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Journal Articles Physical Review E Year : 2023

A four compartment epidemic model with retarded transition rates

Abstract

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 → C → I → R → S remaining in each compartments C, I, and R for certain random waiting time t C , t I , t R , respectively. The waiting times for each compartment are independent and drawn from specific probability density functions (PDFs) introducing memory into the model. The first part of the paper is devoted to the macroscopic SCIRS model. We derive memory evolution equations involving convolutions (time derivatives of general fractional type). We consider several cases. The memoryless case is represented by exponentially distributed waiting times. Cases of long waiting times with fat-tailed waiting time distributions are considered as well where the SCIRS evolution equations take the form of time-fractional ODEs. 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. In the second part, 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. SCIRS type models open a wide field of applications allowing the identification of pertinent parameters governing the phenomenology of epidemic dynamics such as extinction, convergence to a stable endemic equilibrium, or persistent oscillatory behavior.
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Dates and versions

hal-03819827 , version 1 (18-10-2022)

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Téo Granger, Thomas M Michelitsch, Michael Bestehorn, Alejandro P Riascos, Bernard A Collet. A four compartment epidemic model with retarded transition rates. Physical Review E , 2023, 107 (4), pp.044207. ⟨10.1103/PhysRevE.107.044207⟩. ⟨hal-03819827⟩
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