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

Authors: Zhi-Long He; Lin-Fei Nie
Journal: Discrete Dynamics in Nature and Society
 (2015)

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## Abstract

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.


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## Body

## 1. 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., [2–7] 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 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 [13–16] 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., [17–21] 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 Section2, 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.


## 2. Model Formulation and Preliminaries

Wang and Xiao [20] proposed the following SIR epidemic model with media impact:(1)dStdt=μ-βexp⁡-αϵISI-μS,dItdt=βexp⁡-αϵISI-μ+γI,dRtdt=γI-μRwith(2)ϵ=0,σS,I<0,1,σS,I>0,where σ(S,I)=σ(I)=I-Ic. S, I, and R represent the densities or quantities of susceptible, infected, and recovered populations, respectively. All model parameters are positive constants, where μ is the natural birth/death rate, β denotes the basic transmission rate, and γ represents the removed/recovered rate. When I increases and reaches a certain level Ic, 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 βexp⁡(-αI), 0<α<1 to reflect the impact of media coverage to the effective contact rate.From Proposition 2 in [20], authors showed that, for model (1) with ϵ=1, the disease-free equilibrium E0(1,0,0) is globally asymptotically stable if R0=β/(μ+γ)<1, and the unique endemic equilibrium E∗(S∗,I∗,R∗) is globally asymptotically stable if R0>1, where(3)S∗=μ+γβexp⁡αI∗,I∗=μμ+γ-1αLambertWαμβexp⁡αμμ+γ,R∗=1-S∗-I∗,where the LambertW function is defined to be a multivalued inverse of the function z↦zez satisfying Wzexp⁡(W(z))=z.We assume, throughout this paper, thatϵ=1 and R0>1. That is to say, model (1) with ϵ=1 has a unique endemic equilibrium E∗(S∗,I∗,R∗), 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 I reaches the higher threshold value. That is, when the density of the infected individuals reaches the higher hazardous threshold value Hh (Hh∈(0,I∗)) at time ti(Hh) at the ith time, the vaccination and intense treatment are taken, and the densities of susceptible, infected, and recovered individuals turn very suddenly to a great degree to (1-p)S(ti+(Hh)), (1-q1)I(ti+(Hh)), and R(ti+(Hh))+pS(ti+(Hh))+q1I(ti+(Hh)), respectively, where p∈(0,1) and q1∈(0,1) are the vaccination intensity and medication intensification effort, respectively. However, when the density of the susceptible reaches the relatively small threshold value Hl (Hl∈(0,(1-q1)Hh)) at time tj(Hl) at the jth time, according to the minds and behavior of people on the threat of disease, it just needs to enhance the strength of treatment. In this case, the densities of susceptible, infected, and recovered individuals turn very suddenly to a great degree to S(tj+(Hh)), (1-q2)I(tj+(Hl)), and R(tj+(Hh))+q2I(tj+(Hl)), respectively, where q2∈(0,1) is the medication intensification effort.Figure 1
The phase space of model (1) with ϵ=1.Under the above assumptions, it follows from model (1) that we propose the following multiple state dependent pulse control differential equations:(i)
I≠Hh,Hl, or I=Hh,S<S1∗, or I=Hl,S<S2∗: (4)dStdt=μ-βexp⁡-αISI-μS,dItdt=βexp⁡-αISI-μ+γI,dRtdt=γI-μR;(ii)
I=Hh and S≥S1∗:(5)ΔSt=-pSt,ΔIt=-q1It,ΔRt=pSt+q1It;(iii)
I=Hl and S≥S2∗:(6)ΔSt=0,ΔIt=-q2It,ΔRt=q2It,where S1∗>0 and S2∗>0 are the abscissa of intersection of the the horizontal isocline βexp⁡(-αI)S-μ-γ=0 and the lines I=Hh and I=Hl, respectively, and S1∗=μ+γexp⁡(αHh)/β and S2∗=μ+γexp⁡(αHl)/β.LetR=(-∞,∞) and R+3={(x,y,z)∣x>0,y>0,z>0}. The global existence and uniqueness of solution for systems (4)–(6) are guaranteed by the smoothness of the right-hand sides of systems (4)–(6). For more details, we refer to [22].On the positive and ultimate boundedness of solutions of systems (4)–(6), we introduce the following Lemma 1.Lemma 1.
