Propagation dynamics for an epidemic model in RN with time-almost periodic and spatially periodic media

This paper investigates a time-almost periodic and spatially periodic epidemic model with non-monotone incidence in R N . For the model involving noncooperative nonlinearities, we first characterize the critical speed of transition waves in any given direction, and show the existence of transition waves in a given direction whenever the mean speed exceeds the critical speed corresponding to that direction. Under cooperative nonlinearities, it is further proven that the transition waves in a given direction are periodically monotonic with respect to each spatial coordinate direction. Subsequently, in the noncooperative case, upper and lower estimates of the spreading speeds are established under initial conditions with exponential decay in a given direction. Finally, we show that the critical wave speed of the noncooperative epidemic model is minimal, i.e., transition waves with slow exponential decay in a given direction no longer exist when the mean speed is less than the critical speed in that direction.