K.E.A., E.S., and A.P.G. work as consultants for Sanofi Pasteur, MSD. T.V.E. is an employee of GlaxoSmithKline. This does not alter the authors' adherence to all the PLoS ONE policies on sharing data and materials.
Conceived and designed the experiments: VEP BFB TVE CJA WJE CV MMP BTG UDP BAL. Performed the experiments: VEP KEA BFB TVE BAL. Analyzed the data: VEP KEA BFB TVE BAL ES APG WJE CV MMP BTG UDP. Contributed reagents/materials/analysis tools: JPH. Wrote the paper: VEP KEA BFB TVE BAL ES APG WJE CV MMP BTG UDP CJA JPH.
Current address: Department of Epidemiology of Microbial Diseases, Yale School of Public Health, New Haven, Connecticut, United States of America
Early observations from countries that have introduced rotavirus vaccination suggest that there may be indirect protection for unvaccinated individuals, but it is unclear whether these benefits will extend to the long term. Transmission dynamic models have attempted to quantify the indirect protection that might be expected from rotavirus vaccination in developed countries, but results have varied. To better understand the magnitude and sources of variability in model projections, we undertook a comparative analysis of transmission dynamic models for rotavirus. We fit five models to reported rotavirus gastroenteritis (RVGE) data from England and Wales, and evaluated outcomes for short and longterm vaccination effects. All of our models reproduced the important features of rotavirus epidemics in England and Wales. Models predicted that during the initial year after vaccine introduction, incidence of severe RVGE would be reduced 1.8–2.9 times more than expected from the direct effects of the vaccine alone (28–50% at 90% coverage), but over a 5year period following vaccine introduction severe RVGE would be reduced only by 1.1–1.7 times more than expected from the direct effects (54–90% at 90% coverage). Projections for the longterm reduction of severe RVGE ranged from a 55% reduction at full coverage to elimination with at least 80% coverage. Our models predicted shortterm reductions in the incidence of RVGE that exceeded estimates of the direct effects, consistent with observations from the United States and other countries. Some of the models predicted that the shortterm indirect benefits may be offset by a partial shifting of the burden of RVGE to older unvaccinated individuals. Nonetheless, even when such a shift occurs, the overall reduction in severe RVGE is considerable. Discrepancies among model predictions reflect uncertainties about age variation in the risk and reporting of RVGE, and the duration of natural and vaccineinduced immunity, highlighting important questions for future research.
Rotavirus is the leading cause of severe diarrhea in children, representing a major source of morbidity and mortality worldwide. The recent development and licensing of two vaccines, Rotarix (GlaxoSmithKline Biologicals; Rixensart, Belgium) and RotaTeq (Merck & Co; Whitehouse Station, NJ), provide a novel means of controlling rotavirus. Early evidence from developed countries that have introduced rotavirus vaccination into their national immunization program strongly supports the direct and indirect benefits of vaccination
In response to the advent of rotavirus vaccines, there has been a recent surge in the development of mathematical models for the transmission dynamics of rotavirus
Models for the transmission dynamics of rotavirus are structured based on studies of the natural history of infection and immunity
Whereas some models predict that the reduction in RVGE due to vaccination will exceed estimates derived only from the direct effects of vaccination
We brought together five groups that have previously developed dynamic mathematical models for the transmission of rotavirus
The five models for the transmission dynamics of rotavirus we explored follow an SIS (susceptibleinfectioussusceptible) or SIRSlike (susceptibleinfectiousrecoveredsusceptible) compartmental framework (
Model A  Model B  Model C  Model D  Model E 
Risk of infection and severity of RVGE depend on age  Risk of infection and severity depend on the number of previous infections  Risk of infection and severity depends on age and the number of previous infections; short delay between infection and onset of infectiousness  Risk of infection and severity depends on the number of previous infections  Following infection, individuals develop full immunity or become susceptible again 
Temporary immunity following infection  Temporary immunity following infection  Temporary immunity following infection  No period of full immunity following infection  Probability of developing full immunity depends on the number of previous infections 
Severe and mild RVGE are tracked separately and vary in infectiousness; asymptomatic infections do not transmit  After 2 infections, subsequent infections are less infectious and not reported  After 2 infections, subsequent infections are less infectious and not reported  After 4 infections, all individuals develop full immunity (that may wane)  After 4 infections, all individuals develop full immunity (that may wane); asymptomatic infections do not transmit 
Only severe RVGE cases are reported  Only severe RVGE cases are reported  Only severe RVGE cases are reported  Mild and severe RVGE cases are reported; reporting rate depends on age (<5 or ≥5 years old)  Mild and severe RVGE cases are reported 
Parameter  Model A  Model B  Model C  Model D  Model E 
Duration of maternal immunity  13 weeks  13 weeks  13 weeks  13 weeks  13 weeks 
Duration of incubation period  NA  NA  1 day  NA  NA 
Duration of infectiousness  