For anyp∈(0,1), qi∈(0,1) (i=1,2), and each component of the solution (S(t),I(t),R(t)) of systems (4)–(6) with initial value (S(0),I(0),R(0))∈R+3 is positive and ultimately bounded for all t∈[0,+∞).The proof of Lemma1 is similar to Lemma 1 in [11]; hence we omit it here.Since from systems (4)–(6), without loss of generality, the total population is normalized to unity that S(t)+I(t)+R(t)=1, therefore, systems (4)–(6) are equivalent to the following system:(7)dStdt=μ-βexp⁡-αISI-μS,dItdt=βexp⁡-αISI-μ+γI,I≠Hh,HlorI=Hh,S<S1∗orI=Hl,S<S2∗,ΔSt=-pSt,ΔIt=-q1It,I=Hh,S≥S1∗,ΔSt=0,ΔIt=-q2It,I=Hl,S≥S2∗,S0=S0>0,I0=I0>0.Based on the biological background of model (7), we only consider dynamical behavior of model (7) in region Ω={(S,I):S>0,I>0,S+I≤1}.Generally, a semidynamical system(X,π,R+) is denoted by (X,π). For any P∈X, the function πP:R+→X defined as πP(t)=π(P,t) is continuous, and we call πP(t) the trajectory passing through point P. Consider the following general state dependent pulse differential equation:(8)dxdt=fx,y,dydt=gx,y,φx,y≠0,Δx=αx,y,Δy=βx,y,φx,y=0,where (x,y)∈R2. f, g, α, and β are continuous functions mapping R2 into R and M⊂R2 is the set of impulses. According to the denotations in [22], we denote N=I(M), for any P∈M, I(P)=P+∈N, where M={(x,y):(x,y)∈R2,φ(x,y)=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 (X,π,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 2 (semicontinuous dynamical system [22]).
A triple(X,π,R+) is said to be a semidynamical system if X is a metric space, R+ is the set of all nonnegative real, and π(P,t):X×R+→X is a continuous map such that (a)
π(P,0)=P for all P∈X,(b)
π(π(P,t),s)=π(P,t+s) for all P∈X and t,s∈R+.Definition 3 (order-1 periodic solution [23]).
A trajectoryπ(P0,t) is called order-1 periodic solution with period T if there exists a point P0∈M and T>0 such that π(P0,t)=P∈M and I=I(π(P0,T))=P0∈N.Definition 4 (orbitally asymptotically stable [23]).
Supposeπ(P0,t) is an order-1 periodic solution of model (8). If, for any ɛ>0, there must exist δ>0 and t0≥0, such that for any point P1∈U(P0,δ)∩N, one has d(π(P1,t),π(P0,t))<ɛ for t>t0, where U(P0,δ) denotes a δ-neighborhood of point P0∈N and d(π(P1,t),π(P0,t)) is the distance from π(P1,t) to π(P0,t). Then one calls the order-1 periodic solution π(P0,t) orbitally asymptotically stable.Definition 5 (successor function [23]).
Supposeϕ:N→N is a map. For any P(xP,yP)∈N, if there exists a t1>0 such that π(P,t1)=P1(xP1,yP1)∈M,P1+(xP1+,yP1+)=I(P1)∈N, then ϕ(P)=xP1+-xP is called the successor function of point P, and the point P1+ is called the successor point of P.The following lemma and remarks are on the properties of successor functionϕ(P).Lemma 6.
The successor functionϕ(P) is continuous.The proof of Lemma6 is obvious; hence we omit it here.Remark 7.
From Lemma6, it is obvious that model (8) exists as positive order-1 periodic solution if there exist two points P1,P2∈N satisfying ϕ(P1)ϕ(P2)<0.Remark 8.
