Analysis of metapopulation models of the transmission of SARS-CoV-2 in the United States.

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, [Formula: see text] 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 [Formula: see text] from [Formula: see text] 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 [Formula: see text] has captured the popular imagination, the metapopulation [Formula: see text] could be attained in an infinite number of ways even if only one intervention (e.g., vaccination) were capable of reducing [Formula: see text] However, gradients of expressions derived from heterogeneous host population models,[Formula: see text] 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 [Formula: see text] 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.

Implications for infectious disease models of heterogeneous mixing on control thresholds.

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.

Analysis of serological surveys of antibodies to SARS-CoV-2 in the United States to estimate parameters needed for transmission modeling and to evaluate and improve the accuracy of predictions.

Seroprevalence studies can estimate proportions of the population that have been infected or vaccinated, including infections that were not reported because of the lack of symptoms or testing. Based on information from studies in the United States from mid-summer 2020 through the end of 2021, we describe proportions of the population with antibodies to SARS-CoV-2 as functions of age and time. Slices through these surfaces at arbitrary times provide initial and target conditions for simulation modeling. They also provide the information needed to calculate age-specific forces of infection, attack rates, and - together with contact rates - age-specific probabilities of infection on contact between susceptible and infectious people. We modified the familiar Susceptible-Exposed-Infectious-Removed (SEIR) model to include features of the biology of COVID-19 that might affect transmission of SARS-CoV-2 and stratified by age and location. We consulted the primary literature or subject matter experts for contact rates and other parameter values. Using time-varying Oxford COVID-19 Government Response Tracker assessments of US state and DC efforts to mitigate the pandemic and compliance with non-pharmaceutical interventions (NPIs) from a YouGov survey fielded in the US during 2020, we estimate that the efficacy of social-distancing when possible and mask-wearing otherwise at reducing susceptibility or infectiousness was 31% during the fall of 2020. Initialized from seroprevalence among people having commercial laboratory tests for purposes other than SARS-CoV-2 infection assessments on 7 September 2020, our age- and location-stratified SEIR population model reproduces seroprevalence among members of the same population on 25 December 2020 quite well. Introducing vaccination mid-December 2020, first of healthcare and other essential workers, followed by older adults, people who were otherwise immunocompromised, and then progressively younger people, our metapopulation model reproduces seroprevalence among blood donors on 4 April 2021 less well, but we believe that the discrepancy is due to vaccinations being under-reported or blood donors being disproportionately vaccinated, if not both. As experimenting with reliable transmission models is the best way to assess the indirect effects of mitigation measures, we determined the impact of vaccination, conditional on NPIs. Results indicate that, during this period, vaccination substantially reduced infections, hospitalizations and deaths. This manuscript was submitted as part of a theme issue on "Modelling COVID-19 and Preparedness for Future Pandemics."

Assessing the burden of congenital rubella syndrome in China and evaluating mitigation strategies: a metapopulation modelling study.

BACKGROUND: 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. METHODS: 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. FINDINGS: 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. INTERPRETATION: 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. FUNDING: WHO and National Science Foundation.

On the benefits of flattening the curve: A perspective.

The many variations on a graphic illustrating the impact of non-pharmaceutical measures to mitigate pandemic influenza that have appeared in recent news reports about COVID-19 suggest a need to better explain the mechanism by which social distancing reduces the spread of infectious diseases. And some reports understate one benefit of reducing the frequency or proximity of interpersonal encounters, a reduction in the total number of infections. In hopes that understanding will increase compliance, we describe how social distancing (a) reduces the peak incidence of infections, (b) delays the occurrence of this peak, and (c) reduces the total number of infections during epidemics. In view of the extraordinary efforts underway to identify existing medications that are active against SARS-CoV-2 and to develop new antiviral drugs, vaccines and antibody therapies, any of which may have community-level effects, we also describe how pharmaceutical interventions affect transmission.

Modeling the waning and boosting of immunity from infection or vaccination.

