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.     

Coalescent inference for infectious disease: meta-analysis of hepatitis C

Genetic analysis of pathogen genomes is a powerful approach to investigating the population dynamics and epidemic history of infectious diseases. However, the theoretical underpinnings of the most widely used, coalescent methods have been questioned, casting doubt on their interpretation. The aim of this study is to develop robust population genetic inference for compartmental models in epidemiology. Using a general approach based on the theory of metapopulations, we derive coalescent models under susceptible–infectious (SI), susceptible–infectious–susceptible (SIS) and susceptible–infectious–recovered (SIR) dynamics. We show that exponential and logistic growth models are equivalent to SI and SIS models, respectively, when co-infection is negligible. Implementing SI, SIS and SIR models in BEAST, we conduct a meta-analysis of hepatitis C epidemics, and show that we can directly estimate the basic reproductive number (R₀) and prevalence under SIR dynamics. We find that differences in genetic diversity between epidemics can be explained by differences in underlying epidemiology (age of the epidemic and local population density) and viral subtype. Model comparison reveals SIR dynamics in three globally restricted epidemics, but most are better fit by the simpler SI dynamics. In summary, metapopulation models provide a general and practical framework for integrating epidemiology and population genetics for the purposes of joint inference.     

Stability analysis and optimal vaccination of an SIR epidemic model

Almost all mathematical models of diseases start from the same basic premise: the population can be subdivided into a set of distinct classes dependent upon experience with respect to the relevant disease. Most of these models classify individuals as either a susceptible individual S, infected individual I or recovered individual R. This is called the susceptible-infected-recovered (SIR) model. In this paper, we describe an SIR epidemic model with three components; S, I and R. We describe our study of stability analysis theory to find the equilibria for the model. Next in order to achieve control of the disease, we consider a control problem relative to the SIR model. A percentage of the susceptible populations is vaccinated in this model. We show that an optimal control exists for the control problem and describe numerical simulations using the Runge-Kutta fourth order procedure. Finally, we describe a real example showing the efficiency of this optimal control.     

An SIR epidemic model with partial temporary immunity modeled with delay

The SIR epidemic model for disease dynamics considers recovered individuals to be permanently immune, while the SIS epidemic model considers recovered individuals to be immediately resusceptible. We study the case of temporary immunity in an SIR-based model with delayed coupling between the susceptible and removed classes, which results in a coupled set of delay differential equations. We find conditions for which the endemic steady state becomes unstable to periodic outbreaks. We then use analytical and numerical bifurcation analysis to describe how the severity and period of the outbreaks depend on the model parameters.      

A Simulation Study Comparing Epidemic Dynamics on Exponential Random Graph and Edge-Triangle Configuration Type Contact Network Models

We compare two broad types of empirically grounded random network models in terms of their abilities to capture both network features and simulated Susceptible-Infected-Recovered (SIR) epidemic dynamics. The types of network models are exponential random graph models (ERGMs) and extensions of the configuration model. We use three kinds of empirical contact networks, chosen to provide both variety and realistic patterns of human contact: a highly clustered network, a bipartite network and a snowball sampled network of a “hidden population”. In the case of the snowball sampled network we present a novel method for fitting an edge-triangle model. In our results, ERGMs consistently capture clustering as well or better than configuration-type models, but the latter models better capture the node degree distribution. Despite the additional computational requirements to fit ERGMs to empirical networks, the use of ERGMs provides only a slight improvement in the ability of the models to recreate epidemic features of the empirical network in simulated SIR epidemics. Generally, SIR epidemic results from using configuration-type models fall between those from a random network model (i.e., an Erdős-Rényi model) and an ERGM. The addition of subgraphs of size four to edge-triangle type models does improve agreement with the empirical network for smaller densities in clustered networks. Additional subgraphs do not make a noticeable difference in our example, although we would expect the ability to model cliques to be helpful for contact networks exhibiting household structure.     

Theory of early warning signals of disease emergenceand leading indicators of elimination

Anticipating infectious disease emergence and documenting progress in disease elimination are important applications for the theory of critical transitions. A key problem is the development of theory relating the dynamical processes of transmission to observable phenomena. In this paper, we consider compartmental susceptible–infectious–susceptible (SIS) and susceptible–infectious–recovered (SIR) models that are slowly forced through a critical transition. We derive expressions for the behavior of several candidate indicators, including the autocorrelation coefficient, variance, coefficient of variation, and power spectra of SIS and SIR epidemics during the approach to emergence or elimination. We validated these expressions using individual-based simulations. We further showed that moving-window estimates of these quantities may be used for anticipating critical transitions in infectious disease systems. Although leading indicators of elimination were highly predictive, we found the approach to emergence to be much more difficult to detect. It is hoped that these results, which show the anticipation of critical transitions in infectious disease systems to be theoretically possible, may be used to guide the construction of online algorithms for processing surveillance data.     

