Abstract
The main purpose of this paper is to study the propagation dynamics for a class of time‐periodic reaction–diffusion systems with network structures. In the first part, by using the persistence theory, we obtain threshold results for the extinction and uniform persistence of the corresponding periodic ordinary differential system. The second part is concerned with the asymptotic speed of spread and traveling wave solutions. The uniform boundedness of solutions is proved by employing the refined high‐dimensional local ‐estimate and abstract periodic evolution theories and the spreading properties of the corresponding solutions are established. We also prove the existence of the critical periodic traveling wave with wave speed by using a delicate limitation argument. Finally, these results are applied to a multistage epidemiological model.