Spatial propagation for a reaction-diffusion SI epidemic model with vertical transmission (Q2092063)

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.