Optimal Strategies for COVID-19 Control in a Stochastic Process

Abstract

First, the necessary mathematical tools are recalled regarding the concepts of stochastic control processes, including stochastic optimal control and one of its fundamental principles, the minimization principle. Then, a new controlled stochastic model of COVID-19 dynamics is formulated and represented by a stochastic differential equation with a 6-dimensional random vector of state variables (susceptible, exposed, mildly symptomatic, severely symptomatic, recovered for humans and surface concentration of SARS-CoV-2 coronavirus) and a vector of external control functions. The objective is to control the evolution and diffusion of SARS-COV-2 in a stochastic process in order to determine the optimal strategies to combat the spread of COVID-19. The stochastic analysis of the model focuses on the positivity and boundedness of the solutions, as well as the global behavior of the intermediate system, under the sole condition of stability of the disease-free equilibria and the endemic equilibria of the stochastic model. The results of the analysis reveal, provided that the disease-free equilibria of the stochastic model are exponentially p-stable and globally asymptotically stable; finally, that the endemic equilibria are locally asymptotically stable. An optimal control problem is formulated with the aim of eradicating the evolution of COVID-19; using its fundamental principle, the Pontryagin minimum principle, this problem is solved numerically with the optimal strategies to be adopted among the scenarios designed to control COVID-19 in a stochastic process.