A Delayed Epidemic Model with Pulse Vaccination

Abstract

A delayed SEIRS epidemic model with pulse vaccination and nonlinear incidence rate is proposed. We analyze the dynamical behaviors of this model and point out that there exists an infection-free periodic solution which is globally attractive if 𝑅₁<1, 𝑅₂>1, and the disease is permanent. Our results indicate that a short period of pulse or a large pulse vaccination rate is the sufficient condition for the eradication of the disease. The main feature of this paper is to introduce time delay and impulse into SEIRS model and give pulse vaccination strategies.

1. Introduction

Infectious diseases are usually caused by pathogenic microorganisms, such as bacteria, viruses, parasites, or fungi; the diseases can be spread directly or indirectly. The severe and sudden epidemics of infectious diseases have a great influence on the human life and socioeconomy, which compel scientists to design and implement more effective control and preparedness pro- grams. Pulse vaccination is an effective method to use in attempts to control infectious diseases.

In recent years, epidemic mathematical models of ordinary differential equations have been studied by many authors (e.g., [1–3]). In most of the research literatures, authors always assume that the disease incubation is negligible, therefore, once infected, each susceptible in- dividual becomes infectious instantaneously and later recovers with a temporary acquired im- munity. An epidemic model based on these assumptions is customarily called SIR (susceptible, infectious, recovered) model. However, many diseases incubate inside the hosts for a period of time before the hosts become infectious. We assume that a susceptible individual first goes through a latent period after infection before becoming infectious. The resulting model is called SEIRS (susceptible, exposed, infectious, recovered) model. The SEIRS infections disease model is a very important biologic model and has been studied by many authors (e.g., [4–6]).

Bilinear and standard incidence rates have been frequently used in classical epidemic models [7]. Simple dynamics of these models seem related to such functions. These different incidence rates have been proposed by researchers. Anderson et al. pointed out that standard incidence is more suitable than bilinear incidence [8–10]. Levin et al. have adopted an incidence form like 𝛽𝑆^𝑞𝐼^𝑝 or 𝛽𝑆^𝑞𝐼^𝑝/𝑁 which depends on different infective disease and environments [11]. L. S. Chen and J. Chen [12] set forth transmission effect like the saturation effect 𝛽𝑆(𝑡)/(1+𝑎𝑆(𝑡)) as the infection rate. In this paper, we will adopt the infection rate 𝛽𝑆(𝑡)/(1+𝑎𝑆(𝑡)) because it includes the behavioral change and crowding effect of the infective individuals and prevents the unboundedness of the contact rate by choosing suitable parameters.

On the one hand, the newborns of the infectious may already be infected with the disease at birth such as hepatitis and phthisis, and so forth. This is called vertical transmission. On the other hand, some diseases may be spread from one individual to another via horizontal contacting transmission. Some epidemic models with vertical transmission were studied by many authors. However, only a few literatures [13] deal with the analysis of disease with pulse vaccination, vertical and horizontal transmissions.

Most of the research literature on these epidemic models are established by ODE, delayed ODE or impulsive ODE. However, impulsive equations with time delay are not many [14, 15]. In this paper, we establish a delayed SEIRS epidemic disease model with pulse vaccination and nonlinear incidence rate. We study their dynamic behaviors, establish sufficient condition for disease-eradication, as well as investigate the role of incubation in disease transmission. The main feature of this paper is to introduce time delay and pulse vaccination into epidemic model and obtain some important qualitative properties with valid pulse vaccination strategy.

The organization of this paper is as follows. In the next section, we introduce the delayed SEIRS model with pulse vaccination. To prove our main results, we also give several definitions, notations, and lemmas. In Section 3, we investigate the dynamic behavior of the model with nonlinear incidence and the sufficient condition is obtained for the global attractivity of infection-free periodic solution and the permanence of the model. In the final section, we try to interpret our mathematical results in terms of their ecological implication and also point out some future research directions.

2. Model Formulation and Preliminary

In the following model, we study a population that is partitioned into four classes, the susceptible, exposed, infectious, and recovered, with sizes denoted by S, E, I, and R, respectively, and we consider pulse vaccination strategy in the delayed SEIRS epidemic model with nonlinear incidence rate 𝛽(𝑆/(1+𝑎𝑆))𝐼, the following mathematical model is formulated: 𝑆^(𝑡)=𝐴−𝛽𝑆(𝑡)𝐼(𝑡)𝐸1+𝑎𝑆(𝑡)−𝜇𝑆(𝑡)−(1−𝑝)𝜇𝐼(𝑡)+𝛼𝑅(𝑡),𝑡≠𝑛𝑇,_(𝑡)=𝛽𝑆(𝑡)𝐼(𝑡)1+𝑎𝑆(𝑡)−𝛽𝑒^−𝜇𝜏𝑆(𝑡−𝜏)𝐼(𝑡−𝜏)𝐼1+𝑎𝑆(𝑡−𝜏)−𝜇𝐸(𝑡)+(1−𝑝)𝜇𝐼(𝑡),𝑡≠𝑛𝑇,_(𝑡)=𝛽𝑒^−𝜇𝜏𝑆(𝑡−𝜏)𝐼(𝑡−𝜏)1+𝑅𝑎𝑆(𝑡−𝜏)−(𝑟+𝑑+𝜇)𝐼(𝑡),𝑡≠𝑛𝑇,_𝑆(𝑡₊(𝑡)=𝑟𝐼(𝑡)−𝜇𝑅(𝑡)−𝛼𝑅(𝑡),𝑡≠𝑛𝑇,)=𝐸((1𝑡₊−𝜃)𝑆(𝑡),𝑡=𝑛𝑇,𝑛=1,2,…𝐼(𝑡₊)=𝐸(𝑡),𝑡=𝑛𝑇,𝑛=1,2,…𝑅(𝑡₊)=𝐼(𝑡),𝑡=𝑛𝑇,𝑛=1,2,…)=𝑅(𝑡)+𝜃𝑆(𝑡),𝑡=𝑛𝑇,𝑛=1,2,….(2.1)

