A measure model for the spread of viral infections with mutations

Abstract

<p style='text-indent:20px;'>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 <inline-formula><tex-math id="M1">\begin{document}$ S $\end{document}</tex-math></inline-formula> and removed <inline-formula><tex-math id="M2">\begin{document}$ R $\end{document}</tex-math></inline-formula> populations by ODEs and the infected <inline-formula><tex-math id="M3">\begin{document}$ I $\end{document}</tex-math></inline-formula> population by a MDE comprised of a probability vector field (PVF) and a source term. In addition, the ODEs for <inline-formula><tex-math id="M4">\begin{document}$ S $\end{document}</tex-math></inline-formula> and <inline-formula><tex-math id="M5">\begin{document}$ R $\end{document}</tex-math></inline-formula> contains terms that are related to the measure <inline-formula><tex-math id="M6">\begin{document}$ I $\end{document}</tex-math></inline-formula>. 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 case of constant or time-dependent parameters as special cases.</p>