Control of transmission with two types of infection

The effectiveness of a vaccination strategy to control transmission of an infectious disease depends on the way vaccine doses are distributed to individuals in a community of households. Here we show that this dependence is more complicated when acquisition and severity of illness are determined by the size of the infecting dose, as is thought to be the case for measles and varicella. Two alternative formulations for the way vaccination changes an individual's susceptibility and infectivity show that vaccination coverage, the nature of the vaccine response and the distribution of household size also have a big impact on which strategy is more effective. These judgements are made by comparing the post-vaccination reproduction numbers corresponding to different vaccination strategies.     

On vaccine efficacy and reproduction numbers

We consider the impact of a vaccination programme on the transmission potential of the infection in large populations. We define a measure of vaccine efficacy against transmission which combines the possibly random effect of the vaccine on individual susceptibility and infectiousness. This definition extends some previous work in this area to arbitrarily heterogeneous populations with one level of mixing, but leads us to question the usefulness of the concept of vaccine efficacy against infectiousness. We derive relationships between vaccine efficacy against transmission, vaccine coverage and reproduction numbers, which generalize existing results. In particular we show that the projected reproduction number RV does not depend on the details of the vaccine model, only on its overall effect on transmission. Explicit expressions for RV and the basic reproduction number R0 are obtained in a variety of settings. We define a measure of projected effectiveness of a vaccination programme PE=1-(RV/R0) and investigate its relationship with efficacy against transmission and vaccine coverage. We also study the effective reproduction number Re(t) at time t. Monitoring Re(t) over time is an important aspect of programme surveillance. Programme effectiveness PE is less sensitive than RV or the critical vaccination threshold to model assumptions. On the other hand Re(t) depends on the details of the vaccine model.     

Estimating individual and household reproduction numbers in an emerging epidemic

Reproduction numbers, defined as averages of the number of people infected by a typical case, play a central role in tracking infectious disease outbreaks. The aim of this paper is to develop methods for estimating reproduction numbers which are simple enough that they could be applied with limited data or in real time during an outbreak. I present a new estimator for the individual reproduction number, which describes the state of the epidemic at a point in time rather than tracking individuals over time, and discuss some potential benefits. Then, to capture more of the detail that micro-simulations have shown is important in outbreak dynamics, I analyse a model of transmission within and between households, and develop a method to estimate the household reproduction number, defined as the number of households infected by each infected household. This method is validated by numerical simulations of the spread of influenza and measles using historical data, and estimates are obtained for would-be emerging epidemics of these viruses. I argue that the household reproduction number is useful in assessing the impact of measures that target the household for isolation, quarantine, vaccination or prophylactic treatment, and measures such as social distancing and school or workplace closures which limit between-household transmission, all of which play a key role in current thinking on future infectious disease mitigation.     

An intuitive formulation for the reproductive number for the spread of diseases in heterogeneous populations

The thresholds for mathematical epidemiology models specify the critical conditions for an epidemic to grow or die out. The reproductive number can provide significant insight into the transmission dynamics of a disease and can guide strategies to control its spread. We define the mean number of contacts, the mean duration of infection, and the mean transmission probability appropriately for certain epidemiological models, and construct a simplified formulation of the reproductive number as the product of these quantities. When the spread of the epidemic depends strongly upon the heterogeneity of the populations, the epidemiological models must account for this heterogeneity, and the expressions for the reproductive number become correspondingly more complex. We formulate several models with different heterogeneous structures and demonstrate how to define the mean quantities for an explicit expression for the reproductive number. In complex heterogeneous models, it seems necessary to define the reproductive number for each structured subgroup or cohort and then use the average of these reproductive numbers weighted by their heterogeneity to estimate the reproductive number for the total population.     

