Spatial Spread of Infectious Diseases Under Saturated Incidence: A Diffusive SIR Framework

We study a delayed reaction–diffusion susceptible–infectious–recovered (SIR) epidemic model with a Holling type-II incidence function. The model incorporates spatial diffusion and a discrete delay in the infectious input to the transmission term. We first formulate the model and discuss the existence and uniqueness of nonnegative solutions under suitable nonnegative initial histories. We then investigate the disease-free and endemic equilibria and derive the epidemic threshold by linearizing the infected equation at the disease-free equilibrium. For this normalised model the threshold quantity is given by R0 the normalisation is such that the disease free susceptible level is equal to one. We prove the disease-free stability result for the strict subthreshold case R0<1 and separate the critical case R0=1 from this regime. The existence of a positive endemic equilibrium for R0>1 is characterised. The stability of this equilibrium in the presence of a time delay requires that the characteristic equation and the global stability argument be consistent with the delay-dependent dynamics. The effects of diffusion, saturated incidence and delay are studied by numerical simulations. Transient oscillations are distinguished from sustained oscillations. The results presented here elucidate the assumptions and limitations of the proposed spatial epidemic model.