Bimodal Epidemic Size Distributions for Near-Critical SIR with Vaccination

We introduce a recursive algorithm which enables the computation of the distribution of epidemic size in a stochastic SIR model for very large population sizes. In the important parameter region where the model is just slightly supercritical, the distribution of epidemic size is decidedly bimodal. We find close agreement between the distribution for large populations and the limiting case where the distribution is that of the time a Brownian motion hits a quadratic curve. The model includes the possibility of vaccination during the epidemic. The effects of the parameters, including vaccination level, on the form of the epidemic size distribution are explored.     

Pulse vaccination strategy in the SIR epidemic model

Theoretical results show that the measles ‘pulse’ vaccination strategy can be distinguished from the conventional strategies in leading to disease eradication at relatively low values of vaccination. Using the SIR epidemic model we showed that under a planned pulse vaccination regime the system converges to a stable solution with the number of infectious individuals equal to zero. We showed that pulse vaccination leads to epidemics eradication if certain conditions regarding the magnitude of vaccination proportion and on the period of the pulses are adhered to. Our theoretical results are confirmed by numerical simulations. The introduction of seasonal variation into the basic SIR model leads to periodic and chaotic dynamics of epidemics. We showed that under seasonal variation, in spite of the complex dynamics of the system, pulse vaccination still leads to epidemic eradication. We derived the conditions for epidemic eradication under various constraints and showed their dependence on the parameters of the epidemic. We compared effectiveness and cost of constant, pulse and mixed vaccination policies.     

Network-based analysis of stochastic SIR epidemic models with random and proportionate mixing

In this paper, we outline the theory of epidemic percolation networks and their use in the analysis of stochastic susceptible-infectious-removed (SIR) epidemic models on undirected contact networks. We then show how the same theory can be used to analyze stochastic SIR models with random and proportionate mixing. The epidemic percolation networks for these models are purely directed because undirected edges disappear in the limit of a large population. In a series of simulations, we show that epidemic percolation networks accurately predict the mean outbreak size and probability and final size of an epidemic for a variety of epidemic models in homogeneous and heterogeneous populations. Finally, we show that epidemic percolation networks can be used to re-derive classical results from several different areas of infectious disease epidemiology. In an Appendix, we show that an epidemic percolation network can be defined for any time-homogeneous stochastic SIR model in a closed population and prove that the distribution of outbreak sizes given the infection of any given node in the SIR model is identical to the distribution of its out-component sizes in the corresponding probability space of epidemic percolation networks. We conclude that the theory of percolation on semi-directed networks provides a very general framework for the analysis of stochastic SIR models in closed populations.     

Multiple Lattice Model for Influenza Spreading

Behavioral differences among age classes, together with the natural tendency of individuals to prefer contacts with individuals of similar age, naturally point to the existence of a community structure in the population network, in which each community can be identified with a different age class. Data on age-dependent contact patterns also reveal how relevant is the role of the population age structure in shaping the spreading of an infectious disease. In the present paper we propose a simple model for epidemic spreading, in which a contact network with an intrinsic community structure is coupled with a simple stochastic SIR model for the epidemic spreading. The population is divided in 4 different age-communities and hosted on a multiple lattice, each community occupying a specific age-lattice. Individuals are allowed to move freely to nearest neighbor empty sites on the age-lattice. Different communities are connected with each other by means of inter-lattices edges, with a different number of external links connecting different age class populations. The parameters of the contact network model are fixed by requiring the simulated data to fully reproduce the contact patterns matrices of the Polymod survey. The paper shows that adopting a topology which better implements the age-class community structure of the population, one gets a better agreement between experimental contact patterns and simulated data, and this also improves the accordance between simulated and experimental data on the epidemic spreading.     

Equilibria of an Epidemic Game with Piecewise Linear Social Distancing Cost

Around the world, infectious disease epidemics continue to threaten people’s health. When epidemics strike, we often respond by changing our behaviors to reduce our risk of infection. This response is sometimes called “social distancing.” Since behavior changes can be costly, we would like to know the optimal social distancing behavior. But the benefits of changes in behavior depend on the course of the epidemic, which itself depends on our behaviors. Differential population game theory provides a method for resolving this circular dependence. Here, I present the analysis of a special case of the differential SIR epidemic population game with social distancing when the relative infection rate is linear, but bounded below by zero. Equilibrium solutions are constructed in closed-form for an open-ended epidemic. Constructions are also provided for epidemics that are stopped by the deployment of a vaccination that becomes available a fixed-time after the start of the epidemic. This can be used to anticipate a window of opportunity during which mass vaccination can significantly reduce the cost of an epidemic.     

