In this talk, I will derive and analyze an infectious disease model containing a fixed latency and non-local infection caused by the mobility of the latent individuals in a continuous bounded domain. The model is given by a spatially non-local reaction–diffusion system carrying a discrete delay associated with the zero-flux condition on the boundary. By applying some existing abstract results in dynamical systems theory, we prove the existence of a global attractor for the model system. By appealing to the theory of monotone dynamical systems and uniform persistence, we show that the model has the global threshold dynamics which can be described either by the principal eigenvalue of a linear non-local scalar reaction diffusion equation or equivalently by the basic reproduction number R0 for the model. Such threshold dynamics predicts whether the disease will die out or persist. We identify the next generation operator, the spectral radius of which defines basic reproduction number. When all model parameters are constants, we are able to find explicitly the principal eigenvalue and R0. In addition to computing the spectral radius of the next generation operator, we also discuss an alternative way to compute R0.

This research was realised in the framework of the TÁMOP 4.2.4.A/2-11-1-2012-0001 "National Excellence Programme - Elaborating and operating an inland student and researcher personal support system convergence programme". The project was subsidised by the European Union and co-financed by the European Social Fund.