Phylodynamic inference for structured epidemiological models

Abstract

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.

Figure 1. Prevalence and transmission rates estimated from a representative genealogy simulated under the three-stage SIR model.

(A) Stage-specific prevalence estimates with the 95% credible intervals shaded and the posterior medians shown as solid lines. Dashed lines show the true prevalence. (B–D) The marginal posterior densities of the stage-specific transmission rates. (E–G) The corresponding pairwise joint densities of the transmission rates, which were constructed from the MCMC samples using nonparametric kernel density estimation.

Figure 2. Posterior densities of parameters inferred from one HIV genealogy.

Solid red lines mark the median values and dashed lines indicate the 95% credible intervals. The estimated parameters are the early stage transmission rate , the chronic stage transmission rate , the AIDS stage transmission rate , the incidence scaler and the initial introduction time of HIV into Detroit.

Figure 3. Population dynamics inferred from the Detroit HIV genealogies.

(A) Stage-specific prevalence estimates from one genealogy with shaded regions showing the 95% credible intervals and lines the median of the posterior densities. (B) Estimated total yearly incidence and the estimated incidence attributable to early stage infections. The dashed black line shows the incidence back-calculated from Michigan Department of Community Health surveillance data. (C) Total incidence estimated from 10 randomly sampled HIV genealogies.

Figure 4. Parameter and prevalence estimates for the two-population model.

Mixing rates between the two populations were varied from low (), medium () to high (). (A–C) Joint posterior densities for the transmission rate and the mixing parameter . (D–F) Prevalence estimates for the two populations with the 95% credible intervals shaded and the posterior medians shown as solid lines. Dashed lines show the true prevalence. Initial conditions for the number of susceptible and infected individuals in each population were fixed at their true values for these simulations.

Figure 5. Genealogies simulated under different mixing rates.

Mixing rates between the red and blue population were varied from low (), medium () to high (). (A–C) The true lineage states mapped onto the genealogy. (D–F) Lineage state probabilities given with respect to the probability that the lineage is in the red state. (G–I) Entropy in the lineage states, which shows how much uncertainty there is in the lineage states. For each lineage , the entropy .

Figure 6. Likelihood profiles for the strength of coupling obtained from genealogies simulated under different values of .

Red lines correspond to the true value of . The likelihoods were computed from genealogies with 100 samples in (A–C), 500 samples in (D–F) and 1000 samples in (G–I). These sample sizes correspond, respectively, to approximately 0.2%, 1.0% and 2% of all infected individuals being sampled.