Abstract
In this paper, we propose and investigate a fractional diffusive model for hepatitis B virus (HBV) infection with capsids and immune response presented by cytotoxic T lymphocyte (CTL) cells. We derive the conditions for global asymptotic stability of the steady states of the model in terms of the basic reproduction numberR0and the immune response reproduction numberR1. By constructing appropriate Lyapunov functionals, it is shown that the infection-free equilibrium is globally asymptotically stable whenR0≤ 1, the immune-free infection equilibrium is globally asymptotically stable whenR1≤ 1 <R0and the infection equilibrium with CTL immune response is globally asymptotically stable whenR1> 1. Numerical simulations are performed to illustrate the analytical results.
Highlights
Many mathematical models used fractional differential equations (FDEs) have been proposed and analyzed to gain insights into the dynamics of hepatitis B virus (HBV) infection in vivo
We have studied a time-fractional diffusion model for HBV infection with capsids, cytotoxic T lymphocyte (CTL) immune response and spatial diffusion in capsids and virus
We have shown that the global dynamics of the model is completely determined by two threshold parameters that are the basic reproduction number R0 and the reproduction numbers for cellular immunity R1
Summary
Many mathematical models used fractional differential equations (FDEs) have been proposed and analyzed to gain insights into the dynamics of HBV infection in vivo. Bachraoui et al [2] proposed a fractional order-model for HBV infection with capsids and CTL immune response that improved and generalized the mathematical models formulated by ordinary differential equations (ODEs) in [19, 20] and the FDE models introduced in [6, 27, 32] by considering the Hattaf’s incidence rate [11] that includes the common types such as the bilinear incidence rate, the saturated incidence rate and the Beddington-De Anglis functional response [4, 9]. Manna and Hattaf [16] studied an HBV infection model which contains two arms of immunity, three time delays, capsids, general incidence rate, and allow the movement of capsids and viruses by diffusion.
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