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Vaccination programmes to prevent outbreaks after introductions of infectious people aim to maintain the average number of secondary infections per infectious person at one or less. We aimed to assess heterogeneity in vaccine uptake and other characteristics that, together with non-random mixing, could increase this number and to evaluate strategies that could mitigate their impact. Because most US children attend elementary school in their own neighbourhoods, surveys of children entering elementary school (age 5 years before Sept 1) allow assessment of spatial heterogeneity in the proportion of children immune to vaccine-preventable diseases. We used data from a 2008 school-entry survey by the Immunization Division of the California Department of Public Health to obtain school addresses; numbers of students enrolled; proportions of enrolled students who had received one or two doses of the measles, mumps, and rubella (MMR) vaccine; and proportions with medical or personal-belief exemptions. Using a mixing model suitable for spatially-stratified populations, we projected the expected numbers of secondary infections per infectious person for measles, mumps, and rubella. We also mapped contributions to this number for measles in San Diego County's 638 elementary schools and its largest district, comprising 200 schools (31%). We then modelled the effect on measles' realised reproduction number (RV) of the following plausible interventions: vaccinating all children with personal-belief exemptions, increasing uptake by 10% to 50% in all low-immunity schools (<90% of students immune) or in only influential (effective daily contact rates >3 or contacts inter-school >30%) low-immunity schools, and increasing private school uptake to the public school average. In 2008, 39 132 children began elementary school in San Diego County, CA, USA. At entry to school, 97% had received at least one dose of the MMR vaccine, with 2·5% having personal-belief exemptions. We note substantial heterogeneity in immunity throughout the county. Although the average population immunities for measles, mumps, and rubella (92%, 87%, 92%) were similar to the population-immunity thresholds in homogeneous, randomly-mixing populations (91%, 88%, 76%), after accounting for heterogeneity and non-random mixing, the basic reproduction numbers increased by 70%, meaning that introduced pathogens could cause outbreaks. The impact of our modelled interventions ranged from negligible to a nearly complete reduction in the outbreak potential of measles. The most effective intervention to lower the realised reproduction number (RV 3·39) was raising immunity by 50% in 114 schools with low immunity (RV 1·02), but raising immunity by this level in only influential, low-immunity schools also was effective (RV 2·02). The effectiveness of vaccinating the 972 children with personal-belief exemptions was similar to that of targeting all low-immunity schools (RV 1·11). Targeting only private schools had little effect. Our findings suggest that increasing vaccine uptake could prevent outbreaks such as that of measles in San Diego in 2008. Vaccinating children with personal-belief exemptions was one of the most effective interventions that we modelled, but further research on mixing in heterogeneous populations is needed. None.

In 2006 the United States experienced the largest nationwide mumps epidemic in 20 years, primarily affecting college dormitory residents. Unexpected elements of the outbreak included very abrupt time course (75% of cases occurred within 90 days), geographic focality (85% of cases occurred in eight rural Midwestern states), rapid upward and downward shift in peak age-specific attack rate (5–9-year olds to 18–24-year olds, then back), and two-dose vaccine failure (63% of case-patients had received two doses). To construct a historical context in which to understand the recent outbreak, we reviewed US mumps surveillance data, vaccination coverage estimates, and relevant peer-reviewed literature for the period 1917–2008. Many of the unexpected features of the 2006 mumps outbreak had been reported several times previously in the US, e.g., the 1986–1987 mumps resurgence had extremely abrupt onset, rural geographic focality, and an upward-then-downward age shift. Evidence suggested recurrent mumps outbreak patterns were attributable to accumulation of susceptibles in dispersed situations where the risk of endemic disease exposure was low and were triggered when this susceptible population was brought together in crowded living conditions. The 2006 epidemic followed this pattern, with two unique variations: it was preceded by a period of very high vaccination rates and very low disease incidence and was characterized by two-dose failure rates among adults vaccinated in childhood. Data from the past 80 years suggest that preventing future mumps epidemics will depend on innovative measures to detect and eliminate build-up of susceptibles among highly vaccinated populations.

Historically, annual rotavirus activity in the United States has started in the southwest in late fall and ended in the northeast 3 months later; this trend has diminished in recent years. Traveling waves of infection or local environmental drivers cannot account for these patterns. A transmission model calibrated against epidemiological data shows that spatiotemporal variation in birth rate can explain the timing of rotavirus epidemics. The recent large-scale introduction of rotavirus vaccination provides a natural experiment to further test the impact of susceptible recruitment on disease dynamics. The model predicts a pattern of reduced and lagged epidemics postvaccination, closely matching the observed dynamics. Armed with this validated model, we explore the relative importance of direct and indirect protection, a key issue in determining the worldwide benefits of vaccination.

