The Effect of State-Dependent Control for an SIRS Epidemic Model with Varying Total Population

Based on the mechanism of prevention and control of infectious disease, we propose, in this paper, an SIRS epidemic model with varying total population size and statedependent control, where the fraction of susceptible individuals in population is as the detection threshold value. By the Poincaré map, theory of differential inequalities and differential equation geometry, the existence and orbital stability of the diseasefree periodic solution are discussed. Theoretical results show that by state-dependent pulse vaccination we can make the proportion of infected individuals tend to zero, and control the transmission of disease in population.


Introduction
It is generally known that the spread of infectious diseases has been a threat to healthy of human beings and other species.In order to prevent and control the transmission of disease (such as hepatitis C, malaria, influenza), pulse vaccination as an effective strategy has been widely studied by many scholars in the study of mathematical epidemiology.In the classical research literature it is usually assumed that the pulse vaccination occurs at fixed moment intervals and total population size remains constant [1] [2], and so on.Although fixed time pulse vaccination strategy is better than the traditional vaccination strategies (continuous vaccination), it has a few disadvantages.For these reasons, a new vaccination strategies, state-dependent pulse vaccination is proposed when the number of the susceptible individuals or infected individuals reaches a critical value.Clearly, the latter control strategies are more rational for disease control because of its efficiency, economy, and feasibility.In recent years, mathematical models with state-dependent pulse control strategies have been extensively applied to research fields of applied science, such as pest management model [3], tumor model [4], predator-prey model [5], and others.Particularly, Nie et al. [6] investigated an SIR epidemic model with state-dependent pulse vaccination.In it, authors obtained the existence and stability of positive order-1 and order-2 periodic solution.Tang et al. [7] proposed an SIR epidemic model with state-dependent pulse control strategies.Authors demonstrated that the combination of pulse vaccination and treatment is optimal in terms of cost under certain conditions, and studied the existence and stability of periodic solution.
On the other hand, the population sizes of all epidemic models with state-dependent pulse control are constant.These types of models have been studied extensively since they are easier to analyze than variable population size models.Obviously, the assumption that the total population size which remains constant is reasonable if negligible mortality rate and the disease spread quickly through the population.However, it fails to hold for diseases that are endemic in communities with changing populations, and for diseases which raise the mortality rate substantially.In such situation, we can hardly expect a population remaining constant, and hence more complicated epidemic models with varying population size should be considered.In fact, studies of this type of models have been become a major topic in mathematical epidemiology.For example, an general epidemiological model with vaccination and varying total population was discussed by Yang et al. [8], in which the global dynamics of this model and it's corresponding proportionate model are investigated.The conditions between the two models in terms of disease eradication and persistence are obtained.Hui et al. [9] introduced an SEIS epidemic model with total population which is not stationary.
Results are obtained in terms of three threshold which respectively determines whether or not the disease dies out and dynamics of epidemic model when births of population are throughout a year.At same time, they also discussed the existence of disease-free periodic solution when births of population are birth pulse.More related literature, we also can be found in [10] [11], and the references therein.
As far as we know, epidemic model with varying total population and state-dependent feedback control strategies had never been done in the literatures.Hence, in this paper, the dynamical behavior of an SIRS epidemic model with varying total population and state-dependent pulse control strategy is studied.The main aim is to explore how the state-dependent pulse control strategy affects the transmission of diseases.The remaining part of this paper is organized as follows.In the next section, an SIRS control model is constructed and some preliminaries are introduced, which are useful for the latter discussion.In section 3, we will focus our attention on the existence and orbital stability of disease-free periodic.Finally, some concluding remarks are presented in the last section.

Models and Preliminaries
In the study of the dynamic properties of infectious diseases, it was found that when the popularity of disease for a long time total population size change this factor should be considered.In this case, Busenberg et al. [12] proposed the following SIRS epidemic model with varying total population size.