In Lemma6, if ϕ(P)=0, then trajectory π(P0,t) with initial point P0 is an order-1 periodic solution of model (8).The following Lemma9 is on the orbitally asymptotical stability of periodic solution of model (8), which comes from Corollary 2 of Theorem 1 of [24].Lemma 9 (analogue of Poincaré Criterion [24]).
TheT-periodic solution (ϕ(t),ψ(t)) of system (8) is orbitally asymptotically stable if the Floquet multiplier μ satisfies the condition μ<1, where(9)μ=∏j=1nΔjexp⁡∫0T∂fϕt,ψt∂x+∂gϕt,ψt∂ydt,Δj=∂η/∂y∂φ/∂x-∂η/∂x∂φ/∂y+∂φ/∂xf++∂ξ/∂x∂φ/∂y-∂ξ/∂y∂φ/∂x+∂φ/∂yg+∂φ/∂xf+∂φ/∂ygand f, g, ∂ξ/∂x, ∂ξ/∂y, ∂η/∂x, ∂η/∂y, ∂φ/∂x, and ∂φ/∂y are calculated at the point (ϕ(τj),ψ(τj)), f+=f(ϕ(τj+),ψ(τj+)), g+=g(ϕ(τj+),ψ(τj+)), and τj(j∈N) is the time of the jth jump.To discuss the dynamical behaviors of model (7), we denote two pulse sets(10)Σ1≔S,I:S1∗≤S≤1-Hh,I=Hh,Σ2≔S,I:S2∗≤S≤1-Hl,I=Hland two phase sets(11)Σq1≔I1Σ1=S,I:1-pS1∗≤S≤1-p1-Hh,I=1-q1Hh,Σq2≔I2Σ2=S,I:S2∗≤S≤1-Hl,I=1-q2Hl,where I1:(S,Hh)∈Σ1→((1-p)S,(1-q1)Hh)∈Ω and I2:(S,Hl)∈Σ2→(S,(1-q2)Hl)∈Ω are continuous functions.


## 3. Main Results

Since the endemic equilibriumE∗(S∗,I∗) of model (7) without pulse effect is globally asymptotically stable, then any positive solutions of model (7) without pulse effect will eventually tend to E∗. Therefore, region Ω is divided into four different domains with the vertical isocline dS/dt=0 and the horizontal isocline dI/dt=0 of model (7), where(12)D1≔S,I∈Ω:dSdt>0,dIdt<0,D2≔S,I∈Ω:dSdt>0,dIdt>0,D3≔S,I∈Ω:dSdt<0,dIdt>0,D4≔S,I∈Ω:dSdt<0,dIdt<0.For convenience, we denote theI-axis intersect line I=Hl at point M(0,Hl). Suppose that the horizontal isocline line βexp⁡(-αI)S-μ-γ=0 intersects lines I=Hh, I=(1-q1)Hh, and I=(1-q2)Hl at points O1(S1∗,Hh), A(SA,(1-q1)Hh), and G(SG,(1-q2)Hl), respectively. Suppose that the vertical isocline line μ-βexp⁡(-αI)S-μS=0 intersects lines I=(1-q1)Hh and I=(1-q2)Hl at points C(SC,(1-q1)Hh) and K(SK,(1-q2)Hl), respectively.According to the uniqueness of solution to initial value, we know there exists a unique trajectoryπN(t) starting from the initial point N(0,IN)(IN>Hl) and tangent to line I=Hl at point O2=(S2∗,Hl). Let(13)Ω1=S,I∈Ω:Hl<I≤Hh,Ω2=S,I∈Ω:I≤Hland let Ω3 be a bounded domain by the phase trajectory NO2~ and segments O2M¯ and MN¯. Obviously, if Hh<I∗, then any trajectory with initial value (S0,I0)∈Ω1/Ω3 will reach pulse set Σ1, and any trajectory with initial value (S0,I0)∈Ω2∪Ω3 will reach pulse set Σ2. So, in this section, we discuss the existence and stability of positive order-1 periodic solutions of model (7) in cases of initial values (S0,I0)∈Ω1/Ω3 and (S0,I0)∈Ω2∪Ω3, respectively.Firstly, the following result is on the existence and stability of positive order-1 periodic solution for model (7).Theorem 10.
