Estimation for diffusion processes partially observed with measurement errors. Applications to epidemics.

Estimating the parameters governing epidemic dynamics, such as the transmission rate, from available data is a major issue in order to provide reliable predictions of these dynamics and of the
impact of control strategies. In this context, several difficulties occur: all the components of the system dynamics are not observed, and data are available at discrete times with measurement errors. Diffusion processes with small diffusion coefficient are a convenient set-up for modelling epidemics ([1], [2]), the small diffusion coefficient being related to the population size $N$ , i.e. $\varepsilon = 1/ \sqrt{N}$ . To estimate the key epidemic parameters, we therefore propose to consider time-dependent diffusion processes on $\mathbb{R}^p$ satisfying the stochastic differential equation
$dX(t) = b(α; t, X(t))dt + σ(β; t, X(t))dB(t); X(0) = ξ$,
where $(B(t) t≤0)$ is a $p$-dimensional standard Brownian motion defined on a probability space
$\mathbb{P} = (Ω, (F_t ) t≤0 , P)$, independent of $ξ, b(β; t, ·) : \mathbb{R}^p → \mathbb{R}^p$, $σ(β; t, ·) : \mathbb{R}^p → \mathbb{R}^p × \mathbb{R}^p$ and where $\varepsilon → 0$ in the asymptotics.
In practical applications on epidemic dynamics, it often occurs that some coordinates of $X(t)$ are
not observed and, when observed, measurement errors are systematically present. We are then
concerned with the estimation of $(α, β, ξ)$ when the diffusion process is discretely observed with
noise and with sampling step $δ$ on a finite time interval $[0, T]$, and when some components of $X(t)$ cannot be observed. We propose a procedure derived from Kalman filtering approaches to compute estimates of the parameters based on approximate likelihoods. Our approach is original because it combines the framework of diffusions with small diffusion coefficient with approximate likelihood methods and Kalman filtering, the latter being little exploited for the inference of epidemic dynamics partially observed and with errors.
We carry out simulation studies to assess the performances of the proposed methods. An application to real epidemic data of seasonal human influenza is still in progress.

References
[1] Guy, R., Larédo, C., Vergu, E. (2014) Parametric inference for discretely observed multidimen-
sional diffusions with small diffusion coefficient., Stoch Proc Appl, 124, 51–80.
[2] Guy, R., Larédo, C., Vergu, E. (2015) Approximation of epidemic models by diffusion processes
and their statistical inference., J Math Biol, 70, 721–746.