Mathematical analysis of a cholera model with public health interventions

Cholera, an acute gastro-intestinal infection and a waterborne disease continues to emerge in developing countries and remains an important global health challenge. We formulate a mathematical model that captures some essential dynamics of cholera transmission to study the impact of public health educational campaigns, vaccination and treatment as control strategies in curtailing the disease. The education-induced, vaccination-induced and treatment-induced reproductive numbers R(E), R(V), R(T) respectively and the combined reproductive number R(C) are compared with the basic reproduction number R(0) to assess the possible community benefits of these control measures. A Lyapunov functional approach is also used to analyse the stability of the equilibrium points. We perform sensitivity analysis on the key parameters that drive the disease dynamics in order to determine their relative importance to disease transmission and prevalence. Graphical representations are provided to qualitatively support the analytical results.     

Modelling the proportion of influenza infections within households during pandemic and non-pandemic years

Background

The key epidemiological difference between pandemic and seasonal influenza is that the population is largely susceptible during a pandemic, whereas, during non-pandemic seasons a level of immunity exists. The population-level efficacy of household-based mitigation strategies depends on the proportion of infections that occur within households. In general, mitigation measures such as isolation and quarantine are more effective at the population level if the proportion of household transmission is low.

Methods/Results

We calculated the proportion of infections within households during pandemic years compared with non-pandemic years using a deterministic model of household transmission in which all combinations of household size and individual infection states were enumerated explicitly. We found that the proportion of infections that occur within households was only partially influenced by the hazard h of infection within household relative to the hazard of infection outside the household, especially for small basic reproductive numbers. During pandemics, the number of within-household infections was lower than one might expect for a given because many of the susceptible individuals were infected from the community and the number of susceptible individuals within household was thus depleted rapidly. In addition, we found that for the value of at which 30% of infections occur within households during non-pandemic years, a similar 31% of infections occur within households during pandemic years.

Interpretation

We suggest that a trade off between the community force of infection and the number of susceptible individuals in a household explains an apparent invariance in the proportion of infections that occur in households in our model. During a pandemic, although there are more susceptible individuals in a household, the community force of infection is very high. However, during non-pandemic years, the force of infection is much lower but there are fewer susceptible individuals within the household.     

Estimating vaccine effects on transmission of infection from household outbreak data

This article is concerned with a method for making inferences about various measures of vaccine efficacy. These measures describe reductions in susceptibility and in the potential to transmit infection. The method uses data on household outbreaks; it is based on a model that allows for transmission of infection both from within a household and from the outside. The use of household data is motivated by the hope that these are informative about vaccine-induced reduction of the potential to transmit infection, as household outbreaks contain some information about the possible source of infection. For illustration, the method is applied to observed data on household outbreaks of smallpox. These data are of the form needed and the number of households is of a size that can be managed in a vaccine trial. It is found that vaccine effects, such as the mean reduction in susceptibility and the mean reduction in the potential to infect others, per infectious contact, can be estimated with precision. However, a more specific parameter reflecting the reduction in infectivity for individuals partially responding to vaccination is not estimated well in the application. An evaluation of the method using artificial data shows that this parameter can be estimated with greater precision when we have outbreak data on a large number of small households.     

Epidemic growth rate and household reproduction number in communities of households,schools and workplaces

In this paper we present a novel and coherent modelling framework for the characterisation of the real-time growth rate in SIR models of epidemic spread in populations with social structures of increasing complexity. Known results about homogeneous mixing and multitype models are included in the framework, which is then extended to models with households and models with households and schools/workplaces. Efficient methods for the exact computation of the real-time growth rate are presented for the standard SIR model with constant infection and recovery rates (Markovian case). Approximate methods are described for a large class of models with time-varying infection rates (non-Markovian case). The quality of the approximation is assessed via comparison with results from individual-based stochastic simulations. The methodology is then applied to the case of influenza in models with households and schools/workplaces, to provide an estimate of a household-to-household reproduction number and thus asses the effort required to prevent an outbreak by targeting control policies at the level of households. The results highlight the risk of underestimating such effort when the additional presence of schools/workplaces is neglected. Our framework increases the applicability of models of epidemic spread in socially structured population by linking earlier theoretical results, mainly focused on time-independent key epidemiological parameters (e.g. reproduction numbers, critical vaccination coverage, epidemic final size) to new results on the epidemic dynamics.     

