Spatial dynamic analysis for COVID-19 epidemic model with diffusion and Beddington-DeAngelis type incidence

Abstract

<p style='text-indent:20px;'>A diffusion SEIAR model with Beddington-DeAngelis type incidence is proposed to characterize the spread of COVID-19 with spatial transmission. First, the well-posedness of solution is studied. Second, the basic reproduction number <inline-formula><tex-math id="M3">\begin{document}$ \mathcal R_{0} $\end{document}</tex-math></inline-formula> is derived and served as a threshold value to determine whether COVID-19 will spread. Meanwhile, we consider the effect of diffusion on the spread of COVID-19 in spatial homogenous environment, by which we can obtain that if <inline-formula><tex-math id="M4">\begin{document}$ \mathcal R_{0}<1 $\end{document}</tex-math></inline-formula>, then the infection-free steady state is globally asymptotically stable, while if <inline-formula><tex-math id="M5">\begin{document}$ \mathcal R_{0}>1 $\end{document}</tex-math></inline-formula>, then the endemic steady state is globally asymptotically stable. Furthermore, according to the official reporting data about COVID-19 in Wuhan, China, the actual value of <inline-formula><tex-math id="M6">\begin{document}$ \mathcal R_{0} $\end{document}</tex-math></inline-formula> is estimated, and comparing with other types of incidence, we find that the estimated peak with Beddington-DeAngelis type incidence is more close to the cases in reality. Finally, by numerical simulations, we can see that the diffusion behavior has evident impact on the spread of COVID-19 in spatial heterogeneity than homogeneity of environment.</p>