Dynamical behaviors of a stochastic HTLV-I infection model with general infection form and Ornstein–Uhlenbeck process

In this study, considering the Ornstein–Uhlenbeck process to perturb the infection rate, we develop a HTLV-I infection model with general infection form. By constructing several suitable Lyapunov functions and a compact set, and then using the strong law of numbers and Fatou’s lemma, we obtain sufficient conditions for the existence and uniqueness of the ergodic stationary distribution $\eta \left(\cdot \right)$ for the stochastic model. This implies long-term persistence of HTLV-I infection in a biological sense. Moreover, by using Itô’s integral stochastic model is transformed into the corresponding linearized system. Then solving the Fokker–Planck equation, we obtain the exact expression of probability density function around the quasi-equilibrium of the stochastic model. In addition, sufficient conditions for the extinction of HTLV-I infection are established. Finally, considering different incidence rate functions, we employ numerical simulations to support our results.