An SIR–like kinetic model tracking individuals' viral load

Rossella Della Marca; Nadia Loy; Andrea Tosin; Rossella Della Marca; Nadia Loy; Andrea Tosin

doi:10.3934/nhm.2022017

Networks and Heterogeneous Media

2022, Volume 17, Issue 3: 467-494. doi: 10.3934/nhm.2022017

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An SIR–like kinetic model tracking individuals' viral load

1.
Mathematics Area, SISSA – International School for Advanced Studies, Via Bonomea 265, I-34136 Trieste, Italy
2.
Department of Mathematical, Physical and Computer Sciences, Università di Parma, Parco Area delle Scienze 53/A, 43124 Parma, Italy
3.
Department of Mathematical Sciences "G. L. Lagrange", Politecnico di Torino, Corso Duca degli Abruzzi 24, 10129 Torino, Italy

Received: 01 June 2021 Revised: 01 January 2022 Published: 31 March 2022
Primary: 35Q20, 37N25; Secondary: 35Q70

In classical epidemic models, a neglected aspect is the heterogeneity of disease transmission and progression linked to the viral load of each infected individual. Here, we investigate the interplay between the evolution of individuals' viral load and the epidemic dynamics from a theoretical point of view. We propose a stochastic particle model describing the infection transmission and the individual physiological course of the disease. Agents have a double microscopic state: a discrete label, that denotes the epidemiological compartment to which they belong and switches in consequence of a Markovian process, and a microscopic trait, measuring their viral load, that changes in consequence of binary interactions or interactions with a background. Specifically, we consider Susceptible–Infected–Removed–like dynamics where infectious individuals may be isolated and the isolation rate may depend on the viral load–sensitivity and frequency of tests. We derive kinetic evolution equations for the distribution functions of the viral load of the individuals in each compartment, whence, via upscaling procedures, we obtain macroscopic equations for the densities and viral load momentum. We perform then a qualitative analysis of the ensuing macroscopic model. Finally, we present numerical tests in the case of both constant and viral load–dependent isolation control.
- Boltzmann–type equations,
- Markov–type jump processes,
- epidemic,
- SIR model,
- basic reproduction number,
- viral load,
- qualitative analysis
Citation: Rossella Della Marca, Nadia Loy, Andrea Tosin. An SIR–like kinetic model tracking individuals' viral load[J]. Networks and Heterogeneous Media, 2022, 17(3): 467-494. doi: 10.3934/nhm.2022017

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Abstract

In classical epidemic models, a neglected aspect is the heterogeneity of disease transmission and progression linked to the viral load of each infected individual. Here, we investigate the interplay between the evolution of individuals' viral load and the epidemic dynamics from a theoretical point of view. We propose a stochastic particle model describing the infection transmission and the individual physiological course of the disease. Agents have a double microscopic state: a discrete label, that denotes the epidemiological compartment to which they belong and switches in consequence of a Markovian process, and a microscopic trait, measuring their viral load, that changes in consequence of binary interactions or interactions with a background. Specifically, we consider Susceptible–Infected–Removed–like dynamics where infectious individuals may be isolated and the isolation rate may depend on the viral load–sensitivity and frequency of tests. We derive kinetic evolution equations for the distribution functions of the viral load of the individuals in each compartment, whence, via upscaling procedures, we obtain macroscopic equations for the densities and viral load momentum. We perform then a qualitative analysis of the ensuing macroscopic model. Finally, we present numerical tests in the case of both constant and viral load–dependent isolation control.

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