Spatial propagation for a reaction-diffusion SI epidemic model with vertical transmission
type: Item
instance of: scholarly article
title: Spatial propagation for a reaction-diffusion SI epidemic model with vertical transmission
zbMATH Open document ID: 1501.92208
author: Lin Zhao
author: Haifeng Huo
published in: Mathematical Biosciences and Engineering
publication date: 2022-11-02T00:00:00Z
review text: This paper studies spatial propagation for a reaction-diffusion SI epidemic model with vertical transmission. Let \(S(t,x)\) and \(I(t,x)\) represent the densities of the susceptible individuals and the infective individuals, and \(\theta\in(0,1)\) denotes the proportion of offspring born from an infective individual who is susceptible at birth. The system is given by \(\partial S(t,x)/\partial t=\Delta S(t,x)-\beta S(t,x)I(t,x)+bS(t,x)+\theta b_II(t,x)-(m+kS(t,x)+kI(t,x))S(t,x)\) for \(t>0, x\in\mathbb{R}^N\); \(\partial I(t,x)/\partial t=\Delta I(t,x)+\beta S(t,x)I(t,x)+(1-\theta)b_II(t,x)-\alpha I(t,x)-(m+kS(t,x)+kI(t,x))I(t,x)\) for \(t>0, x\in\mathbb{R}^N\); \(S(0,x)=S_0(x)\ge0\), \(I(0,x)=I_0(x)\ge0\) for \(x\in\mathbb{R}^N\), where \(\beta\) is the incidence rate, \(b\) and \(b_I\) are the birth rate of the susceptible and infective individuals, \(m\) is the mortality rate of the population, and \(1/\alpha\) is the average infection cycle. The paper shows the asymptotic behavior of the solution of the system and shows that the solution of the system converges to the disease-free equilibrium as time goes to infinity when \(R_0\le1\). Here, \(R_0\) is the basic reproduction number. If \(R_0>1\), there exists the minimal speed \(\bar{c}\) such that if \(\|x\|=ct\) with \(0<c<\bar{c}\), the disease is persistent. If \(\|x\|\ge ct\) with \(c>\bar{c}\), then the infection will die out.
reviewed by: Yilun Shang
Mathematics Subject Classification ID: 92D30
Mathematics Subject Classification ID: 35K57
zbMATH DE Number: 7610920
zbMATH Keywords: non-monotone system
zbMATH Keywords: SI epidemic model
zbMATH Keywords: vertical transmission
zbMATH Keywords: spreading speed
MaRDI profile type: Publication
full work available at URL: mbe.2021301
OpenAlex ID: W3180573301
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DOI: 10.3934/MBE.2021301
description: scientific article; zbMATH DE number 7610920