Dynamical Behavior of Nonautonomous and Stochastic HBV Infection Model

Abstract

Mathematical modeling of population and transmission dynamics of an infectious disease considered a critical theoretical epidemiology method provides a strong understanding of the virus dynamics. This dissertation studies the Hepatitis B Virus Infection dynamical behavior with different approaches using mathematical modeling and dynamic systems theory.

Firstly, we propose an autonomous differential equations system, where all the parameters are constants. We show the basic solution properties, such as the existence and uniqueness of solutions, and as with any population model, we show that the solution is always positive. Next, we show the system has exactly two equilibrium points. We then discuss the stability analysis at each equilibrium point, then we obtain sufficient conditions that make the system exponentially stable by constructing an appropriate Liponouv function.

Secondly, we consider the case where the target cells' production rate is time-dependent, making the system nonautonomous. We use tools from the nonautonomous dynamical systems to show the solution exists, unique, and stay positive for all time. Then we prove that the system has a pullback absorbing and a positively invariant set, which implies the system has a unique global pullback attractor that guarantees the existence of entire solution. However, the results provide sufficient conditions for the existence of nonautonomous attractors and Singleton attractors.

Thirdly, We consider the HBV infection model with stochastic perturbation, and we investigate the longtime dynamics behavior of the stochastics model. First, we show the existence, uniqueness, and positiveness of solutions. For the stability analysis, we prove that if the reproductive number corresponding to the deterministic system $\mathcal{R}_0 <1$ and the parameters satisfy some conditions, then the system is almost surely exponentially stable. Furthermore, we provide sufficient conditions that guarantee that a unique stationary ergodic distribution exists for $\mathcal{R}_0 >1$, which implies the stochastic model's stability around the endemic equilibrium of the corresponding deterministic model by constructing suitable stochastic Lyapunov functions.

Finally, we provide numerical results to illustrate and support the theoretical results of this study. All the simulation codes are written using MATLAB.