Threshold Dynamics Of Two Almost Periodic Epidemic Models

In view of the great threat of infectious disease to human health, the prevention and control of infectious diseases have been a major problem in the world. Due to the influence of seasonal factors, the disease transmission presents obviously periodic feature, and it is more reasonable to consider the disease transmission from periodic perspective. Almost periodic functions as a generalization of periodic functions,almost periodic models can better reveal the influence of seasonal factors on the disease transmission.Firstly, we consider an almost periodic malaria transmission model with the time-delayed input of vector. It is shown that the disease is uniformly persistent when the basic reproduction ratio R0> 1, and it will die out when R0< 1 under the assumption that there exists a small invasion. Furthermore, the global stability of the disease-free almost periodic state is obtained provided that the disease-induced death rate is null. The numerical simulations of the model show that the appropriate protective measures can prompt malaria to die, and the periodic epidemic models may overestimate or underestimate the malaria risk.Secondly, the persistence and extinction of an almost periodic epidemic model in a patchy environment are studied. It is shown that the disease cannot invade the disease-free state if the exponential growth bound is less than zero and can invade if it is greater than zero. It is also shown that there exists an almost periodic solution which is globally attractive when each patch admits the same dispersal rate. Finally,numerical simulations illustrate the above results.Finally, a simple discussion about the study of spatial and temporal heterogenous epidemic models completes this paper.