Effective degree network disease models

An effective degree approach to modeling the spread of infectious diseases on a network is introduced and applied to a disease that confers no immunity (a Susceptible-Infectious-Susceptible model, abbreviated as SIS) and to a disease that confers permanent immunity (a Susceptible-Infectious-Recovered model, abbreviated as SIR). Each model is formulated as a large system of ordinary differential equations that keeps track of the number of susceptible and infectious neighbors of an individual. From numerical simulations, these effective degree models are found to be in excellent agreement with the corresponding stochastic processes of the network on a random graph, in that they capture the initial exponential growth rates, the endemic equilibrium of an invading disease for the SIS model, and the epidemic peak for the SIR model. For each of these effective degree models, a formula for the disease threshold condition is derived. The threshold parameter for the SIS model is shown to be larger than that derived from percolation theory for a model with the same disease and network parameters, and consequently a disease may be able to invade with lower transmission than predicted by percolation theory. For the SIR model, the threshold condition is equal to that predicted by percolation theory. Thus unlike the classical homogeneous mixing disease models, the SIS and SIR effective degree models have different disease threshold conditions.     

A Class of Pairwise Models for Epidemic Dynamics on Weighted Networks

In this paper, we study the SIS (susceptible–infected–susceptible) and SIR (susceptible–infected–removed) epidemic models on undirected, weighted networks by deriving pairwise-type approximate models coupled with individual-based network simulation. Two different types of theoretical/synthetic weighted network models are considered. Both start from non-weighted networks with fixed topology followed by the allocation of link weights in either (i) random or (ii) fixed/deterministic way. The pairwise models are formulated for a general discrete distribution of weights, and these models are then used in conjunction with stochastic network simulations to evaluate the impact of different weight distributions on epidemic thresholds and dynamics in general. For the SIR model, the basic reproductive ratio R ₀ is computed, and we show that (i) for both network models R ₀ is maximised if all weights are equal, and (ii) when the two models are ‘equally-matched’, the networks with a random weight distribution give rise to a higher R ₀ value. The models with different weight distributions are also used to explore the agreement between the pairwise and simulation models for different parameter combinations.     

Efficient Bayesian Sample Size Calculation for Designing a Clinical Trial with Multi‐Cluster Outcome Data

Health care utilization and outcome studies call for hierarchical approaches. The objectives were to predict major complications following percutaneous coronary interventions by health providers, and to compare Bayesian and non‐Bayesian sample size calculation methods. The hierarchical data structure consisted of: (1) Strata: PGY4, PGY7, and physician assistant as providers with varied experiences; (2) Clusters: k_s providers per stratum; (3) Individuals: n_s patients reviewed by each provider. The main outcome event illustrated was mortality modeled by a Bayesian beta‐binomial model. Pilot information and assumptions were utilized to elicit beta prior distributions. Sample size calculations were based on the approximated average length, fixed at 1%, of 95% posterior intervals of the mean event rate parameter. Necessary sample sizes by both non‐Bayesian and Bayesian methods were compared. We demonstrated that the developed Bayesian methods can be efficient and may require fewer subjects to satisfy the same length criterion.     

Additive‐Multiplicative Regression Models for Spatio‐Temporal Epidemics

An extension of the stochastic susceptible–infectious–recovered (SIR) model is proposed in order to accommodate a regression context for modelling infectious disease data. The proposal is based on a multivariate counting process specified by conditional intensities, which contain an additive epidemic component and a multiplicative endemic component. This allows the analysis of endemic infectious diseases by quantifying risk factors for infection by external sources in addition to infective contacts. Inference can be performed by considering the full likelihood of the stochastic process with additional parameter restrictions to ensure non‐negative conditional intensities. Simulation from the model can be performed by Ogata's modified thinning algorithm. As an illustrative example, we analyse data provided by the Federal Research Centre for Virus Diseases of Animals, Wusterhausen, Germany, on the incidence of the classical swine fever virus in Germany during 1993–2004.     

Robust uniform persistence and competitive exclusion in a nonautonomous multi-strain SIR epidemic model with disease-induced mortality

A nonautonomous version of the SIR epidemic model in Ackleh and Allen (2003) is considered, for competition of $n$ infection strains in a host population. The model assumes total cross immunity, mass action incidence, density-dependent host mortality and disease-induced mortality. Sufficient conditions for the robust uniform persistence of the total population, as well as of the susceptible and infected subpopulations, are given. The first two forms of persistence depend entirely on the rate at which the population grows from the extinction state, respectively the rate at which the disease is vertically transmitted to offspring. We also discuss the competitive exclusion among the $n$ infection strains, namely when a single infection strain survives and all the others go extinct. Numerical simulations are also presented, to account for the situations not covered by the analytical results. These simulations suggest that the nonautonomous nature of the model combined with the disease induced mortality allow for many strains to coexist. The theoretical approach developed here is general enough to apply to other nonautonomous epidemic models.     

