An Ensemble Approach to Predicting the Impact of Vaccination on Rotavirus Disease in Niger
Jaewoo Park, Joshua Goldstein, Murali Haran, Matthew Ferrari
AAn Ensemble Approach to Predicting the Impactof Vaccination on Rotavirus Disease in Niger
Jaewoo Park , Joshua Goldstein , Murali Haran ∗ , and Matthew Ferrari Social and Data Analytics Laboratory, 900 N Glebe Rd, Virginia Tech, Arlington, VA 22203. USA Department of Biology, The Pennsylvania State University, University Park, PA 16802, USA
November 7, 2018
Abstract
Recently developed vaccines provide a new way of controlling rotavirus in sub-Saharan Africa.Models for the transmission dynamics of rotavirus are critical both for estimating current burdenfrom imperfect surveillance and for assessing potential effects of vaccine intervention strategies.We examine rotavirus infection in the Maradi area in southern Niger using hospital surveillancedata provided by Epicentre collected over two years. Additionally, a cluster survey of house-holds in the region allows us to estimate the proportion of children with diarrhea who consultedat a health structure. Model fit and future projections are necessarily particular to a givenmodel; thus, where there are competing models for the underlying epidemiology an ensembleapproach can account for that uncertainty. We compare our results across several variants ofSusceptible-Infectious-Recovered (SIR) compartmental models to quantify the impact of model-ing assumptions on our estimates. Model-specific parameters are estimated by Bayesian inferenceusing Markov chain Monte Carlo. We then use Bayesian model averaging to generate ensembleestimates of the current dynamics, including estimates of R , the burden of infection in the region,as well as the impact of vaccination on both the short-term dynamics and the long-term reductionof rotavirus incidence under varying levels of coverage. The ensemble of models predicts that thecurrent burden of severe rotavirus disease is 2.9 to 4.1% of the population each year and that a2-dose vaccine schedule achieving 70% coverage could reduce burden by 37-43%. Keywords: Rotavirus, Vaccine, Niger, Susceptible-Infectious-Recovered models, Bayesian model aver-aging ∗ Email addresses: [email protected] (J. Goldstein), [email protected] (J. Park), [email protected] (M. Haran),[email protected] (M. Ferrari) a r X i v : . [ s t a t . A P ] M a y Introduction
Diarrheal disease is the second leading cause of death around the world for children under 5years of age [6]. Though there are many infectious causes of diarrheal disease in children, rotavirusis the leading cause of gastroenteritis [8, 43, 41]. In many countries, better sanitation, hygiene andaccess to care have reduced the burden of diarrhea [11, 22]. Despite this trend, the proportion ofdiarrheal hospitalizations attributable to rotavirus increased between 2000 and 2004 [30]. The recentdevelopment of new prophylactic vaccines for rotavirus is a promising advance in the prevention ofdiarrheal disease and the reduction of overall childhood mortality [31, 25].Observation of rotavirus dynamics and estimation of the burden of rotavirus disease is limitedboth by non-specific surveillance and under-reporting. The dynamics of rotavirus transmission mustoften be inferred from non-specific temporal surveillance of diarrheal disease that includes multiplecauses. This is analogous to the dynamics of specific influenza strains, which are commonly inferredfrom non-specific time series surveillance of influenza-like illness (ILI) that includes infection bymultiple influenza strains (influenza A and B), as well as additional viral infections, for exampleparainfluenza, coronavirus, rhinovirus [35, 10]. In sub-Saharan Africa, the cause of diarrheal diseaseis often unknown due to a lack of diagnostic capacity [28]. Even when the cause of disease is known,an unknown fraction of cases will occur in the community and never be recorded by the health system,leading to a potentially significant level of under-reporting. Dynamic models in general, and so-calledstate-space models in particular, have been an important tool in the assessment of disease burden fromnon-specific or imperfect surveillance [19, 16, 7]. We estimate the burden of rotavirus in the Maradiregion of Niger by synthesizing two sources of data. We use hospital surveillance data collected byEpicentre for the incident cases over time, including lab-confirmation to assess the likelilhood thata case of severe diarrhea is caused by rotavirus. In addition, we use a cluster survey of householdsconducted to estimate the proportion of diarrheal disease cases in the region seeking care. The latteris used to help estimate the reporting rate.State-space models rely on the temporal correlation in a dynamic model to make the unobservabletrue state of the system, that is, the incidence of the pathogen of interest, estimable from noisyor imperfectly sampled data [21]. Thus, the inference about disease burden is conditional on thestructure of the underlying dynamic model. For pathogens with well characterized epidemiology,such as measles and influenza, the application of state-space models to infer disease burden andtransmission dynamics has become common [19, 9, 7, 38]. The dynamics of rotavirus, which itselfcomprises multiple strains that result in varying levels of cross-protective immunity to other strains,has been variously described by a suite of different models [32]. Therefore, inference about rotavirusburden is limited both by imperfect surveillance of rotavirus infection and uncertainty about the2nderlying transmission dynamics. Rather than condition our analysis on any one model, we fitthe observed time series to a suite of 5 different model structures and assumptions to account foruncertainty in model parameters as well as the dynamics represented in the models themselves.While the development of several novel rotavirus vaccines is a promising advance for the controlof diarrheal disease in children, the potential impact of the introduction of these vaccines at thepopulation-scale is uncertain. The predicted impact of vaccine introduction may depend both on theefficacy of the vaccine and model structure; for example [32] proposed alternative models for boostingof immunity following sequential exposure to rotavirus. Bayesian model averaging (BMA) [4, 18]allows for the integration of predictions of multiple models, weighted by their posterior support, togenerate a single ensemble estimate that accounts for uncertainty in model selection. Here, via BMA,we use the ensemble of fitted models to predict the short-term and long-term impact of vaccinationon rotavirus incidence. We then estimate the predicted impact using the vaccine efficacy from twodifferent studies. Our ensemble approach predicts that the current burden of severe rotavirus diseaseis 2.9 to 4.1% of the population each year and that a 2-dose vaccine schedule achieving 70% coveragecould reduce burden by 37-43%.
