Impulsive vaccination of SEIR epidemic model with time delay and nonlinear incidence rate

Abstract

In this paper, an SEIR epidemic disease model with time delay and nonlinear incidence rate is studied, and the dynamical behavior of the model under pulse vaccination is analyzed. Using the discrete dynamical system determined by the stroboscopic map, we show that there exists an infection-free periodic solution. Further, we show that the infection-free periodic solution is globally attractive when the period of impulsive effect is less than some critical value. Using a new modelling method, we obtain a sufficient condition for the permanence of the epidemic model with pulse vaccination. We show that time delay, pulse vaccination can bring different effects on the dynamic behavior of the model by numerical analysis. Our results also show the time delay is “profitless”. The main feature of this paper is to introduce time delay and impulse into the SEIR epidemic model and to give pulse vaccination strategies.

Introduction

Epidemic models study the transmission dynamics of infectious disease in host populations, aiming at tracing factors that give rise to their occurrence. In recent years, epidemic models of ordinary differential equations (ODE for short) have been studied by many scholars. The dynamics of the SIR and SIRS epidemic model have been extensively analyzed [2], [4], [10], [19], [25]. Many diseases, however, incubate inside the hosts for a period of time before the hosts become infectious. Models that are more general than the SIR or SIRS types need to be studied to investigate the role of an incubation period in disease transmission. Using a compartmental approach, one may assume that a susceptible individual first goes through a latent period (and is said to become exposed or in the class E) after infection, before becoming infectious. The resulting models are of SEIR or SEIRS types, respectively, depending on whether the acquired immunity is permanent or otherwise. Li et al. [15] considered the global stability of an SEIR epidemic model in which the latent and immune state were infective. Greenhalgh [8] considered an SEIR epidemic model for an infectious disease where the death rate depends on the number of individuals in the population. Cooke and vanden Driessche [5] introduced and studied an SEIRS epidemic model with two delays. Recently, Greenhalgh [9] studied hopf bifurcation in epidemic models with a latent period and non-permanent immunity. Li et al. [17] studied the global stability for the SEIR model in epidemiology. Li et al. [13] studied the global dynamics of the SEIR model with a nonlinear incidence rate and with a standard incidence, respectively. Li et al. [18] analyzed the global dynamics of an SEIR model with vertical transmission and a bilinear incidence. Zhang et al. [28] considered the global dynamics of an SEIR model with the saturating contact rate. The basic and important research subjects for these models are the existence of the threshold value which determines the local and global stability of the disease-free equilibrium and the endemic equilibrium.

Infectious diseases have tremendous influence on human life. Every year, millions of people die of various infectious diseases. Controlling infectious diseases has been an increasingly complex issue in recent years. Pulse vaccination is an effective way to control the transmission of diseases and was considered in the literatures [1], [6], [23], [24].

Pulse vaccination has gained in prominence as a result of its highly successful application in the control of poliomyelitis and measles throughout Central and South America [7], [22]. This method has also been used in the United Kingdom. In November 1994, a single dose of combined measles and rubella vaccine was given to children aged 5–16 years. In England and Wales, an average of 92% of these children were vaccinated. This policy caused a significant fall in the number of cases of measles reported to the Office of Population Censuses and Surveys. It was concluded that the application of pulse immunization to all schoolchildren would probably prevent a large rate of morbidity and mortality and would have a marked effect on measles transmission for several years [20].

Generally, consideration of the latent period gives rise to model the incorporation of delay. Then the death during a latent period should be considered, which is called the phenomena of ‘time delay’. So, time delay has important biological meaning in epidemic models. Most of the research literature on SEIR or SEIRS epidemic models are established by ODE [8], [9], [13], [15], [17], [18], [28], delayed ODE [5], [26], or impulsive ODE [1], [6], [23], [24]. However, dynamical behaviors on impulsive equation with time delay have seldom been studied by authors. In this paper, we study the dynamical behavior of the delay model with pulse vaccination. The main purpose is to show that large vaccination rate or short period pulsing or long latent period of the disease implies that diseases die out. The second purpose is to establish a sufficient condition that the disease is uniformly persistent, that is, there is a positive constant p (independent of the choice of the solution) such that $I(t)\ge p$ for all large t.

The organization of this paper is as follows. In Section 2, we introduce the SEIR epidemic model with time delay and pulse vaccination. In Sections 3 Global attractivity of infection-free periodic solution, 4 Permanence, we investigate the dynamical behaviors of the model with nonlinear incidence rate and obtain the sufficient condition for global attractivity of infection-free periodic solution and the permanence of the population. Lastly, we show the effect of pulse vaccination rate, period of pulsing, latent period of the disease on the dynamical behaviors of model by numerical analysis.