Global dynamics of an SIS epidemic model with cross-diffusion: applications to quarantine measures

This paper considers an SIS model with a cross-diffusion dispersal strategy for the infected individuals describing the public health intervention measures (like quarantine) during the outbreak of infectious diseases. The model adopts the frequency-dependent transmission mechanism and includes demographic changes (i.e. population recruitment and death) subject to homogeneous Neumann boundary conditions. We first establish the existence of global classical solutions with the uniform-in-time bound. Then, we define the basic reproduction number R₀ by a weighted variational form. Due to the presence of the cross-diffusion on infected individuals, we employ a change of variable and apply the index theory along with the principal eigenvalue theory to establish the threshold dynamics in terms of R₀ based on the fact that the sign of the principal eigenvalue of the weighted eigenvalue problem is the same as that of the corresponding unweighted eigenvalue problem. Furthermore, we obtain the global stability of the unique disease-free equilibrium and constant endemic equilibrium under some conditions. Finally, we discuss some open questions and use numerical simulation to demonstrate the applications of our analytical results, showing that the cross-diffusion dispersal strategy can reduce the value of R₀ and help eradicate the diseases even if the habitat is high-risk in contrast to the situation without cross-diffusion.