For anyp∈(0,1), q1∈(0,1), and q2∈(0,1), model (7) has always an orbitally asymptotically stable positive order-1 periodic solution and which starts from pulse set Σq2.Proof.
Obviously, any trajectory with initial value(S0,I0)∈Ω2∪Ω3 will reach pulse set Σ2 and intersects pulse set Σ2 infinite times due to pulse treatment ΔI(t)=I(t+)-I(t)=-q2I(t). The trajectory πG(t) passing through point G which is tangent to phase set Σq2 at point G intersects with pulse set Σ2 at point G1(SG1,Hl) and then jumps to point G1+(SG1+,(1-q2)Hl) due to pulse treatment ΔI(t)=I(t+)-I(t)=-q2I(t), where SG1+=SG1. By the geometrical construction of phase region D2, we have that point G1+ is right point G. That is, SG<SG1+. Hence the successor function of point G is ϕ(G)=SG1+-SG>0.
On the other hand, trajectoryπK(t) from the initial point N(SK,(1-q2)Hl) intersects pulse set Σ2 at point K1(SK1,Hl) and next jumps to point K1+(SK1+,(1-q2)Hl)  (SK1+=SK1) on phase set Σq2. It follows from the geometrical construction of region D3 that we have that point K1+ is right point K. That is, SK1+<SK. Therefore, we have ϕ(K)=SK1+-SK<0.
By Lemma6, we know that there exists a positive order-1 periodic solution of model (7), which starts from pulse set Σq2 (for more details, see Figure 2(a)).
Next, we discuss the orbital stability of positive order-1 periodic solution which starts from pulse setΣq2. Since trajectories starting from any point (S,I)∈Ω1∪Ω3 will enter the set G1K1¯={(S,I)∈Σq2∣SG1≤S≤SK1} after several times pulse effects at most, then the initial point of the order-1 periodic solution only lies in G1+K1+¯. The set G1+K1+¯ is mapped to set G2K2¯={(S,I)∈Σ1∣SG2≤S≤SK2} by the geometrical construction of the phase regions D2 and D3. Subsequently, set G2K2¯ is mapped to set G2+K2+¯={(S,I)∈Σq2∣SG2+≤S≤SK2+} due to the pulse vaccination and pulse treatment. From the geometrical construction of regions D2 and D3, it is easy to know that G1+K1+¯>G2+K2+¯. Repeating the abovementioned process, we can get two sequences {Gn+(SGn+,(1-q2)Hl)} and {Kn+(SGn+,(1-q2)Hl)} (n=1,2,…) on set Σq2 and satisfy(14)SG<SG1+<⋯<SGn+<⋯<SKn+<SK1+<SK,GK¯>G1+K1+¯>⋯>Gn+Kn+¯>⋯.Therefore, the sequence {Gn+Kn+¯} is convergent monotonously and limn→∞Gn+Kn+¯=0. It implies that there exists a unique point P0+∈[SGn+,SK1+] such that g(P0+)=0. Furthermore, we have limn→∞SGn+=limn→∞SKn+=SP0+. For any point Q(SQ,(1-q2)Hl)∈Σq2, where(15)SQ∈SG,SK1+∪SGn+,SGn+1+∪SKn+1+,SKn+∪SK1+,SK+,n=1,2,….Without loss of generality, let SQ∈[SGn+,SGn+1+), the trajectory πQ(t) starting from the initial point Q(SQ,(1-q2)Hl)∈Σq2 intersects pulse set Σ2 at point Q1(SQ1,Hl) (SQ1∈[SGn+1,SGn+2)) and next jumps to point Q1+(SQ1+,(1-q2)Hl)∈Σq2 due to pulse vaccination and pulse treatment, where SQ1+∈[SGn+1+,SGn+2+). Repeating the process, we obtain a sequence {Qn+(SQn+,(1-q2)Hl)} of set Σq2, where SQn+∈[SGn+k+1+,SGn+k+), k=1,2,…. Since limn→∞SGn+=SP0+, then limn→∞SQn+=SP0+. That is, limn→∞Qn+=P0+. Similarly, if SQ∈[SKn+1+,SKn+), we also can get limn→∞Qn+=P0+. Thus trajectory starts from any point Q of set Σq2, which ultimately tends to the positive order-1 periodic solution P0+P0P0+^.
