A four compartment epidemic model with retarded transition rates

• Main Authors: Granger, Teo, Michelitsch, Thomas M, Bestehorn, Michael, Riascos, Alejandro P, Collet, Bernard A
• Format: Journal Article
• Language: English
• Published: 23-03-2023
• Subjects: Physics - Statistical Mechanics; Quantitative Biology - Populations and Evolution
• DOI: 10.48550/arxiv.2210.09912

Physical Review E 107, 044207 (2023) We study an epidemic model for a constant population by taking into account
four compartments of the individuals characterizing their states of health.
Each individual is in one of the compartments susceptible (S); incubated -
infected yet not infectious (C), infected and infectious (I), and recovered -
immune (R). An infection is 'visible' only when an individual is in state I.
Upon infection, an individual performs the transition pathway S to C to I to R
to S remaining in each compartments C, I, and R for certain random waiting
times, respectively. The waiting times for each compartment are independent and
drawn from specific probability density functions (PDFs) introducing memory
into the model. We derive memory evolution equations involving convolutions
(time derivatives of general fractional type). We obtain formulae for the
endemic equilibrium and a condition of its existence for cases when the waiting
time PDFs have existing means. We analyze the stability of healthy and endemic
equilibria and derive conditions for which the endemic state becomes
oscillatory (Hopf) unstable. We implement a simple multiple random walker's
approach (microscopic model of Brownian motion of Z independent walkers) with
random SCIRS waiting times into computer simulations. Infections occur with a
certain probability by collisions of walkers in compartments I and S. We
compare the endemic states predicted in the macroscopic model with the
numerical results of the simulations and find accordance of high accuracy. We
conclude that a simple random walker's approach offers an appropriate
microscopic description for the macroscopic model.