On the convergence of an approximation scheme of fractional steps type, associated to a nonlinear second-order anisotropic reaction-diffusion SEIRD epidemic model

In the present paper, we concern a second-order system of coupled PDEs, endowed with nonlinear anisotropic diffusion and reaction term of cubic nonlinearity type, as well as in-homogeneous Neumann boundary conditions. The PDEs system in question implements a SEIRD (Susceptible, Exposed, Infected, Recovered, Deceased) epidemic model. We start our study by investigating the solvability in the class $ W_p^{1, 2}(Q), p\ge 2 $, of the nonlinear problem associated to the unknown function $ S(t, x) $ from the coupled system.The second goal is to build an iterative splitting scheme in order to numerically treat the cubic nonlinearity corresponding to the nonlinear reaction-diffusion equation in $ S(t, x) $. In this regard, we propose a first-order fractional-steps scheme. Results concerning the convergence of the numerical scheme are also established. Moreover, we formulate a conceptual algorithm in order to approximate the solution to the nonlinear parabolic problem in $ S(t, x) $. Experimental results, intended to implement the conceptual algorithm, are also provided.