Optimal treatment of an SIR epidemic model with time delay

In this paper the optimal control strategies of an SIR (susceptible–infected–recovered) epidemic model with time delay are introduced. In order to do this, we consider an optimally controlled SIR epidemic model with time delay where a control means treatment for infectious hosts. We use optimal control approach to minimize the probability that the infected individuals spread and to maximize the total number of susceptible and recovered individuals. We first derive the basic reproduction number and investigate the dynamical behavior of the controlled SIR epidemic model. We also show the existence of an optimal control for the control system and present numerical simulations on real data regarding the course of Ebola virus in Congo. Our results indicate that a small contact rate(probability of infection) is suitable for eradication of the disease (Ebola virus) and this is one way of optimal treatment strategies for infectious hosts.     

A discrete-time model with vaccination for a measles epidemic.

A discrete-time, age-independent SIR-type epidemic model is formulated and analyzed. The effects of vaccination are also included in the model. Three mathematically important properties are verified for the model: solutions are nonnegative, the population size is time-invariant, and the epidemic concludes with all individuals either remaining susceptible or becoming immune (a property typical of SIR models). The model is applied to a measles epidemic on a university campus. The simulated results are in good agreement with the actual data if it is assumed that the population mixes nonhomogeneously. The results of the simulations indicate that a rate of immunity greater than 98% may be required to prevent an epidemic in a university population. The model has applications to other contagious diseases of SIR type. Furthermore, the simulated results of the model can easily be compared to data, and the effects of a vaccination program can be examined.     

How does the resistance threshold in spatially explicit epidemic dynamics depend on the basic reproductive ratio and spatial correlation of crop genotypes?

We examined the fraction of resistant cultivars necessary to prevent a global pathogen outbreak (the resistance threshold) using a spatially explicit epidemiological model (SIR model) in a finite, two-dimensional, lattice-structured host population. Infectious diseases in our model could be transmitted to susceptible nearest-neighbour sites, and the infected site either recovered or died after an exponentially distributed infectious period. Threshold behaviour of this spatially explicit SIR model cannot be reduced to that of bond percolation, as was previously noted in the literature, unless extreme assumptions (synchronized infection events with a fixed lag) are imposed on infection process. The resistance threshold is significantly lower than that of conventional mean-field epidemic models, and is even lower if the spatial configuration of resistant and susceptible crops are negatively correlated. Finite size scaling applied to the resistance threshold for a finite basic reproductive ratio ρ of pathogen reveals that its difference from static percolation threshold (0.41) is inversely proportional to ρ. Our formula for the basic reproductive ratio dependency of the resistance threshold produced an estimate for the critical basic reproductive ratio (4.7) in a universally susceptible population, which is much larger than the corresponding critical value (1) in the mean-field model and nearly three times larger than the critical growth rate of a basic contact process (SIS model). Pair approximation reveals that the resistance threshold for preventing a global epidemic is factor 1/(1−η) greater with spatially correlated planting than with random planting, where η is initial correlation in host genotypes between nearest-neighbour sites. Thus the eradication is harder with a positive spatial correlation (η>0) in mixed susceptible/resistant plantings, and is easier with a negative correlation (η<0). The effect of finite field size (L), which corresponded to the mean distance between sources of infections, is given by the increased resistance threshold (by the amount L^−0.75) from its infinite size limit. Implications of these results on effective planting strategies in multi-line control plans are discussed.     

Model hierarchies in edge-based compartmental modeling for infectious disease spread

We consider the family of edge-based compartmental models for epidemic spread developed in Miller et al. (J R Soc Interface 9(70):890–906, 2012). These models allow for a range of complex behaviors, and in particular allow us to explicitly incorporate duration of a contact into our mathematical models. Our focus here is to identify conditions under which simpler models may be substituted for more detailed models, and in so doing we define a hierarchy of epidemic models. In particular we provide conditions under which it is appropriate to use the standard mass action SIR model, and we show what happens when these conditions fail. Using our hierarchy, we provide a procedure leading to the choice of the appropriate model for a given population. Our result about the convergence of models to the mass action model gives clear, rigorous conditions under which the mass action model is accurate.     