Here, all coefficients are positive constants, 𝐴 denotes the influx or recruitment of the susceptible and the exposed. The death rate for disease and physical disease rate are 𝑑 and 𝜇, respectively. 𝑟 is the recovery rate of infectious individual. 𝜃(0<𝜃<1) is the proportion of those vaccinated successfully, which is called impulsive vaccination rate. 𝜏 is the latent period of the disease. Consider the death of exposed individuals during latent period of disease, that is, 𝛽𝑒^−𝜇𝜏(𝑆(𝑡−𝜏)𝐼(𝑡−𝜏)/(1+𝑎𝑆(𝑡−𝜏))) term. The disease is propagated both vertically and horizontally, 𝑝𝜇𝐼(0<𝑝<1) is the number of newborns of infectious who transfer to the susceptible class, and (1−𝑝)𝜇𝐼 is the number of newborns of the infectious who are infected vertically.

The total population size 𝑁(𝑡)=𝑆(𝑡)+𝐸(𝑡)+𝐼(𝑡)+𝑅(𝑡) can be determined by the differential equation𝑁^(𝑡)=𝐴−𝜇𝑁(𝑡)−𝑑𝐼(𝑡),(2.2)which is derived by adding all equations in system (2.1). So we have 𝐴−(𝜇+𝑑)𝑁(𝑡)≤𝑁^(𝑡)≤𝐴−𝜇𝑁(𝑡). It follows that𝐴𝜇+𝑑≤lim^𝑡→∞inf𝑁(𝑡)≤lim_𝑡→∞sup𝑁(𝑡)≤𝐴𝜇.(2.3)

Before going to any detail, we simplify model (2.1) and mainly discuss the following model:𝑆^(𝑡)=𝐴−𝛽𝑆(𝑡)𝐼(𝑡)𝐼1+𝑎𝑆(𝑡)−𝜇𝑆(𝑡)−(1−𝑝)𝜇𝐼(𝑡)+𝛼𝑅(𝑡),𝑡≠𝑛𝑇,_(𝑡)=𝛽𝑒^−𝜇𝜏𝑆(𝑡−𝜏)𝐼(𝑡−𝜏)1+𝑅𝑎𝑆(𝑡−𝜏)−(𝑟+𝑑+𝜇)𝐼(𝑡),𝑡≠𝑛𝑇,_(𝑁𝑡)=𝑟𝐼(𝑡)−𝜇𝑅(𝑡)−𝛼𝑅(𝑡),𝑡≠𝑛𝑇,_𝑆(𝑡₊(𝑡)=𝐴−𝜇𝑁(𝑡)−𝑑𝐼(𝑡),𝑡≠𝑛𝑇,)=𝐼((1𝑡₊−𝜃)𝑆(𝑡),𝑡=𝑛𝑇,𝑛=1,2,…𝑅(𝑡₊)=𝐼(𝑡),𝑡=𝑛𝑇,𝑛=1,2,…)=𝑁𝑅(𝑡₊(𝑡)+𝜃𝑆(𝑡),𝑡=𝑛𝑇,𝑛=1,2,…)=𝑁(𝑡),𝑡=𝑛𝑇,𝑛=1,2,….(2.4)The initial condition of (2.4) is given as𝜙(𝜉)=(𝜙₁(𝜉),𝜙₂(𝜉),𝜙₃(𝜉),𝜙₄(𝜉))∈𝐶₊,𝜙_𝑖(0)>0,𝑖=1,2,3,4,(2.5)where 𝐶₊=𝐶([−𝜏,0],𝑅⁴⁺). From biological considerations, we discuss system (2.4) in the closed setΩ=(𝑆,𝐼,𝑅,𝑁)∈𝑅⁴⁺∣0≤𝑆+𝐼+𝑅≤𝐴𝜇,𝑁≤𝐴𝜇,(2.6)where 𝑅⁴⁺ denotes the nonnegative cone of 𝑅⁴ including its lower dimensional faces. It is easy to show that Ω is positively invariant with respect to (2.4).

