On a nonlocal and nonlinear second-order anisotropic reaction-diffusion system with in-homogeneous Cauchy-Neumann boundary conditions. Applications on epidemic infection spread

Abstract

<p style='text-indent:20px;'>In our current paper we are following the results obtained by Pavăl et al. in [<xref ref-type="bibr" rid="b36">36</xref>] and study a nonlocal form of the system they propose. First we are performing a qualitative analysis for the equivalent non-local second-order system of coupled PDEs, equipped with nonlinear anisotropic diffusion and cubic nonlinear reaction. As in [<xref ref-type="bibr" rid="b36">36</xref>] our PDEs system is also implementing a SEIRD (Susceptible, Exposed, Infected, Recovered, Deceased) epidemic model. In order to be able to compare with the before mentioned results we use the same 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 $\end{document}</tex-math></inline-formula>, and we prove the well-posedness of a classical solution in <inline-formula><tex-math id="M8">\begin{document}$ C((0,T],C(\Omega)) $\end{document}</tex-math></inline-formula>, extending the types already proven by other authors.</p><p style='text-indent:20px;'>Secondly we construct the implicit-explicit (IMEX) numerical approximation scheme which allows to compute the solution of the system of coupled PDEs. The results are then compared with the ones obtained by [<xref ref-type="bibr" rid="b36">36</xref>].</p>