S t p S t + = −
where p ( ) is the proportion by which the susceptible individuals numbers is reduced by pulse vaccination.
Taking into account pulse vaccination as state-dependent feedback control strategies, model ( 1) can be extend to the following state-dependent pulse differential equation.
where the critical threshold 0 H > is a constant.The meaning of model ( 2) as following: once the fraction of the susceptible individuals in the population reaches the critical value H at time ( ) i t H , vaccination control strategies are carried out which lead to the number of susceptible and recovered individuals abruptly turn to ( ) ( ) The equation for the total population size ( ) It means that total population size ( ) N t is not constant.In such situations, to discuss the dynamics behavior of model ( 2) we need to consider the fraction of individuals in the three epidemiological classes, namely It following from (3) that we can transforms model ( 2) into the following model for these new variables Define three threshold parameter as follows ( ) On the dynamics of model (4) without pulse effect has been studied in [12].Relevant conclusions can be summarized as the following Theorem 1.
Theorem 1.For model (4) without pulse control, the following result hold true.
1) The disease-free equilibrium ( ) 0 1, 0, 0 E always exists and is globally asymptotically stable in the feasibility region , and unable when 0 1 >  .
2) When 0 1 >  , there exist a unique endemic equilibrium 3) The total population ( ) N t has the asymptotic behavior ( ) ( ) , the total infected population has the asymptotic behavior ( ) ( ) Based on the above discussions, we just need to discuss cases (a) and (b) in Table 1.
Considering the similarities of cases (a) and (b), throughout of this paper, we discuss only the case (a).That is, in a increasing population, the number of infected individuals is converges to infinity, while the fraction of infected individuals in population is tending to a nonzero constant e y .

t b bx t x t y t x t e x t y t t x t H y t x t y t y t y t x t y t b c y t t x t p x t x t H y t y t
λ ε δ By the biological background, we only focus on model (5) in the biological meaning region . Besides, the globally existence and uniqueness properties of solution of model ( 5) are guaranteed by the smoothness of f, which is the mapping defined by right-side of model ( 5), for details see [13].
 be an arbitrary nonempty set and 2 0 P ∈  be an arbitrary point.The distance between 0 P and  is defined by ( ) x t y t = be a solution of model ( 5) starting from initial point We define the positive orbit as follows Firstly, on the positivity of solutions of model ( 5), we have the following Lemma 1.
Lemma 1. Supposing that is a solution of model (5) with the initial condition  1) The solution ( ) ( ) ( ) , x t y t ∈ intersects with line 1 L finitely many times.
For this case, due to the endemic equilibrium ( ) x y is globally asymptotically Table 1.Threshold criteria and asymptotic behavior.
For second situation, assume that solution ( ) ( ) ( )  x t = and ( ) * 0 y t > .For this case, it follows from the first and third equation of model ( 5) that which contradicts the fact that ( ) The other case is that ( ) * 0 x t > and ( ) * 0 y t = .In this regard, it follows from the second and fourth equation of model ( 5) which lead to a contradiction with ( ) * 0 y t = .Therefore, according to above discussion, we can obtain that ( ) 0 x t ≥ and ( ) 0 y t ≥ for all 0 t t ≥ .This proof is complete.
In order to address the dynamical behaviors of model ( 5), we could construct two sections to the vector field of model (

H x t y t x t H y t H
Choosing section p Σ as a Poincaré section.Assume that for any point , H P H y ∈ Σ , the trajectory ( ) , P H y intersects section H Σ infinitely many times.That is, trajectory ( ) From the definition of Poincaré map  , it easy to get that , , , , , , , , . of model ( 5) is said to be order-k periodic solution.

Main Results
Our main purpose in this section is to investigate the existence and orbital stability of periodic solution of model (5).From the geometrical construction of phase space of model ( 5), we note that the trajectory For this case, it will prove that model ( 5) possesses a disease-free periodic solution, which is orbitally asymptotically stable.
For this case, ( 8) is a disease-free periodic solution of model ( 5), and the proof of stability is similar to the proof of Theorem 2, we therefore omit here.
the effective per capita contract rate of infective individuals.All parameter values are assumed to be nonnegative and , 0 b c > .Since the susceptible individuals are immunity toward certain infectious diseases in the crowd, once infected individuals get into the susceptible groups, this will lead to the outbreak of the diseases.For this reason, we propose a pulse vaccination function as follows ( ) ( ) ( )

1
t ∈ , we will discuss all possible cases by the relation of the solution Obviously, function  is continuously differential according to the Cauchy-Lipschitz theorem.If there exist positive integer k such that k =  .On the stability of this disease-free periodic solution ( to the different positions of point 1 E + we has the following results.model(5) exists a positive order-1 periodic solution.Further, if x t y t ∈ intersects with line 1 L