Before starting our main results, we give the following lemmas. Lemma 2.1 (See [16]). Consider the following delay differential equation: 𝑥^(𝑡)=𝑎𝑥(𝑡−𝜏)−𝑏𝑥(𝑡),(2.7)where 𝑎,𝑏,𝜏>0 and 𝑥(𝑡)>0 for 𝑡∈[−𝜏,0]. The following hold:
(i)if 𝑎<𝑏, then lim_𝑡→∞𝑥(𝑡)=0,(ii)if 𝑎>𝑏, then lim_𝑡→∞𝑥(𝑡)=+∞.Lemma 2.2. Consider the following impulsive differential equations: 𝑢𝑢(^𝑡(₊𝑡)=𝑎−𝑏𝑢(𝑡),𝑡≠𝑛𝑇,)=(1−𝜃)𝑢(𝑡),𝑡=𝑛𝑇,𝑛∈𝑁,(2.8)where 𝑎>0,𝑏>0,0<𝜃<1. Then there exists a unique positive periodic solution of (2.8): 𝑢_𝑒(𝑡)=𝑎𝑏+𝑢^∗−𝑎𝑏𝑒^{−𝑏(𝑡−𝑘𝑇)},𝑘𝑇<𝑡<(𝑘+1)𝑇,(2.9)which is globally asymptotically stable, where 𝑢^∗=𝑎(1−𝜃)(1−𝑒^−𝑏𝑇)/𝑏(1−(1−𝜃)𝑒^−𝑏𝑇).

3. Global Attractivity of Infection-Free Periodic Solution

In this section, we study the existence of the infection-free periodic solution of system (2.4), in which infectious individuals are entirely absent from the population permanently, that is, 𝐼(𝑡)=0 for all 𝑡≥0. Under this condition, system (2.4) becomes the following impulsive system without delay: 𝑅𝑆^(𝑡)=𝐴−𝜇𝑆(𝑡)+𝛼𝑅(𝑡),𝑡≠𝑛𝑇,_𝑁(𝑡)=𝜇𝑅(𝑡)−𝛼𝑅(𝑡),𝑡≠𝑛𝑇,_𝑆(𝑡(₊𝑡)=𝐴−𝜇𝑁(𝑡),𝑡≠𝑛𝑇,𝑅(𝑡₊)=(1−𝜃)𝑆(𝑡),𝑡=𝑛𝑇,𝑛=1,2,…𝑁(𝑡₊)=𝑅(𝑡)+𝜃𝑆(𝑡),𝑡=𝑛𝑇,𝑛=1,2,…)=𝑁(𝑡),𝑡=𝑛𝑇,𝑛=1,2,….(3.1) From the third and sixth equations of system (3.1), we have lim_𝑡→∞𝑁(𝑡)=𝐴/𝜇. Further, if 𝐼(𝑡)=0, it follows that lim_𝑡→∞𝐸(𝑡)=0 from the second and sixth equations of system (2.1). In the following, we show that the susceptible population 𝑆 and recovered population 𝑅 oscillate with period 𝑇, in synchronization with the periodic impulsive vaccination. Consider the following limit system of system (3.1): 𝑅(𝑡)=𝐴𝑆𝜇−𝑆(𝑡),_(𝑡)=(𝛼+𝜇)𝐴𝑆(𝑡₊𝜇−𝑆(𝑡),𝑡≠𝑛𝑇,)=(1−𝜃)𝑆(𝑡),𝑡=𝑛𝑇,𝑛∈𝑁.(3.2)By Lemma 2.2, we know that the periodic solution of system (3.2), 𝑆_𝑒(𝑡)=𝐴𝜇+𝑆^∗−𝐴𝜇𝑒^{−(𝛼+𝜇)(𝑡−𝑛𝑇)},𝑛𝑇<𝑡≤(𝑛+1)𝑇,(3.3)is globally asymptotically stable, where 𝑆^∗=𝐴(1−𝜃)(1−𝑒^{−(𝛼+𝜇)𝑇})/𝜇(1−(1−𝜃)𝑒^{−(𝛼+𝜇)𝑇}).

Denote 𝑅₁=𝛽𝑒^−𝜇𝜏𝛿/(1+𝑎𝛿)(𝑟+𝑑+𝜇), where 𝛿=𝐴(1−𝑒^{−(𝛼+𝜇)𝑇})/𝜇(1−(1−𝜃)𝑒^{−(𝛼+𝜇)𝑇}).