Dynamics Analysis of a Multi-strain Cholera Model with an Imperfect Vaccine

A new two-strain model, for assessing the impact of basic control measures, treatment and dose-structured mass vaccination on cholera transmission dynamics in a population, is designed. The model has a globally-asymptotically stable disease-free equilibrium whenever its associated reproduction number is less than unity. The model has a unique, and locally-asymptotically stable, endemic equilibrium when the threshold quantity exceeds unity and another condition holds. Numerical simulations of the model show that, with the expected 50 % minimum efficacy of the first vaccine dose, vaccinating 55 % of the susceptible population with the first vaccine dose will be sufficient to effectively control the spread of cholera in the community. Such effective control can also be achieved if 50 % of the first vaccine dose recipients take the second dose. It is shown that a control strategy that emphasizes the use of antibiotic treatment is more effective than one that emphasizes the use of basic (non-pharmaceutical) anti-cholera control measures only. Numerical simulations show that, while the universal strategy (involving all three control measures) gives the best outcome in minimizing cholera burden in the community, the combined basic anti-cholera control measures and treatment strategy also has very effective community-wide impact.     

Real-time investigation of measles epidemics with estimate of vaccine efficacy

As part of measles elimination effort, evaluation of the vaccination program and real-time assessment of the epidemic dynamics constitute two important tasks to improve and strengthen the control. The present study aimed to develop an epidemiological modeling method which can be applied to estimating the vaccine efficacy at an individual level while conducting the timely investigation of the epidemic. The multivariate renewal process model was employed to describe the temporal evolution of infection by vaccination history, jointly estimating the time-dependent reproduction number and the vaccine efficacy. Analyzing the enhanced surveillance data of measles in Aichi prefecture, Japan from 2007-08, the vaccine efficacy was estimated at 96.7% (95% confidence interval: 95.8, 97.4). Using an age structured model, the vaccine efficacy among those aged from 5-19 years was shown to be smaller than that among those from 0-4 years. The age-dependent vaccine efficacy estimate informs the age-groups to be targeted for revaccination. Because the estimation method can rest on readily available epidemiological data, the proposed model has a potential to be integrated with routine surveillance.     

Efficiency of estimating vaccine efficacy for susceptibility and infectiousness: randomization by individual versus household

In designing vaccine efficacy studies based on the secondary attack rate (SAR) or transmission probability in which both vaccine efficacy for susceptibility, VE(S), and vaccine efficacy for infectiousness, VE(I), are estimated, the allocation of vaccine and placebo within transmission units has an important influence on the efficiency of the study. We compared the following randomization schemes that result in different allocations of vaccine and placebo within two-member households: (1) randomization by individual for a mixed allocation, (2) randomization by transmission unit for concordant allocation, and (3) randomization of only one individual in each transmission unit to either vaccine or placebo. There is a complex interaction among the VE(S), VE(I), and the SAR that determines which allocation of vaccine and placebo within households provides the most information. In general, individual randomization with a mixed allocation of vaccine and placebo is better for estimating both VE(S) and VE(I) than is randomizing by household. However, for estimation of VE(I), at very low SARs and low VE(S), randomization by household is slightly more efficient than randomization by individual.     

Estimating the immunity coverage required to prevent epidemics in a community of households

An estimation of the immunity coverage needed to prevent future outbreaks of an infectious disease is considered for a community of households. Data on outbreak size in a sample of households from one epidemic are used to derive maximum likelihood estimates and confidence bounds for parameters of a stochastic model for disease transmission in a community of households. These parameter estimates induce estimates and confidence bounds for the basic reproduction number and the critical immunity coverage, which are the parameters of main interest when aiming at preventing major outbreaks in the future. The case when individuals are homogeneous, apart from the size of their household, is considered in detail. The generalization to the case with variable infectivity, susceptibility and/or mixing behaviour is discussed more briefly. The methods are illustrated with an application to data on influenza in Tecumseh, Michigan.     

