Stochastic compartment model of epidemic spreading in complex networks with mortality and resetting

We propose an epidemic compartment model which includes mortality caused by the disease, but excludes demographic birth and death processes. Individuals are represented by random walkers which are in one of the states (compartments) S (susceptible for infection), E (exposed: infected but not infectious corresponding to the latency period), I (infected and infectious), R (recovered, immune), D (dead). The disease is transmitted with a certain probability at contacts of I to S walkers. The compartmental sojourn times are independent random variables drawn from specific (here Gamma-) distributions. We implement this model into random walk simulations. Each walker performs an independent simple Markovian random walk on a graph, where we consider a Watts-Strogatz (WS) network. Only I walkers may die. For zero mortality we prove the existence of an endemic equilibrium for basic reproduction number ${\cal R}_0>1$ and for which the disease free (globally healthy) state is unstable. We explore the effects of long-range-journeys (stochastic resetting) and mortality. Our model allows for various interpretations, such as certain chemical reactions, the propagation of wildfires, and in population dynamics.