We use data from two sources: a time series of clinic admissions for diarrheal disease and acommunity based survey of health-seeking behavior. Clinic surveillance covers a collection of healthcenters and district hospitals from four districts in the Maradi region of Niger including Aguie, GuidanRoumdji, Madarounfa, and the city of Maradi. A total of 9,590 cases of diarrhea in children under 5were recorded from December 23, 2009 to March 31, 2012 (118 weeks). For each patient age in months,village of origin, date of consultation were recorded. Also noted were potential symptoms includingtemperature, duration of diarrhea before consultation, presence of blood in the stool, presence andduration of vomiting, and level of dehydration. In each case a rapid test was administered for detectingrotavirus. 2,921 cases tested positive for rotavirus via the rapid test. A subset of 378 cases testingpositive for rotavirus were also genotyped. While 32 separate strains were identified, more than twothirds of positive cases were of strains G12P[8] or G2P[4].The distributed nature of Niger’s healthcare system is a challenge for surveillance. Roughly athird of all health centers in these districts were included. Notably absent were the many local healthposts staffed by community health workers. To estimate both the fraction of cases seeking care at ahealth center, and the fraction seeking any level of care, a second source of data is needed. We use acommunity survey [29] in the region of children under 5 to get estimates of these reporting rates.A total of 2940 children under 5 were selected for inclusion in the cluster survey from households3cross the four districts. Clusters were allotted according to the population of each village fromcensus data. Sampling weights accounted for household composition and the relative populations ofthe districts. Among those surveyed, 1099 caregivers reported at least one episode of diarrhea duringthe recall period of 27 days. Respondents reported whether they sought care at a health structure.We use the reporting rate of severe diarrhea, which is defined as the presence of acute watery diarrheaand the presence of two or more of the signs of loss of consciousness, sunken eyes, and an incapacityto drink or drinking very little.From the cluster survey we determine that an estimated total of 42.9% of caregivers who reportedsevere diarrhea consulted at a health center (95%CI : (33 . , . We consider a range of dynamic models for rotavirus transmission. Information linking individual-level data on the course of infection to the between-person transmission of rotavirus is lacking, leadingto variation in the structure of mathematical models for rotavirus [32]. Using a range of differentmodels allows us to account for the uncertainty in estimation due to model choice. The five modelswe consider are SIR-like compartmental models of transmission, building upon the models in [32].We incorporate age into the model with separate compartments for ages from 0-1 month, 2-3 months,4-5 months, 6-11 months, 12-23 months, and 24-59 months. Fixed parameters are estimated fromEngland and Wales data as described in [32].Here we very briefly outline the main features of five models, Models A through E, based on theSIR framework. Details of the model and inferential procedure are described in the supplementarymaterial. Model A tracks severe and mild rotavirus separately. Severe infections have larger force ofinfections than mild infections. Unlike Model A, Models B-E assume successive infections and immu-nity are obtained through repeated infections. Subsequent infections will have a reduced susceptibilityto infection and level of infectiousness. Model C allows for an incubation period of infections as well.In Model D there is no temporary immunity during successive infections and immunity is grantedafter all repeated infections. Model E assumes that full immunity can be obtained during succes-sive infections. All model parameters are estimated via Markov chain Monte Carlo (MCMC) andestimated burden over time were obtained from each model.4 .2 Vaccination
We assume vaccination imparts immunity comparable to a natural infection, and consider astrategy wherein a first dose is administered at 2 months of age and a second dose is administeredat 4 months. The vaccine was assumed to confer protection comparable to protection conferred byprimary infection following the first dose. The second dose confers additional protection comparableto that conferred by secondary infection. For Model A, where the risk of infection does not decreasebased previous number of infections, a separate input parameter is used for the vaccine efficacy. Thevaccine efficacy is set to be equal to the predicted efficacy for Models B-E (see supplementary materialfor details). We study the effect of the vaccine under varying levels of coverage. The short-term effectof vaccination is assessed by looking at incidence over a five year period following introduction ofthe vaccine. The long-term effect is measured by the yearly reduction in incident cases of Rotavirusgastroenteritis (RVGE) measured 20 years after introduction of the vaccine. Field efficacy of a multi-dose rotavirus vaccination strategy is uncertain. To reflect this uncertainty, we investigate the impactof vaccination using the value of efficacy from two different studies. First based on the results of[24], for low income countries, we assume a seroconversion rate of 63%. Second, a recent study of a3-dose vaccination strategy in Niger [20] estimated efficacy of 66.7% with all doses. The details ofrepresenting these two estimates of efficacy in the 5 models are presented in the supplement.
We fit each model independently and estimate parameters. Then we calculate ensemble esti-mates using Bayesian model averaging (BMA) [4, 18] to formalize uncertainty in model selection(see supplementary material for details). Posterior model probabilities (PMP) measures how mucheach model is supported by the data. Following the BMA approach, based on these probabilities, weprovide a weighted average of estimates from five different models. There is significant discordanceacross models in the measures of model fit (Table 1). Model C, the model with incubation periodsperforms the best. Notably, Model A, the only model that does not allow for successive infectionswith decreased levels of infectiousness, performs significantly worse as measured by posterior modelprobability.