Theorem 3.1. If 𝑅₁<1, then the infection-free periodic solution (𝑆_𝑒(𝑡),0,𝐴/𝜇−𝑆_𝑒(𝑡),𝐴/𝜇) of system (2.4) is globally attractive.Proof. Since 𝑅₁<1, we can choose 𝜀>0 small enough such that𝛽𝑒^−𝜇𝜏(𝛿+𝜀)1+𝑎(𝛿+𝜀)<𝑟+𝑑+𝜇.(3.4)From the first equation of system (2.4), it follows that 𝑆^(𝑡)≤(𝛼+𝜇)((𝐴/𝜇)−𝑆(𝑡)). Thus consider the following comparison impulsive differential system:𝑧^(𝑡)=(𝛼+𝜇)𝐴𝑧(𝑡₊𝜇−𝑧(𝑡),𝑡≠𝑛𝑇,)=(1−𝜃)𝑧(𝑡),𝑡=𝑛𝑇,𝑛∈𝑁.(3.5)By (3.2), we know that the periodic solution of system (3.5),𝑧_𝑒(𝑡)=𝑆_𝑒(𝑡)=𝐴𝜇+𝑆^∗−𝐴𝜇𝑒^{−(𝛼+𝜇)(𝑡−𝑛𝑇)},𝑛𝑇<𝑡≤(𝑛+1)𝑇,(3.6)is globally asymptotically stable, where 𝑆^∗=𝐴(1−𝜃)(1−𝑒^{−(𝛼+𝜇)𝑇})/𝜇(1−(1−𝜃)𝑒^{−(𝛼+𝜇)𝑇}). Let (𝑆(𝑡),𝐼(𝑡),𝑅(𝑡),𝑁(𝑡)) be the solution of system (2.4) with initial condition (2.5) and 𝑆(0⁺)=𝑆₀>0,𝑧(𝑡) be the solution of system (3.5) with initial value 𝑧(0⁺)=𝑆₀. By the comparison theorem for impulsive differential equations [17], there exists an integer 𝑛₁>0 such that𝑆(𝑡)<𝑧(𝑡)<𝑧_𝑒(𝑡)+𝜀,𝑛𝑇<𝑡≤(𝑛+1)𝑇,𝑛>𝑛₁,(3.7)that is,𝑆(𝑡)<𝑧_𝑒(𝑡)+𝜀≤𝐴1−𝑒^{−(𝛼+𝜇)𝑇}𝜇1−(1−𝜃)𝑒^{−(𝛼+𝜇)𝑇}+𝜀≐𝜂.(3.8)Further, from the second equation of system (2.4), we have that, for all 𝑡>𝑛𝑇+𝜏,𝑛>𝑛₁,𝐼^(𝑡)≤𝛽𝜂𝑒^−𝜇𝜏1+𝑎𝜂𝐼(𝑡−𝜏)−(𝑟+𝑑+𝜇)𝐼(𝑡).(3.9)Consider the following comparison equation:𝑦^(𝑡)=𝛽𝜂𝑒^−𝜇𝜏1+𝑎𝜂𝑦(𝑡−𝜏)−(𝑟+𝑑+𝜇)𝑦(𝑡).(3.10)From (3.4), we have that𝛽𝜂𝑒^−𝜇𝜏1+𝑎𝜂<(𝑟+𝑑+𝜇).(3.11)According to Lemma 2.1, we obtain that lim_𝑡→∞𝑦(𝑡)=0.
Set (𝑆(𝑡),𝐼(𝑡),𝑅(𝑡),𝑁(𝑡)) be the solution of system (2.4) with initial condition (2.5) and 𝐼(𝜉)=𝜙(𝜉)>0(𝜉∈[−𝜏,0]),𝑦(𝑡) be the solution of (3.10) with initial condition 𝑦(𝜉)=𝜙(𝜉)>0(𝜉∈[−𝜏,0]). By the comparison theorem in differential equation and the positivity of solution (with 𝐼(𝑡)≥0), we have thatlim_𝑡→∞𝐼(𝑡)=0.(3.12)Therefore, for any 𝜀₁>0 (sufficiently small), there exists an 𝑛₂(𝑛₂𝑇>𝑛₁𝑇+𝜏) such that 0<𝐼(𝑡)<𝜀₁ for all 𝑡>𝑛₂𝑇. By the fourth equation of system (2.4), we have𝑁^(𝑡)>𝐴−𝜇𝑁(𝑡)−𝑑𝜀₁for𝑡>𝑛₂𝑇.(3.13)Consider the following comparison equation: 𝑧^1(𝑡)=(𝐴−𝑑𝜀₁)−𝜇𝑧₁(𝑡).
It is clear that lim_𝑡→∞𝑧₁(𝑡)=(𝐴−𝑑𝜀₁)/𝜇; by the comparison theorem, we have that there exists an integer 𝑛₃>𝑛₂ such that for all 𝑡>𝑛₃𝑇,𝑁(𝑡)≥(𝐴−𝑑𝜀₁)/𝜇−𝜀₁. Since 𝜀₁ is arbitrarily small, we havelim_𝑡→∞𝑁(𝑡)=𝐴𝜇.(3.14)It follows from (3.12) and (3.14) that, there exists 𝑛₄>𝑛₃ such that𝐼(𝑡)<𝜀₁,𝑁(𝑡)>𝐴𝜇−𝜀¹for𝑡>𝑛₄𝑇.(3.15)From the second equation of system (2.1), we have 𝐸^(𝑡)≤𝐴𝛽𝜀¹𝜇+𝑎𝐴+(1−𝑝)𝜇𝜀¹−𝜇𝐸(𝑡)for𝑡>𝑛⁴𝑇.(3.16) It is easy to obtain that there exists an 𝑛₅>𝑛₄ such that 𝐸(𝑡)<𝛿₁+𝜀₁for𝑡>𝑛₅𝑇,(3.17)where 𝛿₁=(𝐴𝛽𝜀₁+(𝜇+𝑎𝐴)(1−𝑝)𝜇𝜀₁)/𝜇(𝜇+𝐴𝑎). So from the first equation of system (2.4), we have 𝑆^(𝑡)≥𝐴+𝛼𝐴𝜇−(1−𝑝)𝜇𝜀¹−𝛼𝛿₁−3𝛼𝜀₁−𝛽𝜀¹+𝜇+𝛼𝑆(𝑡).(3.18)Consider the following comparison impulsive differential equations for 𝑡>𝑛₅𝑇 and 𝑛>𝑛₅: 𝑢^(𝑡)=𝐴+𝛼𝐴𝜇−(1−𝑝)𝜇𝜀¹−𝛼𝛿₁−3𝑢𝛼(𝑡𝜀₁₊−𝛽𝜀¹+𝜇+𝛼𝑢(𝑡),𝑡≠𝑛𝑇,)=(1−𝜃)𝑢(𝑡),𝑡=𝑛𝑇,𝑛∈𝑁.(3.19)By Lemma 2.2, we know that the periodic solution of system (3.19) is𝑢_𝑒(𝑡)=Θ+(𝑢^∗−Θ)𝑒^{−(𝛼+𝜇+𝛽𝜀1)(𝑡−𝑛𝑇)},𝑛𝑇<𝑡≤(𝑛+1)𝑇,(3.20)which is globally asymptotically stable, where Θ=𝐴+𝐴𝛼/𝜇−𝛼𝛿¹−3𝛼𝜀₁−(1−𝑝)𝜇𝜀₁𝛼+𝜇+𝛽𝜀₁𝑢_∗,=Θ(1−𝜃)1−𝑒^{−(𝛼+𝜇+𝛽𝜀1)𝑇}1−(1−𝜃)𝑒^{−(𝛼+𝜇+𝛽𝜀1)𝑇}.(3.21)By using the comparison theorem of impulsive differential equation [17], there exists an 𝑛₆>𝑛₅ such that𝑆(𝑡)>𝑢_𝑒(𝑡)−𝜀₁,𝑛𝑇<𝑡≤(𝑛+1)𝑇,𝑛>𝑛₆.(3.22)Let 𝜀₁→0, then it follows from (3.8) and (3.22) that 𝑆_𝑒(𝑡)=𝐴𝜇1−1−𝜃(𝑒1−𝜃)𝑒^{−(𝛼+𝜇)(𝑡−𝑛𝑇)−(𝛼+𝜇)𝑇},𝑛𝑇<𝑡≤(𝑛+1)𝑇,(3.23)is globally attractive, that is,lim_𝑡→∞𝑆(𝑡)=𝑆_𝑒(𝑡).(3.24)By the positivity of 𝐸(𝑡) and sufficiently small 𝜀₁, it follows from (3.17) that lim_𝑡→∞𝐸(𝑡)=0.(3.25) From the restriction 𝑁(𝑡)=𝑆(𝑡)+𝐸(𝑡)+𝐼(𝑡)+𝑅(𝑡), we have lim_𝑡→∞𝑅(𝑡)=𝐴/𝜇−𝑆_𝑒(𝑡). Therefore, the infection-free periodic solution (𝑆_𝑒(𝑡),0,𝐴/𝜇−𝑆_𝑒(𝑡),𝐴/𝜇) is globally attractive. This completes the proof.Corollary 3.2. In system (2.4), the following states are true.
(i)If 𝐴𝛽𝑒^−𝜇𝜏<(𝑟+𝑑+𝜇)(𝜇+𝐴𝑎),then infection-free periodic solution (𝑆_𝑒(𝑡),0,𝐴/𝜇−𝑆_𝑒(𝑡),𝐴/𝜇) is globally attractive.(ii)If 𝐴𝛽𝑒^−𝜇𝜏>(𝑟+𝑑+𝜇)(𝜇+𝐴𝑎) and 𝑇<𝑇^∗, then infection-free periodic solution (𝑆_𝑒(𝑡),0,𝐴/𝜇−𝑆_𝑒(𝑡),𝐴/𝜇) is globally attractive, where 𝑇^∗=1/(𝛼+𝜇)ln(1+(𝜃(𝑟+𝑑+𝜇)𝜇)/(𝐴𝛽𝑒^−𝜇𝜏−𝑎𝐴(𝑟+𝑑+𝜇)−(𝑟+𝑑+𝜇)𝜇).(iii)If 𝜃>𝜃^∗, then infection-free periodic solution (𝑆_𝑒(𝑡),0,𝐴/𝜇−𝑆_𝑒(𝑡),𝐴/𝜇) is globally attractive, where 𝜃^∗=(𝑟+𝑑𝜇)𝑒^{−(𝛼+𝜇)𝑇}−(𝑟+𝑑+𝜇)𝜇+(𝐴𝛽𝑒^−𝜇𝜏−𝑎(𝑟+𝑑+𝜇)𝐴)(1−𝑒^{−(𝛼+𝜇)𝑇})/𝜇(𝑟+𝑑+𝜇)𝑒^{−(𝛼+𝜇)𝑇}.

