Propagation of Epidemics Along Lines with Fast Diffusion
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Propagation of Epidemics Along Lines with Fast Diffusion Henri Berestycki1,2
· Jean-Michel Roquejoffre3 · Luca Rossi1
Received: 6 June 2020 / Accepted: 23 October 2020 © Society for Mathematical Biology 2020
Abstract It has long been known that epidemics can travel along communication lines, such as roads. In the current COVID-19 epidemic, it has been observed that major roads have enhanced its propagation in Italy. We propose a new simple model of propagation of epidemics which exhibits this effect and allows for a quantitative analysis. The model consists of a classical SIR model with diffusion, to which an additional compartment is added, formed by the infected individuals travelling on a line of fast diffusion. The line and the domain interact by constant exchanges of populations. A classical transformation allows us to reduce the proposed model to a system analogous to one we had previously introduced Berestycki et al. (J Math Biol 66:743–766, 2013) to describe the enhancement of biological invasions by lines of fast diffusion. We establish the existence of a minimal spreading speed, and we show that it may be quite large, even when the basic reproduction number R0 is close to 1. We also prove here further qualitative features of the final state, showing the influence of the line. Keywords COVID-19 · Epidemics · SIR model · Reaction-diffusion system · Line of fast diffusion · Spreading speed
1 Context and Motivation of this Study In the present context of the COVID-19 pandemic, a worldwide scientific effort is currently under way to develop the modelling of its dynamics and propagation. Such an endeavour is of essential value to monitor, and forecast the propagation of the epidemic.
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Henri Berestycki [email protected]
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Ecole des Hautes Etudes en Sciences Sociales, CNRS, Centre d’Analyse et Mathématiques Sociales, 54 boulevard Raspail, 75006 Paris, France
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HKUST Jockey Club Institute for Advanced Study, Hong Kong University of Science and Technology, Clear Water Bay, Hong Kong
3
Université Paul Sabatier, Institut de Mathématiques de Toulouse, 118 route de Narbonne, 31062 Toulouse Cedex 4, France 0123456789().: V,-vol
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H. Berestycki et al.
Most of the models that are used rely on various extensions of the classical SIR cornerstone model of epidemiology. That is, they use population compartmental models that contain various additional compartments to the SIR ones to account for segments of the populations that are exposed, asymptomatic, presymptomatic, treated, etc. Such models yield the evolution of the infected population at a given level of territorial granularity (whole countries, regions, counties or cities). The spatial interplay aspect, mostly overlooked, is included by involving transfer matrices of populations and infected between various patches each of which being considered as uniform. Yet, the propagation of COVID-19 exhibits remarkable spatial structure properties. Indeed, the spatial organization and spreading of epidemics in general reveal important features of the
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