Our fitted models allow us to construct estimates of the burden in these four districts (Table1). Of children under five, an approximate 3.5% per year develop severe RVGE as estimated byModels B-D, though this estimate is lower for Model E and significantly larger for Model A. The5odel Probability R BurdenA 0 30.7 (25.8,34.3) 9.2 (8.1,10.1)B 0.01 13.9 (12.7,15.4) 3.5 (3.1,3.8)C 0.92 13.4 (11.7,15.3) 3.5 (3.1,3.9)D 0.03 11.2 (9.4,12.7) 3.6 (3.2,4.1)E 0.04 10.3 (9.5,12.6) 3.2 (2.9,3.5)BMA 13.4 (10.3,15.4) 3.5 (2.9,4.2)Table 1: For each model we provide posterior model probability (PMP), the basic reproductive number R , and estimated burden. Burden corresponds to yearly cases with severe RVGE (% of population).The last row corresponds to the model-averaged (via Bayesian model averaging) versions of theseestimates.basic reproductive number R is found as the largest eigenvalue of the next-generation matrix [14]and significantly larger for Model A. BMA for burden and R are close to those of Model C, whichhas the highest weight. In Figure 1, we plot our model projections with uncertainty for reported casesof rotavirus as well as for all cases of severe RVGE. We also note that Models B-E predict a steepdecline in cases in children over 1y of age following the epidemic peak; cases in infants under 1y, bycontrast, are predicted to decline more slowly.Figure 2 shows the BMA-based model projections which are close to those of Model C. Howeverwe note that BMA-based projections have wider confidence intervals because averaged projectionsincorporate model uncertainty. 6 Dec. May Sept. Feb. Jun. Nov. Apr. llllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllll
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Month C a s e s ( R epo r t ed ) C a s e s ( A ll ) C a s e s ( B y A ge ) Model A Model B Model C Model D Model E
Figure 1: Burden estimates under the five fitted models. Dashed lines denote 95% confidence interval.Top: weekly reported cases of RVGE and model projections. Middle: model projections of all severeRVGE cases. Bottom: model projections of RVGE incidence by age. Lines are model projectionswhile points represent observed cases.
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Dec. May Sept. Feb. Jun. Nov. Apr. C a s e s ( A ll ) Dec. May Sept. Feb. Jun. Nov. Apr. < 1yr> 1yr C a s e s ( B y A ge ) Figure 2: Model-averaged (BMA) burden estimates from the five fitted models. Dashed lines denote95% confidence interval. Left: weekly reported cases of RVGE and model projections. Middle: modelprojections of all severe RVGE cases. Right: model projections of RVGE incidence by age. Lines aremodel projections while points represent observed cases.All of the fitted models are able to successfully capture the observed age distribution of cases(Figure 3), though Models C and E predict noticeably more cases than observed for older children(2-5 years). The models vary in their ability to capture the temporal dynamics. During the secondyear of hospital surveillance we can see a secondary peak in the number of cases that is not capturedby our fitted model, although we did find that the model dynamics can produce this double peakthrough an interaction of a high birth rate and seasonal variation when the seasonal forcing is stronger7han that estimated here. BMA shows a similar trend as the Model C, which has the highest weight.
Here we investigate the impact of vaccination based on the seroconversion rate for low socio-economic settings [24]. In the supplementary material we provide the impact of vaccination using adifferent value of efficacy which is measured based on a 3-dose strategy [20]. This was qualitativelysimilar, but quantitatively small compared to the results in the main paper. Vaccination causesa noticeable shift in the age distribution across Models B-E (Figure 3), with a higher proportion ofRVGE cases occurring in older children. This has significant benefits when considering the age-specificmortality of rotavirus is higher for children under 2 years of age [27]. The BMA-based burden showsa similar trend. l l l l l . . . . . . . Age P r opo r t i on o f R V G E c a s e s i n age g r oup l BMApost vaccinationModel Apost vaccinationModel Bpost vaccinationModel Cpost vaccinationModel Dpost vaccinationModel Epost vaccinationobserved age distribution
Figure 3: Distribution of cases across age groups observed in the data (black dots), predicted by themodels (red lines), and predicted after vaccination has been introduced at 70% coverage (blue lines).Over the short term, Models A-E predict an overall decline in total burden, but an increase in themagnitude of peak incidence (Figure 4). 8 . . . . . . −2 −1 0 1 2 3 4 5 . . . . . . −2 −1 0 1 2 3 4 5 . . . . . . −2 −1 0 1 2 3 4 5 . . . . . . −2 −1 0 1 2 3 4 5 . . . . . . −2 −1 0 1 2 3 4 5 Year R e l a t i v e i n c i den c e o f s e v e r e R V G E Model A Model B Model C Model D Model E
Figure 4: Relative incidence of severe RVGE after vaccination has been introduced into the modelsassuming 70% coverge, out to five years after vaccination has been introduced. The vaccination hasbeen introduced at 0 year.Figure 5 provides short term and long term impact of vaccination which are model-averaged valuesfrom five different models. The short term trend of vaccination impacts based on BMA is similar tothat of Model C. At equilibrium (long term), we can observe the reduction in severe rotavirus caseswith higher levels of coverage. For a fixed (70%) level of coverage, we predict 38.9% reduction ofsevere RVGE (99%CI : (37 . , . . , . . . . . . . −2 −1 0 1 2 3 4 5 R e l a t i v e i n c i den c e o f s e v e r e R V G E Year Coverage Coverage . . . . . . . l l l l l l l l Long − t e r m r edu c t i on o f s e v e r e R V G E ( p r opo r t i on ) l l l l l l l l Long − t e r m r edu c t i on o f s e v e r e R V G E ( c a s e s ) Figure 5: Relative incidence of severe RVGE (Left), percent (Middle) and absolute (Right) long termreduction in cases by coverage for Bayesian model averaging from the five fitted models. Dashedlines denote 99% confidence interval. The vaccination has been introduced at 0 year. Variation inreduction for a fixed (70%) level of coverage is demonstrated.