Immunity following natural infection or immunization may wane, increasing susceptibility to infection with time since infection or vaccination. Symptoms, and concomitantly infectiousness, depend on residual immunity. We quantify these phenomena in a model population composed of individuals whose susceptibility, infectiousness, and symptoms all vary with immune status. We also model age, which affects contact, vaccination and possibly waning rates. The resurgences of pertussis that have been observed wherever effective vaccination programs have reduced typical disease among young children follow from these processes. As one example, we compare simulations with the experience of Sweden following resumption of pertussis vaccination after the hiatus from 1979 to 1996, reproducing the observations leading health authorities to introduce booster doses among school-aged children and adolescents in 2007 and 2014, respectively. Because pertussis comprises a spectrum of symptoms, only the most severe of which are medically attended, accurate models are needed to design optimal vaccination programs where surveillance is less effective.

Influence of demographically-realistic mortality schedules on vaccination strategies in age-structured models.

Because demographic realism complicates analysis, mathematical modelers either ignore demography or make simplifying assumptions (e.g., births and deaths equal). But human populations differ demographically, perhaps most notably in their mortality schedules. We developed an age-stratified population model with births, deaths, aging and mixing between age groups. The model includes types I and II mortality as special cases. We used the gradient approach (Feng et al., 2015, 2017) to explore the impact of mortality patterns on optimal strategies for mitigating vaccine-preventable diseases such as measles and rubella, which the international community has targeted for eradication. Identification of optimal vaccine allocations to reduce the effective reproduction number Rv under various scenarios is presented. Numerical simulations of the model with various types of mortality are carried out to ascertain the long-term effects of vaccination on disease incidence. We conclude that both optimal vaccination strategies and long-term effects of vaccination may depend on demographic assumptions.

Evaluating vaccination policies to accelerate measles elimination in China: a meta-population modelling study.

BACKGROUND: Measles is among the most highly infectious human diseases. By virtue of increasingly effective childhood vaccination, together with targeted supplemental immunization activities (SIAs), health authorities in the People's Republic of China have reduced measles' reproduction number from about 18 to 2.3. Despite substantial residual susceptibility among young adults, more in some locales than others, sustained routine childhood immunization likely would eliminate measles eventually. To support global eradication efforts, as well as expedite morbidity and mortality reductions in China, we evaluated alternative SIAs via mechanistic mathematical modelling. METHODS: Our model Chinese population is stratified by immune status (susceptible to measles infection; infected, but not yet infectious; infectious; and recovered or immunized), age (0, 1-4, 5-9, …, 65+ years) and location (31 provinces). Contacts between sub-populations are either empirical or a mixture of preferential and proportionate with respect to age and decline exponentially with distance between locations at age-dependent rates. We estimated initial conditions and most parameters from recent cross-sectional serological surveys, disease surveillance and demographic observations. Then we calculated the reproduction numbers and gradient of the effective number with respect to age- and location-specific immunization rates. We corroborated these analytical results by simulating adolescent and young adult SIAs using a version of our model in which the age-specific contact rates vary seasonally. RESULTS: Whereas the gradient indicates that vaccinating young adults generally is the optimal strategy, simulations indicate that a catch-up campaign among susceptible adolescent schoolchildren would accelerate elimination, with timing dependent on uptake. CONCLUSIONS: These results are largely due to indirect effects (i.e. fewer infections than immunized people might otherwise cause), which meta-population models with realistic mixing are uniquely capable of reproducing accurately.

ESTIMATING AGE-SPECIFIC HAZARD RATES OF INFECTION FROM CROSS-SECTIONAL OBSERVATIONS.

Mathematical models of pathogen transmission in age-structured host populations, can be used to design or evaluate vaccination programs. For reliable results, their forces or hazard rates of infection (FOI) must be formulated correctly and the requisite contact rates and probabilities of infection on contact estimated from suitable observations. Elsewhere, we have described methods for calculating the probabilities of infection on contact from the contact rates and FOI. Here, we present methods for estimating the FOI from cross-sectional serological surveys or disease surveillance in populations with or without concurrent vaccination. We consider both continuous and discrete age, and present estimates of the FOI for vaccine-preventable diseases that confer temporary or permanent immunity.

