Spatiotemporal Monitoring of Epidemics via Solution of a Coefficient Inverse Problem

Abstract: Let S,I and R be susceptible, infected and recovered populations in a city affected by an epidemic. The SIR model of Lee, Liu, Tembine, Li and Osher, \emph{SIAM J. Appl. Math.},~81, 190--207, 2021 of the spatiotemoral spread of epidemics is considered. This model consists of a system of three nonlinear coupled parabolic Partial Differential Equations with respect to the space and time dependent functions S,I and R. For the first time, a Coefficient Inverse Problem (CIP) for this system is posed. The so-called \textquotedblleft convexification" numerical method for this inverse problem is constructed. The presence of the Carleman Weight Function (CWF) in the resulting regularization functional ensures the global convergence of the gradient descent method of the minimization of this functional to the true solution of the CIP, as long as the noise level tends to zero. The CWF is the function, which is used as the weight in the Carleman estimate for the corresponding Partial Differential Operator. Numerical studies demonstrate an accurate reconstruction of unknown coefficients as well as S,I,R functions inside of that city. As a by-product, uniqueness theorem for this CIP is proven. Since the minimal measured input data are required, then the proposed methodology has a potential of a significant decrease of the cost of monitoring of epidemics.