Given the above, we obtain that model (7) has a positive order-1 periodic solution, which starts from the initial value (S0,I0)∈Ω2∪Ω3 and is orbitally asymptotically stable. This completes the proof.Figure 2
(a) The illustration on the existence of order-1 periodic solution of model (7) starting from pulse set Σq2; (b) The illustration of existence of order-1 periodic solution of model (7) starting from pulse set Σq1.
(a)
(b)Theorem 11.
IfIN≤(1-q1)Hh, for any p∈(0,1), q1∈(0,1), and q2∈(0,1), then model (7) has two positive order-1 periodic solutions in region Ω1∪Ω2: one starts from pulse set Σq1, and the other starts from Σq2. Furthermore, let (ϕ(t),ψ(t)) be an order-1 periodic solution which starts from the pulse set Σq1 of model (7) with period T; if(16)μ=1-pβe-α1-q1Hh1-pϕT-μ-γβe-αHhϕT-μ-γ·exp⁡-∫0Tμ+1+αϕtβe-αψtψtdt<1,then (ϕ(t),ψ(t)) is orbitally asymptotically stable. Thus, trajectory of model (7) with initial value (S0,I0)∈Ω1∪Ω3 will tend to the stable positive order-1 periodic solution which starts from pulse set Σq2, and trajectory of model (7) with initial value (S0,I0)∈Ω1/Ω3 will tend to the stable positive order-1 periodic solution which starts from Σq1.Proof.
The proof of the existence and stability of the positive order-1 periodic solution starting from pulse setΣq2 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 Σq1.
Suppose that the trajectory passing through pointA(SA,1-q1Hh) intersects with pulse set Σ1 at point A1(SA1,Hh). Since point A1∈Σ1, then pulse occurs at A1; supposing point A1 is subject to pulse effects to point A1+(SA1+,(1-q1)Hh), where SA1+=(1-p)SA1, the position of A1+ has the following three cases (for more details, see Figure 2(b)):(i)
If pointA1+ coincides with A, that is, SA=SA1+, then the successor function of A is that f(A)=SA1+-SA=0. Thus, model (7) has a positive order-1 periodic solution which starts from pulse set Σq1.(ii)
If pointA1+ is left point A, that is, SA>SA1+, thus the successor function of A is that f(A)=SA1+-SA<0. On the other hand, choose a point B(ɛ,(1-q1)Hh)∈Σq1 which is left point A1+, where ɛ∈(0,(1-p)S1∗) is small enough. Suppose that trajectory from the initial point B(ɛ,(1-q1)Hh) intersects pulse set Σ1 at point B1(SB1,Hh) and next jumps to point B1+(SB1+,(1-q1)Hh) on phase set Σq1 due to pulse effects. According to the existence and uniqueness of solution for pulse differential equation, point B1 is right point A1 and point B1+ is right point A1+. Therefore, the successor function of point B is f(B)=SB1+-SB>0. By Lemma 6, we know that there exists a positive order-1 periodic solution of model (7), which starts from pulse set Σq1.(iii)
If pointA1+ is right point A, that is, SA<SA1+, thus, the successor function of A is that f(A)=SA1+-SA>0. In the meanwhile, suppose that trajectory from the initial point C(SC,(1-q1)Hh) intersects pulse set Σ1 at point C1(SC1,Hh) and next jumps to point C1+(SC1+,(1-q1)Hh) on phase set Σq1 again, where SC1+=(1-P)SC1. From the geometrical construction of phase region D3 of model (7), we have SC1<SC. Therefore, point C1+ is left point C. That is, SC1+<SC. Thus, the successor function of A is f(C)=SC1+-SC<0. By Lemma 6, we know that model (7) admits a positive order-1 periodic solution, which starts from pulse set Σq1.
