Evolution systems containing fractional derivatives can result to suitable mathematical models for describing better and important physical phenomena. In this paper, we consider a multi-components nonlinear fractional-in-space reaction-diffusion equations consisting of an improved deterministic model which describe the spread of hepatitis B virus disease in areas of high endemic communities. The model is analyzed. We give some useful biological results to show that the disease-free equilibrium is both locally and globally asymptotically stable when the basic reproduction number is less than unity. Our findings of this paper strongly recommend a combination of effective treatment and vaccination as a good control measure, is important to record the success of HBV disease control through a careful choice of parameters. Some simulation results are presented to support the analytical findings.
Keywords: Disease free equilibrium; Exponential integrator; Fourier spectral method; Fractional reaction–diffusion; Nonlinear PDEs; Numerical simulations; Reproduction number.