Diarrheal disease is a major source of childhood morbidity and mortality. However, the multi-etiology nature of diarrheal disease means that it is difficult, in the absence of lab confirmation, toinfer total burden or project the consequences of novel interventions. We have rich but short-term9ata with which to understand the dynamic process; in combination with survey data on health-seeking behavior; however, we can bring additional information to bear on the observation rate tointerpret the patterns from the non-specific clinic surveillance.For rotavirus, the uncertainty inherent in imperfectly observed incidence is compounded by thelack of a generally accepted model and debate about the underlying mechanisms that drive theepidemiological response [32]. This motivates an ensemble approach, using a combination of differentmodels along with quantitative surveillance to get practical measures of burden and projections aboutthe operational impact of controls. This multi-model ensemble approach is common in geosciences[26, 40, 42], where different assumptions on complex underlying processes can produce different climateprojections, which motivates a probabilistic forecast from a variety of models. A competing modelsapproach has been adapted to epidemiological problems as well, such as choosing an optimal strategyfor measles vaccination [36] and assessing the impact of control actions for foot and mouth diseaseoutbreaks [23].Here we formally address these two sources of uncertainty, using a state-space model to addressthe problem of incidence from non-specific surveillance data, and comparing the inference from anensemble of proposed models to address the uncertainty in the underlying dynamics. Our ensembleapproach suggests robust support for some general patterns of rotavirus dynamics. The peak trans-mission is well estimated, with a maximum in early March, with little variation between models.Rainfall, which is a primary driver of seasonality in the region, peaks in August. [5] found that earlyMarch, when urban population density is at its maximum due to seasonal rural-urban migration,was the peak season for transmission of measles. Though measles is transmitted through aerosolizeddroplets, the similarity in the peak seasonality suggests that higher population density may alsofacilitate transmission of rotavirus.We find the SEIRS structure in Model C (model with incubation period) best explains the observeddata. In this model, subsequent infections have decreased levels of infectiousness and lower risk ofinfection compared to the initial infection. All models except for Model A, which offers the worst fitto the data, include this dynamic. The estimated basic reproductive number is fairly robust acrossModels B-E. In particular, point estimates for models B-E vary from 10.3 to 13.9 in Table 1, thoughModel A has a much larger R .There is an observed double peak in incidence (Figure 1) during the second year of observationwhich our fitted models do not capture. However, this may be an anomaly, as the double peak is notseen strongly during the first and third years. We note that our models are capable of reproducingthis behavior when the seasonal variation in transmission is stronger than the best fit estimate, via aninteraction between seasonal effects and the high birth rate in the region. More complex explanations10or such double peaks have been observed elsewhere. In cholera, similar to rotavirus in transmission,local ecological variations were responsible for bimodal incidence [13].Our estimate of overall burden of severe RVGE is robust across Models B-E. In spite of thefact that the full epidemiological processes are unknown, we can be fairly sure that the total yearlyburden among children under 5 is in the vicinity of 3.5% (Table 1). Model A predicts a 3-fold greaterincidence of severe RGVE; however, this model has the weakest support and model-averaged burdenis similar to Models B-E.While uncertainty in retrospective dynamics and disease burden can be characterized using dif-ferent models, additional uncertainty about the efficacy of proposed interventions limits the abilityto predict future dynamics and disease burden. [2] estimated that rotavirus vaccine could result in2.46 million childhood deaths between 2011-2030. Of course, uncertainty in the seroconversion rate[24] and achievable vaccination coverage means that the true benefit of these vaccines is unknown.Here, we used the ensemble prediction to project the potential impact of rotavirus vaccination in theNiger setting under two scenarios for vaccine efficacy; thus integrating both dynamic uncertainty dueto different models and sensitivity to the realized effectiveness of a vaccine program. Using a vaccineefficacy derived from [24] we estimate that 70% coverage could result in 37-43% reduction in severeRVGE in children under 5. [20] reported a lower efficacy from a 3-dose schedule in Niger; this wouldlower the projected reduction of severe RVGE to 28-33%. Notably, although BMA estimates a totalreduction in yearly cases using both the efficacy reported in [24] and [20], it also predicts higher peakswhere more cases are observed than pre-vaccination. This short-term difference in cycle amplitudefor these models is a phenomenon anticipated by [33]. Anticipation of this shift in dynamic regimecaused by vaccination may be critical to the interpretation of short-term surveillance as the observa-tion of higher peak incidence following the introduction of vaccination may be wrongly interpretedas a failure in the vaccination program.Dynamic models are a powerful tool to interpret disease surveillance data and anticipate thepotential consequences of interventions. The method we describe here addresses two main sourcesof uncertainty: imperfectly observed data and scientific uncertainty about epidemiological dynamics.Our methods also allow us to identify key epidemiological interpretations – transmission seasonalityand the proportional impact of vaccination – that are robust to model choice, and those that aremodel dependent, that is, R and the annual burden of severe RVGE. By assessing the fit of theobserved surveillance to each model, we find that these latter measures are robust within the subsetof well supported models. 11 Acknowledgments
The authors are grateful to Epicentre for providing the data sets for this research project. MFis funded by a grant from the Ecology and Evolution of Infectious Disease program of the NSF/NIH(award number 1 R01 GM105247-01).
