Bifurcation analysis of an SIRS epidemic model with generalized incidence.

*(English)*Zbl 1088.34035The authors consider an SIRS epidemic model with generalized incidence
\[
\begin{cases} \frac{dS}{dt}=\Pi-\beta[1+f(I;v)]\frac{IS^p}{N}-\mu S+\delta R,\\ \frac{dI}{dt}=\beta[1+f(I;v)]\frac{IS^p}{N}-(\mu+\alpha)I,\\ \frac{dR}{dt}=\alpha I-(\mu+\delta)R.\end{cases}
\]

Extending previous investigations, it is assumed that the natural immunity acquired by infection is not permanent but wanes with time. The nonlinearity of the functional form of the incidence of infection, which is subject only to a few general conditions, is biologically justified. The stability analysis of the associated equilibria is carried out, and the threshold quantity (\(\operatorname{Re}_0\)) that governs the disease dynamics is derived. It is shown that \(\operatorname{Re}_0\), called the basic reproductive number, is independent of the functional form of the incidence. Local bifurcation theory is applied to explore the rich variety of dynamical behavior of the model. Normal forms are derived for the different types of bifurcation that the model undergoes, including Hopf, saddle-node, and Bogdanov-Takens. The first Lyapunov coefficient is computed to determine various types of Hopf bifurcation, such as forward or backward, subcritical or supercritical. The existence of a saddle-node bifurcation, at the turning point of backward bifurcation, is established by applying Sotomayor’s theorem. The Bogdanov-Takens normal form is used to formulate the local bifurcation curve for a family of homoclinic orbits arising when a Hopf and a saddle-node bifurcation merge. These theoretical results are detailed and numerically illustrated for two different kinds of incidence, corresponding to unbounded and saturated contact rates. The coexistence of two limit cycles, due to the occurrence of a backward subcritical Hopf bifurcation, is also demonstrated. These results lead to the determination of ranges for the periodicity behavior of the model based on two critical parameters: the basic reproductive number and the rate of loss of natural immunity.

Extending previous investigations, it is assumed that the natural immunity acquired by infection is not permanent but wanes with time. The nonlinearity of the functional form of the incidence of infection, which is subject only to a few general conditions, is biologically justified. The stability analysis of the associated equilibria is carried out, and the threshold quantity (\(\operatorname{Re}_0\)) that governs the disease dynamics is derived. It is shown that \(\operatorname{Re}_0\), called the basic reproductive number, is independent of the functional form of the incidence. Local bifurcation theory is applied to explore the rich variety of dynamical behavior of the model. Normal forms are derived for the different types of bifurcation that the model undergoes, including Hopf, saddle-node, and Bogdanov-Takens. The first Lyapunov coefficient is computed to determine various types of Hopf bifurcation, such as forward or backward, subcritical or supercritical. The existence of a saddle-node bifurcation, at the turning point of backward bifurcation, is established by applying Sotomayor’s theorem. The Bogdanov-Takens normal form is used to formulate the local bifurcation curve for a family of homoclinic orbits arising when a Hopf and a saddle-node bifurcation merge. These theoretical results are detailed and numerically illustrated for two different kinds of incidence, corresponding to unbounded and saturated contact rates. The coexistence of two limit cycles, due to the occurrence of a backward subcritical Hopf bifurcation, is also demonstrated. These results lead to the determination of ranges for the periodicity behavior of the model based on two critical parameters: the basic reproductive number and the rate of loss of natural immunity.

Reviewer: Chen Lan Sun (Beijing)

##### MSC:

34C23 | Bifurcation theory for ordinary differential equations |

92D30 | Epidemiology |

34C20 | Transformation and reduction of ordinary differential equations and systems, normal forms |

34C25 | Periodic solutions to ordinary differential equations |