The Effect of Pulse Vaccination and Treatment on SIR Epidemic Model with Media Impact

We propose a novel SIR epidemic dynamical control model with media impact, where the state dependent pulse vaccination and medication treatment control strategies are being introduced to prevent the spread of disease at different control threshold values. By using the geometry theory of differential equation and method of successor function, the existence of positive order-1 periodic solution is studied. Further, some sufficient conditions of the orbitally asymptotical stability for positive order-1 periodic solution are given by the analog Poincaré criterion. Furthermore, numerical simulations are carried to illustrate the feasibility of our main results presented here.


Introduction
Millions of human beings suffer from or die of various infectious diseases every year.For example, malaria, dengue, AIDS, SARS, cholera, Ebola, and avian influenza have a tremendous influence on human health at the last few years.Therefore, controlling infectious diseases has been an increasingly complex issue worldwide.It is well known that vaccination is widely regarded as the most effective measure in preventing such viral infections as rabies, yellow fever, poliovirus, hepatitis B, parotitis, and encephalitis B. The vaccination strategies lead to infectious diseases eradication if the proportion of the successfully vaccinated individuals is larger than a certain critical value, for example, which is approximately equal to 95% for measles [1].However, in practice, it is both difficult and expensive to implement vaccination for such a large population coverage.
Recently, pulse vaccination has gained prominent achievement as a result of its highly successful application in the control of poliomyelitis and measles throughout Central and South America.In viewing of this, epidemiological models with pulse vaccination control strategies have been set up and investigated in many literatures (see, e.g., [2][3][4][5][6][7] and the references therein).Particularly, a theoretical result in this context was obtained by Shulgin et al. [8].They showed that the infection-free solution can exist and be stable, which implies the disease could be eradicated.d'Onofrio [5] proposed a SEIR epidemic model with pulse vaccination strategy and discussed the local and global asymptotic stabilities of the periodic eradication solution.Röst and Vizi [9] investigated a SIVS model with pulse vaccination strategy, and their main result is that nontrivial endemic periodic solutions are bifurcating from the disease-free periodic solution as a parameter is passing through the threshold value one.
In a real world application, however, the eradication of a disease is sometimes difficult both practically and economically in a short time.So, it is necessary to keep the density of infections at a low level to avoid the spread of the disease.Motivated by this idea, the state dependent pulse control strategy is applied widely to the control of spread of infectious disease due to its economic high efficiency and feasibility nature.For example, a simple SIR model with state dependent pulse control strategies was first considered by Tang et al. [10], and theoretical results showed that the combination of pulse vaccination and treatment (or isolation) is optimal in terms of cost under certain conditions, which depends on the RL (where RL is defined as the number of infected patients such that control actions must be taken in order to avoid economic and social damage), and the existence and stability of periodic solution with the maximum value of the infective being no Discrete Dynamics in Nature and Society larger than RL are obtained.This implies that disease can be successfully controlled in a local area.Further, Nie et al. [11,12] proposed SIR and SIRS models with state dependent pulse vaccination and analyzed the existence and stability of positive periodic solution using the Poincaré map and the method of qualitative analysis.Additionally, the state dependent pulse control strategy also can be found in many other areas like agricultural production and fishery industry, where the control measures (such as catching, poisoning, releasing the natural enemy, and harvesting) are taken only when the number of populations reaches a threshold value.We refer some of them to [13][14][15][16] and the references therein.
On the other hand, we note that people's response to the threat of disease is often relied on the public and private information disseminated widely by the media, such as broadcast reports and network information.Massive news coverage and fast information flow can generate a profound psychological impact on the public.A lot of press coverage and fast information flow about the risk of disease can affect the psychological quality of the masses and further affect people's daily behavior.Therefore, media communications have played an important role in affecting the outcome of infectious disease outbreaks (see, e.g., [17][18][19][20][21] and the references therein).
In this paper, according to the different minds and behaviors of people at the different threat levels and different stages of disease, we propose a novel SIR epidemic model with media coverage by combination of state dependent pulse vaccination for the susceptibles and treatment of the infected at different control threshold values.This paper is structured as follows.In Section 2, a SIR epidemic model with media coverage and state dependent pulse control strategies is constructed, and some basic definitions, preliminaries, and lemmas are given.In Section 3, the existence and stability of positive periodic solution of this model are examined.In Section 4, some numerical simulations are given to illustrate our results.Some concluding remarks are presented in the last section.

