Modeling rotavirus strain dynamics in developed countries to understand the potential impact of vaccination on genotype distributions

Abstract

Understanding how immunity shapes the dynamics of multistrain pathogens is essential in determining the selective pressures imposed by vaccines. There is currently much interest in elucidating the strain dynamics of rotavirus to determine whether vaccination may lead to the replacement of vaccine-type strains. In developed countries, G1P[8] strains constitute the majority of rotavirus infections most years, but occasionally other genotypes dominate for reasons that are not well understood. We developed a mathematical model to examine the interaction of five common rotavirus genotypes. We explored a range of estimates for the relative strength of homotypic vs. heterotypic immunity and compared model predictions against observed genotype patterns from six countries. We then incorporated vaccination in the model to examine its impact on rotavirus incidence and the distribution of strains. Our model can explain the coexistence and cyclical pattern in the distribution of genotypes observed in most developed countries. The predicted frequency of cycling depends on the relative strength of homotypic vs. heterotypic immunity. Vaccination that provides strong protection against G1 and weaker protection against other strains will likely lead to an increase in the relative prevalence of non-G1 strains, whereas a vaccine that provides equally strong immunity against all strains may promote the continued predominance of G1. Overall, however, disease incidence is expected to be substantially reduced under both scenarios and remain below prevaccination levels despite the possible emergence of new strains. Better understanding of homotypic vs. heterotypic immunity, both natural and vaccine-induced, will be critical in predicting the impact of vaccination.

Differences in the strength of immunity and cross-immunity among strains are thought to play an important role in shaping the epidemiological and evolutionary patterns of infectious diseases (1). Models for the transmission dynamics of multistrain pathogens have helped elucidate the mechanisms behind empirical patterns, including multiannual oscillations in the incidence of influenza (2–4) and dengue cases (5–10), antigenic drift within influenza subtypes (11–14), and cyclical patterns in the predominance of different strains of influenza (3, 15), respiratory syncytial virus (16), dengue (5–10), malaria (17), and cholera (18). Vaccines that elicit a stronger immune response to specific strains may impact the distribution of genotypes in unforeseen ways, which in turn may affect overall disease incidence. The response of multistrain pathogens to selective pressures imposed by vaccines is a concern for many newly introduced vaccines, including human papillomavirus, pneumococcal, and rotavirus vaccines (15, 19, 20).

Rotavirus is a major cause of severe gastroenteritis in both humans and animals (21). More than half a million deaths each year are attributed to rotavirus, most occurring in children <5 y of age (22). Models for the transmission dynamics have aided our understanding of the important factors driving epidemic patterns of rotavirus diarrhea in the community (23). These models can be used to estimate the expected direct and indirect effects of rotavirus vaccines (23–27) and have generated predictions that agree with early observations of the impact of vaccination in the United States (23). However, heretofore all models for the transmission dynamics of rotavirus have ignored any complexity relating to the different strains causing infections.

There are seven rotavirus groups (A–G), but the majority of human disease is caused by rotavirus A (21). Group A rotavirus strains can be differentiated both molecularly and serologically according to their VP7 and VP4 surface proteins into G and P types (21). Historically, four common rotavirus genotypes (G1P[8], G2P[4], G3P[8], and G4P[8]) have cocirculated along with several less common genotypes (28). In developed countries, G1 strains constitute the majority of infections in most years, but occasionally other genotypes dominate for reasons that are not well understood. Over the past 2 decades, G9 strains emerged and spread across the globe and are now considered to be the fourth most prevalent genotype (28, 29). Efforts to monitor the distribution of genotypes in different countries have increased in anticipation of the introduction of rotavirus vaccines.

Evidence from observational studies and vaccine trials indicates that natural rotavirus infection and vaccination confer both homotypic [i.e., against the genotype causing natural infection or genotype(s) included in the vaccine] and heterotypic [i.e., against genotypes other than the infecting genotype or genotype(s) included in the vaccine] protection. However, the nature and strength of protection may vary depending on the number of previous infections and/or type of vaccine (e.g., animal, reassortant, or human strain), and even homotypic protection is incomplete. Prospective cohort studies that have identified and typed sequential infections in infants suggest that second infections are more likely to be caused by a different genotype than the one causing first infection, but second infections with the same genotype can occur (30–34). Although first infections elicit a predominantly homotypic, serum-neutralizing antibody response, subsequent infections generally elicit a broader, cross-reactive response (35, 36), providing one rationale for administering multiple doses of vaccine.

