Finite-time stability and optimal impulsive control for age-structured HIV model with time-varying delay and Lévy noise

This paper investigates the finite-time stability and optimal impulsive control for stochastic age-structured HIV model with time-varying delay. A stochastic noise is introduced by using the Levy process to characterize the phenomenon of discontinuous jumps in virus transmission, which cannot be described by a continuous stochastic process (e.g., Brownian motion). By employing the comparison theorem and the bounded impulsive interval method, we obtain the sufficient conditions of finite-time stability for a stochastic HIV system. The effects of impulse, delay and Levy noise on the finite-time stability are considered in our sufficient conditions. Furthermore, optimal impulsive control is studied to seek the optimal and cost-effective strategy for HIV treatments, which shows that control strategies play an important role in HIV virus transmissions. Numerical simulations are performed to illustrate the validity of our results.