Model Formulation and Preliminaries
Wang and Xiao [20] proposed the following SIR epidemic model with media impact: with where (, ) = () =  −   ., , and  represent the densities or quantities of susceptible, infected, and recovered populations, respectively.All model parameters are positive constants, where  is the natural birth/death rate,  denotes the basic transmission rate, and  represents the removed/recovered rate.When  increases and reaches a certain level   , mass media start to report information about the disease, including ways of transmission and number of infected individuals, and then the public tries their best to avoid being infected.This consequently lowers the effective contact, resulting in a reduction in transmission rate which is usually represented by  exp(−), 0 <  < 1 to reflect the impact of media coverage to the effective contact rate.
We assume, throughout this paper, that  = 1 and  0 > 1.That is to say, model (1) with  = 1 has a unique endemic equilibrium  * ( * ,  * ,  * ), which is globally asymptotically stable (see Figure 1).To keep the infected density at a low level, we propose a state dependent pulse vaccination for the susceptible patients and treatment for the infected at different control threshold values.Comparing to the disease cycles, the medication for some infectious diseases is relatively short; we suppose that the procedure of medication takes pulse effect when the number of group  reaches the higher threshold value.
On the positive and ultimate boundedness of solutions of systems ( 4)-( 6), we introduce the following Lemma 1.
The proof of Lemma 1 is similar to Lemma 1 in [11]; hence we omit it here.
Generally, a semidynamical system (, , R + ) is denoted by (, ).For any  ∈ , the function   : R + →  defined as   () = (, ) is continuous, and we call   () the trajectory passing through point .Consider the following general state dependent pulse differential equation: where (, ) ∈ R 2 ., , , and  are continuous functions mapping R 2 into R and M ⊂ R 2 is the set of impulses.
According to the denotations in [22], we denote N = I(M), for any  ∈ M, I() =  + ∈ N, where M = {(, ) : (, ) ∈ R 2 , (, ) = 0} is the set of impulses, I is the pulse function, and N is the set of phase after impulses.Obviously, the solution mapping of system ( 8) is a semicontinuous dynamical system, which is denoted by (, , M, I).Obviously, in model ( 7), we discuss in the paper, which is a semicontinuous dynamical system.For the sake of investigating the existence and stability of periodic solution of model ( 7), we give the following definitions and lemmas.
Definition 5 (successor function [23]).Suppose  : 1 −   is called the successor function of point , and the point  + 1 is called the successor point of .
The following lemma and remarks are on the properties of successor function ().Lemma 6.The successor function () is continuous.
The proof of Lemma 6 is obvious; hence we omit it here.
The following Lemma 9 is on the orbitally asymptotical stability of periodic solution of model (8), which comes from Corollary 2 of Theorem 1 of [24].

