Optimal Control Problem of an SIR Model with Random Inputs Based on a Generalized Polynomial Chaos Approach

Abstract

This paper studies the optimal control problem of a susceptible-infectious-recovered (SIR) epidemic model with random inputs. We prove the existence and uniqueness of a solution to the SIR random differential equation (RDE) model and investigate the numerical solution to the model by using a generalized polynomial chaos (gPC) approach. We formulate the optimal control problem of the SIR RDE model and consider the gPC Galerkin method to convert the problem into an optimal control problem with high-dimensional ordinary differential equations. Numerical simulations show that to effectively control an epidemic, vaccination should be given at the highest rate in the first few days, and after that, vaccination should be stopped completely. In addition, we observe that the optimal control function and the corresponding states are very robust to the uncertainty of random inputs.