A measure model for the spread of viral infections with mutations

Xiaoqian Gong; Benedetto Piccoli; Xiaoqian Gong; Benedetto Piccoli

doi:10.3934/nhm.2022015

Networks and Heterogeneous Media

2022, Volume 17, Issue 3: 427-442. doi: 10.3934/nhm.2022015

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A measure model for the spread of viral infections with mutations

Xiaoqian Gong ^{1
,},
Benedetto Piccoli ^{2
,
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1.
School of Mathematical and Statistical Science, Arizona State University, Tempe, AZ, 85281, USA
2.
Department of Mathematical Sciences and Center for Computational and Integrative Biology, Rutgers University, Camden, NJ, 08102, USA

Received: 01 May 2021 Revised: 01 September 2021 Published: 31 March 2022
Primary: 58F15, 58F17; Secondary: 53C35

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 a MDE comprised of a probability vector field (PVF) and a source term. In addition, the ODEs for $ S $ and $ R $ contains 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 case of constant or time-dependent parameters as special cases.
- measure differential equation,
- generalized Wasserstein distance,
- SIR model,
- viral infections,
- mutations
Citation: Xiaoqian Gong, Benedetto Piccoli. A measure model for the spread of viral infections with mutations[J]. Networks and Heterogeneous Media, 2022, 17(3): 427-442. doi: 10.3934/nhm.2022015

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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 a MDE comprised of a probability vector field (PVF) and a source term. In addition, the ODEs for $ S $ and $ R $ contains 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 case of constant or time-dependent parameters as special cases.

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