In this paper, we propose a time-fractional derivative model, building upon previous works, to describe the transmission dynamics of hepatitis C virus (HCV), integrating time-fractional derivative, spatial diffusion and cellular immune response. The model is structured in four compartments: \(X\) representing healthy hepatocytes, \(Y\) infected hepatocytes, \(V\) free viral particles and \(Z\) cytotoxic T cells (CTLs) involved in the immune response. We rigorously establish the existence, uniqueness, positivity and boundedness of the model solution. The basic reproduction number \(R_0\) and the immune reproduction number \(R_{\textrm{CTL}}\) are determined to analyze the dynamics evolution of infection and immune response. The model has three equilibrium states: the equilibrium without infection \(E_f\) , the immune equilibrium \(E_1\) and the endemic equilibrium \(E_2\) . By applying the LaSalle principle of invariance, we identify the conditions that ensure the global stability of these equilibria. It was demonstrated that when \(R_0 < 1\) , the infection-free equilibrium point is globally asymptotically stable. The immune-free equilibrium point \(E_1\) is globally asymptotically stable if \(R_0>1\) and \(R_{\textrm{CTL}}<1\) . In addition, the endemic equilibrium point \(E_2\) is globally asymptotically stable if and only if both reproduction numbers are greater than unity. Using the Euler scheme for the fractional derivative, numerical results are presented to study the effect of the model parameters and confirm the theoretical results regarding the global stability of the solutions.
