  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.
