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.     

Estimating the basic reproductive number from viral sequence data

Epidemiological processes leave a fingerprint in the pattern of genetic structure of virus populations. Here, we provide a new method to infer epidemiological parameters directly from viral sequence data. The method is based on phylogenetic analysis using a birth-death model (BDM) rather than the commonly used coalescent as the model for the epidemiological transmission of the pathogen. Using the BDM has the advantage that transmission and death rates are estimated independently and therefore enables for the first time the estimation of the basic reproductive number of the pathogen using only sequence data, without further assumptions like the average duration of infection. We apply the method to genetic data of the HIV-1 epidemic in Switzerland.     

Coexistence of multiple pathogen strains in stochastic epidemic models with density-dependent mortality

Stochastic differential equations that model an SIS epidemic with multiple pathogen strains are derived from a system of ordinary differential equations. The stochastic model assumes there is demographic variability. The dynamics of the deterministic model are summarized. Then the dynamics of the stochastic model are compared to the deterministic model. In the deterministic model, there can be either disease extinction, competitive exclusion, where only one strain persists, or coexistence, where more than one strain persists. In the stochastic model, all strains are eventually eliminated because the disease-free state is an absorbing state. However, if the population size and the initial number of infected individuals are sufficiently large, it may take a long time until all strains are eliminated. Numerical simulations of the stochastic model show that coexistence cases predicted by the deterministic model are an unlikely occurrence in the stochastic model even for short time periods. In the stochastic model, either disease extinction or competitive exclusion occur. The initial number of infected individuals, the basic reproduction numbers, and other epidemiological parameters are important determinants of the dominant strain in the stochastic epidemic model.     

Inferring epidemiological parameters on the basis of allele frequencies

In this article, I develop a methodology for inferring the transmission rate and reproductive value of an epidemic on the basis of genotype data from a sample of infected hosts. The epidemic is modeled by a birth-death process describing the transmission dynamics in combination with an infinite-allele model describing the evolution of alleles. I provide a recursive formulation for the probability of the allele frequencies in a sample of hosts and a Bayesian framework for estimating transmission rates and reproductive values on the basis of observed allele frequencies. Using the Bayesian method, I reanalyze tuberculosis data from the United States. I estimate a net transmission rate of 0.19/year [0.13, 0.24] and a reproductive value of 1.02 [1.01, 1.04]. I demonstrate that the allele frequency probability under the birth-death model does not follow the well-known Ewens' sampling formula that holds under Kingman's coalescent.     

Uncovering epidemiological dynamics in heterogeneous host populations using phylogenetic methods

Host population structure has a major influence on epidemiological dynamics. However, in particular for sexually transmitted diseases, quantitative data on population contact structure are hard to obtain. Here, we introduce a new method that quantifies host population structure based on phylogenetic trees, which are obtained from pathogen genetic sequence data. Our method is based on a maximum-likelihood framework and uses a multi-type branching process, under which each host is assigned to a type (subpopulation). In a simulation study, we show that our method produces accurate parameter estimates for phylogenetic trees in which each tip is assigned to a type, as well for phylogenetic trees in which the type of the tip is unknown. We apply the method to a Latvian HIV-1 dataset, quantifying the impact of the intravenous drug user epidemic on the heterosexual epidemic (known tip states), and identifying superspreader dynamics within the men-having-sex-with-men epidemic (unknown tip states).     

Likelihood Inference of Non-Constant Diversification Rates with Incomplete Taxon Sampling