Los modelos matemáticos de transmisión de patógenos en poblaciones de huéspedes estructuradas por edad pueden usarse para diseñar o evaluar programas de vacunación. Para obtener resultados confiables, sus fuerzas o tasas de riesgo de infección (FOI) deben formularse correctamente y las tasas de contacto requeridas y las probabilidades de infección en contacto deben estimarse a partir de observaciones adecuadas. En otros lugares, hemos descrito métodos para calcular las probabilidades de infección por contacto a partir de las tasas de contacto y FOI. Aquí, presentamos métodos para estimar el FOI a partir de encuestas serológicas transversales o vigilancia de enfermedades en poblaciones con o sin vacunación concurrente. Consideramos tanto la edad continua como la discreta, y presentamos estimaciones del FOI para enfermedades prevenibles por vacunación que confieren inmunidad temporal o permanente.

Analysis of an epidemiological model structured by time-since-last-infection.

Modeling time-since-last-infection (TSLI) provides a means of formulating epidemiological models with fewer state variables (or epidemiological classes) and more flexible descriptions of infectivity after infection and susceptibility after recovery than usual. The model considered here has two time variables: chronological time (t) and the TSLI (τ), and it has only two classes: never infected ( N ) and infected at least once (i). Unlike most age-structured epidemiological models, in which the i equation is formulated using ( ∂ ∂ τ + ∂ ∂ t ) i ( τ , t ) , ours uses a more general differential operator. This allows weaker conditions for the infectivity and susceptibility functions, and thus, is more generally applicable. We reformulate the model as an age dependent population problem for analysis, so that published results for these types of problems can be applied, including the existence and regularity of model solutions. We also show how other coupled models having two types of time variables can be stated as age dependent population problems.

Constrained minimization problems for the reproduction number in meta-population models.

The basic reproduction number ([Formula: see text]) 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 [Formula: see text] 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 ([Formula: see text]), which consists of partial derivatives of [Formula: see text] with respect to the proportions immune [Formula: see text] 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 [Formula: see text] sub-populations are obtained, and the bounds for optimal solutions are derived for [Formula: see text] 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 [Formula: see text] and [Formula: see text] are bounded below and above by their corresponding expressions when mixing is proportionate and isolated, respectively.

Vaccination and unexplained sudden death risk in Taiwanese infants.

PURPOSE: In March 1992, eight infants who had died within 36 hours of receiving whole-cell pertussis vaccine (diphtheria, tetanus, and whole-cell pertussis [DTwP]) prompted the Taiwan health authorities to suspend its use. We conducted an investigation of vaccination and sudden unexplained infant death (SUID) and repeated it more recently after Taiwan switched to acellular pertussis vaccine (diphtheria, tetanus, and acellular pertussis [DTaP]) in 2010. METHODS: All SUIDs aged 31-364 days during 1990-1992 and 1996-2013 were selected from the death registration databases. The case-control investigation matched each case to two controls on clinic, sex, and birth date, whereas the follow-up self-controlled case series study compared risk of death during the 30-day post-vaccination risk periods with those in the control periods within the same case. RESULTS: Sudden unexplained infant death was associated with never receiving DTwP (odds ratio 2.28, 95% confidence interval 1.25-4.15) in the case-control investigation. The odds ratios within 0-1, 2-7, 8-14, and 15-30 days of DTwP administration were 1.18, 0.26, 0.50, and 0.77. In the 1996-2013 self-controlled case series studies, this temporal shift between DTwP and SUID was consistently observed for female (incidence rate ratio 1.70, 0.75, 1.01, and 0.84) but not male or DTaP recipients. A pooled analysis showed significant risk within 2 days of receiving DTwP in female infants (incidence rate ratio 1.66, 95% confidence interval 1.05-2.60). CONCLUSIONS: Being unvaccinated and recent receipt of DTwP in female infants was significantly associated with SUID; the latter was consistent with a temporal shift pattern without overall increase in risk. The currently used pertussis vaccine, DTaP, did not increase risk of SUID. Copyright © 2016 John Wiley & Sons, Ltd.

