Global dynamics of a differential-difference diffusive SIR model with Dirichlet boundary conditions

In this work, we investigate the global asymptotic behavior of an age-structured epidemic model with diffusion, under Dirichlet boundary conditions and a protection phase of limited duration. We consider a bounded n-dimensional spatial domain with homogeneous Dirichlet boundary conditions. The system is directly transformed to a coupled system of reaction-diffusion equations and a continuous difference equation with a time-delay and a non-local spatial term due to mobility during the protection phase. The Basic Reproduction Number (BRN) is determined for this system and some properties are obtained according to this parameter. Using Schauder's fixed point theorem, it is shown that disease-free equilibrium always exists and is globally attractive when BRN is less than one. Otherwise, we use the method of monotonic iterations for a particular elliptic problem to show that the endemic steady-state exists when BRN is greater than one. This condition on BRN follows naturally from the construction of an upper- and a lower-solution. Moreover, we prove that the system is uniformly strongly persistent. Finally, we perform numerical simulations that confirm and complete our theoretical results.