical data to test these relationships.

##### 4.3. Comparison to Other Models of Epidemic Cycling

Oscillations in incidence were predicted in many previous epidemic models with different underlying mechanisms, for example, in one-strain SIR models [4, 8–11], two-strain SIR models [68], and two-strain SIRS models [12] where the seasonal forcing was thought to be the mechanism. Castillo-Chavez et al. [53] established that age-structure and cross-immunity generate sustained oscillations in a two-strain SIR model while Nuño et al. [66] showed that host isolation and cross-immunity lead to periodic epidemic outbreaks. Gupta et al. [59] investigated a multistrain SIR model in which coinfections included in the force of infection to reflect the generation of new strains through recombination are not set up as separate compartments and do not contribute to transmission. This is equivalent to the situation of and with within our model. Gupta et al. [59] showed that cyclical and chaotic epidemic is generated at intermediate cross-immunity but disappeared at strong cross-immunity. This is in agreement with our results shown in Figure 4(a): cyclical and chaotic epidemics are prohibited when simultaneous transmission of and partial transmission from coinfections are absent. Ferguson et al. [38] and others [39, 62, 75] considered a similar SIR model and showed the enhancement of transmission during subsequent infection (i.e., so-called antibody-dependent enhancement) induces the cyclical and chaotic epidemics. Wearing and Rohani [84] and Bianco et al. [85] both consider multistrain models for dengue disease and combine antibody-dependent enhancement and temporal cross-immunity after recovering from primary infection. In contrast to Ferguson et al. [38] and other similar models, they can generate sustained oscillation in incidence even without antibody-dependent enhancement. However, the underlying mechanism for the cyclical epidemics is similar to what was shown by Nuño et al. [66]: the temporal cross-immunity class acts as the isolation class. All these dengue diseases models considered strain interaction during sequential infection, which is different from the concurrent infection we study here. Vasco et al. [61] considered a complicated variant of two-pathogen SIR model. It can produce sustained cyclical epidemics through a complicated combination of interactions between pathogens: competition due to cross-immunity, quarantine, or disease-induced mortality and cooperation due to immunosuppression and cross-enhancement. However, their model is different from ours in that the interaction between pathogens is associated with the changed susceptibility only and two pathogens cannot be simultaneously transmitted. Among these different mechanisms of epidemic cycling, which should be the truth? For example, inclusion of either age-structure [53, 57] or strain interaction within coinfection enables two-strain models to produce sustained oscillations in incidence. To discern which is correct, we need epidemiological data that include the additional information of age and antigenic diversity of infected people. These valuable data might be possible in near future, for example, for influenza viruses.

##### 4.4. Simplifications and Possible Developments

Our model is simplified in several aspects. For instance, the transmission rate is assumed to be constant, which is not biologically true. The transmissibility depends on infectiousness of disease agents, contact patterns, and susceptibility of host [1, 73, 86, 87] and these factors alter with environmental conditions. It has been shown that the inclusion of a stochastic environmental transmission in a one-strain SIR model can induce periodic outbreaks of infections [15, 88]. Actually, much effort has been made to search for the seasonal forcing that was induced from the variation in contact patterns and susceptibility of host populations [4, 8, 19]. These external mechanisms are certainly important, especially for explaining the consistent peak timing of cyclical epidemics, which appears as a weakness of the model presented here. However, they have difficulty in explaining the globally quick spread of influenza [13] and especially for the cyclical dynamics of syphilis [15]. It has been suggested that the mechanisms for the common oscillation of infectious diseases must be multiple [6, 10, 19]. The true mechanism might lie in the interplay of the intrinsic (as suggested by this investigation and Grassly et al. [15]) and external periodic forcing [4, 8].

For simplicity, we assume complete cross-immunity during subsequent infection. In reality, cross-immunity is more likely to be partial [43–45, 89] and therefore infection by different strains is likely to occur sequentially. Relaxing these assumptions will improve the predictions but will not change the essential findings: coinfections facilitate pathogen diversity, and simultaneous transmission from coinfections collaborates with cross-immunity to induce cyclical epidemics.

#### Appendix

#### A. Two-Strain SIRS Model

In this appendix we consider a special scenario with but allow two strains to differ in epidemiological properties (i.e., so-called asymmetric two-strain model). From the model systems (1) and (2), the two-strain SIRS model is given by where , , and are the proportions in infections with strain alone, strain alone, and both. , , and are transmission rate of strain , strain , and both, respectively, and , , and are their corresponding recovery rates.

There are four equilibria (see Table 3); in the following we use stability analysis to establish conditions for successful invasion and coexistence and the condition for Hopf bifurcation from steady to cyclical epidemics.

