TITLE:
Asymptotic Periodicity in the Fecally-Orally Epidemic Model in a Heterogeneous Environment
AUTHORS:
Abdelrazig K. Tarboush, Zhengdi Zhang
KEYWORDS:
Fecally-Orally Epidemic Model, Basic Reproduction Number, Time Periodicity, Asymptotic Behavior
JOURNAL NAME:
Journal of Applied Mathematics and Physics,
Vol.7 No.5,
May
16,
2019
ABSTRACT: 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.