7 days (severe)  7 days  7 days  7 days  7 days 

3.5 days (mild)  3.5 days  3.5 days  3.5 days  7 days 
Relative risk of infection following:  

NA  0.62  0.62  0.62  0.62 

0.37  0.37  0.37  0.37  

0.37  0.37  0.37  0.37  
Proportion of infections with any RVGE (severe RVGE)  

0.76 (0.24) for  0.47 (0.13)  0.47 (0.13)  0.47 (0.13)  0.47 (0.13) 

<5 yr olds  0.25 (0.03)  0.25 (0.03)  0.25 (0.03)  0.25 (0.03) 

Estimated for ≥5  0.20 (0)  0.20 (0)  0.32 (0)  0.32 (0) 

yr olds  NA  NA  0.20 (0)  0.20 (0) 
Relative infectiousness (compared to first infection)  

0.5 for mild vs severe RVGE;  0.5  0.5  0.5  Only individuals with any RVGE 

asymptomatic infections do not transmit  0.2  0.2  0.2  transmit (see above) 
Duration of complete immunity  1 year  1 year  1 year  NA  NA 
Type of cases reported  Severe RVGE  Severe RVGE  Severe RVGE  Any RVGE  Any RVGE 
Parameter  Symbol  Model A  Model B  Model C  Model D  Model E 
Duration of immunity to symptomatic infection  1/ 
NA 
280 years  A = 7.31e9, B = 0.228 
833 years  201 years 
Amplitude of seasonality in transmission 

0.064  0.057  0.040  0.046  0.052 
Seasonal offset 

0.089  0.377  0.209  0.014  0.237 
Agespecific risk of infection  0.083 (<1 y), 0.065 (1 y), 0.017 (2 y), 0.006 (3 y), 0.003 (4–65 y), 0.025 (≥65 y)  0.291 (all age groups)  0.402 (all age groups)  0.562 (<1 y), 0.718 (1 y), 0.344 (2 y), 0.144 (3 y), 0.077 (4 y), 0.068 (≥5 y)  0.890 (all age groups)  
Proportion of cases with severe RVGE in ≥5 yr olds 

0.015  NA  NA  NA  NA 
Reporting fraction 

0.064  0.122  0.123  0.024 (<5 y), 0.005 (≥5 y)  Fixed at 0.029 
Basic reproductive number 

