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.