Stability and Hopf bifurcation for a logistic SIR model with a stage-structure (Q2788460)

The author considers the stage-structured SIR-model NEWLINE\[NEWLINE \left\{\begin{align*}{s^\prime(t)&=r\,s(t)\left(1-\frac{s(t)+i(t)}{K}\right)-i(t)\,p(s(t)), \cr i^\prime(t)&=e^{-\mu \tau}i(t-\tau)\,p(s(t-\tau))-(\sigma+\mu)\,i(t), \cr R^\prime(t)&=\sigma\,i(t)-\mu\,R(t),}\end{align*}\right. NEWLINE\]NEWLINE where \(r,K,\mu\), and \(\sigma\) are all positive reals, \(\tau\geq 0\), and \(p\) is a strictly increasing differentiable function with \(p(0)=0\) and with the property that \(p(s)/s\) is bounded as \(s> 0\). As the first two differential equations are decoupled from the last one, only the system consisting of the first two equations is studied for initial data \(\phi\in C([-\tau,0],\mathbb{R}_+^2)\). This reduced system has always the stationary points \(E_0:=(0,0)\) and \(E_1:=(K,0)\) in \(C([-\tau,0],\mathbb{R}_+^2)\). Additionally, there is a further inner stationary point \(E^*\), provided the basic reproduction number NEWLINE\[NEWLINE R_0=\frac{e^{-\mu\tau}}{\sigma+\mu}\,p(K) NEWLINE\]NEWLINE is greater than \(1\). The author discusses the asymptotic stability of the stationary point \(E_1\) in case \(R_0\leq 1\). On the other hand, in the situation \(R_0>1\), the permanence of the system, the local stability of \(E^*\) and some sufficient conditions for the occurrence of Hopf bifurcation at \(E^*\) are studied. Moreover, in the later case, the author addresses also the questions about the direction of the Hopf bifurcation and about the stability of the bifurcating periodic orbits. Finally, some numerical simulations are performed.