To sum up the above discussion, model (7) exists as an order-1 periodic solution (ϕ(t),ψ(t)) which starts from pulse set Σq1.
Next, we show the orbitally asymptotical stability of this order-1 periodic solution(ϕ(t),ψ(t)). According to Lemma 9, suppose that (ϕ(t),ψ(t)) intersects phase set Σq1 and pulse set Σ1 at points P+((1-p)ϕ(T),(1-q1)Hh) and P(ϕ(T),Hh), respectively. Comparing with model (8), we have(17)fS,I=μ-βe-αISI-μS,gS,I=βe-αISI-μ+γI,ξ(S,I)=-pS, η(x,y)=-q1Hh, and φ(S,I)=I-Hh. Thus,(18)∂f∂S=-βe-αII-μ,∂g∂I=-αβe-αISI+βe-αIS-μ-γ,∂ξ∂S=-p,∂η∂I=-q1,∂φ∂I=1,∂ξ∂I=∂η∂S=∂φ∂S=0.Furthermore, it follows that(19)Δ1=∂η/∂I∂φ/∂S-∂η/∂S∂φ/∂I+∂φ/∂Sf++∂ξ/∂S∂φ/∂I-∂ξ/∂I∂φ/∂S+∂φ/∂Ig+∂φ/∂Sf+∂φ/∂Ig=1-pg+ϕT+,ψ2T+gϕ1T,ψ2T=1-p1-q1βe-α1-q1Hh1-pϕT-μ-γβe-αHhϕT-μ-γ,μ=Δ1exp⁡∫0T∂f∂Sϕt,ψt+∂g∂Iϕt,ψtdt=Δ1exp⁡∫0T-βe-αψtψt-μ-αβe-αψtϕtψt+βe-αψtϕt-μ-γdt.On the other hand, we integrate both sides of the second equation of model (7) along the orbit P+P~, and we have(20)ln⁡11-q1=∫1-q1HhHhdII=∫0Tβe-αψtϕt-μ-γdt.Then, we have(21)μ=11-q11-p1-q1βe-α1-q1Hh1-pϕT-μ-γβe-αHhϕT-μ-γ·exp⁡-∫0Tμ+1+αϕtβe-αψtψtdt=1-pβe-α1-q1Hh1-pϕT-μ-γβe-αHhϕT-μ-γ·exp⁡-∫0Tμ+1+αϕtβe-αψtψtdt.
By condition (16), we note that model (7) satisfies all conditions of Lemma 9. Therefore, the order-1 periodic solution (ϕ(t),ψ(t)) starting from the pulse set Σq1 is orbitally asymptotically stable. This completes the proof.From the proof of Theorem11, integrating both sides of the first equation of model (7) along the orbit P+P~, we obtain that(22)ln⁡1-1-pϕT1-ϕT=∫1-pϕTϕTdS1-S≤μT.This shows that(23)exp⁡-∫0Tμ+1+αϕtβe-αψtψtdt<exp⁡-μT≤1-ϕT1-1-pϕT.The following Corollary12 is a direct consequence of Theorem 11.Corollary 12.
