Canard phenomenon for an SIS epidemic model with nonlinear incidence (Q401039)

The authors analyze the susceptible-infected-susceptible (SIS) epidemic model which is expressed after rescaling of the variables and parameters in the form of a singularly perturbed system NEWLINE\[NEWLINE\begin{aligned} \frac{du}{ds}&=\left[(v-u)(1+u)-(\epsilon\lambda+\gamma^2)\right]u, \\ \frac{dv}{ds}&=\epsilon\left[v(1-kv)-\mu u\right], \end{aligned}NEWLINE\]NEWLINE where \(u>0,\) \(v>0,\) the parameters \(\mu,\) \(\gamma,\) \(k\) are positive and \(0<\epsilon\ll 1.\)NEWLINENEWLINEIt is shown that for each \(\gamma>1\) there exists a unique value \(\mu(\gamma)\) such thatNEWLINENEWLINE\ (1) from the slow-fast cycle at most one canard limit cycle can bifurcate if \(\mu\neq\mu(\gamma),\) the limit cycle is hyperbolic (when exists), and stable for \(\mu>\mu(\gamma)\) and unstable for \(\mu<\mu(\gamma);\)NEWLINENEWLINE\ (2) at most two hyperbolic limit cycles (or at most one semi-stable limit cycle) can bifurcate if \(\mu=\mu(\gamma),\) and in the case of two limit cycles the outside one is stable and the inner one is unstable.NEWLINENEWLINEFinally, the authors summarize the main results, and show the epidemiological meaning of existence of canard cycles.