Estimation of effective reproduction numbers for infectious diseases using serological survey data

The effective reproduction number of an infection, denoted Re, may be used to monitor the impact of a vaccination programme. If Re is maintained below 1, then sustained endemic transmission of the infection cannot occur. In this paper we discuss methods for estimating Re from serological survey data, allowing for age and individual heterogeneity. We describe semi-parametric and parametric models, and obtain an upper bound on Re when vaccine coverage and efficacy are not known. The methods are illustrated using data on mumps and rubella in England and Wales.     

Social Contact Networks and Disease Eradicability under Voluntary Vaccination

Certain theories suggest that it should be difficult or impossible to eradicate a vaccine-preventable disease under voluntary vaccination: Herd immunity implies that the individual incentive to vaccinate disappears at high coverage levels. Historically, there have been examples of declining coverage for vaccines, such as MMR vaccine and whole-cell pertussis vaccine, that are consistent with this theory. On the other hand, smallpox was globally eradicated by 1980 despite voluntary vaccination policies in many jurisdictions. Previous modeling studies of the interplay between disease dynamics and individual vaccinating behavior have assumed that infection is transmitted in a homogeneously mixing population. By comparison, here we simulate transmission of a vaccine-preventable SEIR infection through a random, static contact network. Individuals choose whether to vaccinate based on infection risks from neighbors, and based on vaccine risks. When neighborhood size is small, rational vaccinating behavior results in rapid containment of the infection through voluntary ring vaccination. As neighborhood size increases (while the average force of infection is held constant), a threshold is reached beyond which the infection can break through partially vaccinated rings, percolating through the whole population and resulting in considerable epidemic final sizes and a large number vaccinated. The former outcome represents convergence between individually and socially optimal outcomes, whereas the latter represents their divergence, as observed in most models of individual vaccinating behavior that assume homogeneous mixing. Similar effects are observed in an extended model using smallpox-specific natural history and transmissibility assumptions. This work illustrates the significant qualitative differences between behavior–infection dynamics in discrete contact-structured populations versus continuous unstructured populations. This work also shows how disease eradicability in populations where voluntary vaccination is the primary control mechanism may depend partly on whether the disease is transmissible only to a few close social contacts or to a larger subset of the population.     

Optimal Control and Sensitivity Analysis of an Influenza Model with Treatment and Vaccination

We formulate and analyze the dynamics of an influenza pandemic model with vaccination and treatment using two preventive scenarios: increase and decrease in vaccine uptake. Due to the seasonality of the influenza pandemic, the dynamics is studied in a finite time interval. We focus primarily on controlling the disease with a possible minimal cost and side effects using control theory which is therefore applied via the Pontryagin’s maximum principle, and it is observed that full treatment effort should be given while increasing vaccination at the onset of the outbreak. Next, sensitivity analysis and simulations (using the fourth order Runge-Kutta scheme) are carried out in order to determine the relative importance of different factors responsible for disease transmission and prevalence. The most sensitive parameter of the various reproductive numbers apart from the death rate is the inflow rate, while the proportion of new recruits and the vaccine efficacy are the most sensitive parameters for the endemic equilibrium point.     

Epidemiological impact of vaccination on the dynamics of two childhood diseases in rural Senegal

Measles and pertussis are ubiquitous vaccine-preventable diseases, which remain an important public health problem in developing countries. Hence, developing a deep understanding of their transmission dynamics remains imperative. To achieve this, we compared the impact of vaccination at both individual and population levels in a Senegalese rural community. This study represents the first such comparative study in tropical conditions and constitutes a point of comparison with other studies of disease dynamics in developed countries. Changes in the transmission rates of infections are reflected in their mean ages at infection and basic reproductive ratio calculated before and after vaccination. We explored persistence of both infections in relation to population size in each village and found the inter-epidemic period for the whole area using wavelets analysis. As predicted by epidemiological theory, we observed an increase in the mean age at infection and a decrease in the reproductive ratio of both diseases. We showed for both the pre- vaccination and vaccine eras that persistence depends on population size. After vaccination, persistence decreased and the inter-epidemic period increased. The observed changes suggest that vaccination against measles and pertussis induced a drop in their transmission. Similarities in disease dynamics to those of temperate regions such as England and Wales were also observed.     

