Modelling contagious viral dynamics: a kinetic approach based on mutual utility

Giulia Bertaglia; Lorenzo Pareschi; Giuseppe Toscani; Giulia Bertaglia; Lorenzo Pareschi; Giuseppe Toscani

doi:10.3934/mbe.2024187

Mathematical Biosciences and Engineering

2024, Volume 21, Issue 3: 4241-4268. doi: 10.3934/mbe.2024187

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Modelling contagious viral dynamics: a kinetic approach based on mutual utility

1.
Department of Environmental and Prevention Sciences, University of Ferrara, Ferrara, Italy
2.
Maxwell Institute and Department of Mathematics, Heriot-Watt University, Edinburgh, UK
3.
Department of Mathematics and Computer Science, University of Ferrara, Ferrara, Italy
4.
Department of Mathematics, University of Pavia, Pavia, Italy
5.
IMATI, Institute for Applied Mathematics and Information Technologies "Enrico Magenes", Pavia, Italy

Academic Editor: Raluca Eftimie

Received: 31 December 2023 Revised: 05 February 2024 Accepted: 18 February 2024 Published: 26 February 2024

The temporal evolution of a contagious viral disease is modelled as the dynamic progression of different classes of population with individuals interacting pairwise. This interaction follows a binary mechanism typical of kinetic theory, wherein agents aim to improve their condition with respect to a mutual utility target. To this end, we introduce kinetic equations of Boltzmann-type to describe the time evolution of the probability distributions of the multi-agent system. The interactions between agents are defined using principles from price theory, specifically employing Cobb-Douglas utility functions for binary exchange and the Edgeworth box to depict the common exchange area where utility increases for both agents. Several numerical experiments presented in the paper highlight the significance of this mechanism in driving the phenomenon toward endemicity.
- kinetic models,
- epidemic models,
- irreversible processes,
- Cobb-Douglas utility function,
- Edgeworth box,
- Boltzmann-type equations
Citation: Giulia Bertaglia, Lorenzo Pareschi, Giuseppe Toscani. Modelling contagious viral dynamics: a kinetic approach based on mutual utility[J]. Mathematical Biosciences and Engineering, 2024, 21(3): 4241-4268. doi: 10.3934/mbe.2024187

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Abstract

The temporal evolution of a contagious viral disease is modelled as the dynamic progression of different classes of population with individuals interacting pairwise. This interaction follows a binary mechanism typical of kinetic theory, wherein agents aim to improve their condition with respect to a mutual utility target. To this end, we introduce kinetic equations of Boltzmann-type to describe the time evolution of the probability distributions of the multi-agent system. The interactions between agents are defined using principles from price theory, specifically employing Cobb-Douglas utility functions for binary exchange and the Edgeworth box to depict the common exchange area where utility increases for both agents. Several numerical experiments presented in the paper highlight the significance of this mechanism in driving the phenomenon toward endemicity.

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