Networks, epidemics and vaccination through contact tracing

We consider a (social) network whose structure can be represented by a simple random graph having a pre-specified degree distribution. A Markovian susceptible-infectious-removed (SIR) epidemic model is defined on such a social graph. We then consider two real-time vaccination models for contact tracing during the early stages of an epidemic outbreak. The first model considers vaccination of each friend of an infectious individual (once identified) independently with probability ρ. The second model is related to the first model but also sets a bound on the maximum number an infectious individual can infect before being identified. Expressions are derived for the influence on the reproduction number of these vaccination models. We give some numerical examples and simulation results based on the Poisson and heavy-tail degree distributions where it is shown that the second vaccination model has a bigger advantage compared to the first model for the heavy-tail degree distribution.     

Determination of optimal vaccination strategies using an orbital stability threshold from periodically driven systems

We analyse a periodically driven SIR epidemic model for childhood related diseases, where the contact rate and vaccination rate parameters are considered periodic. The aim is to define optimal vaccination strategies for control of childhood related infections. Stability analysis of the uninfected solution is the tool for setting up the control function. The optimal solutions are sought within a set of susceptible population profiles. Our analysis reveals that periodic vaccination strategy hardly contributes to the stability of the uninfected solution if the human residence time (life span) is much larger than the contact rate period. However, if the human residence time and the contact rate periods match, we observe some positive effect of periodic vaccination. Such a vaccination strategy would be useful in the developing world, where human life spans are shorter, or basically in the case of vaccination of livestock or small animals whose life-spans are relatively shorter.     

Optimizing Real-Time Vaccine Allocation in a Stochastic SIR Model

Real-time vaccination following an outbreak can effectively mitigate the damage caused by an infectious disease. However, in many cases, available resources are insufficient to vaccinate the entire at-risk population, logistics result in delayed vaccine deployment, and the interaction between members of different cities facilitates a wide spatial spread of infection. Limited vaccine, time delays, and interaction (or coupling) of cities lead to tradeoffs that impact the overall magnitude of the epidemic. These tradeoffs mandate investigation of optimal strategies that minimize the severity of the epidemic by prioritizing allocation of vaccine to specific subpopulations. We use an SIR model to describe the disease dynamics of an epidemic which breaks out in one city and spreads to another. We solve a master equation to determine the resulting probability distribution of the final epidemic size. We then identify tradeoffs between vaccine, time delay, and coupling, and we determine the optimal vaccination protocols resulting from these tradeoffs.     

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.     

Imperfect vaccine aggravates the long-standing dilemma of voluntary vaccination

Achieving widespread population immunity by voluntary vaccination poses a major challenge for public health administration and practice. The situation is complicated even more by imperfect vaccines. How the vaccine efficacy affects individuals' vaccination behavior has yet to be fully answered. To address this issue, we combine a simple yet effective game theoretic model of vaccination behavior with an epidemiological process. Our analysis shows that, in a population of self-interested individuals, there exists an overshooting of vaccine uptake levels as the effectiveness of vaccination increases. Moreover, when the basic reproductive number, R0, exceeds a certain threshold, all individuals opt for vaccination for an intermediate region of vaccine efficacy. We further show that increasing effectiveness of vaccination always increases the number of effectively vaccinated individuals and therefore attenuates the epidemic strain. The results suggest that 'number is traded for efficiency': although increases in vaccination effectiveness lead to uptake drops due to free-riding effects, the impact of the epidemic can be better mitigated.     