1.23  18.2  17.6  5.03  26.2 
Number of parameters estimated 

10  5  6  12  4 

71,990  84,977  76,303  83,912  66,697 
NA = Not applicable.
An exponential distribution was used to describe increasing probability for reported symptomatic infection with age. With probability p(a) = A*exp(a*B) exposed individuals in “later infection" are moved to exposed group of second infection. The remaining 1p(a) continues to the “later" infection group. The age a was chosen as the midpoint of the various age groups.
Model A is based on Shim et al (2009)
Model B is based on Pitzer et al (2009)
Model C is based on de Blasio et al (2010)
Model D is based on Van Effelterre et al (2009)
Model E is based on Atchison et al. (2010)
We estimated several (nonfixed) model parameters and statistically validated and compared our models by fitting each model to agestratified reports of laboratoryconfirmed rotavirus infection from England and Wales
We calculated the loglikelihood of the data under each model by assuming that the reported number of weekly RVGE cases in each age group was Poisson distributed with a mean equal to the modelpredicted number of cases (see
We incorporated vaccination into each of the bestfitting models and explored both the short and longterm effects of vaccination under a variety of assumptions about vaccine coverage and efficacy. Our analysis focused on model projections for the incidence of severe RVGE (Vesikari score ≥11); results for any RVGE are presented in
Most of the models assume the effect of vaccination is comparable to natural immunity from rotavirus infection, as observed in natural history studies, such that successive doses of the vaccine confer immunity comparable to that conferred by one or more natural infections. Thus, vaccination was assumed to confer some protection against infection and stronger protection against infectiousness and disease given infection. This approach yields predicted vaccine efficacies against RVGE similar to those measured during clinical trials conducted in developed countries (see below)
It is unclear whether successive doses of a vaccine confer additional protection, as is observed with natural infections. To address this uncertainty, we explored two possible scenarios: (1) vaccination confers protection comparable to that conferred by primary infection following the first dose administered at 2 months of age (with further doses providing no additional benefit), and (2) vaccination confers protection comparable to that following primary infection when the first dose is given at 2 months of age
In preliminary analyses, we also explored the effect of vaccination assuming that protection is comparable to that conferred by primary infection and only occurs after the second vaccine dose at 4 months of age (weakest effect), or allowing for additional vaccineinduced protection following each dose including a third vaccine dose administered at 6 months of age (strongest effect). However, model projections for the short and longterm impact of vaccination under these scenarios did not differ substantially from those under scenarios 1 and 2, respectively.
For all models, we evaluated the shortterm effect of vaccination on the dynamics of rotavirus over a fiveyear period following the introduction of the vaccine. For each scenario, we assumed that the vaccine was introduced into the population prior to the start of the rotavirus season in October (week 40) at a coverage level of either 70% or 90% of all eligible infants; coverage was maintained at a constant level following vaccine introduction. We examined model projections for the incidence of severe RVGE and any RVGE in children <5 years of age (an age range in which 95% of reported RVGE cases occur prior to vaccination) and in individuals ≥5 years of age to examine a possible shift in the burden of illness to older age groups following vaccine introduction.
We also examined the shortterm percent reduction in the cumulative incidence of severe RVGE cases predicted by the models over one, two, and five years after the introduction of the vaccine (evaluated starting from week 1 of the year following vaccine introduction) and compared this to the direct effect of vaccination.
The longterm effect of vaccination was measured by the percent reduction in the mean incidence of severe or any RVGE over the full range of vaccine coverage levels (0–100%) during a 10 year period beginning 10 years after vaccine introduction, i.e. during years 10–19 postintroduction. Again, we compared this to the direct effect of the vaccine.
The direct effect of vaccination
All the fitted models were qualitatively similar to the E&W rotavirus data (
(A) Mean annual size and timing of rotavirus epidemics in individuals (i) <5 years of age and (ii) ≥5 years of age. The solid black line represents the mean number of RVGE cases per week. Dashed lines show the minimum and maximum number of cases each week. Colored lines represent the fitted models: Model A (blue), Model B (yellow), Model C (green), Model D (purple), Model E (red). (B) Age distribution of reported rotavirus cases (bars) and the fitted models (colored lines).
The five models predicted a range of possible shortterm dynamics following vaccine introduction (
