A State Dependent Pulse Control Strategy for a SIRS Epidemic System
Lin-Fei Nie, Zhi-Dong Teng & Bao-Zhu Guo
Bulletin of Mathematical Biology A Journal Devoted to Research at the Junction of Computational, Theoretical and Experimental Biology Official Journal of The Society for Mathematical Biology ISSN 0092-8240 Volume 75 Number 10 Bull Math Biol (2013) 75:1697-1715 DOI 10.1007/s11538-013-9865-y
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Author's personal copy Bull Math Biol (2013) 75:1697–1715 DOI 10.1007/s11538-013-9865-y O R I G I N A L A RT I C L E
A State Dependent Pulse Control Strategy for a SIRS Epidemic System Lin-Fei Nie · Zhi-Dong Teng · Bao-Zhu Guo
Received: 19 December 2012 / Accepted: 30 May 2013 / Published online: 28 June 2013 © Society for Mathematical Biology 2013
Abstract With the consideration of mechanism of prevention and control for the spread of infectious diseases, we propose, in this paper, a state dependent pulse vaccination and medication control strategy for a SIRS type epidemic dynamic system. The sufficient conditions on the existence and orbital stability of positive order-1 or order-2 periodic solution are presented. Numerical simulations are carried out to illustrate the main results and compare numerically the state dependent vaccination strategy and the fixed time pulse vaccination strategy. Keywords SIRS epidemic model · State dependent pulse · Order-k periodic solution · Orbital stability
1 Introduction The control and hence eradication of infectious disease is one of the major concerns in the study of mathematical epidemiology. In the last several decades, the epidemic dynamical models which are described by the differential equations have played a crucial role in control and eradication of the infectious diseases. Perhaps an earliest classical epidemic dynamical model was developed by Kermack and Mckendrick L.-F. Nie () · Z.-D. Teng College of Mathematics and Systems Science, Xinjiang University, Urumqi 830046, P.R. China e-mail:
[email protected] B.-Z. Guo Academy of Mathematics and Systems Science, Academia Sinica, Beijing 100190, P.R. China B.-Z. Guo School of Computational and Applied Mathematics, University of the Witwatersrand, Wits 2050, Johannesburg, South Africa
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(1927) in the late 1920s. In this model, the total population N is divided into susceptible individuals, infected individuals, and recovered individuals, which are represented shorthand by S, I , and R respectively. Many kinds of epidemic dynamical models have been developed subsequently to analyze the spread and control of infectious diseases. We refer some of them to (Anderson and May 1979; Capasso 1993; Diekmann and Heesterbeek 2000; Ruan and Wang 2003) and the references therein. Based on these dynamical models, many researchers studied the evolution of an infectious disease from different perspectives. These include the existence of the threshold value which is the index for the evolution and extinction of an infectious disease; the local or global stability of the disease-free equilibrium and endemic equilibrium; the existence of periodic solutions; and the persistence and extinction of the disease, name just a few. In the past years, with the progress of the globalization, the control of infectious diseases has been concerned increasingly interdisciplinary in different contexts. Of many strategies, the efficient ways for elimination or control of infectious diseases are still immunization and medication (Sabin 1991; Ramsay et al. 1994). One of the medications is the pulse vaccination and medication, which has become a major topic in mathematical biology and mathematical epidemiology (see, e.g., Gao et al. 2006, 2007; Terry 2010; Wang et al. 2010; Liu et al. 2009; Pei et al. 2009, and the references therein). Using a SIR epidemic model, Shulgin et al. (1998) showed that under a planned pulse vaccination region, the sate of system converges to a stable sate with which the size of infectious population is zero. This shows that the pulse vaccination may lead to the eradication of infectious disease provided that the magnitude of vaccination keeps a rational proportion and the period of pulses is sustained. D’Onofrio (2002) proposed a SEIR epidemic model based pulse vaccination strategy by which the local and global asymptotic stabilities of the periodic eradication solution are analyzed. Gao et al. (2011) proposed an impulsive SIRS epidemic model with periodic saturation incidence and vertical transmission. The effects of periodic varying contact rate and mixed vaccination strategy on eradication of infectious disease are studied. In all these studies, the impulses are supposed in a fixed time interval, for which we may call the fixed-time pulse vaccination and medication strategy (FTPS for short). The theoretical results and statistical data show, however, FTPS is very different from the conventional strategies in leading to disease (viral infections, such as rabies, yellow fever, poliovirus, hepatitis B, and encephalitis B) eradication at relatively low values of vaccination and medication (Agur et al. 1983). A noticed fact is that the transmission of different infectious diseases has different ways. For instance, measles, cholera and influenza, HIV/AIDS, pulmonary tuberculosis, hepatitis E virus and bacillary dysentery are transmitted irregularly, or transmitted through particular mechanism. The practice shows that these diseases cannot be eradicated in a short time. A natural idea is therefore to keep the density of infections at a low level to avoid the spread of the disease. In this regard, exploring an effective and easily implemented control measure to keep the level of spread of diseases is significant both theoretically and practically. Recently, the state dependent impulsive feedback control strategy is applied widely to the control of spread of infectious disease due to its economic, high ef-
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ficiency, and feasibility nature. This idea 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 population reaches a threshold value. In this regard, it is different from the fixed-time pulse control strategy. Many works have been focused on the analysis of mathematical models described by ordinary differential equations with state dependent pulse effects. In (Nie et al. 2010, 2009; Jiang and Lu 2006, 2007; Tang et al. 2005), the dynamic behaviors of predator–prey models with state dependent pulse effects are considered and the existence and stability of positive periodic solution by the Poincaré map, properties of the Lambert W function, analogue of Poincaré criterion are obtained. In addition, Ross (1911) proposed a mathematical model to study the spread between human beings and mosquitoes for malaria in earlier 1911, where a concept of threshold density is introduced and it is concluded that “. . . in order to counteract malaria anywhere we need not banish Anopheles there entirely—we need only to reduce their numbers below a certain figure.” Motivated by these facts, we propose, in this paper, a state dependent pulse vaccination and medication control for a SIRS epidemic system. The difference of the dynamic behaviors between our model and the fixed-time pulse effect model is illustrated in Sect. 2. Some sufficient conditions are presented in Sect. 3 for the existence and stability of positive periodic solutions. The numerical simulations are carried out in Sect. 4 for illustration. Through the numerical comparison with the fixed-time pulse control strategy, it is shown that the state dependent pulse vaccination and medication strategy is more effective and easily implemented to prevent the spread level of the disease. Some concluding remarks are presented in Sect. 5.