Theorem 3.1 determines the global attractivity of system (2.4) in Ω for the case 𝑅₁<1. From Corollary 3.2, we can see that a short pulse periodic (with 𝑇) or a large pulse vaccination rate (with 𝜃) is the sufficient condition for the global attractivity of infection-free periodic solution (𝑆_𝑒(𝑡),0,𝐴/𝜇−𝑆_𝑒(𝑡),𝐴/𝜇).

4. Permanence

In this section, it is noted that the disease is endemic if the infectious population persists above a certain threshold for sufficiently large time. The endemicity of the disease can be well captured and studied through the notion of uniform persistence and permanence.Definition 4.1. System (2.4) is said to be uniformly persistent if there exists an 𝑚>0 (independent of the initial data) such that every solution (𝑆(𝑡),𝐼(𝑡),𝑅(𝑡),𝑁(𝑡)) with initial conditions (2.5) of system (2.4) satisfieslim_𝑡→∞inf𝑆(𝑡)≥𝑚,lim_𝑡→∞inf𝐼(𝑡)≥𝑚,lim_𝑡→∞inf𝑅(𝑡)≥𝑚,lim_𝑡→∞inf𝑁(𝑡)≥𝑚.(4.1)Definition 4.2. System (2.4) is said to be permanent if there exists a compact region Ω₀∈intΩ such that every solution of system (2.4) with initial data (2.5) will eventually enter and remain in region Ω₀.

Denote𝑅₂=𝛽𝑒^−𝜇𝜏/(𝑟+𝑑+𝜇)−𝑎𝐴(1−𝜃)1−𝑒^−𝜇𝑇𝐼_∗𝜇1−(1−𝜃)𝑒^−𝜇𝑇,=𝐴𝜇𝑅²−1𝐴𝛽+(1−𝑝)𝜇²𝑅₂.(4.2)

