Likelihood-based estimation of continuous-time epidemic models from time-series data: application to measles transmission in London

Abstract

We present a new statistical approach to analyse epidemic time-series data. A major difficulty for inference is that (i) the latent transmission process is partially observed and (ii) observed quantities are further aggregated temporally. We develop a data augmentation strategy to tackle these problems and introduce a diffusion process that mimics the susceptible-infectious-removed (SIR) epidemic process, but that is more tractable analytically. While methods based on discrete-time models require epidemic and data collection processes to have similar time scales, our approach, based on a continuous-time model, is free of such constraint. Using simulated data, we found that all parameters of the SIR model, including the generation time, were estimated accurately if the observation interval was less than 2.5 times the generation time of the disease. Previous discrete-time TSIR models have been unable to estimate generation times, given that they assume the generation time is equal to the observation interval. However, we were unable to estimate the generation time of measles accurately from historical data. This indicates that simple models assuming homogenous mixing (even with age structure) of the type which are standard in mathematical epidemiology miss key features of epidemics in large populations.

Figure 1

Examples of trajectories for the number of infectives I_t consistent with U_k=4, where U_k is the number of new infections occurring during time period ]kT,(k+1)T]: (a) I_kT=2, I_(k+1)T=2 and (b) I_kT=4, I_(k+1)T=7.

Figure 2

Convergence of the MCMC algorithm for the SIR epidemic simulated with 1/γ=7 days. (a) Recovery rate, 1/γ; (b) reporting rate, ρ; (c) initial number of susceptibles, S₀; (d) log-likelihood; (e) transmission rate for bi-week 1; (f) transmission rate for bi-week 10.

Figure 3

Posterior mean (solid line), 95% credible interval (dotted lines) and simulation value (dashed line) of the ratio β/γ for the SIR epidemic simulated with 1/γ=14 days. β is the transmission rate (seasonal variations with period=1 year) and 1/γ is the mean infectious period.

Figure 4

Seasonality and trend of the transmission rate and the effective reproduction number for measles epidemics in London, estimated under the assumption that the mean infectious period is equal to 14 days. (a) Seasonality of the transmission rate (solid line, posterior mean; dashed line, 95% credible interval; grey line, posterior mean of the daily transmission rate r_k/14 estimated with the TSIR approach). (b) Trend of the transmission rate β/N. (c) Trend of the effective reproduction number (R). The transmission rate for period k is β_k/N_k where N_k is the size of the core group (children with age below 4 years). The formula for the effective reproduction number is given in the main text. We correct for the fact that the measles latent period is 8 days (Anderson & May 1991).

Figure 5

Model checking. (a) Number of cases of measles reported in London between 1948 and 1964. (b) Variations in the number of susceptibles. (c) Spectral density. The solid line is the observed curve and the dashed line is the predicted curve (average of 40 epidemics simulated from the model, with parameters equal to their posterior mean). The mean infectious period is assumed to be equal to 14 days.