##### A.1. Invasion Conditions and Stability Analysis of Equilibria

The threshold condition for pathogen strain to successfully invade a naive host population was that its basic reproductive number () must be larger than one where is defined as the average number of secondary infections that result from the introduction of a single infectious individual into an entirely susceptible population. Similarly, the critical condition for strain to invade and persist in a host population in which strain was established at equilibrium () is its invasion (or effective) reproductive number being larger than unit; that is, [57]. Applying the method of van den Driessche and Watmough [90], we obtain where In expressions (A.3)-(A.4), is the fraction of infective individuals with strain at equilibrium ; represents the basic reproductive number of double infection. It is worth mentioning that when ; thus, when , . To demonstrate how the interaction between strains determines the successful invasion and coexistence, we rearrange the inequality into with . Under the situation where and which equivalently requires that the infectious period is much shorter than the duration of immunity and life span, we have , and the above inequality is approximated as follows:

By the symmetry between strain and strain , can be obtained by swapping the indexes and in the above equations. When and hold simultaneously, two strains coexist. To illustrate these relationships, two examples are shown: one for how strain interaction affects the invasion and coexistence (Figure 7) and the other for distribution of four equilibria and the effect of simultaneous transmission (Figure 8).

The stability conditions for equilibria can be obtained by analysing the eigenvalues of Jacobian matrix of the system (A.1) at equilibria. Consider the stability condition of equilibrium of single infection with strain ; its characteristic equation can be reduced to
Here and are given in Table 3. Matrix** C** always has two negative eigenvalues while matrix** D** has negative eigenvalues only if the following inequalities hold:
Inequality (A.9) is equivalent to and it can also be expressed as follows:
Combining this with inequality (A.8), we have , , and . These are stability conditions listed in Table 3. Similarly we can obtain the stability conditions of equilibria and . Due to the complexity, the stability condition for coexistence equilibrium is hard to obtain analytically though in principle it can be determined through the negativeness of the eigenvalues of the characteristic equation (see below). Interestingly, numerical simulations (data not shown) demonstrate that even when and , conditions that plus and can guarantee the coexistence of both strains (cf. [90]).

Some special situations are listed in Table 4 to illustrate the effect of coinfections on the successful invasion. Without simultaneous transmission of coinfections, strain can invade only when it has a higher basic reproductive number (see also Figure 8). However, strain interactions within coinfections change this outcome. If , strain can invade if the efficiency of transmitting strain from coinfection is larger than , suggesting that if coinfections have doubled infectious periods, strain can invade if . On the other extreme scenario where is nil, strain can invade only when and the previous single infection will be replaced by coinfections. In particular, it is possible for a less transmissible strain (i.e., ) to invade if both and are strong enough.

Next we consider a symmetric situation where two strains share the same properties: , and so that from (A.1) we have at equilibrium and When is not very strong and so that only the steady coexistence is possible, numerical results show that (Tables 1 and 2); from the second and third equations of (A.1), we obtain the approximation Putting (A.12) into (A.11) gives rise to approximation (6) in the text for the situation of two symmetric strains.

##### A.2. Analysis of Hopf Bifurcation

Because of the complexity in model system (A.1), the analytical expression for the condition of Hopf bifurcation is not obtained. Taking the advantage of R computing language (http://www.r-project.org/), however, we can numerically test and obtain the condition under which the Hopf bifurcation occurs. The characteristic equation at the coexistence equilibrium is as follows: where elements are functions of the interior equilibrium and model parameters. The characteristic equation can be written as follows: The coefficients , , are the functions of matrix elements , .

The quartic equation (A.14) can be solved by Ferrari’s method. To investigate the Hopf bifurcation from a steady state to a periodic cycle in incidence, we examine the eigenvalues against the efficiency of single transmission from the coinfection. In general, the four eigenvalues can be expressed as follows: When the real parts are negative, the steady coexistence equilibrium is stable. However, when values of increase to some critical values , real part of some eigenvalues will increase to zero and the equilibrium loses its stability and a Hopf bifurcation occurs [91], from which oscillation in incidence will be induced. The imaginary part of the dominant eigenvalue, , is the frequency of the oscillation when the real part passes through zero to become positive and the interepidemic period can be calculated as follows: One numerical example was shown in Figure 9. As the cooperation between strains within coinfections becomes strong enough, complicated dynamical behaviour such as period-doubling bifurcation and chaotic epidemics can be generated (Figure 10). Similarly, this can also be tested against other model parameters.

Interestingly, the fraction of infections at the peak of a cyclical epidemic is much larger than that at equilibrium of a single strain established alone. As illustrated in Figure 10(b), the peak fraction of single infections with dominant strain is 0.24% while it is 0.04% at equilibrium if it alone infects the host population (from Table 1 and see also Figure 10(a)). Therefore we should be cautious when explaining the meanings and estimating values of the basic reproductive number () for cyclical infections. Moreover, the fraction of single infection with a slightly less transmissible strain is much lower while the fraction of coinfections remains extremely low even at the peak. In Figure 10(b), they are 0.024% and 0.001%, respectively, which indicates the chance for coinfections to be detected being in relation to single infections. Therefore it is understandable that coinfections are practically hard to attract sensible attention, especially if advantageous technology such as genotype typing is not readily available.

