Stability and Hopf bifurcation of a HBV infection model with capsids and CTL immune response delay

In this paper, an in-host HBV model with two infection routes including cell-virus infection and cell-cell infection, intracellular delay \(\tau _1\), virus replication delay \(\tau _2\) and CTL immune response delay \(\tau _3\) are investigated. Firstly, the positivity and boundedness of all solutions for the model with nonnegative initial values have been established. Next, two key parameters with significant biological importance, the virus reproductive number \(R_0\) and the CTL immune reproductive number \(R_1\), are derived. Subsequently, the stability of equilibria is analyzed by constructing appropriate Lyapunov functionals and applying LaSalle’s invariance principle. The results indicate that intracellular delay \(\tau _1\) and virus replication delay \(\tau _2\) do not affect the stability of the three equilibria. However, a positive CTL immune response delay \(\tau _3\) can lead to stability switches at the endemic equilibrium. By treating the CTL immune response delay as a bifurcation parameter, certain conditions for these stability switches are determined. It is concluded that as the CTL immune response delay increases, the endemic equilibrium becomes unstable, resulting in a Hopf bifurcation in the system. Finally, numerical simulations further validate the theoretical findings.