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

Dynamics analysis of a novel fractional HBV system with immune delay