Potential conflict of interest statement: none of the authors has conflicts of interest.12 upplementary Material for An EnsembleApproach to Predicting the Impact of Vaccinationon Rotavirus Disease in Niger
Joshua Goldstein, Jaewoo Park, Murali Haran, and Matthew FerrariWe provide details below about the five different dynamic models along with information aboutcomputational methods used to perform inference for each of them. In addition, we describe theimplemention of the Bayesian model averaging approach used in the manuscript.
A Model Details
Figure 6: Structure of the compartmental models adapted from [32].The structure of the models is given in Figure 6, which we explain in detail below. Commonto each of the models we describe, we assume a time-varying transmission rate with a period of oneyear to account for seasonality, β i ( t ) = β i (cid:18) b cos (cid:18) πt − φ (cid:19)(cid:19) , t is time in weeks, β i is the baseline rate for age class i , and b and φ are the amplitude andoffset of the seasonal variation.We also assume the birth rate µ ( t ) varies with time. The mean weekly birthrate is estimated by¯ µ = 1 / (5 × r .Table 2: Seasonal variation in birth rate in Niger, estimated from 1980-2000 using Demographic andHealth Surveys. [15] An amplitude of − .
17 for January tells us the birth rate is 17% below the mean.Month Jan Feb Mar Apr May Jun Jul Aug Sep Oct Nov DecAmplitude -.17 .01 .03 .25 .12 .03 -.01 .09 .01 .13 -.31 -.17We describe in detail the dynamics of each of the five models outlined in Figure 6. Model A [37, 3]is an SIRS model in which severe and mild rotavirus are tracked separately. Severe infections have alonger duration and contribute more to the overall force of infection. Following infection, there is aperiod of temporary immunity that wanes over time. The model is age structured with age groups0-1 month, 2-3 months, 4-5 months, 6-11 months, 1 year, and 2-5 years indexed by i . The differentialequations describing the model dynamics are:Model A dM i dt = α i − M ( i − − α i M i + µN − δM i (1) dS i dt = α i − S ( i − − α i S i + δM i − λ i S i + τ R i (2) dI s , idt = α i − I s, ( i − − α i I s,i + λ s,i S i − γ s I s,i (3) dI m , idt = α i − I m, ( i − − α i I m,i + λ m,i S i − γ m I m,i (4)Movement between age classes occurs at rates dependent on the length of the interval in weeks, α = (cid:26) , , , , , (cid:27) . The force of infection for age class i is given by λ i = (cid:88) j =1 β j ( t ) C ij ( I s + 0 . I m ) N j ,assuming that relative infectiousness for mild infections is less than for severe RVGE. Here C ij repre-sents the frequency of contact from age class i onto class j [17], and satisfies f i C ij = f j C ji where f i isthe fraction of the population in class i . We make the simplifying assumption that contact betweenage groups is homogeneous. With the absence of data on rotavirus infections for children over 5 andadults, we also assume the population of children under 5 is closed and consider child-child trans-mission only. Infection with rotavirus is typically asymptomatic [32] or unreported for older childrenand adults, but could potentially play a role in transmission. The contact matrix is14 = . The differences in our age groups means that the contact matrix is not symmetric, for examplewe assume the population from 2-5 years is 18 times larger than the population from 0-1 months.After a period of maternal immunity ( M i ), individuals can be susceptible ( S i ), infected with eithermild ( I m,i ) or severe ( I s,i ) rotavirus, or recovered ( R i ). These represent the number of individualsin each class. In (1) we see how the number of children protected by maternal immunity changeover time. Newborns are added to this class at rate µ and individuals leave this class when maternalimmunity wanes with rate δ , where the mean period of maternal immunity is assumed to be 13 weeks( δ = ).When maternal immunity wanes children are susceptible to rotavirus infection. In (2), we see thatindividuals enter the susceptible class when maternal immunity wanes. They become infected at arate given by the force of infection λ i . After recovery, individuals may reenter the susceptible classat rate τ , where the mean period of immunity following infection is fixed at one year ( τ = 1 / λ s,i = 0 . λ i . Individuals leavethe infected with severe rotavirus for a mean period of one week ( γ s = 1) following which theyare considered to be recovered. Similarly, (4) tracks the total infections with mild rotavirus, with λ m,i = 0 . λ i and a mean infectious period of just half a week ( γ m = 2).Only a fraction of infections with rotavirus develop RVGE (fixed at 24%), and we assume onlysevere cases are reported, so the expected number of reported cases for age class i is given by 0 . ρλ i S i where ρ is the reporting rate. We make the simplifying assumption for all models that ρ is constantacross time and does not vary by age group.Model B [33] is an SIRS model allowing for successive infections in which a second, third orsubsequent infection will have a reduced susceptibility to infection and level of infectiousness. Thisrepresents partial immunity granted through repeated infections. Only a fraction of individuals inthe a first or second infectious class are assumed to develop severe RVGE. The model dynamics aredescribed by as follows. 