Global asymptotic stability of endemic equilibria for a diffusive SIR epidemic model with saturated incidence and logistic growth

Abstract

<p style='text-indent:20px;'>This paper deals with the diffusive epidemic model with saturated incidence and logistic growth, </p><p style='text-indent:20px;'><disp-formula> <label/> <tex-math id="FE1"> \begin{document}$ \begin{align*} \begin{cases} \dfrac{\partial S}{\partial t} = d_S \Delta S - \dfrac{\beta S I}{1+\alpha I} + rS\left(1- \dfrac{S}{K} \right), &x \in \Omega, \ t>0, \\ \dfrac{\partial I}{\partial t} = d_I \Delta I + \dfrac{\beta S I}{1+\alpha I} - \gamma I, &x \in \Omega, \ t>0, \end{cases} \end{align*} $\end{document} </tex-math></disp-formula></p><p style='text-indent:20px;'>where <inline-formula><tex-math id="M1">\begin{document}$ \Omega \subset \mathbb{R}^N $\end{document}</tex-math></inline-formula> <inline-formula><tex-math id="M2">\begin{document}$ (N \in \mathbb{N}) $\end{document}</tex-math></inline-formula> is a bounded domain with smooth boundary and <inline-formula><tex-math id="M3">\begin{document}$ d_S, d_I, K, r, \alpha, \beta, \gamma >0 $\end{document}</tex-math></inline-formula> are constants. Setting <inline-formula><tex-math id="M4">\begin{document}$ \mathcal{R}_0: = \frac{K \beta}{\gamma} $\end{document}</tex-math></inline-formula>, Avila-Vales et al. [<xref ref-type="bibr" rid="b1">1</xref>] succeeded in showing that if <inline-formula><tex-math id="M5">\begin{document}$ \mathcal{R}_0\leq1 $\end{document}</tex-math></inline-formula>, then the disease-free equilibrium <inline-formula><tex-math id="M6">\begin{document}$ (K, 0) $\end{document}</tex-math></inline-formula> of the model with saturated treatment is globally asymptotically stable, whereas in the case <inline-formula><tex-math id="M7">\begin{document}$ \mathcal{R}_0>1 $\end{document}</tex-math></inline-formula> the model admits a constant endemic equilibrium <inline-formula><tex-math id="M8">\begin{document}$ (S^*, I^*) $\end{document}</tex-math></inline-formula> (<inline-formula><tex-math id="M9">\begin{document}$ S^*, I^*>0 $\end{document}</tex-math></inline-formula>), and it is unknown whether <inline-formula><tex-math id="M10">\begin{document}$ (S^*, I^*) $\end{document}</tex-math></inline-formula> is globally asymptotically stable or not. The purpose of this paper is to establish that the constant endemic equilibrium of the above model is globally asymptotically stable by constructing a strict Lyapunov functional. The construction is carried out by optimizing a function of two real variables through straightforward calculations, division into some cases and arrangement of several conditions. Moreover, to show that the functional is strict, some auxiliary function is introduced.</p>