A stage-structured SEIR model with time-dependent delays in an almost periodic environment

In this paper, we propose and investigate an almost periodic SEIR model with stage structure and latency, in which time-dependent maturation and incubation periods are incorporated. Two threshold parameters for the persistence and extinction of population and disease are introduced: the basic reproduction ratio $\hat{R}_{0}$ for population and the basic reproduction ratio $R_{0}$ for disease. If $\hat{R}_{0}<1 the="" population="" extinction="" state="" is="" globally="" attractive="" in="" the="" case="" where="" hat="" r="" _="" 0="">1$, it is shown that the disease tends to die out if $R_{0}<1 while="" remains="" persistent="" if="" r_="" 0="">1$. By virtue of numerical simulations, we verify the analytic results and investigate the effects of the fluctuations of maturation and incubation periods on disease transmission.