Large-scale phylogenies provide a valuable source to study background diversification rates and investigate if the rates have changed over time. Unfortunately most large-scale, dated phylogenies are sparsely sampled (fewer than 5% of the described species) and taxon sampling is not uniform. Instead, taxa are frequently sampled to obtain at least one representative per subgroup (e.g. family) and thus to maximize diversity (diversified sampling). So far, such complications have been ignored, potentially biasing the conclusions that have been reached. In this study I derive the likelihood of a birth-death process with non-constant (time-dependent) diversification rates and diversified taxon sampling. Using simulations I test if the true parameters and the sampling method can be recovered when the trees are small or medium sized (fewer than 200 taxa). The results show that the diversification rates can be inferred and the estimates are unbiased for large trees but are biased for small trees (fewer than 50 taxa). Furthermore, model selection by means of Akaike''s Information Criterion favors the true model if the true rates differ sufficiently from alternative models (e.g. the birth-death model is recovered if the extinction rate is large and compared to a pure-birth model). Finally, I applied six different diversification rate models – ranging from a constant-rate pure birth process to a decreasing speciation rate birth-death process but excluding any rate shift models – on three large-scale empirical phylogenies (ants, mammals and snakes with respectively 149, 164 and 41 sampled species). All three phylogenies were constructed by diversified taxon sampling, as stated by the authors. However only the snake phylogeny supported diversified taxon sampling. Moreover, a parametric bootstrap test revealed that none of the tested models provided a good fit to the observed data. The model assumptions, such as homogeneous rates across species or no rate shifts, appear to be violated.     

How well can the exponential-growth coalescent approximate constant-rate birth–death population dynamics?

One of the central objectives in the field of phylodynamics is the quantification of population dynamic processes using genetic sequence data or in some cases phenotypic data. Phylodynamics has been successfully applied to many different processes, such as the spread of infectious diseases, within-host evolution of a pathogen, macroevolution and even language evolution. Phylodynamic analysis requires a probability distribution on phylogenetic trees spanned by the genetic data. Because such a probability distribution is not available for many common stochastic population dynamic processes, coalescent-based approximations assuming deterministic population size changes are widely employed. Key to many population dynamic models, in particular epidemiological models, is a period of exponential population growth during the initial phase. Here, we show that the coalescent does not well approximate stochastic exponential population growth, which is typically modelled by a birth–death process. We demonstrate that introducing demographic stochasticity into the population size function of the coalescent improves the approximation for values of R₀ close to 1, but substantial differences remain for large R₀. In addition, the computational advantage of using an approximation over exact models vanishes when introducing such demographic stochasticity. These results highlight that we need to increase efforts to develop phylodynamic tools that correctly account for the stochasticity of population dynamic models for inference.     

A maximum pseudo-likelihood approach for estimating species trees under the coalescent model

Background  

Several phylogenetic approaches have been developed to estimate species trees from collections of gene trees. However, maximum likelihood approaches for estimating species trees under the coalescent model are limited. Although the likelihood of a species tree under the multispecies coalescent model has already been derived by Rannala and Yang, it can be shown that the maximum likelihood estimate (MLE) of the species tree (topology, branch lengths, and population sizes) from gene trees under this formula does not exist. In this paper, we develop a pseudo-likelihood function of the species tree to obtain maximum pseudo-likelihood estimates (MPE) of species trees, with branch lengths of the species tree in coalescent units.     

Inference of past population expansion from the timing of coalescence events in a gene genealogy

We investigate the expected coalescent in populations growing exponentially. The distribution of expected times to coalescence events may show a linear relationship with a number of ancestral lineages, when the latter is subjected to the "epidemic transformation". However, in a number of viral populations, upward curves are created when the epidemically transformed number of ancestral lineages is plotted against time. We consider possible causes of such upward curves. These include the possibility that a curved line is created through a transformation failure due to a sample size that is too large. We suggest a new formula for predicting such failure. The second cause is a population size increasing at an accelerating rate. However, the combination of recent coalescent events and an upward curve is created by an accelerating population increase only under restricted conditions. Specifically, such a pattern is expected only when, were population growth not to have accelerated, the transformation would have failed anyway. The third cause of nonlinearity arises in the estimated coalescent, as distinct from the real coalescent, if the mutation rate is small. However, coalescence times estimated from data typically give a straight line following epidemic transformation, but the rate of exponential increase, or r value, will be underestimated.     

