Threshold Dynamics Of Lyme And Tuberculosis Epidemic Models In An Almost Periodic Environment

Infectious disease is one of the major diseases which threatens human’s health.Its prevention,immunization and treatment have attracted the attention of many mathematicians.Considering the influence of seasonal factors,many scholars have studied the periodic epidemic models.Evidences indicate that the rate of infection,cure,birth,death and other parameters of the model aways don’t share a common period because of the effect of temperature,precipitation and other factors.Espe-cially,if the periods of these periodic coefficients have no common integer multiple,the model isn’t a periodic system.Mathematically,we can treat such a model as an almost periodic system.In fact,almost periodic functions are a generalization of periodic functions,then almost periodic model make it more general and reasonable for characterizing the transmission of Lyme and other diseases.We give the definition of basic reproduction number R0 for the model of Lyme disease under almost periodic case and also prove that it’s an important threshold parameter for global extinction or uniform persistence of the disease.That is to say,if R0<1,the disease free almost periodic solution is globally attractive.If R0>1,the system admits a unique almost periodic positive solution which is globally attractive.Considering the effects of seasonal changes and spatial diffusion of the popula-tion on Tuberculosis infectious disease,we study the reaction-diffusion SEIR model under almost periodic case.Meanwhile,we give the definition of the basic repro-duction number RO and also prove that it’s an important threshold parameter for extinction and uniform persistence of the disease.