Mixing among sub-populations, as well as heterogeneity in characteristics affecting their reproduction numbers, must be considered when evaluating public health interventions to prevent or control infectious disease outbreaks. In this overview, we apply a linear algebraic approach to re-derive some well-known results pertaining to preferential within- and proportionate among-group contacts in compartmental models of pathogen transmission. We give results for the meta-population effective reproduction number ([Formula: see text]) assuming different levels of vaccination in the sub-populations. Specifically, we unpack the dependency of [Formula: see text] on the fractions of contacts reserved for individuals within one's own subgroup and, by obtaining implicit expressions for the partial derivatives of [Formula: see text], we show that these increase as this preferential-mixing fraction increases in any sub-population.

Rubella and congenital rubella syndrome (CRS) continue to be important health problems in many countries. In June 2004, the World Health Organization Steering Committee on Research Related to Measles and Rubella Vaccines and Vaccination met to evaluate data from research and operational activities and to identify critical scientific issues and gaps in knowledge that need to be addressed to improve the global control of rubella and CRS. Information about surveillance for rubella, natural and vaccine-induced immunity to rubella, laboratory diagnosis, the molecular epidemiological profile of rubella virus, and mathematical modeling to assess the burden of CRS and the impact of rubella vaccination was reviewed. This report summarizes the presentations and recommendations for future research

A rubella vaccine was licensed in China in 1993 and added to the Expanded Programme on Immunization in 2008, but a national cross-sectional serological survey during 2014 indicates that many adolescents remain susceptible. Maternal infections during the first trimester often cause miscarriages, stillbirths, and, among livebirths, congenital rubella syndrome. We aimed to evaluate possible supplemental immunisation activities (SIAs) to accelerate elimination of rubella and congenital rubella syndrome. We analysed residual samples from the national serological survey done in 2014, data from monthly rubella surveillance reports from 2005 and 2016, and additional publications through a systematic review. Using an age-structured population model with provincial strata, we calculated the reproduction numbers and evaluated the gradient of the metapopulation effective reproduction number with respect to potential supplemental immunisation rates. We corroborated these analytical results and estimated times-to-elimination by simulating SIAs among adolescents (ages 10–19 years) and young adults (ages 20–29 years) using a model with regional strata. We estimated the incidence of rubella and burden of congenital rubella syndrome by simulating transmission in a relatively small population lacking only spatial structure. By 2014, childhood immunisation had reduced rubella's reproduction number from 7·6 to 1·2 and SIAs among adolescents were the optimal elimination strategy. We found that less than 10% of rubella infections were reported; that although some women with symptomatic first-trimester infections might have elected to terminate their pregnancies, 700 children could have been born with congenital rubella syndrome during 2014; and that timely SIAs would avert outbreaks that, as susceptible adolescents reached reproductive age, could greatly increase the burden of this syndrome. Our findings suggest that SIAs among adolescents would most effectively reduce congenital rubella syndrome as well as eliminate rubella, owing both to fewer infections in the immunised population and absence of infections that those immunised would otherwise have caused. Metapopulation models with realistic mixing are uniquely capable of assessing such indirect effects. WHO and National Science Foundation.

The basic reproduction number (\\[\\mathcal {R}_0\\]) can be considerably higher in an SIR model with heterogeneous mixing compared to that from a corresponding model with homogeneous mixing. For example, in the case of measles, mumps and rubella in San Diego, CA, Glasser et al. (Lancet Infect Dis 16(5):599–605, 2016. https://doi.org/10.1016/S1473-3099(16)00004-9), reported an increase of 70% in \\[\\mathcal {R}_0\\] when heterogeneity was accounted for. Meta-population models with simple heterogeneous mixing functions, e.g., proportionate mixing, have been employed to identify optimal vaccination strategies using an approach based on the gradient of the effective reproduction number (\\[\\mathcal {R}_v\\]), which consists of partial derivatives of \\[\\mathcal {R}_v\\] with respect to the proportions immune \\[p_i\\] in sub-groups i (Feng et al. in J Theor Biol 386:177–187, 2015. https://doi.org/10.1016/j.jtbi.2015.09.006; Math Biosci 287:93–104, 2017. https://doi.org/10.1016/j.mbs.2016.09.013). These papers consider cases in which an optimal vaccination strategy exists. However, in general, the optimal solution identified using the gradient may not be feasible for some parameter values (i.e., vaccination coverages outside the unit interval). In this paper, we derive the analytic conditions under which the optimal solution is feasible. Explicit expressions for the optimal solutions in the case of \\[n=2\\] sub-populations are obtained, and the bounds for optimal solutions are derived for \\[n>2\\] sub-populations. This is done for general mixing functions and examples of proportionate and preferential mixing are presented. Of special significance is the result that for general mixing schemes, both \\[\\mathcal {R}_0\\] and \\[\\mathcal {R}_v\\] are bounded below and above by their corresponding expressions when mixing is proportionate and isolated, respectively.