Currently licensed rotavirus vaccines consist of a single live attenuated human G1P[8] strain (Rotarix; GlaxoSmithKline) or a pentavalent human–bovine reassortant strain containing the G1, G2, G3, G4, and P[8] surface proteins (RotaTeq; Merck) (22). Although both vaccines have demonstrated similar efficacy in clinical trials, they differ with respect to the immunological mechanisms underlying protection (SI Appendix). The relative importance of the various mechanisms of immunity is not entirely understood (37).

Vaccination could impact the distribution of rotavirus strains by increasing selection pressure on certain genotypes, potentially leading to decreased vaccine effectiveness (20). During vaccine trials, small discrepancies in efficacy were noted; however, these trials typically lacked sufficient power to detect significant differences in vaccine efficacy against homotypic vs. heterotypic strains. Although there is speculation that rotavirus vaccines may be impacting strain distributions in countries with routine immunization, there are not yet enough data to establish robust conclusions (38–42).

Given the importance of this question to the ultimate success of rotavirus vaccination programs, we sought to extend our model for the transmission dynamics of rotavirus to incorporate the interaction of five strains (23). We compared observed patterns of genotype cycling with those predicted by our model, exploring a range of possible values for the strength of homotypic vs. heterotypic immunity. We then incorporated vaccination into the model to explore the potential impact of vaccines that offer differential protection against circulating strains and considered the consequences of the emergence of a new strain. We conclude by setting these results in the context of pathogen strain dynamics in general.

Results

Empirical Patterns.

To better understand the prevaccination cycling of rotavirus genotypes and calibrate our strain-specific model, we analyzed published time series of annual genotype distributions from six developed countries (43–49). Fourier analysis suggests that the predominant rotavirus strains cycle with periods (T) ranging from 3 to 11 y (Fig. 1). We found significant signals between T = 6.1 and 11.6 y for G1–G4 strains in Italy, G4 in Hungary, G3 in the United States and Australia, and G9 in Spain (Fig. 1B). In all countries, G1 strains exhibited strong cycles between T = 5.8 and 10.7 y, although some countries also exhibited significant signals at shorter periods (e.g., T = 2.8–3.2 y for G1 strains in Spain, the United States, and Japan and T = 2.4–3.2 for G2 in Australia and G9 in the United States). Wavelet analysis of genotype distributions in Melbourne, Australia revealed similar periodicities but did not suggest any strong temporal trends (SI Appendix).

Fig. 1.

Analysis of genotype oscillations observed in six countries. (A) Genotype distributions (percentage of typeable rotavirus-positive samples) for G1–G4 and G9, as described in refs. 43–49. (B) Fourier analysis of cyclical patterns for each of the five genotypes. Fourier amplitudes for each genotype are plotted on a log scale for periods ranging from 2 to 12 y. Asterisks represent significant signals according to bootstrap analysis.

Modeling Prevaccination Dynamics.

We developed a mathematical model for the transmission dynamics of rotavirus that incorporated the interaction of five different strains. These strains are meant to represent the five predominant G-types circulating in most populations (G1–G4 and G9). We assumed the fifth strain emerged 5 y into our simulations, in line with the recent worldwide emergence of G9 (28, 29). Our model was able to reproduce the major patterns of genotype predominance, coexistence, and cycling observed in the prevaccination era. When immunity against second infection with a homotypic strain was assumed to be much stronger than immunity against heterotypic strains (Fig. 2A, upper left corner), the model predicted a quicker cycling of the predominant genotype (T ≈ 3 y), whereas when there was only slightly stronger protection against homotypic compared with heterotypic strains, the predicted period of oscillations was considerably greater (T ≈ 8 y). These oscillations were also present in the incidence data and were sustained regardless of whether we accounted for seasonality in the transmission rate (SI Appendix). When immunity against homotypic and heterotypic strains was equal or near equal, there was no cycling of genotypes (Fig. 2A, lower right corner).

Fig. 2.

Model-predicted patterns for different strengths of homotypic and heterotypic immunity. The colorbars indicate (A) the dominant period of oscillations for G1 strains (as indicated by the maximum Fourier amplitude) and (B) the mean proportion of severe diarrhea cases due to G1 strains over an 80-y period, for relative risks of second infection with homotypic strains ranging from 0.01 to 0.5 (and relative infectiousness ranging from 0.1 to 0.5) and relative risks of second infection with heterotypic strains ranging from 0.5 to 1.0 (with corresponding relative infectiousness).