The cyclical epidemics become hard to generate when immunity period becomes either too short or too long (Figure 11). Given low reproductive numbers for two different strains but relatively high transmissibility for double infection as in Figure 11, the critical efficiency of single transmission from double infection for cyclical epidemics is for immunity that lasts for one month or for lifelong immunity. This may be too high for most pathogens. However, when the duration of immunity takes some intermediate values, a low efficiency of single transmission from double infection should be sufficient to maintain the cyclical epidemics. When oscillation is induced, the interepidemic period increases with the efficiency of single transmission from coinfections (cf. Figures 10(b) and 10(c)). While it first increases with the simultaneous transmissibility , after reaching peak at a critical value, it decreases (Figure 12).

#### Conflict of Interests

The authors declare that there is no conflict of interests regarding the publication of this paper.

#### Acknowledgments

Xu-Sheng Zhang was funded by the Public Health England and Ke-Fei Cao was funded by the National Natural Science Foundation of China (NSFC) (Grant nos. 11365023 and 10565004) and the Specialized Research Fund for the Doctoral Program of Higher Education of China (SRFDP) (Grant no. 20050673001). The authors would like to thank the anonymous reviewers for their helpful comments and constructive suggestions on the paper.

#### References

- R. M. Anderson and R. M. May,
*Infectious Diseases of Humans: Dynamics and Control*, Oxford University Press, Oxford, UK, 1991. - N. J. Dimmock, A. J. Easton, and K. N. Leppard,
*Introduction to Modern Virology*, Blackwell Publishing, Oxford, UK, 6th edition, 2007. - C. J. E. Metcalf, O. N. Bjørnstad, B. T. Grenfell, and V. Andreasen, “Seasonality and comparative dynamics of six childhood infections in pre-vaccination Copenhagen,”
*Proceedings of the Royal Society B: Biological Sciences*, vol. 276, no. 1676, pp. 4111–4118, 2009. View at Publisher · View at Google Scholar · View at Scopus - S. Altizer, A. Dobson, P. Hosseini, P. Hudson, M. Pascual, and P. Rohani, “Seasonality and the dynamics of infectious diseases,”
*Ecology Letters*, vol. 9, no. 4, pp. 467–484, 2006. View at Publisher · View at Google Scholar · View at Scopus - V. E. Pitzer, C. Viboud, B. A. Lopman, M. M. Patel, U. D. Parashar, and B. T. Grenfell, “Influence of birth rates and transmission rates on the global seasonality of rotavirus incidence,”
*Journal of the Royal Society Interface*, vol. 8, no. 64, pp. 1584–1593, 2011. View at Publisher · View at Google Scholar · View at Scopus - E. Lofgren, N. H. Fefferman, Y. N. Naumov, J. Gorski, and E. N. Naumova, “Influenza seasonality: underlying causes and modeling theories,”
*Journal of Virology*, vol. 81, no. 11, pp. 5429–5436, 2007. View at Publisher · View at Google Scholar · View at Scopus - H. Broutin, J.-F. Guégan, E. Elguero, F. Simondon, and B. Cazelles, “Large-scale comparative analysis of pertussis population dynamics: periodicity, synchrony, and impact of vaccination,”
*The American Journal of Epidemiology*, vol. 161, no. 12, pp. 1159–1167, 2005. View at Publisher · View at Google Scholar · View at Scopus - N. C. Grassly and C. Fraser, “Seasonal infectious disease epidemiology,”
*Proceedings of the Royal Society B: Biological Sciences*, vol. 273, no. 1600, pp. 2541–2550, 2006. View at Publisher · View at Google Scholar · View at Scopus - Y. A. Kuznetsov and C. Piccardi, “Bifurcation analysis of periodic SEIR and SIR epidemic models,”
*Journal of Mathematical Biology*, vol. 32, no. 2, pp. 109–121, 1994. View at Google Scholar · View at Scopus - K. Koelle and M. Pascual, “Disentangling extrinsic from intrinsic factors in disease dynamics: a nonlinear time series approach with an application to cholera,”
*The American Naturalist*, vol. 163, no. 6, pp. 901–913, 2004. View at Publisher · View at Google Scholar · View at Scopus - M. J. Ferrari, R. F. Grais, N. Bharti et al., “The dynamics of measles in sub-Saharan Africa,”