Estimating the impact of vaccination on acute simian-human immunodeficiency virus/simian immunodeficiency virus infections

The dynamics of HIV infection have been studied in humans and in a variety of animal models. The standard model of infection has been used to estimate the basic reproductive ratio of the virus, calculated from the growth rate of virus in acute infection. This method has not been useful in studying the effects of vaccination, since, for the vaccines developed so far, early growth rates of virus do not differ between control and vaccinated animals. Here, we use the standard model of viral dynamics to derive the reproductive ratio from the peak viral load and nadir of target cell numbers in acute infection. We apply this method to data from studies of vaccination in SHIV and SIV infection and demonstrate that vaccination can reduce the reproductive ratio by 2.3- and 2-fold, respectively. This method allows the comparison of vaccination efficacies among different viral strains and animal models in vivo.     

Analysis of rabies in China: transmission dynamics and control

Human rabies is one of the major public-health problems in China. The number of human rabies cases has increased dramatically in the last 15 years, partially due to the poor understanding of the transmission dynamics of rabies and the lack of effective control measures of the disease. In this article, in order to explore effective control and prevention measures we propose a deterministic model to study the transmission dynamics of rabies in China. The model consists of susceptible, exposed, infectious, and recovered subpopulations of both dogs and humans and describes the spread of rabies among dogs and from infectious dogs to humans. The model simulations agree with the human rabies data reported by the Chinese Ministry of Health. We estimate that the basic reproduction number for the rabies transmission in China and predict that the number of the human rabies is decreasing but may reach another peak around 2030. We also perform some sensitivity analysis of in terms of the model parameters and compare the effects of culling and immunization of dogs. Our study demonstrates that (i) reducing dog birth rate and increasing dog immunization coverage rate are the most effective methods for controlling rabies in China; and (ii) large scale culling of susceptible dogs can be replaced by immunization of them.     

Key strategies for reducing spread of avian influenza among commercial poultry holdings: lessons for transmission to humans

Recent avian flu epidemics (A/H5N1) in Southeast Asia and case reports from around the world have led to fears of a human pandemic. Control of these outbreaks in birds would probably lead to reduced transmission of the avian virus to humans. This study presents a mathematical model based on stochastic farm-to-farm transmission that incorporates flock size and spatial contacts to evaluate the impact of control strategies. Fit to data from the recent epidemic in the Netherlands, we evaluate the efficacy of control strategies and forecast avian influenza dynamics. Our results identify high-risk areas of spread by mapping of the farm level reproductive number. Results suggest that an immediate depopulation of infected flocks following an accurate and quick diagnosis would have a greater impact than simply depopulating surrounding flocks. Understanding the relative importance of different control measures is essential for response planning.     

The Interaction between Seasonality and Pulsed Interventions against Malaria in Their Effects on the Reproduction Number

The basic reproduction number (R ₀) is an important quantity summarising the dynamics of an infectious disease, as it quantifies how much effort is needed to control transmission. The relative change in R ₀ due to an intervention is referred to as the effect size. However malaria and other diseases are often highly seasonal and some interventions have time-varying effects, meaning that simple reproduction number formulae cannot be used. Methods have recently been developed for calculating R ₀ for diseases with seasonally varying transmission. I extend those methods to calculate the effect size of repeated rounds of mass drug administration, indoor residual spraying and other interventions against Plasmodium falciparum malaria in seasonal settings in Africa. I show that if an intervention reduces transmission from one host to another by a constant factor, then its effect size is the same in a seasonal as in a non-seasonal setting. The optimal time of year for drug administration is in the low season, whereas the best time for indoor residual spraying or a vaccine which reduces infection rates is just before the high season. In general, the impact of time-varying interventions increases with increasing seasonality, if carried out at the optimal time of year. The effect of combinations of interventions that act at different stages of the transmission cycle is roughly the product of the separate effects. However for individual time-varying interventions, it is necessary to use methods such as those developed here rather than inserting the average efficacy into a simple formula.