It was necessary to assume that G1 strains were slightly more transmissible than other genotypes to fully explain why G1 is the predominant genotype in temperate countries (SI Appendix). When we assumed that the relative transmissibility of G2–G4 and G9 strains was 90% that of G1 strains, the model predicted that G1 strains accounted for 41–100% of severe diarrhea cases, depending on the strength of homotypic and heterotypic immunity (Fig. 2B). When homotypic and heterotypic immunity were nearly equal, only G1 strains persisted in the population (Fig. 2B, lower right corner); it is possible to demonstrate mathematically that a second strain is not able to invade (SI Appendix).

Epidemiologic studies suggest that the relative risk of second infection with any strain, regardless of genotype, is 0.62 (95% confidence interval, 0.5–0.83) (34). We considered this to be a lower bound for the relative risk of second infection with a heterotypic strain. Thus, for subsequent analyses, we examined the model-predicted patterns assuming σ_he = 0.65 and σ_ho = 0.35. This yields a predicted period of genotype oscillations of ≈6 y, and G1 strains account for 73% of severe diarrhea cases, which is consistent with the observed data (Fig. 1) (28). We performed sensitivity analyses with other immunity parameters to account for the shorter (≈3-y) period also observed, but overall conclusions were similar (SI Appendix).

When we allowed for the extinction and stochastic reintroduction of strains, we found that the rarer genotypes (G2–G4, G9) occasionally faded out and did not reappear for a number of years (Fig. 3A). Variability in the period of oscillations increased, particularly for non-G1 strains.

Fig. 3.

Model-predicted genotype distributions and incidence of severe rotavirus diarrhea. The proportion of cases in a given year attributable to each genotype (Left), the weekly incidence of severe rotavirus diarrhea (Center), and the mean genotype distributions for years 21–40 (Right) are plotted for the following situations: (A) prevaccination, (B) 50% coverage with a vaccine that provides strong protection against G1 and weaker protection against other genotypes, (C) 80% coverage with such a vaccine, (D) 50% coverage with a vaccine that provides strong protection against all genotypes, and (E) 80% coverage with such a vaccine. Vaccination is introduced in year 10 in B–E. The results represent single realizations of a model allowing for fadeout and stochastic reintroduction of genotypes.

Impact of Vaccination.

Vaccination that provides strong protection against G1 and weaker protection against other strains results in a large decrease in the proportion of infections caused by G1 (Fig. 3B). There is a corresponding increase in the proportion of infections caused by other genotypes, and quicker cycling of the predominant genotype; a different genotype caused the majority of cases every 2 to 3 y. The quicker cycling is due to the assumption that non-G1 strains are comparable in terms of transmissibility, and hence there is more frequent replacement. The periodicity of the individual genotypes actually increased, as expected because vaccination is similar to a reduction in birth rate (SI Appendix). It is possible that G1 strains could be eliminated at high coverage levels (Fig. 3C). However, the overall incidence is substantially reduced despite the relative increase in the proportion of infections caused by non-G1 strains.

If instead the vaccine is assumed to provide equally strong protection against all five genotypes, vaccination is predicted to lead to a slight increase in the proportion of infections caused by G1 and a lengthening of the period of oscillations at intermediate coverage levels, assuming G1 strains are slightly more transmissible (Fig. 3D). At higher coverage, the pattern of strain oscillations becomes somewhat irregular as epidemics begin to occur biennially and genotypes become more likely to fade out during the epidemic troughs, but G1 remains the dominant genotype (Fig. 3E).

Postvaccination Strain Emergence.

The emergence of a new strain after vaccination is predicted to have different effects depending on the type of vaccine and whether it confers some or no protection against the emergent strain (SI Appendix, Table S2). When vaccination is assumed to protect primarily against G1 and provide weaker protection against all other strains including the new strain (scenario 1), the new strain is expected to take a few years to become established in the population, at which time it causes a normal-sized epidemic (similar to postvaccination incidence) and proceeds to cycle along with the other genotypes (Fig. 4A). If the vaccine provides no protection against the emergent strain (scenario 2), the new strain may cause an epidemic when it first emerges that is similar in size or slightly larger than the average prevaccination epidemic; but after ≈3 y, incidence is expected to decrease to slightly below prevaccination levels (Fig. 4B). At intermediate coverage levels, only the new strain and one other genotype (typically G1) are maintained in the population, and these genotypes cycle every 4–6 y.

Fig. 4.

