SIR model with social gatherings

TL;DR

This paper extends the classic SIR epidemiological model by incorporating social gatherings, providing a probabilistic foundation and analyzing how gathering size influences disease spread through a mean-field limit approach.

Contribution

It introduces a novel SIR model with social gatherings, linking gathering size to infection dynamics and establishing a rigorous probabilistic foundation for the model.

Findings

Basic reproduction number depends quadratically on gathering size

Model generalizes existing models with a probabilistic basis

Provides insights into social gathering restrictions and disease control

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 ODEs where the force of 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 to understand 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.