(A) Weekly incidence of severe RVGE predicted for individuals <5 years of age, scaled by peak prevaccination incidence, for the following scenarios: (i) 70% coverage with a vaccine that confers immunity comparable to primary infection following first dose at 2 months of age (82% efficacy) (scenario 1); (ii) 90% coverage under scenario 1; (iii) 70% coverage with a vaccine that confers immunity comparable to one natural infection following each dose at 2 and 4 months of age (99% efficacy) (scenario 2); and (iv) 90% coverage under scenario 2. (B) Incidence of severe RVGE predicted in individuals ≥5 years of age under coverage scenarios (i–iv) for Model A (blue), Model B (yellow), and Model C (green), Model D (purple), Model E (red).
The incidence of severe RVGE in individuals older than 5 years exhibited similar timing to the incidence in children <5 years of age, but some of the models suggested that the relative incidence of severe RVGE in older individuals could increase following vaccine introduction (
The reduction in the cumulative incidence of severe RVGE during the first year after vaccine introduction was similar across the five models, particularly for the <5 year old population where most of the cases occur (
The relative cumulative incidence of severe RVGE after versus before vaccine introduction in individuals (i) <5 years of age, (ii) ≥5 years of age, and (iii) all age groups over the first (
However, differences in the level of indirect protection predicted by the different models were accentuated when examining the reduction in cumulative incidence of severe RVGE over 2 to 5 years following vaccine introduction (
With a vaccine efficacy of 82% against severe RVGE assumed under scenario 1, the predicted effects of vaccination range from a longterm reduction in the incidence of severe RVGE that is slightly less than would be expected from the direct effect of the vaccine alone to indirect protection that could eliminate rotavirus from the population at 100% coverage. Four of the five models, however, predicted that the vaccine would not provide longterm indirect protection in the population as a whole (
The reduction in the incidence of severe RVGE during a 10year period beginning 10 years after vaccine introduction, as compared to the mean prevaccination incidence, is plotted for coverage levels from 0 to 100%. The panels represent the reduction in incidence of severe RVGE under (A) scenario 1: vaccination is assumed to confer immunity comparable to primary infection following the first dose at 2 months of age (82% efficacy), and (
The longterm impact of vaccination predicted by the models under scenario 2 was similar to that under scenario 1. However, because the efficacy of the vaccine against severe RVGE is assumed to be 99% under scenario 2 compared to 82% under scenario 1, the reduction in the incidence of severe RVGE due to the direct effect of vaccination is expected to be greater (
Our results reveal several interesting discrepant findings among the model projections for the short and longterm impact of vaccination that shed light on some of the important questions about rotavirus epidemiology. Direct comparison of differences in model structure and estimated parameters allows us to understand the reasons for the variation in model projections. Fundamentally, these variations in model projections reflect gaps in our understanding of the mechanisms of rotavirus infection, natural immunity, epidemiology, and the biological nature of vaccine protection. Much of the uncertainty regarding the expected indirect effects of vaccination in the literature stems from different model assumptions for why severe RVGE is rare among older children and adults, and to what extent natural and vaccineinduced immunity wanes over time.
The five models we analyzed reproduce the seasonality and age distribution of rotavirus incidence in E&W, providing an important source of model validation. We compared the statistical fit of these models to week and agestratified laboratory reports of confirmed rotavirus cases from E&W, and found that all five models had AIC values between 66,697 and 84,997. Given the large number of data points we attempted to fit (547 weeks×19 age groups = 10,393 data points), the relatively large AIC values and the variation among models with different numbers of estimated parameters is not surprising. Model E provided the best fit to the prevaccination E&W RVGE reports, but the relative ranking of models based on AIC should be interpreted cautiously when extrapolating to the ability of the models to predict the impact of vaccination. Furthermore, we only explored a single set of fixed parameters and fit to a single data set; there is additional uncertainty regarding the parameter values that was not accounted for in our analysis, and which would presumably affect the fit differently for different models.
The models make different assumptions to explain why most RVGE cases occur among children <5 years of age (
Models B, C, and E are similar to Model D in that they also assume the progressive buildup of natural immunity to both infection and symptoms of infection explains the greater concentration of RVGE in children <5 years of age. Immunity is assumed to wane on a time scale that does not affect the <5 year old population (
Part of the difficulty in deciding which model best represents the underlying epidemiology of rotavirus infection is lack of knowledge about the reporting pyramid for rotavirus. In other words, what fraction of rotavirus infections are symptomatic, what fraction of symptomatic infections present to the healthcare system, and what fraction of those presenting get properly diagnosed as rotavirus? A few studies have attempted to elucidate the reporting pyramid for rotavirus infections in E&W, but they were underpowered to understand the reporting fraction in adults, how reporting varies over time, and how it correlates with the severity of symptoms
