A reaction-diffusion SIS epidemic model in a time-periodic environment (Q2888619)

In this paper the authors consider the following SIS reaction-diffusion model for a population in a continuous spatial habitat NEWLINE\[NEWLINE\begin{cases} S_{t}-d_S\Delta S=-\beta(x,t)SI/(S+I)+\gamma(x,t)I, & x\in\Omega, \;t>0 \\ I_{t}-d_I\Delta I=\beta(x,t)SI/(S+I)-\gamma(x,t)I, & x\in\Omega, \;t>0 \\ \partial S/\partial \nu=\partial I/\partial \nu =0, & x\in \partial\Omega, \;t>0 \\ S(x,0)=S_0(x), I(x,0)=I_0(x), & x\in \Omega. \end{cases}NEWLINE\]NEWLINE Here \(\Omega\) is a bounded domain with smooth boundary, \(\beta(x,t)\) and \(\gamma(x,t)\) represent the disease transmission rate and disease recovery rate respectively. They are assumed to be Hölder continuous and nonnegative on \(\overline\Omega\times \mathbb{R}\), and periodic with period \(\omega>0\). The initial data \(S_0(x)\) and \(I_0(x)\) are continuous and nonnegative on \(\overline{\Omega}\), and \(I_0(x)\) satisfy the condition \(\int_{\Omega}I_0(x)dx>0\).NEWLINENEWLINEIt is shown that the total population size \(N=\int_{\Omega}[S(x,t)+I(x,t)]dx\) is a constant for all \(t\geq 0\). The authors define a basic reproduction number \(R_0\) and establish a threshold-type result in terms of \(R_0\) on the global dynamics of the nonnegative solutions of the above system. In particular, If \(R_0<1\), then \((S,I)\) approaches the unique disease-free solution \((N/|\Omega|,0)\) uniformly on \(\overline{\Omega}\) as \(t\rightarrow \infty\). If \(R_0>1\), then the system has at least one endemic \(\omega\)-periodic solution which is uniformly persistent on \(\overline{\Omega}\). The global attractivity of the solutions and the biological implications of the results are also discussed.