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In this work, we propose a time‐delayed reaction–diffusion model to describe the propagation of infectious viral diseases like COVID‐19. The model is a two‐dimensional system of partial differential equations that describes the interactions between disjoint groups of a human population. More precisely, we assume that the population is conformed by individuals who are susceptible to the virus, subjects who have been exposed to the virus, members who are infected and show symptoms, asymptomatic infected individuals, and recovered subjects. Various realistic assumptions are imposed upon the model, including the consideration of a time‐delay parameter which takes into account the effects of social distancing and lockdown. We obtain the equilibrium points of the model and analyze them for stability. Moreover, we examine the bifurcation of the system in terms of one of the parameters of the model. To simulate numerically this mathematical model, we propose a time‐splitting nonlocal finite‐difference scheme. The properties of the model are thoroughly established, including its capability to preserve the positivity of solutions, its consistency, and its stability. Some numerical experiments are provided for illustration purposes. [ABSTRACT FROM AUTHOR]