Evaluating targeted interventions via meta-population models with multi-level mixing.

Among the several means by which heterogeneity can be modeled, Levins' (1969) meta-population approach preserves the most analytical tractability, a virtue to the extent that generality is desirable. When model populations are stratified, contacts among their respective sub-populations must be described. Using a simple meta-population model, Feng et al. (2015) showed that mixing among sub-populations, as well as heterogeneity in characteristics affecting sub-population reproduction numbers, must be considered when evaluating public health interventions to prevent or control infectious disease outbreaks. They employed the convex combination of preferential within- and proportional among-group contacts first described by Nold (1980) and subsequently generalized by Jacquez et al. (1988). As the utility of meta-population modeling depends on more realistic mixing functions, the authors added preferential contacts between parents and children and among co-workers (Glasser et al., 2012). Here they further generalize this function by including preferential contacts between grandparents and grandchildren, but omit workplace contacts. They also describe a general multi-level mixing scheme, provide three two-level examples, and apply two of them. In their first application, the authors describe age- and gender-specific patterns in face-to-face conversations (Mossong et al., 2008), proxies for contacts by which respiratory pathogens might be transmitted, that are consistent with everyday experience. This suggests that meta-population models with inter-generational mixing could be employed to evaluate prolonged school-closures, a proposed pandemic mitigation measure that could expose grandparents, and other elderly surrogate caregivers for working parents, to infectious children. In their second application, the authors use a meta-population SEIR model stratified by 7 age groups and 50 states plus the District of Columbia, to compare actual with optimal vaccination during the 2009-2010 influenza pandemic in the United States. They also show that vaccination efforts could have been adjusted month-to-month during the fall of 2009 to ensure maximum impact. Such applications inspire confidence in the reliability of meta-population modeling in support of public health policymaking.

Targeting pediatric versus elderly populations for norovirus vaccines: a model-based analysis of mass vaccination options.

BACKGROUND: Noroviruses are the leading cause of acute gastroenteritis and foodborne diarrheal disease in the United States. Norovirus vaccine development has progressed in recent years, but critical questions remain regarding which age groups should be vaccinated to maximize population impact. METHODS: We developed a deterministic, age-structured compartmental model of norovirus transmission and immunity in the U.S. POPULATION: The model was fit to age-specific monthly U.S. hospitalizations between 1996 and 2007. We simulated mass immunization of both pediatric and elderly populations assuming realistic coverages of 90% and 65%, respectively. We considered two mechanism of vaccine action, resulting in lower vaccine efficacy (lVE) between 22% and 43% and higher VE (hVE) of 50%. RESULTS: Pediatric vaccination was predicted to avert 33% (95% CI: 27%, 40%) and 60% (95% CI: 49%, 71%) of norovirus episodes among children under five years for lVE and hVE, respectively. Vaccinating the elderly averted 17% (95% CI: 12%, 20%) and 38% (95% CI: 34%, 42%) of cases in 65+ year olds for lVE and hVE, respectively. At a population level, pediatric vaccination was predicted to avert 18-21 times more cases and twice as many deaths per vaccinee compared to elderly vaccination. CONCLUSIONS: The potential benefits are likely greater for a pediatric program, both via direct protection of vaccinated children and indirect protection of unvaccinated individuals, including adults and the elderly. These findings argue for a clinical development plan that will deliver a vaccine with a safety and efficacy profile suitable for use in children.

Mathematical models of Ebola-Consequences of underlying assumptions.

