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• A reaction-diffusion fractional-order model for M1 oncolytic virotherapy with CTL immune response is presented. • Threshold parameters and equilibria are determined. • Stability analysis are rigorously established. • Numerical simulations are given to illustrate the analytical results. Despite many years of research, cancer continues to be a major worldwide concern. Several researchers proposed their mathematical models to understand and describe the dynamics of tumor cells under some cancer treatments. In this paper, we present a fractional partial differential equation system to describe the dynamics in space and time of the concentration of normal cells, tumor cells, nutrient, M1 virus, and cytotoxic T lymphocytes (CTL) cells, as well as the interaction between them. We study and analyze the equilibrium points of the presented system. The global stability of these equilibrium points is proved by constructing adequate Lyapunov functional for each equilibrium point. Furthermore, we give some numerical simulations to illustrate our results. [ABSTRACT FROM AUTHOR]

• We propose a new model of fractional-order for the hepatitis C virus (HCV) infection. • The results obtained in this paper are a kind of the power-law and the exponential-law kernel. • The numerical solutions based upon the fundamental theorem of fractional calculus as well as the Lagrange polynomial interpolation. • We investigate the effect of the variation of the fractional-order. • We obtain chaotic behaviors for (HCV) infection. Our main aim in this article is to introduce and investigate a new model of fractional-order model for hepatitis C virus (HCV) infection with immune response, in which the effect of dendritic cells (DC) and cytotoxic T lymphocytes (CTL) on HCV infection is considered. In the first stage, the HCV model was extended to Liouville-Caputo (LC) fractional derivative and Caputo-Fabrizio fractional derivative in the Caputo sense (CFC). In particular, we evaluate the approximate solutions of this model by means of the power law and the exponential law kernel. The approximate solutions are based upon the fundamental theorem of fractional calculus as well as the Lagrange polynomial interpolation. We study the effect of the variation the fractional-order on the behavior of the solutions due to the presence of definitions of new fractional-calculus operators. We observe that the chaotic behavior and illustrate the chaotic of a chronic HCV infection model with immune response by plotting the solutions in the plane. Finally, some numerical simulations using with the aid of Mathematica were done for different values of fractional order. [ABSTRACT FROM AUTHOR]