Switching vaccination schemes for epidemic models with distributed time delay and two types of noise

Abstract

Abstract Novel switched HIV/AIDS (human immunodeficiency virus/acquired immune deficiency syndrome) epidemic models with distributed time delay and bounded noise and Gaussian white noise are developed and investigated using stochastic Itô’s lemma and the Lyapunov–Razumikhin method. New criteria depending on these factors are established to confirm that the disease-free equilibrium of the model is stochastically asymptotically stable as the threshold parameter is less than unity, which implies that the disease eventually disappears theoretically. Otherwise, the disease persists weakly. Further, the main results show that the threshold values are related to two types of noise and time delay. Pulse control strategies are then applied to two types of the infected population, the susceptible population, and the infected population, respectively. More precisely, the effects of each control strategy on the stochastic solution of the model are evaluated to justify the relation between control parameters and threshold parameters of the model. In comparison with the basic reproduction number of the model with pulse control, it is easily found that the main results in these references are improved and extended. Finally, four examples are presented to support the main results, and one future research direction is suggested.

Keywords

• Switched epidemic model
• Distributed time delay
• Bounded noise and Gaussian white noise
• Stochastic asymptotic stability
• Lyapunov–Razumikhin method
• Pulse vaccination strategies