*Nature*, vol. 451, no. 7179, pp. 679–684, 2008. View at Publisher · View at Google Scholar · View at Scopus - J. Truscott, C. Fraser, S. Cauchemez et al., “Essential epidemiological mechanisms underpinning the transmission dynamics of seasonal influenza,”
*Journal of the Royal Society Interface*, vol. 9, no. 67, pp. 304–312, 2012. View at Publisher · View at Google Scholar · View at Scopus - S. F. Dowell, “Seasonal variation in host susceptibility and cycles of certain infectious diseases,”
*Emerging Infectious Diseases*, vol. 7, no. 3, pp. 369–374, 2001. View at Google Scholar · View at Scopus - J. Dushoff, J. B. Plotkin, S. A. Levin, and D. J. D. Earn, “Dynamical resonance can account for seasonality of influenza epidemics,”
*Proceedings of the National Academy of Sciences of the United States of America*, vol. 101, no. 48, pp. 16915–16916, 2004. View at Publisher · View at Google Scholar · View at Scopus - N. C. Grassly, C. Fraser, and G. P. Garnett, “Host immunity and synchronized epidemics of syphilis across the United States,”
*Nature*, vol. 433, no. 7024, pp. 417–421, 2005. View at Publisher · View at Google Scholar · View at Scopus - H. W. Hethcote, “Mathematics of infectious diseases,”
*Society for Industrial and Applied Mathematics Review*, vol. 42, no. 4, pp. 599–653, 2000. View at Google Scholar · View at Scopus - H. W. Hethcote and S. A. Levin, “Periodicity in epidemiological models,” in
*Applied Mathematical Ecology*, S. A. Levin, T. G. Hallam, and L. J. Gross, Eds., vol. 18 of*Biomathematics*, pp. 193–211, Springer, Berlin, Germany, 1989. View at Google Scholar - Z. Feng and H. R. Thieme, “Recurrent outbreaks of childhood diseases revisited: the impact of isolation,”
*Mathematical Biosciences*, vol. 128, no. 1-2, pp. 93–130, 1995. View at Publisher · View at Google Scholar · View at Scopus - J. Tamerius, M. I. Nelson, S. Z. Zhou, C. Viboud, M. A. Miller, and W. J. Alonso, “Global influenza seasonality: reconciling patterns across temperate and tropical regions,”
*Environmental Health Perspectives*, vol. 119, no. 4, pp. 439–445, 2011. View at Publisher · View at Google Scholar · View at Scopus - B. Grenfell and O. Bjørnstad, “Sexually transmitted diseases: epidemic cycling and immunity,”
*Nature*, vol. 433, no. 7024, pp. 366–367, 2005. View at Publisher · View at Google Scholar · View at Scopus - G. Snounou, S. Viriyakosol, W. Jarra, S. Thaithong, and K. N. Brown, “Identification of the four human malaria parasite species in field samples by the polymerase chain reaction and detection of a high prevalence of mixed infections,”
*Molecular and Biochemical Parasitology*, vol. 58, no. 2, pp. 283–292, 1993. View at Publisher · View at Google Scholar · View at Scopus - B. Briseño-García, H. Gómez-Dantés, E. Argott-Ramírez et al., “Potential risk for dengue hemorrhagic fever: the isolation of serotype dengue-3 in Mexico,”
*Emerging Infectious Diseases*, vol. 2, no. 2, pp. 133–135, 1996. View at Google Scholar - S. Toda, R. Okamoto, T. Nishida et al., “Isolation of influenza A/H3 and B viruses from an influenza patient: confirmation of co-infection by two influenza viruses,”
*Japanese Journal of Infectious Diseases*, vol. 59, no. 2, pp. 142–143, 2006. View at Google Scholar · View at Scopus - L. Denoeud, C. Turbelin, S. Ansart, A.-J. Valleron, A. Flahault, and F. Carrat, “Predicting pneumonia and influenza mortality from morbidity data,”
*PLoS ONE*, vol. 2, no. 5, article e464, 2007. View at Publisher · View at Google Scholar · View at Scopus - M. Lipsitch and J. J. O'Hagan, “Patterns of antigenic diversity and the mechanisms that maintain them,”
*Journal of the Royal Society Interface*, vol. 4, no. 16, pp. 787–802, 2007. View at Publisher · View at Google Scholar · View at Scopus - J. Parkhill and B. W. Wren, “Bacterial epidemiology and biology—lessons from genome sequencing,”
*Genome Biology*, vol. 12, no. 10, article 230, 2011. View at Publisher · View at Google Scholar · View at Scopus - L. O. Bakaletz, “Developing animal models for polymicrobial diseases,”
*Nature Reviews Microbiology*, vol. 2, no. 7, pp. 552–568, 2004. View at Publisher · View at Google Scholar · View at Scopus - M. Peacey, R. J. Hall, S. Sonnberg et al., “Pandemic (H1N1) 2009 and seasonal influenza A (H1N1) co-infection, New Zealand, 2009,”
