Dynamics on Hepatitis B Virus Infection In Vivo with Interval Delay

In view of the molecular biological mechanism of the cytotoxic T lymphocytes proliferation induced by hepatitis B virus infection in vivo, a novel dynamical model with interval delay is proposed. The interval delay is determined by two delay parameters, namely delay center and delay radius. We derive the basic reproduction number R-0 for the viral infection and obtain that the virus-free equilibrium (VFE) is globally asymptotically stable if R-0 < 1. When R-0 > 1, besides VFE, the unique virus-present equilibrium (VPE) exists and the conditions of its asymptotical stability are obtained. Moreover, we study the Hopf bifurcations induced by the two delay parameters. Although there is no mitotic term in the target-cell dynamics, the results indicate that both these delay parameters can lead to periodic fluctuations at VPE, but only the smaller delay radius will destabilize the system, which is different from the classical discrete delay or distributed delay. Numerical simulations indicate that the proposed model can capture the profiles of the clinical data of two untreated chronic hepatitis B patients. The ability of interval delay to destabilize the system is between discrete delay and distributed delay, and the delay center plays the primary role. Pharmaceutical treatment can affect the stability of VPE and induce the fast-slow periodic phenomenon.