Consequences of recombination on traditional phylogenetic analysis

We investigate the shape of a phylogenetic tree reconstructed from sequences evolving under the coalescent with recombination. The motivation is that evolutionary inferences are often made from phylogenetic trees reconstructed from population data even though recombination may well occur (mtDNA or viral sequences) or does occur (nuclear sequences). We investigate the size and direction of biases when a single tree is reconstructed ignoring recombination. Standard software (PHYLIP) was used to construct the best phylogenetic tree from sequences simulated under the coalescent with recombination. With recombination present, the length of terminal branches and the total branch length are larger, and the time to the most recent common ancestor smaller, than for a tree reconstructed from sequences evolving with no recombination. The effects are pronounced even for small levels of recombination that may not be immediately detectable in a data set. The phylogenies when recombination is present superficially resemble phylogenies for sequences from an exponentially growing population. However, exponential growth has a different effect on statistics such as Tajima's D. Furthermore, ignoring recombination leads to a large overestimation of the substitution rate heterogeneity and the loss of the molecular clock. These results are discussed in relation to viral and mtDNA data sets.     

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.     

Maximum tree: a consistent estimator of the species tree

We propose a model based approach to use multiple gene trees to estimate the species tree. The coalescent process requires that gene divergences occur earlier than species divergences when there is any polymorphism in the ancestral species. Under this scenario, speciation times are restricted to be smaller than the corresponding gene split times. The maximum tree (MT) is the tree with the largest possible speciation times in the space of species trees restricted by available gene trees. If all populations have the same population size, the MT is the maximum likelihood estimate of the species tree. It can be shown the MT is a consistent estimator of the species tree even when the MT is built upon the estimates of the true gene trees if the gene tree estimates are statistically consistent. The MT converges in probability to the true species tree at an exponential rate.     

Bayesian Inference of Sampled Ancestor Trees for Epidemiology and Fossil Calibration

Phylogenetic analyses which include fossils or molecular sequences that are sampled through time require models that allow one sample to be a direct ancestor of another sample. As previously available phylogenetic inference tools assume that all samples are tips, they do not allow for this possibility. We have developed and implemented a Bayesian Markov Chain Monte Carlo (MCMC) algorithm to infer what we call sampled ancestor trees, that is, trees in which sampled individuals can be direct ancestors of other sampled individuals. We use a family of birth-death models where individuals may remain in the tree process after sampling, in particular we extend the birth-death skyline model [Stadler et al., 2013] to sampled ancestor trees. This method allows the detection of sampled ancestors as well as estimation of the probability that an individual will be removed from the process when it is sampled. We show that even if sampled ancestors are not of specific interest in an analysis, failing to account for them leads to significant bias in parameter estimates. We also show that sampled ancestor birth-death models where every sample comes from a different time point are non-identifiable and thus require one parameter to be known in order to infer other parameters. We apply our phylogenetic inference accounting for sampled ancestors to epidemiological data, where the possibility of sampled ancestors enables us to identify individuals that infected other individuals after being sampled and to infer fundamental epidemiological parameters. We also apply the method to infer divergence times and diversification rates when fossils are included along with extant species samples, so that fossilisation events are modelled as a part of the tree branching process. Such modelling has many advantages as argued in the literature. The sampler is available as an open-source BEAST2 package (https://github.com/CompEvol/sampled-ancestors).     