*Emerging Infectious Diseases*, vol. 16, no. 10, pp. 1618–1620, 2010. View at Publisher · View at Google Scholar · View at Scopus - W. Liu, Z.-D. Li, F. Tang et al., “Mixed infections of pandemic H1N1 and seasonal H3N2 viruses in 1 outbreak,”
*Clinical Infectious Diseases*, vol. 50, no. 10, pp. 1359–1365, 2010. View at Publisher · View at Google Scholar · View at Scopus - S. Räsänen, S. Lappalainen, S. Kaikkonen, M. Hämäläinen, M. Salminen, and T. Vesikari, “Mixed viral infections causing acute gastroenteritis in children in a waterborne outbreak,”
*Epidemiology and Infection*, vol. 138, no. 9, pp. 1227–1234, 2010. View at Publisher · View at Google Scholar · View at Scopus - H. Tanner, E. Boxall, and H. Osman, “Respiratory viral infections during the 2009-2010 winter seasons in Central England, UK: incidence and patterns of multiple virus co-infections,”
*European Journal of Clinical Microbiology and Infectious Diseases*, vol. 31, no. 11, pp. 3001–3006, 2012. View at Google Scholar - O. Balmer and M. Tanner, “Prevalence and implications of multiple-strain infections,”
*The Lancet Infectious Diseases*, vol. 11, no. 11, pp. 868–878, 2011. View at Publisher · View at Google Scholar · View at Scopus - E. C. Griffiths, A. B. Pedersen, A. Fenton, and O. L. Petchey, “The nature and consequences of coinfection in humans,”
*Journal of Infection*, vol. 63, no. 3, pp. 200–206, 2011. View at Publisher · View at Google Scholar · View at Scopus - M. Lipsitch, C. Colijn, T. Cohen, W. P. Hanage, and C. Fraser, “No coexistence for free: neutral null models for multistrain pathogens,”
*Epidemics*, vol. 1, no. 1, pp. 2–13, 2009. View at Publisher · View at Google Scholar · View at Scopus - T. DaPalma, B. P. Doonan, N. M. Trager, and L. M. Kasman, “A systematic approach to virus-virus interactions,”
*Virus Research*, vol. 149, no. 1, pp. 1–9, 2010. View at Publisher · View at Google Scholar · View at Scopus - X.-S. Zhang, J. Holt, and J. Colvin, “Synergism between plant viruses: a mathematical analysis of the epidemiological implications,”
*Plant Pathology*, vol. 50, no. 6, pp. 732–746, 2001. View at Publisher · View at Google Scholar · View at Scopus - J. Saldaña, S. F. Elena, and R. V. Solé, “Coinfection and superinfection in RNA virus populations: a selection-mutation model,”
*Mathematical Biosciences*, vol. 183, no. 2, pp. 135–160, 2003. View at Publisher · View at Google Scholar · View at Scopus - N. Ferguson, R. Anderson, and S. Gupta, “The effect of antibody-dependent enhancement on the transmission dynamics and persistence of multiple-strain pathogens,”
*Proceedings of the National Academy of Sciences of the United States of America*, vol. 96, no. 2, pp. 790–794, 1999. View at Publisher · View at Google Scholar · View at Scopus - D. A. T. Cummings, I. B. Schwartz, L. Billings, L. B. Shaw, and D. S. Burke, “Dynamic effects of antibody-dependent enhancement on the fitness of viruses,”
*Proceedings of the National Academy of Sciences of the United States of America*, vol. 102, no. 42, pp. 15259–15264, 2005. View at Publisher · View at Google Scholar · View at Scopus - A. L. Graham, I. M. Cattadori, J. O. Lloyd-Smith, M. J. Ferrari, and O. N. Bjørnstad, “Transmission consequences of coinfection: cytokines writ large?”
*Trends in Parasitology*, vol. 23, no. 6, pp. 284–291, 2007. View at Publisher · View at Google Scholar · View at Scopus - S. Gorman, N. L. Harvey, D. Moro et al., “Mixed infection with multiple strains of murine cytomegalovirus occurs following simultaneous or sequential infection of immunocompetent mice,”
*Journal of General Virology*, vol. 87, no. 5, pp. 1123–1132, 2006. View at Publisher · View at Google Scholar · View at Scopus - B. M. Davis, A. E. Aiello, S. Dawid, P. Rohani, S. Shrestha, and B. Foxman, “Influenza and community-acquired pneumonia interactions: the impact of order and time of infection on population patterns,”
*The American Journal of Epidemiology*, vol. 175, no. 5, pp. 363–367, 2012. View at Publisher · View at Google Scholar · View at Scopus - S. L. Epstein, “Prior H1N1 influenza infection and susceptibility of Cleveland Family Study participants during the H2N2 pandemic of 1957: an experiment of nature,”
