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<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}&lt;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}&gt;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>

In this work, we consider a stage-structured cannibalism model with two delays. One delay characterizes the lag effect of negative feedback of the prey species, the other has the effect of gestation of the adult predator population. Firstly, criteria for the local stability of feasible equilibria are established. Meanwhile, by choosing delay as a bifurcation parameter, the criteria on the existence of Hopf bifurcation are established. Furthermore, by the normal form theory and center manifold theorem, we derive the explicit formulas determining the properties of periodic solutions. Finally, the theoretical results are illustrated by numerical simulations, from which we can see that the predator’s gestation time delay can make the chaotic phenomenon disappear and maintain periodic oscillation, and that a large feedback time delay of prey can make predators extinct and prey form a periodic solution.