Maximum likelihood estimation for a stochastic SEIR system with a COVID-19 application

Abstract

In this paper, we propose a stochastic model for epidemiology data. The proposed model is obtained as a random perturbation of a suitable parameter in a deterministic SEIR system. This perturbation allows us to obtain a set of coupled stochastic differential equations (SDEs) that still have the conservation law. Afterward, by using Girsanov's Theorem, we calculate the maximum likelihood estimation (MLE) for parameters that represent the symptomatic infection rate, asymptomatic infection rate, and the proportion of symptomatic individuals. These parameters are crucial to obtain information about the dynamic of the disease. We prove the consistency of the MLE for a fixed time observation window, in which the disease is in its growth phase. The proposed stochastic SEIR model improves the uncertainty quantification of an overestimated MCMC scheme based on its deterministic model to count reported-confirmed COVID-19 cases of Mexico City. Using a particular mechanism to manage missing data, we developed MLE for some parameters of the stochastic model that improves the description of variance of the actual data.