Chronic immune activation and viral latency are among the main factors behind the persistence of Human Immunodeficiency Virus (HIV) and other viral diseases. Immune activation is chiefly lead by dendritic cells which detect the presence of Virus (HIV) and notify other immune cells particularly the helper T cells via the process called antigen presentation. However, antigen presentation process increases the chances of T cells to be infected by the viral particles trapped by dendritic cells. In this paper, a time-fractional diffusion model is proposed to investigate the impact of dendritic cells on the spread of HIV among susceptible T cells with respect to time and anomalous diffusion posed by cells crowding. The equilibrium points of the model are obtained and analysed. The disease free steady state proved to be stable both in spatially homogeneous case and in the presence diffusion and chemotaxis. The endemic steady state is stable in the absence of diffusion and chemotaxis, however the presence of diffusion induces instability if a thresshold value is exceeded. A priori estimates were also obtained in the appropriate Sobolev spaces. Furthermore, the numerical experiments were conducted to examine the dynamic behaviour of cells densities with respect to time and the sub-diffusion parameter.