Spreading dynamics for a time-periodic nonlocal dispersal epidemic model with delay and vaccination

It is known that vaccination plays an important strategy in eliminating infectious diseases. In this paper, we investigate the spreading dynamics for a time-periodic nonlocal dispersal epidemic model with delay and vaccination. We first establish the spreading speed of the model and an abstract framework on the existence of time-periodic traveling waves, which will help us to derive the existence of the super-critical and critical time-periodic traveling waves. Then we show that the spreading speed coincides with the minimal waves speed of time-periodic traveling waves. Further, we consider the effects of delay, periodicity, nonlocality and vaccination on the spreading speed. In the absence of delay, we find a large class of the time-periodic systems that have the same spreading speed. When delay is introduced, some numerical simulations reveal that the spreading speed initially exhibits oscillatory behavior and ultimately converges to a constant as time-period increases. Moreover, we observe that both delay and efficacy of vaccination decrease the spreading speed; both diffusion rate and nonlocality of infectious individuals increase the spreading speed; while the diffusion rates, nonlocalities of susceptible and vaccinated individuals do not affect the spreading speed. In particular, it is worth mentioning that the spreading speed is highly sensitivity to the efficacy of vaccination than the rate of vaccination.