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SIR model with social gatherings

Part of: Markov processes

Published online by Cambridge University Press:  15 January 2024

Roberto Cortez Open the ORCID record for Roberto Cortez [Opens in a new window]
Roberto Cortez*
Affiliation:
Universidad Andrés Bello
*
*Postal address: Sazié 2212, sexto piso, Santiago, Chile.Email: roberto.cortez.m@unab.cl
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Abstract

We introduce an extension to Kermack and McKendrick’s classic susceptible–infected–recovered (SIR) model in epidemiology, whose underlying mechanism of infection consists of individuals attending randomly generated social gatherings. This gives rise to a system of ordinary differential equations (ODEs) where the force of the infection term depends non-linearly on the proportion of infected individuals. Some specific instances yield models already studied in the literature, to which the present work provides a probabilistic foundation. The basic reproduction number is seen to depend quadratically on the average size of the gatherings, which may be helpful in understanding how restrictions on social gatherings affect the spread of the disease. We rigorously justify our model by showing that the system of ODEs is the mean-field limit of the jump Markov process corresponding to the evolution of the disease in a finite population.

Keywords

Epidemiologysocial distancingsocial gatheringsreproduction numberjump Markov processmean-field limit

MSC classification

Primary: 92D30: Epidemiology
Secondary: 60J28: Applications of continuous-time Markov processes on discrete state spaces
Type
Original Article
Information
Journal of Applied Probability , First View , pp. 1 - 18
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Copyright
© The Author(s), 2024. Published by Cambridge University Press on behalf of Applied Probability Trust

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