15odel B dM i dt = α i − M ( i − − α i M i + µN − δM i dS i dt = α i − S i − − α i S i + δM i − λ i S i dI i dt = α i − I i − − α i I i + λ i S i − γ I i dR i dt = α i − R i − − α i R i + γ I i − τ R i dS i dt = α i − S i − − α i S i + τ R i − λ i S i dI i dt = α i − I i − − α i I i + λ i S i − γ i I i dR i dt = α i − R i − − α i R i + γ I i − τ R i dS i dt = α i − S i − − α i S i + τ R i + τ R i − λ i S i dI i dt = α i − I i − − α i I i + λ i S i − γ I i dR i dt = α i − R i − − α i R i + γ I i − τ R i Here in addition to an initial period of maternal immunity, individuals can be in the susceptible,infected, or recovered classes for their first ( S , I , R ), second ( S , I , R ), or third and subsequent( S , I , R ) infections. The force of infection for age class i is given by λ i = (cid:88) j =1 β j ( t ) C ij ( I j + 0 . I j + 0 . I j ) N j ,assuming that relative infectiousness decreases for subsequent infections. We assume the relative riskof infection decreases for subsequent infections, setting λ i = 0 . λ i and λ i = 0 . λ i as in [32]. Only13% of first infections and 3% of second infections are assumed to develop severe RVGE, based ondata from a Mexico cohort study [45]. So the expected number of reported cases for age class i isgiven by ρλ i (0 . S i + 0 . S i ). Following [45], we assume that the mean infectious period for thefirst infection is one week ( γ = 1) and for subsequent infections is half a week ( γ = 2).Model C [12] is an SEIRS model, similar to Model B but allowing for an additional exposed orincubation period. Individuals in the exposed class are infected but not yet infectious. The dynamicequations are given by: 16odel C dM i dt = α i − M ( i − − α i M i + µN − δM i dS i dt = α i − S i − − α i S i + δM i − λ i S i dE i dt = α i − E i − − α i E i + λ i S i − ξE i dI i dt = α i − I i − − α i I i + ξE i − γ I i dR i dt = α i − R i − − α i R i + γ I i − τ R i dS i dt = α i − S i − − α i S i + τ R i − λ i S i dE i dt = α i − E i − − α i E i + λ i S i − ξE i dI i dt = α i − I i − − α i I i + ξE i − γ I i dR i dt = α i − R i − − α i R i + γ I i − τ R i dS i dt = α i − S i − − α i S i + τ R i − λ i S i dE i dt = α i − E i − − α i E i + λ i S i − ξE i dI i dt = α i − I i − − α i I i + ξE i − γ I i dR i dt = α i − R i − − α i R i + γ I i − τ R i The modeling assumptions are the same as Model B but for the addition of an exposed class forthe first, second, or subsequent infections ( E , E , E ). We assume a mean exposed period of 1 day( ξ = 7).Model D [44] is an SIS model which also allows for successive infections with different levels ofinfectiousness, but assumes there is no period of temporary immunity following infection. After fourinfections individuals are assumed to be fully immune to infection. The dynamics are described asfollows. 17odel D dM i dt = α i − M ( i − − α i M i + µN − δM i dS i dt = α i − S i − − α i S i + δM i − λ i S i dI i dt = α i − I i − − α i I i + λ i S i − γ I i dS i dt = α i − S i − − α i S i + γ I i − λ i S i dI i dt = α i − I i − − α i I i + λ i S i − γ i I i dS i dt = α i − S i − − α i S i + γ I i − λ i S i dI i dt = α i − I i − − α i I i + λ i S i − γ I i dS i dt = α i − S i − − α i S i + γ I i − λ i S i dI i dt = α i − I i − − α i I i + λ i S i − γ I i The force of infection is λ i = (cid:88) j =1 β j ( t ) C ij ( I + 0 . I + 0 . I + 0 . I ) N , assuming that relative in-fectiousness decreases for subsequent infections. We also assume the relative risk of infection decreasesfor subsequent infections, setting λ i = 0 . λ i and λ i = λ i = 0 . λ i . Again, we assume only 13%of first infections and 3% of second infections are assumed to develop severe RVGE. So the expectednumber of reported cases in age group i is given by ρλ (0 . S i + 0 . S i ).Finally, Model E [1] is an SIR-SIS hybrid wherein following infection, individuals have a chanceto either return to the susceptible class or gain full immunity. The equations for the dynamics are asfollows. 18odel E dM i dt = α i − M ( i − − α i M i + µN − δM i dS i dt = α i − S i − − α i S i + δM i − λ i S i dI i dt = α i − I i − − α i I i + λ i S i − γ I i dS i dt = α i − S i − − α i S i + κ γ I i − λ i S i dI i dt = α i − I i − − α i I i + λ i S i − γ i I i dS i dt = α i − S i − − α i S i + κ γ I i − λ i S i dI i dt = α i − I i − − α i I i + λ i S i − γ I i dS i dt = α i − S i − − α i S i + κ γ I i − λ i S i dI i dt = α i − I i − − α i I i + λ i S i − γ I i The chance of returning to the susceptible class varies by number of previous infections. Following[1] we fix κ = 0 . κ = 0 . κ = 0 .
