The infected frontier in an SEIR epidemic model with infinite delay (Q379072)

The authors study a SEIR epidemic model with infinite delay as follows: NEWLINE\[NEWLINE\begin{cases} S_t-d_1\Delta S=\Lambda-\beta \frac{S}{N}\int_0^{\infty}k(a)\mu E(x,t-a)e^{-\omega a}da -dS, \\ E_t-d_2\Delta E=\beta \frac{S}{N}\int_0^{\infty}k(a)\mu E(x,t-a)e^{-\omega a}da-\sigma E,\\ I_t-d_3\Delta I=\mu E-\omega I,\\ R_t-d_4\Delta R=rI-dR, \end{cases}\tag{1}NEWLINE\]NEWLINE where \(N=S+E+I+R,\;\omega=d+\delta+r,\;\sigma=\mu+d.\) Two aspects are studied in this article: (1) defined in a fixed and bounded domain with Neumann boundary condition and initial value; (1) with free boundary. If (1) is defined in a fixed and bounded domain, the positivity, boundedness of solutions and the stability of equilibria for the model are discussed. If (1) is defined with a free boundary (the free boundary \(x=h(t)\) satisfies the famous Stefan condition), the existence and uniqueness of solution, and spreading and the vanishing of the system are investigated. More specifically, let \(R_0\) be the basic reproduction number, whether the disease will die out or not depends on \(R_0<1\) or \(R_0>1\). While for free boundary problem, they show that under certain conditions the disease will die out even for \(R_0>1\). That is, besides \(R_0\), the initial size of the infected domain and the diffusivity of the disease in a new region also have a non-negligible influence to the disease transmission.