Model-predicted genotype distributions and incidence of severe rotavirus diarrhea after the emergence of a new strain after vaccination. The proportion of cases in a given year attributable to each genotype (Left) and the weekly incidence of severe rotavirus diarrhea (Right) are plotted for 50% coverage with (A) a vaccine that provides strong protection against G1 and weaker heterotypic immunity to other genotypes, including the new strain (scenario 1), (B) a vaccine targeting G1 that provides no immunity to the new strain (scenario 2), (C) a vaccine that provides strong immunity against G1–G4 but only weak immunity to the new strain (scenario 3), and (D) a vaccine providing strong immunity against G1–G4 and no immunity to the new strain (scenario 4). Vaccination is introduced in year 10, and the new strain is introduced in year 15. The results represent single realizations of a model allowing for fadeout and stochastic reintroduction of genotypes.

When vaccination is assumed to provide strong protection against G1–G4 and weak protection against the emergent strain (scenario 3), the new strain again may take up to a few years to become established and does not lead to an abnormally large epidemic (Fig. 4C). In this situation, the emergence of a new strain may lead to the extinction of other less-transmissible genotypes (G2–G4); the new strain is predicted to cycle along with G1, with T ≈ 8 y. If vaccination provides strong protection against G1-G4 and no protection against the emergent strain (scenario 4), the new strain could lead to a large epidemic when it is first introduced, with a peak incidence greater than prevaccination epidemics (Fig. 4D). The size of the epidemic will depend on when such a strain emerges (SI Appendix). The new strain is expected to become dominant in the population, whereas incidence is expected to stabilize at slightly below prevaccination levels.

Discussion

Models for the transmission dynamics of rotavirus have provided key insights into the epidemiology and impact of vaccination for this important diarrheal pathogen (23–27). However, the models developed thus far have not accounted for the different genotypes that cocirculate in the population. Theory stemming from multistrain models for various pathogens suggests that cross-immunity can lead to cyclical patterns in incidence and the distribution of strains (e.g., refs. 2, 3, 7, 17, and 18). However, most of these models assume that homotypic immunity is complete and lifelong. Including the interaction of multiple strains of rotavirus while accounting for the incomplete nature of immunity makes the model considerably more complicated, and there are only limited data on which to base assumptions about important parameters, such as the strength of homotypic vs. heterotypic immunity. Nevertheless, we are able to make some qualitatively robust inferences that help explain the observed patterns of genotype coexistence and cycling. We then use the model to explore the potential impact of vaccination on genotype patterns, as well as scenarios in which a new strain emerges that partially or fully escapes vaccine protection.

Understanding Observed Genotype Cycling.

By examining fluctuations in genotype distributions from six countries (43–49), we found evidence of 3- to 11-y cycles in the predominant strains. An interepidemic period of ≈5 y in the monthly incidence of rotavirus hospitalizations with G1–G4 strains in Melbourne, Australia has been previously noted (50, 51). The observed variability in the dominant periods of oscillations is likely due to the inherent stochasticity resulting from occasional fadeouts of strains, the short length of available time-series data relative to observed multiannual periodicities, and potential differences in transmission dynamics among countries.

The cycling of rotavirus genotypes can be understood in terms of the build-up of population-level immunity to the prevailing strain. During the period in which G1 strains predominate, the proportion of infants who have been infected with G1 but have yet to experience a second infection gradually increases. These individuals are less susceptible to a second infection with a G1 strain than they are to infection with a heterotypic strain. Thus, even though G1 strains are more prevalent in the population at that time (and may be slightly more transmissible among completely naïve individuals), the rarer genotypes gain a fitness advantage owing to their increased ability to infect those with homotypic immunity to G1. The stronger homotypic immunity is relative to heterotypic immunity, the quicker the cycling is expected to occur, because it will take less time for the rarer genotypes to gain a fitness advantage at the population level.

The rarer genotypes are able to infect both completely susceptible infants as well as slightly older individuals who have experienced only one previous infection. As a result, the average age of cases is expected to increase when there is a change in the predominant strain (SI Appendix, Fig. S6). Indeed, it has been noted that there was a slight shift in the age distribution of rotavirus patients toward a broader spectrum of age groups after changes in the predominant strain (52).

Although G1P[8] strains represent >70% of clinical isolates in North America, Europe, and Australia, countries in South America, Asia, and Africa tend to be characterized by a greater diversity of strains (28). Furthermore, cyclical variations in the predominant genotypes may be more frequent in such countries. G2 strains were found to be dominant or codominant with G1 every 3 to 4 y in South Africa (53). The higher birth rates typical of such countries may help explain these patterns (SI Appendix).