Phylodynamics of Infectious Disease Epidemics

We present a formalism for unifying the inference of population size from genetic sequences and mathematical models of infectious disease in populations. Virus phylogenies have been used in many recent studies to infer properties of epidemics. These approaches rely on coalescent models that may not be appropriate for infectious diseases. We account for phylogenetic patterns of viruses in susceptible–infected (SI), susceptible–infected–susceptible (SIS), and susceptible–infected–recovered (SIR) models of infectious disease, and our approach may be a viable alternative to demographic models used to reconstruct epidemic dynamics. The method allows epidemiological parameters, such as the reproductive number, to be estimated directly from viral sequence data. We also describe patterns of phylogenetic clustering that are often construed as arising from a short chain of transmissions. Our model reproduces the moments of the distribution of phylogenetic cluster sizes and may therefore serve as a null hypothesis for cluster sizes under simple epidemiological models. We examine a small cross-sectional sample of human immunodeficiency (HIV)-1 sequences collected in the United States and compare our results to standard estimates of effective population size. Estimated prevalence is consistent with estimates of effective population size and the known history of the HIV epidemic. While our model accurately estimates prevalence during exponential growth, we find that periods of decline are harder to identify.COALESCENT theory has found wide applications for inference of viral phylogenies (Nee et al. 1996; Rosenberg and Nordborg 2002; Drummond et al. 2005) and estimation of epidemic prevalence (Yusim et al. 2001; Robbins et al. 2003; Wilson et al. 2005), yet there have been few attempts to formally integrate coalescent theory with standard epidemiological models (Pybus et al. 2001; Goodreau 2006). While epidemiological models such as susceptible–infected–recovered (SIR) consider the dynamics of an entire population going forward in time, the coalescent theory operates on a small sample of an infected subpopulation and models the merging of lineages backward in time until a common ancestor has been reached. The original coalescent theory was based on a population of constant size with discrete generations (Kingman 1982a,b). Numerous extensions have been made for populations with overlapping generations in continuous time, exponential or logistic growth (Griffiths and Tavare 1994), and stochastically varying size (Kaj and Krone 2003). However, infectious disease epidemics are a special case of a variable size population, often characterized by early explosive growth followed by decline that leads to extinction or an endemic steady state.If superinfection is rare and if mutation is fast relative to epidemic growth, each lineage in a phylogenetic tree corresponds to a single infected individual with its own unique viral population. An infection event viewed in reverse time is equivalent to the coalescence of two lineages and every transmission of the virus between hosts can generate a new branch in the phylogeny of consensus viral isolates from infected individuals. Recently diverged sequences should represent transmissions in the recent past, and branches close to the root of a tree should represent transmissions from long ago. Consequently, branching patterns provide information about the frequency of transmissions over time (Wilson et al. 2005). The correspondence between transmission and phylogenetic branching is easiest to detect for viruses such as human immunodeficiency virus (HIV) and hepatitis C virus that have a high mutation rate relative to dispersal. Underlying SIR dynamics also apply to other pathogens, although in some cases it may be more difficult to reconstruct the transmission history.We examined the properties of viral phylogenies generated by the most common epidemiological models based on ordinary differential equations (ODEs). We are able to fit epidemiological models to a reconstructed phylogeny for sampled viral sequence data and make inferences regarding the size of the corresponding infected population. Our solution takes the form of an ODE analogous to those used to track epidemic prevalence and thereby provides a convenient link between commonly used epidemiological models and phylodynamics. Virtually all coalescent theory to date has been expressed in terms of integer-valued stochastic processes. Our motivation for using differential equations to describe the coalescent process is a desire to formalize a link with standard epidemiological models that are also expressed in terms of differential equations.We use our method to calculate the distribution of coalescent times for samples of viral sequences, fit SIR models to a viral phylogeny, and calculate median time to the most recent common ancestor (MRCA) of the sample. Our method also provides equations that describe the time evolution of the cluster size distribution (CSD)—the distribution of the number of descendants of a lineage over time. Clusters of related virus are often interpreted as epidemiologically linked. For example, clusters of acute HIV infections may represent short transmission chains between high-risk individuals (Yerly et al. 2001; Hue et al. 2005; Pao et al. 2005; Goodreau 2006; Brenner et al. 2007; Drumright and Frost 2008; Lewis et al. 2008). Because our model reproduces the moments of the cluster size distribution, it can be used to predict the level of clustering as a function of epidemiological conditions. The moments could be directly compared to empirical values or they could be used to reconstruct the entire CSD, whereupon standard statistical tests could be used for comparing distributions.Although our equations describe the macroscopic properties of the population distribution of cluster sizes, we generalize our method to the case of a small cross-sectional sample of sequences. This allows us to develop a likelihood-based approach to fitting SIR models to observed sequences.By considering variable degrees of incidence and the size of the infected population, our solution sheds light on the relationship between coalescent rates and epidemic dynamics. Coalescent rates are low near peak prevalence, but higher when there is a large ratio of incidence to prevalence. This can occur early on, when the epidemic is entering its expansion phase, as well as late if the epidemic has multiple periods of growth.     