Theorem 4.3. If 𝑅₂>1, then there exists a positive constant 𝑚 such that each positive solution (𝑆(𝑡),𝐼(𝑡),𝑅(𝑡),𝑁(𝑡)) of system (2.4) satisfies 𝐼(𝑡)≥𝑚 for 𝑡 large enough.Proof. Note that the second equation of system (2.4) can be rewritten as follows:𝐼^(𝑡)=𝐼(𝑡)𝛽𝑒^−𝜇𝜏𝑆(𝑡)1+𝑎𝑆(𝑡)−(𝑟+𝑑+𝜇)−𝛽𝑒^−𝜇𝜏𝑑𝑑𝑡^{𝑡𝑡−𝜏}𝑆(𝑢)𝐼(𝑢)1+𝑎𝑆(𝑢)𝑑𝑢.(4.3)Define𝑉(𝑡)=𝐼(𝑡)+𝛽𝑒^−𝜇𝜏^{𝑡𝑡−𝜏}𝑆(𝑢)1+𝑎𝑆(𝑢)𝐼(𝑢)𝑑𝑢.(4.4)According to (4.3), we calculate the derivative of 𝑉 along the solution of (2.4):𝑉^(𝑡)=𝐼(𝑡)𝛽𝑒^−𝜇𝜏𝑆(𝑡)=(𝑟+𝑑+𝜇)1𝐼(+𝑡)𝑎𝑆(𝑡)−𝛽(𝑒𝑟+_−𝜇𝜏𝑑+𝜇)𝑆(𝑡)(𝑟+𝑑+𝜇)(1+𝑎𝑆(𝑡))−1.(4.5)Since 𝑅₂>1, then 𝐼^∗>0 and there exists sufficiently small 𝜀>0 such that𝛽𝑒^−𝜇𝜏𝜎(𝑟+𝑑+𝜇)(1+𝑎𝜎)>1,(4.6)where𝜎=𝐴−(1−𝑝)𝜇𝐼^∗(1−𝜃)1−𝑒^{−(𝛽𝐼∗+𝜇)𝑇}(𝛽𝐼^∗+𝜇)1−(1−𝜃)𝑒^{−(𝛽𝐼∗+𝜇)𝑇}−𝜀>0.(4.7)We claim that for any 𝑡₀>0, it is impossible that 𝐼(𝑡)<𝐼^∗ for all 𝑡≥𝑡₀. Otherwise, there is a 𝑡₀>0 such that 𝐼(𝑡)<𝐼^∗ for all 𝑡≥𝑡₀. It follows from the first equation of (2.4) that we have𝑆^(𝑡)>𝐴−(1−𝑝)𝜇𝐼^∗−(𝛽𝐼^∗+𝜇)𝑆(𝑡).(4.8)Consider the following comparison impulsive system for 𝑡≥𝑡₀:𝑣𝑣(^𝑡(₊𝑡)=𝐴−(1−𝑝)𝜇𝐼^∗−(𝛽𝐼^∗+𝜇)𝑣(𝑡),𝑡≠𝑛𝑇,)=(1−𝜃)𝑣(𝑡),𝑡=𝑛𝑇,𝑛∈𝑁.(4.9)According to Lemma 2.2, we obtain that𝑣_𝑒(𝑡)=𝐴−(1−𝑝)𝜇𝐼^∗𝛽𝐼^∗+𝜇+𝑣^∗−𝐴−(1−𝑝)𝜇𝐼^∗𝛽𝐼^∗+𝜇𝑒^{−(𝛽𝐼∗+𝜇)(𝑡−𝑛𝑇)},𝑛𝑇<𝑡≤(𝑛+1)𝑇,(4.10)is the unique globally asymptotically stable positive periodic solution, where 𝑣^∗=
By the comparison theorem for impulsive differential equation [17], we know that there exists (𝐴−(1−𝑝)𝜇𝐼^∗) such that the following inequality holds for (1−𝜃):(1−𝑒^{−(𝛽𝐼∗+𝜇)𝑇})Thus/(𝛽𝐼^∗+𝜇)(1−(1−𝜃)𝑒^{−(𝛽𝐼∗+𝜇)𝑇}).From (4.6), we have 𝑡₁>𝑡₀+𝜏 By (4.5) and (4.12), we have𝑡>𝑡₁Let𝑆(𝑡)>𝑣_𝑒(𝑡)−𝜀.(4.11)We will show that 𝑆(𝑡)>𝑣^∗−𝜀≐𝜎>0for𝑡≥𝑡₁.(4.12) for all 𝛽𝑒^−𝜇𝜏𝜎/(𝑟+𝑑+𝜇)(1+𝑎𝜎)>1.. Otherwise, there is a 𝑉^(𝑡)>(𝑟+𝑑+𝜇)𝐼(𝑡)𝛽𝑒^−𝜇𝜏𝜎(𝑟+𝑑+𝜇)(1+𝑎𝜎)−1for𝑡≥𝑡¹.(4.13) such that 𝐼_𝑙=min_{𝑡∈[𝑡1,𝑡1+𝜏]}𝐼(𝑡).(4.14) for 𝐼(𝑡)≥𝐼_𝑙 and 𝑡≥𝑡₁. However, the second equation of systems (2.4) and (4.12) imply that𝑇₀>0This is a contradiction, thus, 𝐼(𝑡)≥𝐼_𝑙 for all 𝑡∈[𝑡₁,𝑡₁+𝜏+𝑇₀],𝐼(𝑡₁+𝜏+𝑇₀)=𝐼_𝑙,. As a consequence, (4.13) leads to𝐼^(𝑡₁+𝜏+𝑇₀)≤0which implies that 𝐼^𝑡¹+𝜏+𝑇₀≥(𝑟+𝑑+𝜇)𝐼^𝑙𝛽𝑒^−𝜇𝜏𝜎(𝑟+𝑑+𝜇)(1+𝑎𝜎)−1>0.(4.15) as 𝐼(𝑡)≥𝐼_𝑙 This contradicts with 𝑡≥𝑡₁ Hence, for any 𝑉^(𝑡)>(𝑟+𝑑+𝜇)𝐼_𝑙𝛽𝑒^−𝜇𝜏𝜎(𝑟+𝑑+𝜇)(1+𝑎𝜎)−1>0for𝑡≥𝑡¹,(4.16) the inequality 𝑉(𝑡)→∞ cannot hold for all 𝑡→∞. Next, we are left to consider two cases:
(1)𝑉(𝑡)≤(𝐴/𝜇)(1+𝐴𝜏𝛽𝑒^−𝜇𝜏/𝜇). for 𝑡₀>0, large enough;(2)𝐼(𝑡)<𝐼^∗ oscillates about 𝑡≥𝑡₀. for 𝐼(𝑡)≥𝐼^∗ large enough.It is clear that if 𝑡 for 𝐼(𝑡) large enough, then our aim is obtained. So we only need consider the case 𝐼^∗
Let𝑡In the following, we will show that 𝐼(𝑡)≥𝐼^∗ for 𝑡 large enough, let (2). and 𝑚=min𝐼^∗2,𝐼^∗𝑒^{−(𝑟+𝑑+𝜇)𝜏}.(4.17) satisfy 𝐼(𝑡)≥𝑚 and 𝑡 for 𝑡^∗>0 where 𝜄>0 is sufficiently large such that 𝐼(𝑡^∗)=𝐼(𝜄+𝑡^∗)=𝐼^∗, for 𝐼(𝑡)<𝐼^∗ we can conclude that 𝑡^∗<𝑡<𝑡^∗+𝜄, is uniformly continuous since the positive solution of (2.4) is ultimately bounded and 𝑡^∗ is not affected by impulsive effects. Hence there exists a constant 𝑆(𝑡)>𝜎 and 𝑡^∗<𝑡<𝑡^∗+𝜄, is independent of the choice if 𝐼(𝑡) such that 𝐼(𝑡) for 𝑇₁(0<𝑇₁<𝜏, If 𝑇₁our aim is obtained. If 𝑡^∗), since 𝐼(𝑡)>𝐼^∗/2, and 𝑡^∗≤𝑡≤𝑡^∗+𝑇₁., it is obvious that 𝜄≤𝑇₁, for 𝑇₁<𝜄≤𝜏. If 𝐼^(𝑡)>−(𝑟+𝑑+𝜇)𝐼(𝑡) by the second equation of (2.4), we obtain 𝐼(𝑡^∗)=𝐼^∗ for 𝐼(𝑡)≥𝐼^∗𝑒^{−(𝑟+𝑑+𝜇)𝜏} The same arguments can be continued, we can obtain 𝑡^∗<𝑡<𝑡^∗+𝜄 for 𝜄>𝜏;. Since the interval 𝐼(𝑡)≥𝐼^∗𝑒^{−(𝑟+𝑑+𝜇)𝜏} is arbitrarily chosen, we can conclude that 𝑡^∗<𝑡<𝑡^∗+𝜏. for 𝐼(𝑡)≥𝐼^∗𝑒^{−(𝑟+𝑑+𝜇)𝜏} large enough. Based on the above discussions, the choice of 𝑡^∗+𝜏<𝑡<𝑡^∗+𝜄 is independent of the positive solution of (2.4), and we have proved that any positive solution of (2.4) satisfies [𝑡^∗,𝑡^∗+𝜄] for all sufficiently large 𝐼(𝑡)≥𝑚. This completes the proof.Theorem 4.4. If 𝑡 then the system (2.4) is permanent.Proof. Suppose that 𝑚 be any solution of (2.4). From the first equation of (2.4), we have𝐼(𝑡)≥𝑚Similarly, we have𝑡where𝑅₂>1,By Theorem 4.3, the third equation of (2.4) becomes(𝑆(𝑡),𝐼(𝑡),𝑅(𝑡),𝑁(𝑡))It is easy to obtain that𝑆^(𝑡)≥𝑝𝐴−𝛽𝐴𝜇+𝜇𝑆(𝑡).(4.18)
Setlim_𝑡→∞𝑆(𝑡)≥𝑞,(4.19)By Theorem 3.1 and above discussions, we know that the set 𝑞=𝑝𝐴𝜇𝛽𝐴+𝜇²(1−𝜃)1−𝑒^{−(𝛽(𝐴/𝜇)+𝜇)𝑇}1−(1−𝜃)𝑒^{−(𝛽(𝐴/𝜇)+𝜇)𝑇}−𝜀.(4.20) is a global attractor in 𝑅^(𝑡)≥𝑟𝑚−(𝜇+𝛼)𝑅(𝑡).(4.21) and, of course, every solution of system (2.4) with initial condition (2.5) will eventually enter and remain in region 𝑅(𝑡)≥𝑟𝑚𝜇+𝛼−𝜀≐𝜔.(4.22). Hence system (2.4) is permanent. This completes the proof. Denote Ω₀=(𝑆,𝐼,𝑅,𝑁)∣𝑞≤𝑆,𝑚≤𝐼,𝜔≤𝑅,𝑆+𝐼+𝑅≤𝐴𝜇,𝜇𝐴+𝑑−𝜀≤𝑁≤𝐴𝜇.(4.23)Corollary 4.5. The following results are true.
(1)If Ω₀ then system (2.4) is permanent.(2)If Ω then system (2.4) is permanent.