Main Results
Since the endemic equilibrium  * ( * ,  * ) of model ( 7) without pulse effect is globally asymptotically stable, then any positive solutions of model (7) without pulse effect will eventually tend to  * .Therefore, region Ω is divided into four different domains with the vertical isocline d/d = 0 and the horizontal isocline d/d = 0 of model (7), where Figure 2: (a) The illustration on the existence of order-1 periodic solution of model (7) starting from pulse set Σ  2 ; (b) The illustration of existence of order-1 periodic solution of model (7) starting from pulse set Σ  1 .
For convenience, we denote the -axis intersect line  =   at point (0,   ).Suppose that the horizontal isocline line According to the uniqueness of solution to initial value, we know there exists a unique trajectory   () starting from the initial point (0,   ) (  >   ) and tangent to line  =   at point  2 = ( * 2 ,   ). Let and let Ω 3 be a bounded domain by the phase trajectory Ñ 2 and segments  2  and .Obviously, if  ℎ <  * , then any trajectory with initial value ( 0 ,  0 ) ∈ Ω 1 /Ω 3 will reach pulse set Σ 1 , and any trajectory with initial value ( 0 ,  0 ) ∈ Ω 2 ∪ Ω 3 will reach pulse set Σ 2 .So, in this section, we discuss the existence and stability of positive order-1 periodic solutions of model (7) in cases of initial values ( 0 ,  0 ) ∈ Ω 1 /Ω 3 and ( 0 ,  0 ) ∈ Ω 2 ∪ Ω 3 , respectively.Firstly, the following result is on the existence and stability of positive order-1 periodic solution for model (7).
Proof.The proof of the existence and stability of the positive order-1 periodic solution starting from pulse set Σ  2 has appeared in Theorem 10.We just need to prove the existence and stability of the other positive order-1 periodic solution which starts from pulse set Σ  1 .Suppose that the trajectory passing through point (  , (1 −  1 ) ℎ ) intersects with pulse set Σ 1 at point  1 (  1 ,  ℎ ).Since point  1 ∈ Σ 1 , then pulse occurs at  1 ; supposing point  1 is subject to pulse effects to point  + 1 (  + 1 , (1 −  1 ) ℎ ), where   + 1 = (1 − )  1 , the position of  + 1 has the following three cases (for more details, see Figure 2(b)): 1 −   = 0. Thus, model ( 7) has a positive order-1 periodic solution which starts from pulse set ) is small enough.Suppose that trajectory from the initial point (, (1 −  1 ) ℎ ) intersects pulse set Σ 1 at point  1 (  1 ,  ℎ ) and next jumps to point  + 1 (  + 1 , (1 −  1 ) ℎ ) on phase set Σ  1 due to pulse effects.According to the existence and uniqueness of solution for pulse differential equation, point  1 is right point  1 and point  + 1 is right point  + 1 .Therefore, the successor function of point  is () =   + 1 −   > 0. By Lemma 6, we know that there exists a positive order-1 periodic solution of model (7), which starts from pulse set Σ  1 .
To sum up the above discussion, model (7) exists as an order-1 periodic solution ((), ()) which starts from pulse set Σ  1 .
Next, we show the orbitally asymptotical stability of this order-1 periodic solution ((), ()).According to Lemma 9, suppose that ((), ()) intersects phase set Σ q 1 and pulse set Furthermore, it follows that On the other hand, we integrate both sides of the second equation of model (7) along the orbit P+ , and we have Then, we have By condition (16), we note that model (7) satisfies all conditions of Lemma 9. Therefore, the order-1 periodic solution ((), ()) starting from the pulse set Σ  1 is orbitally asymptotically stable.This completes the proof.
From the proof of Theorem 11, integrating both sides of the first equation of model (7) along the orbit P+ , we obtain that This shows that The following Corollary 12 is a direct consequence of Theorem 11.
The proof of the existence and stability of positive order-1 periodic solution starting from pulse set Σ  1 of model ( 7) is similar to the proof of Theorem 11, here omitted.

Concluding Remarks
Seeking a reasonable and valid control strategy to prevent infectious diseases from spreading, a novel SIR epidemic model with media impact and state dependent pulse control strategies is proposed.This model is totally different from the traditional state dependent pulse differential equation, where we consider the influences of media impact on people behaviors.That is, to different threat levels and different stages of disease, people's mind and behaviors are different.Therefore, we introduce novel control strategies which are dependent not only on the state of disease, but also on the mind and behaviors of people.
By using the methods of qualitative and successor function, we have studied the existence and orbital stability of positive order-1 periodic solutions of model ( 7) for various cases.The theoretical results show that model (7) always has order-1 periodic solution starting from pulse set Σ  2 , which is orbitally asymptotically stable for the initial value ( 0 ,  0 ) ∈ Ω 2 ∪ Ω 3 (see Theorem 10), and exists two positive order-1 periodic solutions in region Ω 1 ∪ Ω 2 with condition   ≤ (1 −  1 ) ℎ or   > (1 −  1 ) ℎ .This needs some condition guarantee (see Theorem 11 or Theorem 13).
Theoretical results and numerical simulations, in this paper, show that state dependent pulse control strategies are feasible and effective to prevent and control the spread of infectious disease.We can control the density of infected individuals at a low level over a long period of time by adjusting immune, medication strength or monitoring threshold values.At the same time, numerical simulations also show
, model (31) has two order-1 periodic solutions which are orbitally asymptotically stable by Theorem 11 and Corollary 12. Numerical simulations in Figures 3(a)-3(d) show that model (31) has two positive order-1 periodic