Phylodynamic Inference for Structured Epidemiological Models

Coalescent theory is routinely used to estimate past population dynamics and demographic parameters from genealogies. While early work in coalescent theory only considered simple demographic models, advances in theory have allowed for increasingly complex demographic scenarios to be considered. The success of this approach has lead to coalescent-based inference methods being applied to populations with rapidly changing population dynamics, including pathogens like RNA viruses. However, fitting epidemiological models to genealogies via coalescent models remains a challenging task, because pathogen populations often exhibit complex, nonlinear dynamics and are structured by multiple factors. Moreover, it often becomes necessary to consider stochastic variation in population dynamics when fitting such complex models to real data. Using recently developed structured coalescent models that accommodate complex population dynamics and population structure, we develop a statistical framework for fitting stochastic epidemiological models to genealogies. By combining particle filtering methods with Bayesian Markov chain Monte Carlo methods, we are able to fit a wide class of stochastic, nonlinear epidemiological models with different forms of population structure to genealogies. We demonstrate our framework using two structured epidemiological models: a model with disease progression between multiple stages of infection and a two-population model reflecting spatial structure. We apply the multi-stage model to HIV genealogies and show that the proposed method can be used to estimate the stage-specific transmission rates and prevalence of HIV. Finally, using the two-population model we explore how much information about population structure is contained in genealogies and what sample sizes are necessary to reliably infer parameters like migration rates.     

Combining field epidemiological information and genetic data to comprehensively reconstruct the invasion history and the microevolution of the sudden oak death agent Phytophthora ramorum (Stramenopila: Oomycetes) in California

Understanding the migration patterns of invasive organisms is of paramount importance to predict and prevent their further spread. Previous attempts at reconstructing the entire history of the sudden oak death (SOD) epidemic in California were limited by: (1) incomplete sampling; (2) the inability to include infestations caused by a single genotype of the pathogen; (3) collapsing of non-spatially contiguous yet genetically similar samples into large meta-samples that confounded the coalescent analyses. Here, we employ an intensive sampling coverage of 832 isolates of Phytopthora ramorum (the causative agent of SOD) from 60 California forests, genotyped at nine microsatellite loci, to reconstruct its invasion. By using age of infestation as a constraint on coalescent analyses, by dividing genetically indistinguishable meta-populations into highly-resolved sets of spatially contiguous populations, and by using Bruvo genetic distances for most analyses, we reconstruct the entire history of the epidemic and convincingly show infected nursery plants are the original source for the entire California epidemic. Results indicate that multiple human-mediated introductions occurred in most counties and that further disease sources were represented by large wild infestations. The study also identifies minor introductions, some of them relatively recent, linked to infected ornamental plants. Finally, using archival isolates collected soon after the discovery of the pathogen in California, we corroborate that the epidemic is likely to have resulted form 3 to 4 core founder individuals evolved from a single genotype. This is probably the most complete reconstruction ever completed for an invasion by an exotic forest pathogen, and the approach here described may be useful for the reconstruction of invasions by any clonally reproducing organism with a relatively limited natural dispersal range.     

