A deterministic time-delayed SVIRS epidemic model with incidences and saturated treatment

Abstract

A novel nonlinear delayed susceptible–vaccinated–infected–recovered–susceptible (SVIRS) epidemic model with a Holling type II incidence rate for fully susceptible and vaccinated classes, a saturated treatment rate, and an imperfect vaccine given to susceptibles is proposed herein. Analysis of the model shows that it exhibits two equilibria, namely disease-free and endemic. The basic reproduction number $$R_0$$ is derived, and it is demonstrated that the disease-free equilibrium is locally asymptotically stable when $$R_0<1$$ and linearly neutrally stable when $$R_0=1$$. Furthermore, bifurcation analysis is performed for the undelayed model, revealing that it exhibits backward and forward bifurcation when the basic reproduction number varies from unity. The stability behavior of the endemic equilibrium is also discussed, revealing that oscillatory and periodic solutions may appear via Hopf bifurcation when regarding delay as the bifurcation parameter. Moreover, numerical simulations are carried out to illustrate the theoretical findings.