Epidemic waves for a two-group SIRS model with double nonlocal effects in a patchy environment

We propose a lattice dynamical system that arises in a discrete diffusive two-group epidemic model with latency in a patchy environment. The model considers the SIS form and latency of the disease in group 1, while the SIR form without latency of the disease in group 2. The system includes double nonlocal effects, one effect is the nonlocal diffusion of individuals in isolated patches or niches, while the other effect is the distributed transmission delay representing the incubation of the disease. We demonstrate that there is a threshold value $c^*$ that can determine the persistence or disappearance of the disease. If $c\geq c^*$, then there is an epidemic wave connecting the disease-free equilibrium and endemic equilibrium. In this case, the disease will evolve to endemic. If $0<c<c^*$, then the disease will die out.