Stochastic predator-prey compartment model with cyclic and non-cyclic transition pathways and mortality

Abstract

We study variants of a stochastic SEIRS-SEID compartmental model incorporating presymptomatic infectiousness and disease-related mortality, with independent compartmental sojourn times drawn from arbitrary distributions, in particular Weibull distributions. We consider a large population of agents that occupy the states (compartments) S (susceptible to infection), E (exposed: infected but asymptomatic during the latency period and already infectious, including diseases with presymptomatic infections), I (infected and infectious, with symptoms), R (recovered, immune, and non-infectious), or D (dead). We consider (i) cyclic transmission pathways, where compartmental sojourn times are finite and immunity is temporary, and (ii) non-cyclic pathways, where immunity is permanent. In the absence of mortality, cyclic transmission pathways support an endemic equilibrium above the spreading threshold which is either globally stable or oscillatory (Hopf) unstable, whereas for non-cyclic pathways a stationary disease-free state is emerging. From a stochastic perspective, we elucidate the connection between compartmental sojourn-time distributions and the appearance of fractional operators in the evolution equations: fat-tailed compartment waiting-time distributions necessarily lead to fractional derivatives. For cyclic transition pathways we obtain an explicit expression for the expected number of infection cycles that an agent survives. Our model is validated by a combined Monte Carlo -- Runge -- Kutta simulation scheme and random-walk simulations on Barabasi-Albert networks. It has cross-disciplinary applications beyond epidemiology, including wildfire spreading, certain classes of chemical reactions, population dynamics, finance, economics, and biology.