85. The remaining modeling assumptions are the same as formodels B-D.
A.1 Computational Details
Denote the observed data by C = { C i,t ; t ∈ (1 , ..., t obs ) , i ∈ (1 , ..., } where C i,t is the number ofreported cases in age group i during week t . Cases were observed over t obs = 118 weeks. Denote thenumber of cases in age group i during week t predicted by our models by ξ i ( t ). For Model A, ξ i ( t ) = 0 . ρλ i,t S i,t While for models B-E, ξ i ( t ) = ρλ i,t (0 . S i,t + 0 . S i,t )For each model, the periodic solution to the system of ODEs specified above determines thenumber of reported cases in age group i during a given week. Model dynamics are integrated forwardusing the deSolve [39] package in R until a periodic solution is reached. Solutions have a period ofone year; that is, starting from arbitrary initial conditions, we run the dynamics forward until ourexpected number of cases is identical from one 52 week period to the next, to within a small tolerance;19.e. (cid:88) i =1 t ∗ +52 (cid:88) t = t ∗ | ξ i ( t ) − ξ i ( t − | < (cid:15) = 0 . i ( t ); t ∈ (1 , ..., t obs ) , i ∈ (1 , ..., N i,t ∼ N B (Ξ i ( t ) , r ). The likelihood is L ( C | Θ) = (cid:89) i =1 t obs (cid:89) t =1 f N i,t ( C i,t )The number of observed reported cases is modeled as a Negative Binomial with mean equal to theexpected number of cases and dispersion parameter r .Inference for our model parameters is done via Markov chain Monte Carlo (MCMC) for modelsA-E. At each step of the Markov chain, new parameters Θ (cid:48) are proposed and the model dynamics areintegrated forward until the periodic solution Ξ i ( t ; Θ (cid:48) ) is reached in order to calculate L ( C | Θ (cid:48) ). Theparameters estimated by MCMC are Θ = ( b, φ, r, ρ, β i ; i ∈ (1 , ..., b ,seasonal phase φ , the dispersion r of the Negative Binomial observation process, the reporting rate ρ , and the baseline transmission rate for age class β i .MCMC samples are obtained from the posterior distribution π ( b, φ, r, ρ, β , ..., β | C ) ∝ L ( C | b, φ, r, ρ, β , ..., β ) p ( b ) p ( φ ) p ( r ) p ( ρ ) (cid:89) i =1 p ( β i )where we take priors p ( β i ) = N (20 , p ( b ) = Unif(0 , p ( φ ) = Unif(2 , π + 2), p ( r ) =Gamma(0 . , . p ( ρ ) = N (0 . , . ρ is centeredat 11.7%, determined from the estimated reporting rate from the cluster survey (42.9%) and the esti-mated proportion of the population under 5 in the four districts that is covered by hospital surveillance(27.3%, from 2009 census data). In practice, we find that our estimates are robust to the choice ofstandard deviation of p ( ρ ).Table 3 provides parameter estimates from five different models. Estimates of the strength oftransmission are similar for models B-D, higher for Model E and significantly lower for Model A.The same holds true for the reporting rate (Model A’s estimate of the reporting rate is dramaticallylower, and does not agree with estimates from the cluster survey, evidence that it is performingpoorly). Notably, the estimated phase of the transmission φ is similar across all models (Table 3).For reference, an estimated φ of 7 . b φ r ρ A 0.50 (0.48,0.51) 7.4 (7.3,7.5) 1.5 (1.4,1.5) 0.039 (0.035,0.044)B 0.41 (0.37,0.45) 7.4 (7.3,7.6) 2.6 (2.3,2.8) 0.096 (0.089,0.104)C 0.42 (0.36,0.46) 7.4 (7.3,7.5) 2.5 (2.0,2.8) 0.097 (0.089,0.104)D 0.30 (0.26,0.34) 7.2 (7.0,7.4) 2.7 (2.5,2.8) 0.098 (0.090,0.107)E 0.32 (0.27,0.36) 7.1 (7.0,7.3) 2.6 (2.5,2.7) 0.109 (0.099,0.117)
A.2 Dynamics Accounting for Vaccination
Based on the results of [24] we assume that 63% of vaccinated individuals are successfullyseroconvert after a single dose. Our models with vaccination allow for the red transitions in Figure 6.For example, Model B allows for transitions directly from M i =1 and S ,i =1 to R ,i =2 on the first dose,and from R ,i =2 and S ,i =2 to R ,i =3 on the second dose. The dynamics equations will be modifiedby the following terms: dM i =2 dt = (1 − sc ) α M i =1 + ...dS ,i =2 dt = (1 − sc ) α S ,i =1 + ...dR ,i =2 dt = ( sc ) α M i =1 + ( sc ) α S ,i =1 + ...dR ,i =3 dt = (1 − sc ) α R ,i =2 + ...dS ,i =3 dt = (1 − sc ) α S ,i =2 + ...dR ,i =3 dt = ( sc ) α R ,i =2 + ( sc ) α S ,i =2 + ... Where c is the coverage and s = 0 .
63 is the rate of seroconversion [24] for low socio-economicsettings. This means that an individual who is vaccinated with a single dose has a lower risk ofinfection, comparable to the effect of recovering from a natural infection. Vaccination with a seconddose further reduces risk of infection.In Model A, the risk of infection does not decline with the previous number of infections. Therefore,an additional vaccinated state V i is added to the model for age group i . Two additional inputparameters are required for the vaccine efficacy against severe and mild RVGE. We assume thevaccination happens at 2 months, but the vaccine efficacy is equal to the efficacy predicted undermodels B-E for the two dose strategy, η s = .
796 and η m = . M i =2 dt = (1 − c ) α M i =1 + ...dS ,i =2 dt = (1 − c ) α S i =1 + ...dV i =2 dt = ( c ) α M i =1 + ( c ) α S i =1 + ...dV i> dt = α i − V ( i − − α i V i − ( τ + λ s,i (1 − η s ) + λ m,i (1 − η m )) V i dS i> dt = τ V i + ...dI s,i> dt = λ s,i (1 − η s ) V i + ...dI m,i> dt = λ m,i (1 − η m ) V i + ... Given our vaccination strategy for models B-E, the vaccine efficacy for severe RVGE after twodoses is 79.6%, in line with efficacy studies of rotavirus vaccines. This is calculated by multiplying theproportion of individuals who are successfully immunized twice, once, or zero times by the expectedreduction in RVGE incidence for each case.
