Author(s)

Abdelrazig K. Tarboush^1,2, Zhengdi Zhang¹

¹Faculty of Science, Jiangsu University, Zhenjiang, China.
²Department of Mathematics, Faculty of Education, University of Khartoum, Khartoum, Sudan.

To understand the influence of seasonal periodicity and environmental heterogeneity on the transmission dynamics of an infectious disease, we consider asymptotic periodicity in the fecally-orally epidemic model in a heterogeneous environment. By using the next generation operator and the related eigenvalue problems, the basic reproduction number is introduced and shows that it plays an important role in the existence and non-existence of a positive T-periodic solution. The sufficient conditions for the existence and non-existence of a positive T-periodic solution are provided by applying upper and lower solutions method. Our results showed that the fecally-orally epidemic model in a heterogeneous environment admits at least one positive T-periodic solution if the basic reproduction number is greater than one, while no T-periodic solution exists if the basic reproduction number is less than or equal to one. By means of monotone iterative schemes, we construct the true positive solutions. The asymptotic behavior of periodic solutions is presented. To illustrate our theoretical results, some numerical simulations are given. The paper ends with some conclusions and future considerations.

Fecally-Orally Epidemic Model, Basic Reproduction Number, Time Periodicity, Asymptotic Behavior

Tarboush, A. and Zhang, Z. (2019) Asymptotic Periodicity in the Fecally-Orally Epidemic Model in a Heterogeneous Environment. Journal of Applied Mathematics and Physics, 7, 1027-1042. doi: 10.4236/jamp.2019.75069.