Compartment model with retarded transition rates

TL;DR

This paper models an epidemic using a four-compartment system with delayed transitions, analyzing stability, endemic equilibrium, and the basic reproduction number through a microscopic random walker approach on a lattice.

Contribution

It introduces a novel compartmental epidemic model with delayed transition times and connects microscopic random walk dynamics to epidemic spread analysis.

Findings

Derived conditions for epidemic spread based on $R_0>1$

Established stability criteria for the healthy state

Provided numerical evidence supporting the random walker approach

Abstract

Our study is devoted to a four-compartment epidemic model of a constant population of independent random walkers. Each walker is in one of four compartments (S-susceptible, C-infected but not infectious (period of incubation), I-infected and infectious, R-recovered and immune) characterizing the states of health. The walkers navigate independently on a periodic 2D lattice. Infections occur by collisions of susceptible and infectious walkers. Once infected, a walker undergoes the delayed cyclic transition pathway S $\to$ C $\to$ I $\to$ R $\to$ S. The random delay times between the transitions (sojourn times in the compartments) are drawn from independent probability density functions (PDFs). We analyze the existence of the endemic equilibrium and stability of the globally healthy state and derive a condition for the spread of the epidemics which we connect with the basic reproduction number $R_0>1$. We give quantitative numerical evidence that a simple approach based on random walkers offers an appropriate microscopic picture of the dynamics for this class of epidemics.