V E = 1 − (cid:20) . × . . × (cid:18) .
62 0 . . (cid:19) + 0 . × (cid:18) .
37 00 . (cid:19)(cid:21) = 79 . . (5)We assume following [45] that 47% of first infections and 25% of second infections and 32% of thirdinfections are assumed to develop any RVGE (mild RVGE is unreported). Therefore the vaccineefficacy for all RVGE is V E = 1 − (cid:20) . × . . × (cid:18) .
62 0 . . (cid:19) + 0 . × (cid:18) .
37 0 . . (cid:19)(cid:21) = 60 . . In practice, first model parameters Θ are estimated via MCMC for the models without vaccination.Using the fitted model, the dynamics are then integrated forward at the posterior mean of Θ until theperiodic solution has been reached. Then, the dynamics are modified to allow for transitions betweencompartments by vaccination.
A.2.1 Projections Based on Vaccine Efficacy from a Recent Study
Recently [20] estimated that 3 doses of vaccine had 66.7% efficacy against severe RVGE amongchildren in Niger. Though we do not explicitly account for 3 doses of vaccine, we can calculatethe effective seroconversion rate for our model above that would yield this observed efficacy after acomplete sequence of doses. Thus, we set η s = .
667 and use (5) to calculate the effective seroconversionrate as 49%. Then the vaccine efficacy for all RVGE is η m = . η s , η m and s values with the same dynamic equations.Although we used the two dose strategy, by using different value of the efficacy, our study canaccount for uncertainty in the seroconversion rate. Figures 7-9 are matched to Figures 3-5 in themain paper. Because of the lower seroconversion rate, the projected results were qualitatively similar,quantitatively smaller. Vaccination causes a shift in the age distribution across models (Figure 7),with a higher proportion of RVGE cases occurring for older children. l l l l l . . . . . . . Age P r opo r t i on o f R V G E c a s e s i n age g r oup l BMApost vaccinationModel Apost vaccinationModel Bpost vaccinationModel Cpost vaccinationModel Dpost vaccinationModel Epost vaccinationobserved age distribution
Figure 7: Distribution of cases across age groups observed in the data (black dots), predicted by themodels (red lines), and predicted after vaccination has been introduced at 70% coverage (blue lines).Over the short term, Models A-E predict an overall decline in total burden, but an increase in themagnitude of peak incidence (Figure 8). 23 . . . . . . −2 −1 0 1 2 3 4 5 . . . . . . −2 −1 0 1 2 3 4 5 . . . . . . −2 −1 0 1 2 3 4 5 . . . . . . −2 −1 0 1 2 3 4 5 . . . . . . −2 −1 0 1 2 3 4 5 Year R e l a t i v e i n c i den c e o f s e v e r e R V G E Model A Model B Model C Model D Model E
Figure 8: Relative incidence of severe RVGE after vaccination has been introduced into the modelsassuming 70% coverge, out to five years after vaccination has been introduced. The vaccination hasbeen introduced at 0 year.Figure 9 indicates that the short term trend of vaccination impacts based on BMA is similar tothat of Model C. BMA predicts 29.6% of long term reduction (99%CI : (28 . , . . . . . . . −2 −1 0 1 2 3 4 5 R e l a t i v e i n c i den c e o f s e v e r e R V G E Year Coverage Coverage . . . . . . l l l l l l l l Long − t e r m r edu c t i on o f s e v e r e R V G E ( p r opo r t i on ) l l l l l l l l Long − t e r m r edu c t i on o f s e v e r e R V G E ( c a s e s ) Figure 9: Relative incidence of severe RVGE (Left), percent (Middle) and absolute (Right) long termreduction in cases by coverage for Bayesian model averaging from the five fitted models. Dashedlines denote 99% confidence interval. The vaccination has been introduced at 0 year. Variation inreduction for a fixed (70%) level of coverage is demonstrated.
B Bayesian Model Averaging
For k = 1 , ...,
5, consider M k , the k th model, with prior p (Θ k | M k ) and likelihood function L ( C | Θ k , M k ). Note that we take the uniform model prior for p ( M l ) and model evidence P ( C | M k )is approximated via Bayesian information criterion (BIC) as in [34]. Then the posterior modelprobability (PMP) for M k given the observed data C is p ( M k | C ) = p ( C | M k ) p ( M k ) (cid:80) l =1 p ( C | M l ) p ( M l ) , p ( C | M k ) = (cid:90) L ( C | Θ k , M k ) p (Θ k | M k ) d Θ k is the model evidence for M k which measures how well each model is supported by the observed data.Then the BMA estimate of the burden is E [ ξ ( t ) | C ] = (cid:88) l =1 E [ ξ l ( t ) | C, M l ] p ( M l | C ) . A summary of our implementation of BMA is as follows: (1) We construct a separate MCMCalgorithm for each of the models A-E. (2) For each model, the burden estimate ξ k ( t ) is evaluated forthe MCMC samples of the posterior distribution of that model. (3) The expected burden for model k , E [ ξ k ( t ) | C, M k ], is estimated through the sample mean of the ξ k ( t )s obtained from Step (2). (4)We take the weighted average of the burden across all models, with the weights equal to the posteriormodel probabilities, p ( M k | C ), obtained above. References [1] Atchison, C., Lopman, B., and Edmunds, W. J. (2010). Modelling the seasonality of rotavirusdisease and the impact of vaccination in England and Wales.
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