We propose a population approach to model the beginning of the French COVID-19 epidemic at the regional level. We rely on an extended Susceptible-Exposed-Infectious-Recovered (SEIR) mechanistic model, a simplified representation of the average epidemic process. Combining several French public datasets on the early dynamics of the epidemic, we estimate region-specific key parameters conditionally on this mechanistic model through Stochastic Approximation Expectation Maximization (SAEM) optimization using Monolix software. We thus estimate basic reproductive numbers by region before isolation (between 2.4 and 3.1), the percentage of infected people over time (between 2.0 and 5.9% as of May 11 th , 2020) and the impact of nationwide lockdown on the infection rate (decreasing the transmission rate by 72% toward a R e ranging from 0.7 to 0.9). We conclude that a lifting of the lockdown should be accompanied by further interventions to avoid an epidemic rebound.

Differential equations are commonly used to model dynamical deterministic systems in applications. When statistical parameter estimation is required to calibrate theoretical models to data, classical statistical estimators are often confronted to complex and potentially ill-posed optimization problem. As a consequence, alternative estimators to classical parametric estimators are needed for obtaining reliable estimates. We propose a gradient matching approach for the estimation of parametric Ordinary Differential Equations observed with noise. Starting from a nonparametric proxy of a true solution of the ODE, we build a parametric estimator based on a variational characterization of the solution. As a Generalized Moment Estimator, our estimator must satisfy a set of orthogonal conditions that are solved in the least squares sense. Despite the use of a nonparametric estimator, we prove the root-n consistency and asymptotic normality of the Orthogonal Conditions estimator. We can derive confidence sets thanks to a closed-form expression for the asymptotic variance. Finally, the OC estimator is compared to classical estimators in several (simulated and real) experiments and ODE models in order to show its versatility and relevance with respect to classical Gradient Matching and Nonlinear Least Squares estimators. In particular, we show on a real dataset of influenza infection that the approach gives reliable estimates. Moreover, we show that our approach can deal directly with more elaborated models such as Delay Differential Equation (DDE).