*Journal of Infectious Diseases*, vol. 193, no. 1, pp. 49–53, 2006. View at Publisher · View at Google Scholar · View at Scopus - K. Hancock, V. Veguilla, X. Lu et al., “Cross-reactive antibody responses to the 2009 pandemic H1N1 influenza virus,”
*The New England Journal of Medicine*, vol. 361, no. 20, pp. 1945–1952, 2009. View at Publisher · View at Google Scholar · View at Scopus - M. L. B. Hillaire, S. E. van Trierum, J. H. C. M. Kreijtz et al., “Cross-protective immunity against influenza pH1N1 2009 viruses induced by seasonal influenza A (H3N2) virus is mediated by virus-specific T-cells,”
*Journal of General Virology*, vol. 92, no. 10, pp. 2339–2349, 2011. View at Publisher · View at Google Scholar · View at Scopus - R. A. P. M. Perera, S. Riley, S. K. Ma, H.-C. Zhu, Y. Guan, and J. S. M. Peiris, “Seroconversion to pandemic (H1N1) 2009 virus and cross-reactive immunity to other swine influenza viruses,”
*Emerging Infectious Diseases*, vol. 17, no. 10, pp. 1897–1899, 2011. View at Publisher · View at Google Scholar · View at Scopus - J. M. Barry, C. Viboud, and L. Simonsen, “Cross-protection between successive waves of the 1918-1919 influenza pandemic: epidemiological evidence from US Army camps and from Britain,”
*Journal of Infectious Diseases*, vol. 198, no. 10, pp. 1427–1434, 2008. View at Publisher · View at Google Scholar · View at Scopus - T. E. Walton, M. M. Jochim, T. L. Barber, and L. H. Thompson, “Cross-protective immunity between equine encephalomyelitis viruses in equids,”
*The American Journal of Veterinary Research*, vol. 50, no. 9, pp. 1442–1446, 1989. View at Google Scholar · View at Scopus - R. Porrozzi, A. Teva, V. F. Amaral, M. V. Santos da Costa, and G. Grimaldi Jr., “Cross-immunity experiments between different species or strains of Leishmania in rhesus macaques (Macaca mulatta),”
*The American Journal of Tropical Medicine and Hygiene*, vol. 71, no. 3, pp. 297–305, 2004. View at Google Scholar · View at Scopus - S. Gupta, K. Trenholme, R. M. Anderson, and K. P. Day, “Antigenic diversity and the transmission dynamics of Plasmodium falciparum,”
*Science*, vol. 263, no. 5149, pp. 961–963, 1994. View at Google Scholar · View at Scopus - L. J. Abu-Raddad, B. I. S. van der Ventel, and N. M. Ferguson, “Interactions of multiple strain pathogen diseases in the presence of coinfection, cross immunity, and arbitrary strain diversity,”
*Physical Review Letters*, vol. 100, no. 16, Article ID 168102, 2008. View at Publisher · View at Google Scholar · View at Scopus - K. Dietz, “Epidemiologic interference of virus populations,”
*Journal of Mathematical Biology*, vol. 8, no. 3, pp. 291–300, 1979. View at Google Scholar · View at Scopus - C. Castillo-Chavez, H. W. Hethcote, V. Andreasen, S. A. Levin, and W. M. Liu, “Epidemiological models with age structure, proportionate mixing, and cross-immunity,”
*Journal of Mathematical Biology*, vol. 27, no. 3, pp. 233–258, 1989. View at Publisher · View at Google Scholar · View at Scopus - M. A. Nowak and R. M. May, “Superinfection and the evolution of parasite virulence,”
*Proceedings of the Royal Society B: Biological Sciences*, vol. 255, no. 1342, pp. 81–89, 1994. View at Google Scholar · View at Scopus - R. M. May and M. A. Nowak, “Coinfection and the evolution of parasite virulence,”
*Proceedings of the Royal Society B: Biological Sciences*, vol. 261, no. 1361, pp. 209–215, 1995. View at Publisher · View at Google Scholar · View at Scopus - M. Iannelli, M. Martcheva, and X.-Z. Li, “Strain replacement in an epidemic model with super-infection and perfect vaccination,”
*Mathematical Biosciences*, vol. 195, no. 1, pp. 23–46, 2005. View at Publisher · View at Google Scholar · View at Scopus - M. Martcheva and S. S. Pilyugin, “The role of coinfection in multidisease dynamics,”
*SIAM Journal on Applied Mathematics*, vol. 66, no. 3, pp. 843–872, 2006. View at Publisher · View at Google Scholar · View at Scopus - E. Delgado-Eckert, S. Ojosnegros, and N. Beerenwinkel, “The evolution of virulence in RNA viruses under a competition-colonization trade-off,”
*Bulletin of Mathematical Biology*, vol. 73, no. 8, pp. 1881–1908, 2011. View at Publisher · View at Google Scholar · View at Scopus - S. Gupta, N. Ferguson, and R. Anderson, “Chaos, persistence, and evolution of strain structure in antigenically diverse infectious agents,”