Let(ϕ(t),ψ(t)) be the order-1 periodic solution starting from pulse set Σq1 of model (7) with period T. If(24)μ=1-pβe-α1-q1Hh1-pϕT-μ-γβe-αHhϕT-μ-γ·1-ϕT1-1-pϕT<1,then (ϕ(t),ψ(t)) is orbitally asymptotically stable.ForIN>(1-q1)Hh, in this case, suppose that orbit NO2~ starting from the initial point N(0,IN) will intersect lines I=(1-q1)Hh and I=Hl at points D(S1,(1-q1)Hh) and O2(S2∗,Hl), respectively.Integrating both sides of the second equation of model (7) along orbit NO2~, we have(25)ln⁡Hl1-q1Hh=∫1-q1HhHldII=∫t1t2βe-αIS-μ-γdt>-μ+γt2-t1.That is,(26)t2-t1>1μ+γln⁡1-q1HhHl.Further, integrating both sides of the first equation of model (7) along orbit NO2~, we have(27)S1=S2∗-∫t1t2μ-βe-αISI-μSdt<S2∗-μ-βe-α1-q1HhS2∗1-q1Hh-μS2∗t2-t1.From the geometrical construction of phase regionD1, we have μ-βe-αISI-μS>0. Since the point (S2∗,(1-q1)Hh) in region D1, it yields that(28)μ-βe-α1-q1HhS2∗1-q1Hh-μS2∗>0.This together with (26) and (27) gives(29)S1<S2∗-μ-βe-α1-q1HhS2∗1-q1Hh-μS2∗·1μ+γln⁡1-q1HhHl≔θ∗.Theorem 13.
Assuming thatIN>(1-q1)Hh, for any p∈(0,1), q1∈(0,1), and q2∈(0,1), model (7) has a positive order-1 periodic solution starting from pulse set Σq2 which is orbitally asymptotically stable. Furthermore, if(30)0<p≤1-βμ+γθ∗e-αHh,then model (7) has a positive order-1 periodic solution starting from pulse set Σq1. Let (ϕ(t),ψ(t)) be the order-1 periodic solution of model (7) with period T, which starts from pulse set Σq1. Further, if (16) holds, then (ϕ(t),ψ(t)) is orbitally asymptotically stable. Thus, model (7) has two positive order-1 periodic solutions in region Ω1∪Ω2, which are orbitally asymptotically stable.The proof of the existence and stability of positive order-1 periodic solution starting from pulse setΣq1 of model (7) is similar to the proof of Theorem 11, here omitted.


## 4. Numerical Simulations

To illustrate the theoretical results and the feasibility of state dependent pulse control strategies, some simulations are presented. In systems (4)–(6), we fixed parameters μ=0.1, β=0.8, α=0.5, and γ=0.04. That is, we consider the following SIR epidemic model with state-dependent pulse vaccination and medication:(31)dStdt=0.1-0.8e-0.5ISI-0.1S,dItdt=0.8e-0.5ISI-0.1+0.04I,dRtdt=0.04I-0.1R,I≠Hh,HlorI=Hh,S<S1∗orI=Hl,S<S2∗,ΔSt=-pSt,ΔIt=-q1It,ΔRt=pSt+q1It,I=Hh,S≥S1∗,ΔSt=0,ΔIt=-q2It,ΔRt=q2It,I=Hl,S≥S2∗.By directly calculating, we have R0=β/(μ+γ)≈5.7143>1. It is easy to known that model (31) without pulse control has a unique globally asymptotically stable endemic equilibrium E∗(S∗,I∗,R∗)=(0.2304,0.5497,0.2199).Firstly, we choose control parametersq1=q2=0.3, Hh=0.45<I∗=0.5497, and Hl=0.25<(1-q1)Hh=0.315, respectively. By calculating, we get S1∗=(μ+γ)eαHh/β=0.2192 and S2∗=μ+γexp⁡(αHl)/β=0.1983. From numerical simulation, we know model (31) has a unique trajectory πN(t) starting from the initial point N(0,0.2985) which is tangent to line I=Hl at point O2=(S2∗,Hl), as shown in Figure 3(a). Obviously, IN=0.2985<(1-q1)Hh=0.3150. Now, we take p=0.8; it is easy to calculate that(32)μ=1-ϕT1-1-pϕT1-pβe-α1-q1Hh1-pϕT-μ-γβe-αHhϕT-μ-γ=1-0.27321-1-0.8×0.27321-0.80.8e-0.5×1-0.3×0.451-0.8×0.2732-0.1-0.040.8e-0.5×0.45×0.2732-0.1-0.04≈0.4572<1.Therefore, 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 solutions: one starts from point (0.3936,0.1750)∈Σq2, and the other starts from point (0.0546,0.3150)∈Σq1, which are orbitally asymptotically stable.Figure 3
The existence and orbital stability of order-1 periodic solutions of model (31) for case IN≤(1-q1)Hh, where Hh=0.45, Hl=0.25, q1=q2=0.3, IN=0.2985<(1-q1)Hh=0.3150, and p=0.8, respectively.
