Basic Reproduction Numbers for a Class of Reaction-Diffusion Epidemic Models
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Basic Reproduction Numbers for a Class of Reaction-Diffusion Epidemic Models Chayu Yang1 · Jin Wang1 Received: 20 January 2020 / Accepted: 31 July 2020 © Society for Mathematical Biology 2020
Abstract We study the basic reproduction numbers for a class of reaction-diffusion epidemic models that are developed from autonomous ODE systems. We present a general numerical framework to compute such basic reproduction numbers; meanwhile, the numerical formulation provides useful insight into their characterizations. Using matrix analysis, we show that the basic reproduction numbers are the same for these PDE models and their associated ODE models in several important cases that include, among others, a single infected compartment, constant diffusion rates, uniform diffusion patterns among the infected compartments, and partial diffusion in the system. Keywords Compartmental models · Numerical analysis · Eigenvalues
1 Introduction Mathematical modeling and analysis provide a powerful theoretical tool for epidemiological study. Both ordinary differential equations (ODE) and partial differential equations (PDE) are extensively used. In particular, mathematical models based on reaction-diffusion equations have been frequently employed to investigate the transmission and spread of infectious diseases. During the development of a PDE epidemic model, an ODE system is often established first to describe the spatially homogeneous dynamics of disease transmission. Then, diffusion terms are added to study the spatial spread of the disease. A diffusion process represents the random movement and dispersal of hosts/pathogens over the spatial domain, normally without a directional preference. Incorporation of such spatial movement, generally associated with location-dependent diffusion rates, into epidemiological, ecological and other biological models emphasizes the spatial het-
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Jin Wang [email protected] Department of Mathematics, University of Tennessee at Chattanooga, Chattanooga, TN 37403, USA 0123456789().: V,-vol
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erogeneity of population dynamics (Cantrell and Cosner 1991, 2003), particularly with regard to disease transmission and spread. There have been many studies devoted to reaction-diffusion epidemic models. For example, existence and well posedness of solutions are analyzed in Kim et al. (2013) and Yamazaki and Wang (2016), equilibrium analyses are conducted in Wang et al. (2015), Wu and Zou (2018), and Yu and Zhao (2016), traveling waves are investigated in Wang et al. (2015), Wang et al. (2016), and Zhao et al. (2018), and realistic epidemic simulations are carried out in Bertuzzo et al. (2010) and Rinaldo et al. (2012). Particularly, basic reproduction numbers for such epidemic systems are studied in Allen et al. (2008), Thieme (2009), Wang and Zhao (2012), and Magal et al. (2019). The basic reproduction number, typically denote by R0 , is a critical quantity to measure the transmission risk of an infectious disease. It quantifies the expected number of secondary infe
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