Network epidemic models with two levels of mixing

The study of epidemics on social networks has attracted considerable attention recently. In this paper, we consider a stochastic SIR (susceptible-->infective-->removed) model for the spread of an epidemic on a finite network, having an arbitrary but specified degree distribution, in which individuals also make casual contacts, i.e. with people chosen uniformly from the population. The behaviour of the model as the network size tends to infinity is investigated. In particular, the basic reproduction number R(0), that governs whether or not an epidemic with few initial infectives can become established is determined, as are the probability that an epidemic becomes established and the proportion of the population who are ultimately infected by such an epidemic. For the case when the infectious period is constant and all individuals in the network have the same degree, the asymptotic variance and a central limit theorem for the size of an epidemic that becomes established are obtained. Letting the rate at which individuals make casual contacts decrease to zero yields, heuristically, corresponding results for the model without casual contacts, i.e. for the standard SIR network epidemic model. A deterministic model that approximates the spread of an epidemic that becomes established in a large population is also derived. The theory is illustrated by numerical studies, which demonstrate that the asymptotic approximations work well, even for only moderately sized networks, and that the degree distribution and the inclusion of casual contacts can each have a major impact on the outcome of an epidemic.     

Optimal control of epidemics with limited resources

We extend the existing work on the time-optimal control of the basic SIR epidemic model with mass action contact rate. Previous results have focused on minimizing an objective function that is a linear combination of the cost associated with using control and either the outbreak size or the infectious burden. We instead, provide analytic solutions for the control that minimizes the outbreak size (or infectious burden) under the assumption that there are limited control resources. We provide optimal control policies for an isolation only model, a vaccination only model and a combined isolation–vaccination model (or mixed model). The optimal policies described here contain many interesting features especially when compared to previous analyses. For example, under certain circumstances the optimal isolation only policy is not unique. Furthermore the optimal mixed policy is not simply a combination of the optimal isolation only policy and the optimal vaccination only policy. The results presented here also highlight a number of areas that warrant further study and emphasize that time-optimal control of the basic SIR model is still not fully understood.     

Inferring Epidemiological Dynamics with Bayesian Coalescent Inference: The Merits of Deterministic and Stochastic Models

Estimation of epidemiological and population parameters from molecular sequence data has become central to the understanding of infectious disease dynamics. Various models have been proposed to infer details of the dynamics that describe epidemic progression. These include inference approaches derived from Kingman’s coalescent theory. Here, we use recently described coalescent theory for epidemic dynamics to develop stochastic and deterministic coalescent susceptible–infected–removed (SIR) tree priors. We implement these in a Bayesian phylogenetic inference framework to permit joint estimation of SIR epidemic parameters and the sample genealogy. We assess the performance of the two coalescent models and also juxtapose results obtained with a recently published birth–death-sampling model for epidemic inference. Comparisons are made by analyzing sets of genealogies simulated under precisely known epidemiological parameters. Additionally, we analyze influenza A (H1N1) sequence data sampled in the Canterbury region of New Zealand and HIV-1 sequence data obtained from known United Kingdom infection clusters. We show that both coalescent SIR models are effective at estimating epidemiological parameters from data with large fundamental reproductive number R₀ and large population size S₀. Furthermore, we find that the stochastic variant generally outperforms its deterministic counterpart in terms of error, bias, and highest posterior density coverage, particularly for smaller R₀ and S₀. However, each of these inference models is shown to have undesirable properties in certain circumstances, especially for epidemic outbreaks with R₀ close to one or with small effective susceptible populations.     

Multi-species SIR models from a dynamical Bayesian perspective

Multi-species compartment epidemic models, such as the multi-species susceptible–infectious–recovered (SIR) model, are extensions of the classic SIR models, which are used to explore the transient dynamics of pathogens that infect multiple hosts in a large population. In this article, we propose a dynamical Bayesian hierarchical SIR (HSIR) model, to capture the stochastic or random nature of an epidemic process in a multi-species SIR (with recovered becoming susceptible again) dynamical setting, under hidden mass balance constraints. We call this a Bayesian hierarchical multi-species SIR (MSIR_B) model. Different from a classic multi-species SIR model (which we call MSIR_C), our approach imposes mass balance on the underlying true counts rather than, improperly, on the noisy observations. Moreover, the MSIR_B model can capture the discrete nature of, as well as uncertainties in, the epidemic process.     

Household Epidemics: Modelling Effects of Early Stage Vaccination

A Markovian susceptible → infectious → removed (SIR) epidemic model is considered in a community partitioned into households. A vaccination strategy, which is implemented during the early stages of the disease following the detection of infected individuals is proposed. In this strategy, the detection occurs while an individual is infectious and other susceptible household members are vaccinated without further delay. Expressions are derived for the influence on the reproduction numbers of this vaccination strategy for equal and unequal household sizes. We fit previously estimated parameters from influenza and use household distributions for Sweden and Tanzania census data. The results show that the reproduction number is much higher in Tanzania (6 compared with 2) due to larger households, and that infected individuals have to be detected (and household members vaccinated) after on average 5 days in Sweden and after 3.3 days in Tanzania, a much smaller difference.     

