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Mathematical Analysis of Autonomous and Nonautonomous Hepatitis B Virus Transmission Models

Abdallah Alsammani

TL;DR

The paper develops an improved HBV transmission model incorporating both autonomous and nonautonomous dynamics and treatment effects, with state variables $x$, $y$, and $z$ representing target cells, infected cells, and free virus. It analyzes existence, positivity, and stability of equilibria, deriving a disease-free threshold $R_0$ and establishing local stability via the Jacobian and global stability through a Lyapunov function, linking stability to $R_0$. The nonautonomous extension replaces the constant production rate with a time-varying $\Lambda(t)$ and employs nonautonomous dynamical systems theory to show the existence of a pullback attractor and uniform contraction under suitable conditions. Numerical simulations in MATLAB illustrate stability around the disease-free and endemic equilibria and demonstrate how time-varying interventions can affect HBV dynamics, including scenarios where the autonomous model may exhibit blow-up while the nonautonomous version remains bounded.

Abstract

This study presents an improved mathematical model for Hepatitis B Virus (HBV) transmission dynamics by investigating autonomous and nonautonomous cases. The novel model incorporates the effects of medical treatment, allowing for a more comprehensive understanding of HBV transmission and potential control measures. Our analysis involves verifying unique solutions' existence, ensuring solutions' positivity over time, and conducting a stability analysis at the equilibrium points. Both local and global stability are discussed; for local stability, we use the Jacobian matrix and the basic reproduction number, $R_0$. For global stability, we construct a Lyapunov function and derive necessary and sufficient conditions for stability in our models, establishing a connection between these conditions and $R_0$. Numerical simulations substantiate our analytical findings, offering valuable insights into HBV transmission dynamics and the effectiveness of different interventions. This study advances our understanding of Hepatitis B Virus (HBV) transmission dynamics by presenting an enhanced mathematical model that considers both autonomous and nonautonomous cases.

Mathematical Analysis of Autonomous and Nonautonomous Hepatitis B Virus Transmission Models

TL;DR

The paper develops an improved HBV transmission model incorporating both autonomous and nonautonomous dynamics and treatment effects, with state variables , , and representing target cells, infected cells, and free virus. It analyzes existence, positivity, and stability of equilibria, deriving a disease-free threshold and establishing local stability via the Jacobian and global stability through a Lyapunov function, linking stability to . The nonautonomous extension replaces the constant production rate with a time-varying and employs nonautonomous dynamical systems theory to show the existence of a pullback attractor and uniform contraction under suitable conditions. Numerical simulations in MATLAB illustrate stability around the disease-free and endemic equilibria and demonstrate how time-varying interventions can affect HBV dynamics, including scenarios where the autonomous model may exhibit blow-up while the nonautonomous version remains bounded.

Abstract

This study presents an improved mathematical model for Hepatitis B Virus (HBV) transmission dynamics by investigating autonomous and nonautonomous cases. The novel model incorporates the effects of medical treatment, allowing for a more comprehensive understanding of HBV transmission and potential control measures. Our analysis involves verifying unique solutions' existence, ensuring solutions' positivity over time, and conducting a stability analysis at the equilibrium points. Both local and global stability are discussed; for local stability, we use the Jacobian matrix and the basic reproduction number, . For global stability, we construct a Lyapunov function and derive necessary and sufficient conditions for stability in our models, establishing a connection between these conditions and . Numerical simulations substantiate our analytical findings, offering valuable insights into HBV transmission dynamics and the effectiveness of different interventions. This study advances our understanding of Hepatitis B Virus (HBV) transmission dynamics by presenting an enhanced mathematical model that considers both autonomous and nonautonomous cases.
Paper Structure (18 sections, 9 theorems, 53 equations, 5 figures, 5 tables)

This paper contains 18 sections, 9 theorems, 53 equations, 5 figures, 5 tables.

Table of Contents

  1. Introduction
  2. Model formulation
  3. Properties of Solutions
  4. Stability Analysis
  5. Equilibrium Solutions.
  6. Local Stability
  7. Local Stability of Infection-free Equilibrium
  8. Global Stability
  9. Stability at disease-free equilibrium
  10. Stability at the endemic equilibrium
  11. Nonautonomous HBV Model
  12. Preliminaries of Nonautonomous Dynamical Systems
  13. Model Formulation
  14. Solution Properties
  15. Stability Analysis
  16. ...and 3 more sections

Key Result

theorem thmcountertheorem

For any given $t_0 \in \mathbb{R}$ and $u_0=(x_0,y_0,z_0) \in \mathbb{R}_+ ^3$ there exists $T_{max} = T_{max}(t_0, u_0)$ such that the system from (sys-2) has a solution $(x(t;u_0)$, $y(t;t_0,u_0)$, $z(t;t_0,u_0))$ on $[t_0 , t_0+T_{max})$. Furthermore, If $T_{max}<\infty$ then the solution will bl

Figures (5)

  • Figure 1: The Solution of the model \ref{['sys-2']} around Diseases-free equilibrium.
  • Figure 2: Numerical simulation of the autonomous HBV infection model at the epidemic equilibrium.
  • Figure 3: Numerical simulations of the nonautonomous HBV infection model at the disease-free equilibrium (DFE).
  • Figure 4: Numerical simulations of the nonautonomous HBV infection model (Equations \ref{['NS1']}, \ref{['NS2']}, and \ref{['NS3']}) with time-dependent production number $\Lambda(t)$.
  • Figure 5: The free virus solution $z(t)$ blowup, for the same set of parameters that is used on the nonautonomous case.

Theorems & Definitions (20)

  • theorem thmcountertheorem: Local Existence
  • proof
  • lemma thmcounterlemma
  • proof
  • proof
  • proof
  • theorem thmcountertheorem: Global Existence "Boundedness"
  • proof
  • lemma thmcounterlemma
  • proof
  • ...and 10 more