It is not necessary to invoke molecular changes that alter the fitness of the virus to explain the appearance and disappearance of subdominant strains. Evidence argues against such changes occurring. For example, it has been noted that closely related strains can persist over multiple seasons, and that greater genetic diversity can exist among strains belonging to a single G-type circulating in the same year than strains belonging to that same G-type reemerging 12–15 y later (43, 54). It seems unlikely that such strains lose then regain some “fitness factor” during the intervening years. The build-up of population-level homotypic immunity is a more parsimonious explanation for the cycling of strains both between and within G-types.

Additional Complexities in Modeling the Impact of Vaccination.

Our modeling approach provides a substantial advance over previous efforts for rotavirus in that it considers up to five cocirculating genotypes. However, residual complexities remain. For example, G2P[4] strains share neither the VP7 nor VP4 antigen with the other common genotypes; therefore, heterotypic protection against these strains may be weaker (SI Appendix). The vaccination scenarios we explored represent the breadth of possible immune responses. Currently licensed vaccines probably fall at different points along this spectrum, and we would need to make additional assumptions to fully represent them (SI Appendix).

Our model predictions are for the most part consistent with early observations, such as the predominance of G2P[4] in Brazil and Australian states using the Rotarix vaccine, and the continued predominance of G1P[8] in Australian states using RotaTeq (SI Appendix) (38, 40, 41). However, other observations, such as outbreaks caused by G1P[8] in regions with high Rotarix coverage levels (39, 42), remain unexplained. The populations in different states/countries are not isolated from one another, and it is difficult to attribute short-term changes in genotype patterns to the effect of vaccination in light of the natural cycling of genotypes and regional differences in strain distributions apparent in the prevaccination era.

Implications for Vaccine Escape.

Despite possible changes in the distribution of genotypes after vaccine introduction, our model suggests overall disease incidence is expected to remain below prevaccination levels if the vaccine provides at least some protection against emergent strains. This is to be expected if vaccination mimics natural immunity. The emergence of G9 strains in the late 1980s/early 1990s, as well as the more recent emergence of G12 strains (20), did not result in unusually large epidemics. The age distribution of cases caused by these strains has been similar to that of the common genotypes (29), suggesting that heterotypic immunity was effective at controlling these infections and/or that newly emergent genotypes were slightly less transmissible. In the unlikely scenario that a new strain was to emerge for which vaccination provided no immunity, our model suggests it could lead to a large epidemic initially, particularly when vaccination provides strong immunity against other strains.

Implications for Comparative Strain Dynamics.

More generally, the pattern of cyclical variation in the five common rotavirus genotypes contrasts with the evolutionary dynamics exhibited by influenza A, for example, in which only two subtypes at most cocirculate in the population, and gradual antigenic drift occurs within subtypes. The antigenic drift of influenza strains occasionally leads to changes that necessitate the updating of vaccines (55). Phylogenies of rotavirus VP7 genes do not exhibit such strong patterns of drift, and the mutation rate of the virus is lower because it is double-stranded (43, 44, 54). Thus, it seems unlikely that the composition of rotavirus vaccines will need to be updated as frequently as they are for influenza vaccines. Although homotypic and heterotypic immunity likely play similar roles in both systems and help explain between-subtype cycling, the high transmissibility of rotavirus compared with influenza [R₀ ≈ 25 vs. 1.5 (23, 56)], along with slightly weaker homotypic immunity and less reliance on metapopulation dynamics, may promote the coexistence of rotavirus strains rather than the sequential replacement of strains within subtype (13).

Similar cyclical patterns are also evident among dengue serotypes. Although cross-protective immunity may also account for the cycling of dengue serotypes (5), most mathematical modeling studies have attributed these patterns to antibody-dependent enhancement, whereby second infections with a different serotype are assumed to be more transmissible and/or more likely to occur, and immunity to the strain causing first infection is assumed to be complete and lifelong (6–10). The potential for antibody-dependent enhancement makes the development of vaccines for dengue more complicated, because vaccination must provide strong cross-protective immunity without increasing susceptibility to severe disease (57).

Comparative studies that aid our understanding of the “phylodynamics” (i.e., how features of the immunology and epidemiology of pathogens affect their evolutionary trajectories, and vice versa), and the implications for vaccination programs, are an important avenue for future research (1).

Future Refinements and Data Needs.

