Qualitative and quantitative analysis of a nonlinear second-order anisotropic reaction-diffusion model of an epidemic infection spread

Abstract

<p style='text-indent:20px;'>The paper is concerned with two main topics, as follows. In the first instance, a serious qualitative analysis is performed for a second-order system of coupled PDEs, equipped with nonlinear anisotropic diffusion and cubic nonlinear reaction, as well as in-homogeneous Neumann boundary conditions. The PDEs system is implementing a SEIRD (Susceptible, Exposed, Infected, Recovered, Deceased) epidemic model. Under certain hypothesis on the input data: <inline-formula><tex-math id="M1">\begin{document}$ S_0(x) $\end{document}</tex-math></inline-formula>, <inline-formula><tex-math id="M2">\begin{document}$ E_0(x) $\end{document}</tex-math></inline-formula>, <inline-formula><tex-math id="M3">\begin{document}$ I_0(x) $\end{document}</tex-math></inline-formula> <inline-formula><tex-math id="M4">\begin{document}$ R_0(x) $\end{document}</tex-math></inline-formula>, <inline-formula><tex-math id="M5">\begin{document}$ D_0(x) $\end{document}</tex-math></inline-formula>, <inline-formula><tex-math id="M6">\begin{document}$ f(t,x) $\end{document}</tex-math></inline-formula> and <inline-formula><tex-math id="M7">\begin{document}$ w_{_i}(t,x), i = 1,2,3,4,5 $\end{document}</tex-math></inline-formula>, we prove the well-posedness (the existence, a priori estimates, regularity and uniqueness) of a classical solution in the Sobolev space <inline-formula><tex-math id="M8">\begin{document}$ W^{1,2}_p(Q) $\end{document}</tex-math></inline-formula>, extending the types already proven by other authors. The nonlinear second-order anisotropic reaction-diffusion model considered here is then particularized to monitor the spread of an epidemic infection.</p><p style='text-indent:20px;'>The second topic refers to the construction of the IMEX numerical approximation schemes in order to compute the solution of the system of coupled PDEs. We also show several simulation examples highlighting the present model's capabilities.</p>