Predicting and Evaluating the Epidemic Trend of Ebola Virus Disease in the 2014-2015 Outbreak and the Effects of Intervention Measures

We constructed dynamic Ebola virus disease (EVD) transmission models to predict epidemic trends and evaluate intervention measure efficacy following the 2014 EVD epidemic in West Africa. We estimated the effective vaccination rate for the population, with basic reproduction number (R₀) as the intermediate variable. Periodic EVD fluctuation was analyzed by solving a Jacobian matrix of differential equations based on a SIR (susceptible, infective, and removed) model. A comprehensive compartment model was constructed to fit and predict EVD transmission patterns, and to evaluate the effects of control and prevention measures. Effective EVD vaccination rates were estimated to be 42% (31–50%), 45% (42–48%), and 51% (44–56%) among susceptible individuals in Guinea, Liberia and Sierra Leone, respectively. In the absence of control measures, there would be rapid mortality in these three countries, and an EVD epidemic would be likely recur in 2035, and then again 8~9 years later. Oscillation intervals would shorten and outbreak severity would decrease until the periodicity reached ~5.3 years. Measures that reduced the spread of EVD included: early diagnosis, treatment in isolation, isolating/monitoring close contacts, timely corpse removal, post-recovery condom use, and preventing or quarantining imported cases. EVD may re-emerge within two decades without control and prevention measures. Mass vaccination campaigns and control and prevention measures should be instituted to prevent future EVD epidemics.     

A generalized model of social and biological contagion

We present a model of contagion that unifies and generalizes existing models of the spread of social influences and microorganismal infections. Our model incorporates individual memory of exposure to a contagious entity (e.g. a rumor or disease), variable magnitudes of exposure (dose sizes), and heterogeneity in the susceptibility of individuals. Through analysis and simulation, we examine in detail the case where individuals may recover from an infection and then immediately become susceptible again (analogous to the so-called SIS model). We identify three basic classes of contagion models which we call epidemic threshold, vanishing critical mass, and critical mass classes, where each class of models corresponds to different strategies for prevention or facilitation. We find that the conditions for a particular contagion model to belong to one of the these three classes depend only on memory length and the probabilities of being infected by one and two exposures, respectively. These parameters are in principle measurable for real contagious influences or entities, thus yielding empirical implications for our model. We also study the case where individuals attain permanent immunity once recovered, finding that epidemics inevitably die out but may be surprisingly persistent when individuals possess memory.     

Generation interval contraction and epidemic data analysis

The generation interval is the time between the infection time of an infected person and the infection time of his or her infector. Probability density functions for generation intervals have been an important input for epidemic models and epidemic data analysis. In this paper, we specify a general stochastic SIR epidemic model and prove that the mean generation interval decreases when susceptible persons are at risk of infectious contact from multiple sources. The intuition behind this is that when a susceptible person has multiple potential infectors, there is a "race" to infect him or her in which only the first infectious contact leads to infection. In an epidemic, the mean generation interval contracts as the prevalence of infection increases. We call this global competition among potential infectors. When there is rapid transmission within clusters of contacts, generation interval contraction can be caused by a high local prevalence of infection even when the global prevalence is low. We call this local competition among potential infectors. Using simulations, we illustrate both types of competition. Finally, we show that hazards of infectious contact can be used instead of generation intervals to estimate the time course of the effective reproductive number in an epidemic. This approach leads naturally to partial likelihoods for epidemic data that are very similar to those that arise in survival analysis, opening a promising avenue of methodological research in infectious disease epidemiology.     

Transmission Dynamics of Highly Pathogenic Avian Influenza at Lake Constance (Europe) During the Outbreak of Winter 2005–2006

Highly pathogenic avian influenza virus (HPAI) H5N1 poses a serious threat to domestic animals. Despite the large number of studies on influenza A virus in waterbirds, little is still known about the transmission dynamics, including prevalence, behavior, and spread of these viruses in the wild waterbird population. From January to April 2006, the HPAI H5N1 virus was confirmed in 82 dead wild waterbirds at the shores of Lake Constance. In this study, we present simple mathematical models to examine this outbreak and to investigate the transmission dynamics of HPAI in wild waterbirds. The population dynamics model of wintering birds was best represented by a sinusoidal function. This model was considered the most adequate to represent the susceptible compartment of the SIR model. The three transmission models predict a basic reproduction ratio (R ₀) with value of approximately 1.6, indicating a small epidemic, which ended with the migration of susceptible wild waterbirds at the end of the winter. With this study, we quantify for the first time the transmission of HPAI H5N1 virus at Lake Constance during the outbreak of winter 2005–2006. It is a step toward the improvement of the knowledge of transmission of the virus among wild waterbirds.     

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.