(a)
(b)
(c)
(d)Nextly, we choose control parameterHl=0.3<(1-q1)Hh=0.315 and other parameters fixed as above. It is easy to calculate that S1∗=(μ+γ)eαHh/β=0.2192 and S2∗=(μ+γ)exp(αHl)/β=0.2033. By numerical simulations, model (31) has a trajectory which starts from the initial point (0,0.3631) and is tangent to line I=Hl. This is what show in Figure 4(d). Obviously, IN=0.3631>(1-q1)Hh=0.3150. It can be easily calculated that(33)θ1∗=S2∗-μ-βe-α1-q1HhS2∗1-q1Hh-μS2∗·1μ+γln⁡1-q1HhHl=0.2033-0.1-0.8e-0.5×0.21×0.2033×0.21-0.1×0.203310.14·ln⁡0.7×0.450.3=0.1908,1-βμ+γθ1∗e-αHh=1-0.80.14×0.1908×e-0.5×0.45=0.1294.Further, if we choose p=0.1294, it is easy to calculate that(34)μ=1-ϕT1-1-pϕT1-pβe-α1-q1Hh1-pϕT-μ-γβe-αHhϕT-μ-γ=1-0.27061-0.8706×0.27060.8706×0.8e-0.5×1-0.3×0.45×0.8706×0.2706-0.1-0.040.8e-0.5×0.45×0.2706-0.1-0.04≈0.5309<1.Therefore, model (31) has two positive order-1 periodic solutions which are orbitally asymptotically stable by Theorem 13. The plots in Figures 4(a)–4(d) show that one orbital stable order-1 periodic solution starts from point (0.3543,0.21)∈Σq2, and the other starts from point (0.2356,0.315)∈Σq1.Figure 4
The existence and orbital stability of order-1 periodic solutions of model (31) for case IN>(1-q1)Hh, where Hh=0.45, Hl=0.3, q1=q2=0.3, IN=0.3631>(1-q1)Hh=0.3150, and p=0.1294.
(a)
(b)
(c)
(d)We choose, finally, the vaccination intensityp=0.8>1-βθ1∗e-αHh/(μ+γ)=0.1294. Then, the condition of Theorem 13 cannot be met. The plots in Figure 5(a) show that model (31) only has a positive order-1 periodic solution starting from set Σq2. We choose, however, control parameters to be Hh=0.4, Hl=0.31, q1=0.21, and q2=0.15. From numerical simulation and calculation, we have that IN=0.37635>(1-q1)Hh=0.3160 and θ2∗=0.1994. By Theorem 13, we can know that model (31) has two positive order-1 periodic solutions satisfying 0<p≤1-βθ2∗e-αHh/(μ+γ)=0.0671. In particular, take p=0.4>0.0671 which does not satisfy the condition of Theorem 13, but Figure 5(b) shows that model (31) has two positive order-1 periodic solutions. This implies that the dynamical behaviors of model (31) are complex because of the effects of state dependent pulse control strategies.Figure 5
The complex dynamical behaviors of model (31) due to the effects of state dependent pulse control strategies, where (a) Hh=0.45, Hl=0.3, q1=q2=0.3, and p=0.8; (b) Hh=0.4, Hl=0.31, q1=0.21, q2=0.15, and p=0.4.
(a)
(b)


## 5. 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 Σq2, which is orbitally asymptotically stable for the initial value (S0,I0)∈Ω2∪Ω3 (see Theorem 10), and exists two positive order-1 periodic solutions in region Ω1∪Ω2 with condition IN≤(1-q1)Hh or IN>(1-q1)Hh. 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 that model (7) has richer dynamical behaviors because of the effects of state dependent pulse control strategies.



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