Pulse Vaccination Strategy in an Epidemic Model with Two Susceptible Subclasses and Time Delay

In this paper, an impulsive epidemic model with time delay is proposed, which susceptible population is divided into two groups: high risk susceptibles and non-high risk susceptibles. We introduce two thresholds R1, R2 and demonstrate that the disease will be extinct if 1 1 R  and persistent if 2 1 R  . Our results show that larger pulse vaccination rates or a shorter the period of pulsing will lead to the eradication of the disease. The conclusions are confirmed by numerical simulations.


Introduction
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 [1].Over the last fifty years, many scholars have payed great attention to construct mathematical models to describe the spread of infectious diseases.See the literatures [2][3][4][5][6][7][8], the books [9][10][11] and the references therein.In the classical epidemiological models, a population of total size N is divided into S (susceptible numbers), I (infective numbers), or S, I and R (recovered numbers) or S, E (exposed numbers), I and R, and corresponding epidemiological models such as SI, SIS, SIR, SIRS, SIER and SEIRS are constructed.All these models are extensions of the SIR model elaborated by Kermack and McKendrick in 1927 [12].Anderson and May [5,9] discussed the spreading nature of biological viruses, parasites etc. leading to infectious diseases in human population through several epidemic models.Cooke and Driessche [8] investigated an SEIRS model with the latent period and the immune period.The consideration of the latent period and the immune period gave rise to models with the incorporation of delays and integral equation formulations.
However, owing to the physical health status, age and other factors, susceptibles population show different in-fective to a infectious disease.In this paper, we divide the susceptible population into two groups: nonhigh risk susceptibles (S 1 ) and high risk susceptibles (S 2 ), such that individuals in each group have homogeneous susceptibility, but the susceptibilities of individuals from different groups are distinct.In this paper, we propose a new SIR epidemic model, which two noninteracting susceptible subclasses, the nonlinear incidence p S I


, time delay and pulse vaccination are considered.The main purpose of this paper is to study the dynamical behavior of the model and establish sufficient conditions that the disease will be extinct or not.
The organization of this paper is as follows.In the next section, we construct a delayed and impulsive SIR epidemic model with two noninteracting susceptible subclasses.In Section 3, using the discrete dynamical system determined by the stroboscopic map, we establish sufficient conditions for the global attractivity of infection-free periodic solution.And the sufficient conditions for the permanence of the model are obtained in section 4. Finally, we present some numerical simulations to illustrate our results.

Model Formulation and Preliminaries
Gao etc. [13] proposed a delayed SIR epidemic model with pulse vaccination: In model (1), the authors assumed that the birth rate (  ) is equal to the death rate, and use a bilinear incidence rate.Motivated by [13], in this paper, we assume that there are two cases noninteracting susceptible subclasses.We denote the density of the susceptible individuals that belong to different subclasses, the infected individuals, and the recovered individuals in the population by S 1 , S 2 , I and R, respectively, that is, the total variable population   Adding all the equations in model ( 2), the total variable population size is given by the differential equation and we have Note that the first three equations of system (2) do not depend on the forth equation.Thus, we restrict our attention to the following reduced system: The initial conditions of (3) are where , , , of the first kind and which are continuous from the left, i.e.,   From biological considerations, we discuss system (3) in the closed set.It is easy to verify that  is positively invariant with respect to system (3).

Global Attractivity of Disease-Free Periodic Solution
To prove our main results, we state some notations and lemmas which will be essential to our proofs.Lemma 1 (see [6]) Consider the following impulsive differential equation Then above system exists a unique positive periodic solution given by which is globally asymptotically stable, where Definition 1 (see [14]).Let and for each ii) V is locally Lipschitzian in X. Lemma 2 (see [14]).
r t be the maximal solution of the scalar impulsive differential equation x t is any solution of (3).
In the following we shall demonstrate that the disease-free periodic solution , ,0 S S t t is global attractive.We firstly show the existence of the disease-free periodic solution, in which the infectious individuals are entirely absent from the population permanently, i.e.   0 I t  for all 0 t  .Under this condition, the growth of the i-th   , .
According to Lemma 1, we know that the periodic solution of the system is globally asymptotically stable.Therefore system (3) has a unique disease-free periodic solution , ,0 S S t t of system ( 3) is globally attractive.
Proof.Since 1 1 R  , we can choose 0 From the first equation and the second equation of system (3), we have . Then we consider the following impulsive comparison system According to Lemma 1, we obtain the periodic solution of system ( 7) which is globally asymptotically stable.By the comparison theorem [14], we have that there exists 1 n Z   such that for Furthermore, from the third equation of system (3), we get From ( 6) and ( 8), we have   0 i.e., for any sufficiently small 1 0   , there exists an From the first equation and the second equation of system (3), we have for , ( 1,2).
Then we consider the following impulsive comparison systems , , 1 , .
From Lemma 1, we obtain the periodic solution of system ( 9) which is globally asymptotically stable.In view of the comparison theorem [14], there exists an integer 3 Since  and 1  are sufficiently small, from ( 8) and (10), we know that Hence, disease-free periodic solution , ,0 S S t t of system (3) is globally attractive.The proof is completed.
Next, we give some accounts of the Theorem 1 for a well biological meaning.
By simple calculation, from (5) we get and Theorem 1 determines the global attravtivity of the disease-free periodic solution of system (3) in  for the case 1 1  R  .Its epidemiology implies that the disease will die out.From ( 11) and ( 12), we can see that larger pulse vaccination rates or a shorter period of immune vaccination will make for the disease eradication.

Permanence
In this section, we state the disease is endemic if the in-fectious population persists above a certain positive level for sufficiently large time.The endemicity of the disease can be well captured and studied through the notation of uniform persistence.
Definition 2. System (3) is said to be uniformly persistent if there exist positive constants 0 (both are independent of the initial values), such that every solution , , t S S t I t with positive initial conditions of system (3) satisfies R  , then system (3) is uniformly persistent.
, , S t S t I t be any solution of (3) with initial conditions (4), then it is easy to see that We are left to prove there exist positive constants for all sufficiently large t.Firstly, from the first and second equations of system (3), we have Considering the following comparison equations According to Lemma 1 and the comparison theorem, we know that for any sufficiently small 0    , there exists a 0 t such that for Now, we shall prove there exist a 3 0 m  such that   3 I t m  for all sufficiently large t.For convenience, we prove it through the following two steps: Step I. Since 2 1 R  , there exist sufficiently small * 0 I m  and 0 where Considering the following impulsive comparison systems Similarly, we know that there exists 2 0 t  , such that for Further, the third equations of system (3) can be rewritten as From ( 13), we have . This contradicts the boundedness of ( ) V t .Hence, there exists a 1 0 Step II.According to step I, for any positive solution Thus, there exists a In this case, we shall discuss three possible cases in term of the sizes of , g  and  .

Numerical Simulations
In this section, we give some numerical simulations to illustrate the effects of different probability on population.In system (3), 1 0.25,  

Case 3 .
If g     , we will discuss the following two cases, respectively.proof is completed.

3 T 1
 .Time series are drawn in Figure 1(a) and Figure ing cycles.If we take 1 0.45, R  0.9956 .By Theorem 1, we know that the disease will disappear (see Figure 1(a)).If we let 1 0.10,   2   0.20 , then 2 1.4685 R  .According to Theorem 2, we know that the disease will be permanent (see Figure 1(b)).