*Science*, vol. 280, no. 5365, pp. 912–915, 1998. View at Publisher · View at Google Scholar · View at Scopus - P. Rohani, C. J. Green, N. B. Mantilla-Beniers, and B. T. Grenfell, “Ecological interference between fatal diseases,”
*Nature*, vol. 422, no. 6934, pp. 885–888, 2003. View at Publisher · View at Google Scholar · View at Scopus - D. A. Vasco, H. J. Wearing, and P. Rohani, “Tracking the dynamics of pathogen interactions: modeling ecological and immune-mediated processes in a two-pathogen single-host system,”
*Journal of Theoretical Biology*, vol. 245, no. 1, pp. 9–25, 2007. View at Publisher · View at Google Scholar · View at Scopus - M. Recker, K. B. Blyuss, C. P. Simmons et al., “Immunological serotype interactions and their effect on the epidemiological pattern of dengue,”
*Proceedings of the Royal Society B: Biological Sciences*, vol. 276, no. 1667, pp. 2541–2548, 2009. View at Publisher · View at Google Scholar · View at Scopus - V. Andreasen, J. Lin, and S. A. Levin, “The dynamics of cocirculating influenza strains conferring partial cross-immunity,”
*Journal of Mathematical Biology*, vol. 35, no. 7, pp. 825–842, 1997. View at Google Scholar - J. R. Gog and B. T. Grenfell, “Dynamics and selection of many-strain pathogens,”
*Proceedings of the National Academy of Sciences of the United States of America*, vol. 99, no. 26, pp. 17209–17214, 2002. View at Publisher · View at Google Scholar · View at Scopus - O. Restif and B. T. Grenfell, “Integrating life history and cross-immunity into the evolutionary dynamics of pathogens,”
*Proceedings of the Royal Society B: Biological Sciences*, vol. 273, no. 1585, pp. 409–416, 2006. View at Publisher · View at Google Scholar · View at Scopus - M. Nuño, Z. Feng, M. Martcheva, and C. Castillo-Chavez, “Dynamics of two-strain influenza with isolation and partial cross-immunity,”
*SIAM Journal on Applied Mathematics*, vol. 65, no. 3, pp. 964–982, 2005. View at Publisher · View at Google Scholar · View at Scopus - L. J. Abu-Raddad and N. M. Ferguson, “The impact of cross-immunity, mutation and stochastic extinction on pathogen diversity,”
*Proceedings of the Royal Society B: Biological Sciences*, vol. 271, no. 1556, pp. 2431–2438, 2004. View at Publisher · View at Google Scholar · View at Scopus - M. Kamo and A. Sasaki, “The effect of cross-immunity and seasonal forcing in a multi-strain epidemic model,”
*Physica D: Nonlinear Phenomena*, vol. 165, no. 3-4, pp. 228–241, 2002. View at Publisher · View at Google Scholar · View at Scopus - P. Bharaj, H. S. Chahar, A. Pandey et al., “Concurrent infections by all four dengue virus serotypes during an outbreak of dengue in 2006 in Delhi, India,”
*Virology Journal*, vol. 5, article 1, 2008. View at Publisher · View at Google Scholar · View at Scopus - S. Vaccarella, S. Franceschi, R. Herrero et al., “Clustering of multiple human papillomavirus infections in women from a population-based study in Guanacaste, Costa Rica,”
*Journal of Infectious Diseases*, vol. 204, no. 3, pp. 385–390, 2011. View at Publisher · View at Google Scholar · View at Scopus - J. F. Brundage, “Interactions between influenza and bacterial respiratory pathogens: implications for pandemic preparedness,”
*The Lancet Infectious Diseases*, vol. 6, no. 5, pp. 303–312, 2006. View at Publisher · View at Google Scholar · View at Scopus - X.-S. Zhang, D. De Angelis, P. J. White, A. Charlett, R. G. Pebody, and J. McCauley, “Co-circulation of influenza A virus strains and emergence of pandemic via reassortment: the role of cross-immunity,”
*Epidemics*, vol. 5, no. 1, pp. 20–33, 2013. View at Google Scholar - M. J. Keeling and P. Rohani,
*Modeling Infectious Disease in Humans and Animals*, Princeton University Press, Princeton, NJ, USA, 2007. - P. Minayev and N. Ferguson, “Incorporating demographic stochasticity into multi-strain epidemic models: application to influenza A,”
*Journal of the Royal Society Interface*, vol. 6, no. 40, pp. 989–996, 2009. View at Publisher · View at Google Scholar · View at Scopus - M. Aguiar, B. Kooi, and N. Stollenwerk, “Epidemiology of dengue fever: a model with temporary cross-immunity and possible secondary infection shows bifurcations and chaotic behaviour in wide parameter regions,”
