The Global Behavior of a Periodic Epidemic Model with Travel between Patches

Luosheng Wen,Xin Liang,Fengling Zeng,Bin Long

doi:10.1155/2012/295060

Abstract

We establish an SIS (susceptible‐infected‐susceptible) epidemic model, in which the travel between patches and the periodic transmission rate are considered. As an example, the global behavior of the model with two patches is investigated. We present the expression of basic reproduction ratio R0 and two theorems on the global behavior: if R0 < 1 the disease‐free periodic solution is globally asymptotically stable and if R0 > 1, then it is unstable; if R0 > 1, the disease is uniform persistence. Finally, two numerical examples are given to clarify the theoretical results.

Highlights

Epidemic models have been paid intensive attention for recent decades
The recovery rate of infectious individuals from ith patch who are present in region j is γij satisfies the following basic assumptions for Nip ∈ 0, ∞ : t
A3 Bi t, ∞ < dii t, i 1, 2, . . . , n; 1, 2, . . . , n; and the birth function Bi t, Nip B t /Nip C t can be found in the biological literature. We assume that these coefficients are functions being continuous, positive ω-periodic in t and we can obtain a periodic SIS epidemic model, in which individuals can travel among n patches

Summary

Introduction

Epidemic models have been paid intensive attention for recent decades. In the models, population is divided into several compartments, for example, susceptible S , infected I , and recovery R by individual state. Combining the mobility and seasonality, we consider an SIS epidemic model, in which people can travel among n patches and the transmission rate is a periodic function. The recovery rate of infectious individuals from ith patch who are present in region j is γij satisfies the following basic assumptions for Nip ∈ 0, ∞ :. N; and the birth function Bi t, Nip B t /Nip C t can be found in the biological literature We assume that these coefficients are functions being continuous, positive ω-periodic in t and we can obtain a periodic SIS epidemic model, in which individuals can travel among n patches.

The Basic Reproduction Ratio

The Threshold Dynamics

Numerical Simulations