A measure model for the spread of viral infections with mutations

TL;DR

This paper introduces a novel coupled ODE-MDE model based on the SIR framework to better understand the spread of viral infections with mutations, capturing variant distributions and population dynamics.

Contribution

It develops a new mathematical model combining ODEs and measure differential equations to analyze mutation effects in viral spread, with proven well-posedness.

Findings

Model aligns with classical SIR under certain conditions

Provides analytical tools for studying mutation-driven dynamics

Demonstrates applicability through illustrative examples

Abstract

Genetic variations in the COVID-19 virus are one of the main causes of the COVID-19 pandemic outbreak in 2020 and 2021. In this article, we aim to introduce a new type of model, a system coupled with ordinary differential equations (ODEs), and measure differential equation (MDE), stemming from the classical SIR model for the variants distribution. Specifically, we model the evolution of susceptible $S$ and removed $R$ populations by ODEs and the infected $I$ population by an MDE comprised of a probability vector field (PVF) and a source term. In addition, the ODEs for $S$ and $R$ contain terms that are related to the measure $I$. We establish analytically the well-posedness of the coupled ODE-MDE system by using generalized Wasserstein distance. We give two examples to show that the proposed ODE-MDE model coincides with the classical SIR model in the case of constant or time-dependent