Pulse vaccination on SEIR epidemic model with nonlinear incidence rate

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Abstract

In this paper, we consider an SEIR epidemic model with two time delays and nonlinear incidence rate, and study the dynamical behavior of the model with pulse vaccination. By using the Floquet theorem and comparison theorem, we prove that the infection-free periodic solution is globally attractive when R<1, and using a new modelling method, we obtain a sufficient condition for the permanence of the epidemic model with pulse vaccination when R>1.

Introduction

Pulse vaccination has been testified to be an effective strategy in preventing such viral infections as rabies, yellow fever, poliovirus, hepatitis B, parotitis, and encephalitis B. Eventually, pulse vaccination will probably be applied in preventing malaria, some forms of heart disease, and cancer. Even venereal disease may someday be the target of vaccination programs. Recently, pulse vaccination epidemic models have been the subject of intense theoretical analysis [1], [2], [3], [4]. Theoretical results show that the pulse vaccination strategy can be distinguished from the conventional strategies in leading to disease eradication at relatively low values of vaccination [5].

Epidemiological models with pulse vaccination have been set up and studied for many years, most of the research literatures on these of models are established by impulsive ODE [1], [6], [7], [8], [9], [10]. However, impulsive equations with time delay have seldom been studied. In this paper we formulate an impulsive-delayed ODE model to describe the pulse vaccination process and the latent period of the disease. We study the dynamical behaviors of the delayed model with pulse vaccination. The main purpose of this paper is to show that large vaccination rate or short period pulsing or long latent period of the disease implies the disease dies out. The second purpose of this paper is to establish sufficient conditions that the disease is uniformly persistent, that is, there is a positive constant p (independent of the choice of the solution) such that I(t)>p for all large t. The organization of this paper is as follows: Section 2 formulates the SEIR epidemic model with time delay and pulse vaccination. To prove our main results we also provide some lemmas. In Section 3, the global attractivity condition of the disease-free periodic solution is presented. The sufficient condition for the permanence of the SEIR model is obtained in Section 4. In Section 5, we give some discussion.

Section snippets

Model formulation and preliminary

In the following model, we study a population that is partitioned into four classes, the susceptible, exposed, infections and recovered, with sizes denote by S, E, I and R, respectively, and consider nonlinear incidence βISq, the latent period of disease, the temporary immunity period of the recovered and pulse vaccination strategy.

Motivated by Gao et al. [11], we consider the following mathematical model:S˙(t)=μ(a-S(t))-βSq(t)I(t),E˙(t)=βSq(t)I(t)-βSq(t-ω)I(t-ω)e-μω-μE(t),I˙(t)=βSq(t-ω)I(t-ω)e-

Global attractivity of infection-free periodic solution

In this section, we first demonstrate the existence of an infection-free periodic solution, in which infectious individuals are entirely absent from the population permanently, i.e., I(t)=0,t0. Under this condition, we show below that the susceptible population oscillates with T in synchronization with periodic pulse vaccination, and the growth of the susceptible individuals must satisfy:S˙(t)=μ(a-S(t)),tnT,nZ+,S(t+)=(1-δ)S(t),t=nT,nZ+.By Lemma 2.1, we know that periodic solution of system

Permanence

In this section we say the disease is endemic if the infections population persists above a certain positive level for sufficiently large time. Before starting our results, we give the following definition.

Definition 4.1

System (2.2) is said to be uniformly persistent if there is a p>0 (independent of the initial data) such that every solution (S(t),I(t)) with initial condition (2.3) of system (2.2) satisfies

limtinfS(t)p,limtinfI(t)p.

Definition 4.2

System (2.2) is said to be permanent if there exists a compact region Ω

Discussion

In this paper, we introduce time delay, pulse vaccination and nonlinear incidence rate βISq(q>0) in the SEIR model, and we analyze in theory that the latent period of disease and pulse vaccination bring effects on infection–eradication and the permanence of epidemic disease. Theorem 3.1, Theorem 4.1 imply that the disease dynamics of (2.2) is completely determined by R<1 and R>1. Hence, the vaccination effects depend on whether R can be reduced to be below unity or not. Recalling

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This work is supported by the National Natural Science Foundation of China (No. 10771179), the Young Backbone Teacher Foundation of Xinyang Normal University.

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