During the COVID-19 pandemic, renewal equation estimates of time-varying effective reproduction numbers were useful to policymakers in evaluating the need for and impact of mitigation measures. Our objective here is to illustrate the utility of mechanistic expressions for the basic and effective (or intrinsic and realized) reproduction numbers, R 0 , R E , and related quantities derived from a Susceptible-Exposed-Infectious-Removed (SEIR) model including features of COVID-19 that might affect transmission of SARS-CoV-2, including asymptomatic, pre-symptomatic, and symptomatic infections, with which people may be hospitalized. Expressions from homogeneous host population models can be analyzed to determine the effort needed to reduce R E from R 0 to 1 and contributions of modeled mitigation measures. Our model is stratified by age, 0–4, 5–9, …, 75+ years, and location, the 50 states plus District of Columbia. Expressions from such heterogeneous host population models include subpopulation reproduction numbers, contributions from the above-mentioned infectious states, metapopulation numbers, subpopulation contributions, and equilibrium prevalence. While the population-immunity at which R E = 1 has captured the popular imagination, the metapopulation R E ≤ 1 could be attained in an infinite number of ways even if only one intervention (e.g., vaccination) were capable of reducing R E . However, gradients of expressions derived from heterogeneous host population models, ∇ R E , can be evaluated to identify optimal allocations of limited resources among subpopulations. We illustrate the utility of such analytical results by simulating two hypothetical vaccination strategies, one uniform and other indicated by ∇ R E , as well as the actual program estimated from one of the CDC’s nationwide seroprevalence surveys conducted from mid-summer 2020 through the end of 2021.

BackgroundRotavirus causes approximately one-third to one-half (55,000–70,000 hospitalizations per year) of hospitalizations for acute gastroenteritis (AGE) among US children <5 years of age. We forecasted the potential reduction in the number of hospitalizations for rotavirus disease and AGE in US children during 2006–2015 as a result of the new rotavirus vaccine introduced in 2006 MethodsThe mean number of hospitalizations for AGE by calendar month among US children was determined using the National Hospital Discharge Survey from the period 1993–2005. From these baseline prevaccine estimates, we forecasted the effect of vaccine in reducing the number of hospitalizations for rotavirus disease and AGE during 2006–2015 with use of estimates of vaccine effectiveness and uptake ResultsDuring 2006–2015, ∼313,000 (45%) of an estimated 703,190 hospitalizations for rotavirus disease would be directly prevented by vaccination. A significant reduction in the number of hospitalizations for AGE should be detectable among infants aged 0–11 months during the first quarter of 2009, followed by children aged 12–23 months during 2010, and all children <5 years of age during 2011 ConclusionsVaccination is expected to substantially reduce the health burden of hospitalizations for rotavirus disease among US children during 2006–2015, and the impact of vaccination based on direct protective effects alone was expected to first occur for hospitalizations for AGE among infants during winter 2009

The risk of smallpox reintroduction has motivated preparations in potential target countries. After reproducing the spatiotemporal pattern after the 1972 importation into Yugoslavia via coupled, biologically realistic systems of ordinary differential equations, we developed dynamic population models with current US age distributions and typical spatially distributed social structures. Surveillance and containment (SC) coupled with vaccination of 95% of hospital-based health care workers (HCWs) within 2 days after the first diagnosis (estimated to be 18 days after aerosol release) were modeled after simulated exposure of 10, 50, or 10,000 people in various settings. If 90% of patients were isolated within days after symptom onset and 75% of contacts were vaccinated and monitored, SC would reduce cases by 82%–99%. Preemptive immunization of HCWs, closing of schools, and even vaccination of as many as 80% within 1 week would have small marginal benefits. Preparations should emphasize stockpiling vaccine, training HCWs, improving laboratory capacity, and fostering an understanding of SC.