Balancing Detection and Eradication for Control of Epidemics: Sudden Oak Death in Mixed-Species Stands

Culling of infected individuals is a widely used measure for the control of several plant and animal pathogens but culling first requires detection of often cryptically-infected hosts. In this paper, we address the problem of how to allocate resources between detection and culling when the budget for disease management is limited. The results are generic but we motivate the problem for the control of a botanical epidemic in a natural ecosystem: sudden oak death in mixed evergreen forests in coastal California, in which species composition is generally dominated by a spreader species (bay laurel) and a second host species (coast live oak) that is an epidemiological dead-end in that it does not transmit infection but which is frequently a target for preservation. Using a combination of an epidemiological model for two host species with a common pathogen together with optimal control theory we address the problem of how to balance the allocation of resources for detection and epidemic control in order to preserve both host species in the ecosystem. Contrary to simple expectations our results show that an intermediate level of detection is optimal. Low levels of detection, characteristic of low effort expended on searching and detection of diseased trees, and high detection levels, exemplified by the deployment of large amounts of resources to identify diseased trees, fail to bring the epidemic under control. Importantly, we show that a slight change in the balance between the resources allocated to detection and those allocated to control may lead to drastic inefficiencies in control strategies. The results hold when quarantine is introduced to reduce the ingress of infected material into the region of interest.     

Gene tree quality affects empirical coalescent branch length estimation

Assessing effects of gene tree error in coalescent analyses have widely ignored coalescent branch lengths (CBLs) despite their potential utility in estimating ancestral population demographics and detecting species tree anomaly zones. However, the ability of coalescent methods to obtain accurate estimates remains largely unexplored. Errors in gene trees should lead to underestimates of the true CBL, and for a given set of comparisons, longer CBLs should be more accurate. Here, we furthered our empirical understanding of how error in gene tree quality (i.e., locus informativeness and gene tree resolution) affect CBLs using four datasets comprised of ultraconserved elements (UCE) or exons for clades that exhibit wide ranges of branch lengths. For each dataset, we compared the impact of locus informativeness (assessed using number of parsimony-informative sites) and gene tree resolution on CBL estimates. Our results, in general, showed that CBLs were drastically shorter when estimates included low informative loci. Gene tree resolution also had an impact on UCE datasets, with polytomous gene trees producing longer branches than randomly resolved gene trees. However, resolution did not appear to affect CBL estimates from the more informative exon datasets. Thus, as expected, gene tree quality affects CBL estimates, though this can generally be minimized by using moderate filtering to select more informative loci and/or by allowing polytomies in gene trees. These approaches, as well as additional contributions to improve CBL estimation, should lead to CBLs that are useful for addressing evolutionary and biological questions.     

Estimating mutation parameters,population history and genealogy simultaneously from temporally spaced sequence data

Molecular sequences obtained at different sampling times from populations of rapidly evolving pathogens and from ancient subfossil and fossil sources are increasingly available with modern sequencing technology. Here, we present a Bayesian statistical inference approach to the joint estimation of mutation rate and population size that incorporates the uncertainty in the genealogy of such temporally spaced sequences by using Markov chain Monte Carlo (MCMC) integration. The Kingman coalescent model is used to describe the time structure of the ancestral tree. We recover information about the unknown true ancestral coalescent tree, population size, and the overall mutation rate from temporally spaced data, that is, from nucleotide sequences gathered at different times, from different individuals, in an evolving haploid population. We briefly discuss the methodological implications and show what can be inferred, in various practically relevant states of prior knowledge. We develop extensions for exponentially growing population size and joint estimation of substitution model parameters. We illustrate some of the important features of this approach on a genealogy of HIV-1 envelope (env) partial sequences.     

Fitting parameters of stochastic birth-death models to metapopulation data

Populations that are structured into small local patches are a common feature of ecological and epidemiological systems. Models describing this structure are often referred to as metapopulation models in ecology or household models in epidemiology. Small local populations are subject to demographic stochasticity. Theoretical studies of household disease models without resistant stages (SIS models) have shown that local stochasticity can be ignored for between patch disease transmission if the number of connected patches is large. In that case the distribution of the number of infected individuals per household reaches a stationary distribution described by a birth-death process with a constant immigration term. Here we show how this result, in conjunction with the balancing condition for birth-death processes, provides a framework to estimate demographic parameters from a frequency distribution of local population sizes. The parameter estimation framework is applicable to estimate parameters of disease transmission models as well as metapopulation models.