Dynamics of a deterministic time-delayed fractional order SVIRS epidemic model with logistic growth, saturated incidence and treatment functions

Abstract During an infectious disease outbreak, controlling transmission and managing new cases within a population are inherently challenging. These difficulties, however, can be reduced through vaccination, public awareness initiatives, and effective treatment strategies. In this study, we consider the population’s prior experience with previously encountered diseases and introduce a fractional-order, time-delayed epidemic model that incorporates a vaccinated population with an imperfect vaccine, memory effects, logistic growth of susceptibles, saturated incidence, and treatment rates. The model aims to capture the transmission dynamics of infection and identify potential control mechanisms. The time delay reflects the disease’s incubation period. The model is analyzed mathematically for two possible equilibria: the disease-free equilibrium (DFE) and the endemic equilibrium (EE). The critical threshold parameter, the basic reproduction number R 0 , is derived, and the local stability of both the DFE and EE is examined with and without time delay. In addition, the potential occurrence of a Hopf bifurcation is investigated for varying fractional order α , ( 0 < α ⩽ 1 ). The analysis shows that a Hopf bifurcation arises when the time delay τ exceeds a certain critical value. An explicit formula is further obtained to characterize the stability and direction of the periodic solutions emerging from the bifurcation point for fractional order α = 1 . A sensitivity analysis is also carried out with respect to both the basic reproduction number and the system’s state variables at the EE. Finally, numerical simulations are presented to support and illustrate the theoretical results.