The similarities and differences in model structure are reflected in the projections that each model yields for the indirect protection conferred by vaccination. During the first 5 years after vaccine introduction, the reduction in the cumulative incidence of severe RVGE predicted by each model was similar for children <5 years of age and for the overall population. However, it is more difficult to predict the pattern of epidemics following the introduction of vaccination, as suggested by the different epidemic trajectories predicted by each model. Furthermore, none of the models account for additional complexities such as the interaction among different genotypes of rotavirus, or environmental or local effects. All of the models suggest that the postvaccination timing of rotavirus activity can vary considerably, with possible peaks occurring in the summer and/or fall as opposed to the typical prevaccination peaks occurring in winter/spring. This is consistent with the relatively low amplitude of seasonal forcing (4.3–6.4%) estimated for each model, which suggests that environmental factors such as temperature or humidity only have a small effect on the transmission rate, and that the large RVGE epidemics evident in E&W primarily result from the dynamic interaction between susceptible and infectious individuals
Model projections suggest the shortterm reduction in the incidence of reported RVGE during the first five years after vaccine introduction will exceed estimates that account only for the direct effects of vaccination. This is supported by recent observations of the early impact of vaccination in countries that have introduced routine immunization, where 22–68% decreases in the incidence of RVGE have been reported in age groups not eligible to receive the vaccine
All of the models predict that vaccination will lead to a delay in the average age of first infection with rotavirus and a decrease in the number of infections experienced by a single individual during his/her lifetime (results not shown). For Models B, C, and E which assume that the severity of RVGE is associated only with the number of previous infections, the delay in the time to infection may shift some of the burden of severe RVGE to older individuals, because first and second infections may be more likely to occur after 5 years of age. If infections tend to be less severe and/or less likely to be reported in the ≥5 year old age group than in the <5 year old group, as estimated by Models A and D, the delay in the time to infection is not predicted to increase the burden of RVGE in the ≥5 year old age group. Furthermore, the decrease in the overall number of infections experienced by individuals during their lifetime could translate into a decrease in the number of severe RVGE cases among older individuals, particularly if the lifetime number of infections predicted by the model is relatively small.
Whether or not vaccination provides longterm indirect protection against severe RVGE in children <5 years of age depends on whether vaccineinduced immunity wanes completely after 1 year, or if vaccinated infants remain at reduced risk of RVGE for a prolonged period of time. The discrepancy in results between Model A, which predicts the smallest reduction in severe RVGE, and Model D, which predicts that elimination of RVGE is possible under some scenarios, is due primarily to the different assumptions made regarding the duration of immunity. If one makes the alternative assumption in Model A that vaccineinduced immunity lasts at least 3 years, then in this case Model A also predicts that rotavirus could be eliminated at high coverage rates
An important limitation of this study is that we do not comprehensively explore the influences of parameter uncertainty. Key uncertainties in our fixed parameter assumptions for rotavirus include the duration of natural and vaccineinduced immunity, protection conferred by previous infection(s), and relative infectiousness of primary and subsequent symptomatic and asymptomatic) infections. Further work is needed to characterize and explore the impact of this parameter uncertainty
Overall, the comparison of these different models of rotavirus transmission and vaccination allowed us to examine the impact of structural uncertainty on the robustness of model projections, as well as identify key gaps in our understanding of rotavirus epidemiology. The models we compared suggest vaccination will lead to a 64–100% reduction in the incidence of severe RVGE and a 55–100% reduction in any RVGE at full coverage 10–20 years following vaccine introduction if vaccination confers protection comparable to a single natural infection (
Our comparative analysis of the model projections for the indirect effects of vaccination identified three key questions that should be addressed to improve the accuracy of model predictions:
What is the
What is the effect of
Does
Experimental studies of rotavirus pathogenesis, carefully designed epidemiologic studies, and stronger statistical links between data and models will lead to betterinformed model assumptions and help to discriminate among models. In addition, further validation and fitting of transmission dynamic models to postvaccination data from different countries will help to refine model parameter estimates and improve projections of the longterm impact of vaccination.
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K.E.A. thanks Diana Resasco for help with code optimization.