A spatial SIS model in heterogeneous environments with vary advective rate
This paper studies a spatial SIS epidemic model in heterogeneous environments with vary advective rate. Let \(S(x,t)\) and \(I(x,t)\) be the density of susceptible and infected individuals in a spatial interval \((0,L)\) and \(d_s\) and \(d_I\) be positive constants for the diffusion coefficients for the susceptible and infected ones. The system is described as \(S_t=(d_SS_x-a'(x)S)_x-\beta(x)SI/(S+I)+\gamma(x)I\) for \(x\in(0,L),t>0\); \(I_t=(d_II_x-a'(x)I)_x+\beta(x)SI/(S+I)-\gamma(x)I\) for \(x\in(0,L),t>0\); \(d_SS_x-a'(x)S=d_II_x-a'(x)I=0\) for \(x=0,L,t>0\); \(S(x,0)=S_0(x)\), \(I(x,0)=I_0(x)\) for \(x\in(0,L)\), where \(a'(x)\) is a smooth nonnegative function for the advection speed rate, \(\beta(x)\) and \(\gamma(x)\) are the rates of disease transmission and recovery at location \(x\), which are assumed to be Hölder continuous. By defining the basic reproduction number \(R_0\), it is shown that the disease-free equilibrium is unstable if \(R_0>1\) and it is globally asymptotically stable if \(R_0<1\). The stability and existence of endemic equilibrium of the system have also been studied.
scientific article; zbMATH DE number 7610895
scientific article