*Mathematical Modelling of Natural Phenomena*, vol. 3, no. 4, pp. 48–70, 2008. View at Publisher · View at Google Scholar · View at Scopus - B. W. Kooi, M. Aguiar, and N. Stollenwerk, “Analysis of an asymmetric two-strain dengue model,”
*Mathematical Biosciences*, vol. 248, pp. 128–139, 2014. View at Google Scholar - L. Mier-y-Teran-Romero, I. B. Schwartz, and D. A. T. Cummings, “Breaking the symmetry: immune enhancement increases persistence of dengue viruses in the presence of asymmetric transmission rates,”
*Journal of Theoretical Biology*, vol. 332, pp. 203–210, 2013. View at Google Scholar - N. M. Ferguson, A. P. Galvani, and R. M. Bush, “Ecological and immunological determinants of influenza evolution,”
*Nature*, vol. 422, no. 6930, pp. 428–433, 2003. View at Publisher · View at Google Scholar · View at Scopus - K. B. Blyuss and Y. N. Kyrychko, “On a basic model of a two-disease epidemic,”
*Applied Mathematics and Computation*, vol. 160, no. 1, pp. 177–187, 2005. View at Publisher · View at Google Scholar · View at Scopus - J. R. Gog and J. Swinton, “A status-based approach to multiple strain dynamics,”
*Journal of Mathematical Biology*, vol. 44, no. 2, pp. 169–184, 2002. View at Publisher · View at Google Scholar · View at Scopus - Y. Xia, J. R. Gog, and B. T. Grenfell, “Semiparametric estimation of the duration of immunity from infectious disease time series: influenza as a case-study,”
*Journal of the Royal Statistical Society C: Applied Statistics*, vol. 54, no. 3, pp. 659–672, 2005. View at Publisher · View at Google Scholar · View at Scopus - P.-Y. Boëlle, S. Ansart, A. Cori, and A.-J. Valleron, “Transmission parameters of the A/H1N1 (2009) influenza virus pandemic: a review,”
*Influenza and other Respiratory Viruses*, vol. 5, no. 5, pp. 306–316, 2011. View at Publisher · View at Google Scholar · View at Scopus - H. J. Wearing and P. Rohani, “Estimating the duration of pertussis immunity using epidemiological signatures,”
*PLoS Pathogens*, vol. 5, no. 10, Article ID e1000647, 2009. View at Publisher · View at Google Scholar · View at Scopus - H. J. Wearing and P. Rohani, “Ecological and immunological determinants of dengue epidemics,”
*Proceedings of the National Academy of Sciences of the United States of America*, vol. 103, no. 31, pp. 11802–11807, 2006. View at Publisher · View at Google Scholar · View at Scopus - S. Bianco, L. B. Shaw, and I. B. Schwartz, “Epidemics with multistrain interactions: the interplay between cross immunity and antibody-dependent enhancement,”
*Chaos*, vol. 19, no. 4, Article ID 043123, 2009. View at Google Scholar - O. Diekmann and J. A. P. Heesterbeek,
*Mathematical Epidemiology of Infectious Diseases: Model Building, Analysis, and Interpretation*, John Wiley and Sons, Chichester, UK, 2000. - X.-S. Zhang, “
*Escherichia coli*O157 infection on cattle farms: the formulation of the force of infection and its effect on control effectiveness,”*Epidemiology and Infection*, vol. 140, no. 7, pp. 1215–1226, 2012. View at Google Scholar - R.-H. Wang, Z. Jin, Q.-X. Liu, J. van de Koppel, and D. Alonso, “A simple stochastic model with environmental transmission explains multi-year periodicity in outbreaks of avian flu,”
*PLoS ONE*, vol. 7, no. 2, Article ID e28873, 2012. View at Publisher · View at Google Scholar · View at Scopus - D. Rios-Doria and G. Chowell, “Qualitative analysis of the level of cross-protection between epidemic waves of the 1918-1919 influenza pandemic,”
*Journal of Theoretical Biology*, vol. 261, no. 4, pp. 584–592, 2009. View at Publisher · View at Google Scholar · View at Scopus - P. van den Driessche and J. Watmough, “Reproduction numbers and sub-threshold endemic equilibria for compartmental models of disease transmission,”
*Mathematical Biosciences*, vol. 180, no. 1-2, pp. 29–48, 2002. View at Publisher · View at Google Scholar · View at Scopus - B. D. Hassard, N. D. Kazarinoff, and Y.-H. Wan,
*Theory and Applications of Hopf Bifurcation*, vol. 41 of*London Mathematical Society Lecture Note Series*, Cambridge University Press, Cambridge, UK, 1981.