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.
Highlights
The geographic transmission of infectious diseases is an important issue in mathematical epidemiology and various epidemic models have been proposed and analyzed by many researchers [1] [2] [3]
By using the 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
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
Summary
The geographic transmission of infectious diseases is an important issue in mathematical epidemiology and various epidemic models have been proposed and analyzed by many researchers [1] [2] [3]. The periodicity of environmental factors is realistic and highly important for the dynamics of infectious diseases Taking this fact into account, we extend model (1) to the following reaction-diffusion system. In order to study the attractivity of T-periodic solutions of problem (2), (3), we consider model (2) under the initial condition. Various computation algorithms for numerical solutions of the periodic boundary problem were studied in [11]. It is important to mentioned that there are other standard approaches to derive the periodic solution of a weakly-parabolic systems, Hopf bifurcation theorem and Lyapunov functional [7] [15], numerical methods [11] [16], Poincaré index [17], Schauder fixed point theorem [18] [19], etc.
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