The structural identifiability of susceptible–infective–recovered type epidemic models with incomplete immunity and birth targeted vaccination

This paper considers the implications of a structural identifiability analysis on a series of fundamental three-compartment epidemic model structures, derived around the general SIR (susceptible–infective–recovered) framework. The models represent various forms of incomplete immunity acquired through natural infection, or from administration of a birth targeted vaccination programme. It is shown that the addition of a vaccination campaign has a negative effect on the structural identifiability of all considered models. In particular, the actual proportion of vaccination coverage achieved, an essential parameter, cannot be uniquely estimated from even ideal prevalence data.     

Optimal Vaccination Policies for an SIR Model with Limited Resources

The purpose of the paper is to use analytical method and optimization tool to suggest a vaccination program intensity for a basic SIR epidemic model with limited resources for vaccination. We show that there are two different scenarios for optimal vaccination strategies, and obtain analytical solutions for the optimal control problem that minimizes the total cost of disease under the assumption of daily vaccine supply being limited. These solutions and their corresponding optimal control policies are derived explicitly in terms of initial conditions, model parameters and resources for vaccination. With sufficient resources, the optimal control strategy is the normal Bang–Bang control. However, with limited resources, the optimal control strategy requires to switch to time-variant vaccination.     

Stochastic modeling of nonlinear epidemiology

The objectives of this paper to analyse, model and simulate the spread of an infectious disease by resorting to modern stochastic algorithms. The approach renders it possible to circumvent the simplifying assumption of linearity imposed in the majority of the past works on stochastic analysis of epidemic processes. Infectious diseases are often transmitted through contacts of those infected with those susceptible; hence the processes are inherently nonlinear. According to the classical model of Kermack and McKendrick, or the SIR model, three classes of populations are involved in two types of processes: conversion of susceptibles (S) to infectives (I) and conversion of infectives to removed (R). The master equations of the SIR process have been formulated through the probabilistic population balance around a particular state by considering the mutually exclusive events. The efficacy of the present methodology is mainly attributable to its ability to derive the governing equations for the means, variances and covariance of the random variables by the method of system-size expansion of the nonlinear master equations. Solving these equations simultaneously along with rates associated influenza epidemic data yields information concerning not only the means of the three populations but also the minimal uncertainties of these populations inherent in the epidemic. The stochastic pathways of the three different classes of populations during an epidemic, i.e. their means and the fluctuations around these means, have also been numerically simulated independently by the algorithm derived from the master equations, as well as by an event-driven Monte Carlo algorithm. The master equation and Monte Carlo algorithms have given rise to the identical results.     

具有脉冲出生和脉冲接种的SIR传染病模型

考虑了脉冲出生、脉冲接种、垂直传染、因病死亡等因素,建立了脉冲出生和脉冲接种同时进行的SIR传染病模型,通过分析无病周期解的存在性以及稳定性,得出疾病灭绝的条件.     

Multigeneration Reproduction Ratios and the Effects of Clustered Unvaccinated Individuals on Epidemic Outbreak

An SIR epidemiological community-structured model is constructed to investigate the effects of clustered distributions of unvaccinated individuals and the distribution of the primary case relative to vaccination levels. The communities here represent groups such as neighborhoods within a city or cities within a region. The model contains two levels of mixing, where individuals make more intra-group than inter-group contacts. Stochastic simulations and analytical results are utilized to explore the model. An extension of the effective reproduction ratio that incorporates more spatial information by predicting the average number of tertiary infections caused by a single infected individual is introduced to characterize the system. Using these methods, we show that both the vaccination coverage and the variation in vaccination levels among communities affect the likelihood and severity of epidemics. The location of the primary infectious case and the degree of mixing between communities are also important factors in determining the dynamics of outbreaks. In some cases, increasing the efficacy of a vaccine can in fact increase the effective reproduction ratio in early generations, due to the effects of population structure on the likely initial location of an infection.