Our model is not designed to capture all of the complexities of rotavirus strain variation (SI Appendix) but nevertheless demonstrates some important concepts. Mathematical modeling of rotavirus strain dynamics both provides a tool by which we can gain a better understanding of why the prevalent genotypes tend to cycle, and can be used to speculate about how vaccination may impact genotypes distributions in the future. Long-term surveillance of rotavirus genotype distributions both before and after vaccine introduction is needed to validate model predictions and gain a better understanding of the evolutionary dynamics. Furthermore, our model is based on dynamics of rotavirus in developed countries; the findings may not be generalizable to developing countries, where strain diversity is greater and the protection from natural infection seems to differ (28, 30). More epidemiological studies in birth cohorts are needed to obtain better estimates of the protection conferred by first infection with one genotype against subsequent infection (symptomatic or asymptomatic) with homotypic and heterotypic strains. Understanding the nature of homotypic vs. heterotypic immunity, and how this affects the interaction of rotavirus genotypes as well as other multistrain pathogens, will be critical for understanding observed patterns and predicting the long-term impact of vaccination.

Methods

Analysis of Genotype Time Series.

Observed genotype time series with at least 10 y of consecutive data were gathered from the published literature (43–49) (SI Appendix). We used Fourier analysis to detect the dominant multiannual periodicities with which the various genotypes cycled in six countries. Bootstrap analysis was used to identify significant periods of oscillation; we randomly permuted the time series 1,000 times to identify signals that were significant at the 95% confidence level.

Model Description.

We extended an existing model for the transmission dynamics of rotavirus (23) to include strain-specific infection compartments. Details of the model are presented in SI Appendix. In short, we assumed that susceptible individuals can experience a primary infection with one of five strains, recover and are temporarily immune to infection with all strains, then can be reinfected at a reduced rate with either a homotypic or heterotypic strain. After two infections, individuals are assumed to develop strong heterotypic immunity to all strains (SI Appendix provides sensitivity analysis). We assumed that the transmission rate and seasonality parameters were similar to those estimated for a best-fit model for rotavirus hospitalizations in the United States (23) and a birth rate of 15 live births per 1,000 in a population of 1 million individuals.

There is limited information to parameterize differences in homotypic vs. heterotypic immunity (30–34). Thus, we explored a range of assumptions. The relative risk of second infection with a homotypic strain (σ_ho) was varied from 0.01 (near-complete homotypic immunity) to 0.5, whereas the relative risk of second infection with a heterotypic strain (σ_he) was varied from 0.5 to 1 (no heterotypic immunity). We assumed that the relative infectiousness of second infections was reduced in proportion to the relative risk, with a minimum relative infectiousness of 0.1. We explored whether small differences in the transmissibility of various strains or weaker homotypic immunity against G1 strains (due to immunologically distinct sublineages) can explain the predominance of G1 in temperate countries (SI Appendix). Fourier analysis was used to determine the period of oscillations in model-predicted genotype distributions. We identified a range of immunity parameters consistent with the period of oscillations and proportion of infections caused by G1 in the observed data.

Modeling Vaccination.

We examined the effect of two different vaccines intended to represent the breadth of possible scenarios for immune protection. The first was assumed to confer immunity comparable to primary infection with a G1 strain (i.e., strong immunity to G1 and weaker heterotypic immunity to all other strains). The second was assumed to confer equally strong immunity to all strains present at the time of vaccine introduction (G1–G4 and G9 for the initial analysis). We examined the dynamics predicted over a 20-y period after vaccine introduction at coverage levels of 50% and 80%. We also examined the equilibrium distribution of strains averaged over a 20-y period 10 y after vaccine introduction.

We allowed for the local extinction of strains when there was <0.5 primary infection with a given strain to incorporate stochastic effects. Strains were reintroduced into the population at a mean rate of one imported primary infection per year; we varied this rate seasonally from a maximum of 1.5 introductions per year occurring at the same time as the peak in the transmission rate to 0.5 introductions per year occurring during the nadir of transmission.

For both types of vaccine, we explored the effect of introducing the fifth strain 5 y after vaccine introduction. The new strain was assumed to be 90% as transmissible as G1 (equivalent to G2–G4). Under scenarios 1 and 3, we assumed that the vaccine provided weak heterotypic immunity to the newly emergent strain, whereas under scenarios 2 and 4, we assumed it conferred no immunity to the new strain (SI Appendix, Table S2). Again, we examined the dynamics over a 20-y period after vaccine introduction at 50% coverage.