5. Discussion

In this paper, we introduce the delayed SEIRS epidemic model with pulse vaccination and nonlinear incidence rate of the form Ω₀ As a result, it is observed that nonlinear incidence, the latent period of disease, pulse vaccination rate, and pulse vaccination period bring effects on the dynamics of our model. Theorems 3.1 and 4.4 show that 𝑇_∗=−1𝜇ln1−1𝜃1−𝜃𝛽𝑒^−𝜇𝜏/𝑟+𝑑+𝜇−𝑎𝐴𝜃𝛽𝑒^−𝜇𝜏/𝑟+𝑑+𝜇−𝑎𝐴−𝜇_∗,=1−𝜇𝑒^𝜇𝑇𝐴𝛽𝑒^−𝜇𝜏/𝑟+𝑑+𝜇−𝑎𝑒^𝜇𝑇−1+𝜇.(4.24) or 𝑇>𝑇_∗, implies that the disease will be eradicated, whereas 𝜃<𝜃_∗, or 𝛽(𝑆(𝑡)𝐼(𝑡)/(1+𝑎𝑆(𝑡))). implies that the disease will be epidemic. Our results indicate that a short pulse time or a large pulse vaccinate rate will lead to eradication of the disease. In this paper, we only discuss 𝑅₁<1,𝜃>𝜃^∗, and 𝑇<𝑇^∗ but for closed interval 𝑅₂>1,𝜃<𝜃_∗, the dynamical behaviors of system (2.4) have not been studied, that is, the threshold parameter for the reproducing number between the eradication and the permanence of the disease has not been studied, which will be left in the future research.

Acknowledgment

This work was supported by National Natural Science Foundation of China (10771179).