  In the theoretical research of Hepatitis B virus, mathematical models of its
transmission mechanism have been thoroughly investigated, while the dynamics of
the immune process in vivo have not. At present, nearly all the existing models
are based on integer-order differential equations. However, models restricted
to integer-order cannot depict the complex dynamical behaviors of the virus.
Since fractional-order models possess property of memory and memory is the main
feature of immune response, we propose a fractional-order model of Hepatitis B
with time-delay to further explore the dynamical characters of the HBV model.
First, by using the Caputo fractional differential and its properties, we
obtain the existence and uniqueness of the solution. Then by utilizing
stability analysis of fractional-order system, the stability results of the
disease-free equilibrium and the epidemic equilibrium are studied according to
the value of the basic reproduction number. Moreover, the bifurcation diagram,
the largest Lyapunov exponent diagram, phase diagram, Poincar\'e section and
frequency spectrum are employed to confirm the chaotic characteristics of the
fractional HBV model and to figure out the effects of time-delay and fractional
order. Further, a theory of the asymptotical stability of nonlinear autonomous
system with delay on the basis of Caputo fractional differential is proved. The
results of our work illustrates the rich dynamics of fractional HBV model with
time-delay, and can provide theoretical guidance to virus dynamics study and
clinical practice to some extent.