Mathematical models have been used to study Ebola disease transmission dynamics and control for the recent epidemics in West Africa. Many of the models used in these studies are based on the model of Legrand et al. (2007), and most failed to accurately project the outbreak's course (Butler, 2014). Although there could be many reasons for this, including incomplete and unreliable data on Ebola epidemiology and lack of empirical data on how disease-control measures quantitatively affect Ebola transmission, we examine the underlying assumptions of the Legrand model, and provide alternate formulations that are simpler and provide additional information regarding the epidemiology of Ebola during an outbreak. We developed three models with different assumptions about disease stage durations, one of which simplifies to the Legrand model while the others have more realistic distributions. Control and basic reproduction numbers for all three models are derived and shown to provide threshold conditions for outbreak control and prevention.

The effect of heterogeneity in uptake of the measles, mumps, and rubella vaccine on the potential for outbreaks of measles: a modelling study.

BACKGROUND: 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. METHODS: 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. FINDINGS: 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. INTERPRETATION: 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. FUNDING: None.

COMPUTATION OF ℛ IN AGE-STRUCTURED EPIDEMIOLOGICAL MODELS WITH MATERNAL AND TEMPORARY IMMUNITY.

For infectious diseases such as pertussis, susceptibility is determined by immunity, which is chronological age-dependent. We consider an age-structured epidemiological model that accounts for both passively acquired maternal antibodies that decay and active immunity that wanes, permitting reinfection. The model is a 6-dimensional system of partial differential equations (PDE). By assuming constant rates within each age-group, the PDE system can be reduced to an ordinary differential equation (ODE) system with aging from one age-group to the next. We derive formulae for the effective reproduction number ℛ and provide their biological interpretation in some special cases. We show that the disease-free equilibrium is stable when ℛ < 1 and unstable if ℛ > 1.

An elaboration of theory about preventing outbreaks in homogeneous populations to include heterogeneity or preferential mixing.

The goal of many vaccination programs is to attain the population immunity above which pathogens introduced by infectious people (e.g., travelers from endemic areas) will not cause outbreaks. Using a simple meta-population model, we demonstrate that, if sub-populations either differ in characteristics affecting their basic reproduction numbers or if their members mix preferentially, weighted average sub-population immunities cannot be compared with the proportionally-mixing homogeneous population-immunity threshold, as public health practitioners are wont to do. Then we review the effect of heterogeneity in average per capita contact rates on the basic meta-population reproduction number. To the extent that population density affects contacts, for example, rates might differ in urban and rural sub-populations. Other differences among sub-populations in characteristics affecting their basic reproduction numbers would contribute similarly. In agreement with more recent results, we show that heterogeneous preferential mixing among sub-populations increases the basic meta-population reproduction number more than homogeneous preferential mixing does. Next we refine earlier results on the effects of heterogeneity in sub-population immunities and preferential mixing on the effective meta-population reproduction number. Finally, we propose the vector of partial derivatives of this reproduction number with respect to the sub-population immunities as a fundamentally new tool for targeting vaccination efforts.

Modeling the effect of school closures in a pandemic scenario: exploring two different contact matrices.

BACKGROUND: School closures may delay the epidemic peak of the next influenza pandemic, but whether school closure can delay the peak until pandemic vaccine is ready to be deployed is uncertain. METHODS: To study the effect of school closures on the timing of epidemic peaks, we built a deterministic susceptible-infected-recovered model of influenza transmission. We stratified the U.S. population into 4 age groups (0-4, 5-19, 20-64, and ≥ 65 years), and used contact matrices to model the average number of potentially disease transmitting, nonphysical contacts. RESULTS: For every week of school closure at day 5 of introduction and a 30% clinical attack rate scenario, epidemic peak would be delayed by approximately 5 days. For a 15% clinical attack rate scenario, 1 week closure would delay the peak by 9 days. Closing schools for less than 84 days (12 weeks) would not, however, reduce the estimated total number of cases. CONCLUSIONS: Unless vaccine is available early, school closure alone may not be able to delay the peak until vaccine is ready to be deployed. Conversely, if vaccination begins quickly, school closure may be helpful in providing the time to vaccinate school-aged children before the pandemic peaks.