Inference for epidemic data using diffusion processes with small diffusion coefficient

One of the simplest and most appropriate models for the study of epidemic spread is the SIR(Susceptible-Infectious-Removed) model, in which the successive transitions of individuals betweenstates are described in various mathematical frameworks. Epidemic data being often partially observedand also temporally aggregated, parametric inference through likelihood-based approaches is rarelystraightforward, whatever the mathematical representation used. Although these last years methodsbased on data augmentation, able to deal with different patterns of missingness were developed, theydo not provide systematic solutions since they are limited by the amount of augmented data andhence by computation times. In this context, diffusion processes provide a good approximation ofepidemic dynamics and, due to their analytical power, allow shedding new light on inference problemsof epidemic data. Indeed, the normalization by the population size N of the continuous time Markovjump process leads to an ODE system. Before passing to the limit, the SIR dynamics is described by√a bidimensional diffusion with small diffusion coefficient proportional to = 1/ N .We consider here a multidimensional diffusion (Xt )t≥0 with drift coefficient b(α, Xt ) and diffusioncoefficient σ (β, Xt ). The diffusion is discretely observed at times tk = k∆ on a fixed time interval[0, T ] with T = n∆. We study Minimum Contrast Estimators (MCE) derived from a Gaussian processapproximating (Xt )t≥0 for small . We obtain consistent and asymptotically Gaussian estimators of(α, β ) as → 0 and ∆ = ∆n → 0 under the condition “ 2 n“ bounded , and for fixed ∆ consistent andasymptotically Gaussian estimator of α. Our results extend [GC90] to multidimensional diffusions witharbitrary diffusion coefficient and [GS09] to low frequency data. Finally, we simulate various epidemicsdynamics with Markov jump process, we compare our MCE to other commonly used estimators: leastsquares for noisy observed ODE, maximum likelihood for Markov jump processes. Our findings arevery promising: on low and high frequency data, our estimator is very close to the estimator basedon the complete observation of the Markov jump process . We are currently extended these results tothe more realistic case of partially observed data