Approximation and inference of epidemic dynamics by diffusion processes

Résumé

Epidemic data are often aggregated and partially observed. Parametric inference through likelihood-based approaches is rarely straightforward, whatever the mathematical representation used. Recent data augmentation and likelihood-free methods do not completely circumvent the issues related to incomplete data in practice, mainly due to the size of missing data and to the various tuning parameters to be adjusted. In this context, diffusion processes provide a good approximation of epidemic dynamics and allow shedding new light on inference problems related to epidemic data. In this article we summarize and extend previous work on the elaboration of a statistical framework to deal with epidemic models and epidemic data using multidimensional diffusion processes with small diffusion coefficient. First, we construct multidimensional diffusion processes with small variance as mathematical representations of epidemic dynamics, by approximating Markov jump processes. Second, we introduce an inference method related to the asymptotic of the small diffusion coefficient on a fixed time interval for the parameters of the diffusion processes obtained, when all the coordinates are discretely observed. Consistency and asymptotic normality of estimators for this case are obtained for parameters in drift (high and low frequency observations) and diffusion (high frequency observations) coefficients. Third, as an extension of previous work, the case of incomplete data, when only one coordinate of the system is observed, is considered for high frequency observations. Finally, the performances of our estimators are explored for single outbreaks (SIR model, simulated data) and for recurrent outbreaks (SIRS model, simulated and observed data).