Spatiotemporal Monitoring of Epidemics via Solution of a Coefficient Inverse Problem

Let S,I and R be susceptible, infected and recovered populations in a cityaffected 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 ofepidemics is considered. This model consists of a system of three nonlinearcoupled parabolic Partial Differential Equations with respect to the space andtime dependent functions S,I and R. For the first time, a Coefficient InverseProblem (CIP) for this system is posed. The so-called \textquotedblleftconvexification" numerical method for this inverse problem is constructed. Thepresence of the Carleman Weight Function (CWF) in the resulting regularizationfunctional ensures the global convergence of the gradient descent method of theminimization of this functional to the true solution of the CIP, as long as thenoise level tends to zero. The CWF is the function, which is used as the weightin the Carleman estimate for the corresponding Partial Differential Operator.Numerical studies demonstrate an accurate reconstruction of unknowncoefficients 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 inputdata are required, then the proposed methodology has a potential of asignificant decrease of the cost of monitoring of epidemics.