2 Model Formulation and Preliminaries A traditional autonomous SIRS epidemic model is of the following form: ⎧ dS(t) ⎪ ⎪ = b S(t) + I (t) + R(t) − βS(t)I (t) − bS(t) + αR(t), ⎪ ⎪ dt ⎪ ⎪ ⎨ dI (t) = βS(t)I (t) − bI (t) − γ I (t), ⎪ dt ⎪ ⎪ ⎪ ⎪ ⎪ dR(t) ⎩ = γ I (t) − bR(t) − αR(t), dt
(1)
where S(t), I (t), and R(t) stand for the numbers of susceptible, infected, and recovered individuals to the disease of a population at time t respectively, b > 0 represents the immigration rate with the assumption that all newborns are susceptible, which also represents the death rate of susceptible, infected, and recovered groups, respectively. It is assumed that all susceptible group becomes infected at a rate βI , where β > 0 is the contact rate; the infected group becomes recovered who received lifetime immunity at a constant rate γ > 0. Therefore, 1/γ is the mean infectious period. In addition, the recovered group becomes susceptible group again at a constant rate α > 0, so 1/α is the average immune time.
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The system (1) has been studied extensively, and some of them can be found in Anderson and May (1991), Hethcote (1989) and the references therein. The succeeding results on system (1) are available in Ma et al. (2004). Theorem 1 (i) If parameter R0 = β/(b + γ ) < 1, then system (1) admits only a globally asymptotically stable disease-free equilibrium (1, 0, 0). (ii) If parameter R0 > 1, then system (1) admits an unstable disease-free equilibrium (1, 0, 0) and a unique globally asymptotically stable endemic equilibrium (S ∗ , I ∗ , R ∗ ), where S∗ =
b+γ , β
I∗ =
(b + α)(β − b − γ ) , β(b + α + γ )
R∗ =
γ (β − b − γ ) . β(b + α + γ )
Generally speaking, the immunization and medication control for infectious disease can be modeled in three different forms: the continuous time control where the control measure is taken once the disease is discovered; the fixed-time pulse control aforementioned; and the state dependent pulse control presented in this paper. Among them the continuous time control is not practically implementable. Different with the usual continuous control or fixed-time pulse control strategy, we propose a state dependent pulse feedback control strategy. In addition, since 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 threshold value. We first introduce the following assumption (A) before building a new mathematical model. (A) When the number of the infected individuals reaches the hazardous threshold value H where, 0 < H < 1 − S ∗ at time ti (H ) at the ith time, the vaccination and medication are taken and the numbers of susceptible, infected, and recovered individuals turn very suddenly to a great degree to (1 − m)S(ti+ (H )), ((1 − p)I (ti+ (H ))), and R(ti+ (H )) + mS(ti+ (H )) + pI (ti+ (H )), respectively, where m, p ∈ (0, 1) are the vaccination intensity and medication intensification effort, respectively. Under assumption (A), we propose a new control model of infectious disease which is described by the following ordinary differential equation with the state dependent pulse effects: ⎫ ⎧ dS(t) ⎪ ⎪ ⎪ = b S(t) + I (t) + R(t) − βS(t)I (t) − bS(t) + αR(t), ⎪ ⎪ ⎪ ⎪ ⎪ dt ⎪ ⎪ ⎪ ⎪ ⎬ ⎪ ⎪ dI (t) ⎪ ⎪ I < H, ⎪ = βS(t)I (t) − bI (t) − γ I (t), ⎪ ⎪ ⎪ dt ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ dR(t) ⎪ ⎭ (2) = γ I (t) − bR(t) − αR(t), ⎪ dt ⎪ ⎪ ⎫ ⎪ ⎪ ⎪ S(t) = S t + − S(t) = −mS(t), ⎪ ⎪ ⎪ ⎪ ⎬ ⎪ + ⎪ ⎪ ⎪ − I (t) = −pH, I (t) = I t I = H. ⎪ ⎪ ⎪ ⎪ ⎪ + ⎩ ⎭ R(t) = R t − R(t) = mS(t) + pH
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Remark 1 We point out that a priori time of pulse vaccination and medication is not assumed because it is taken at the time when the number of group I reaches the threshold value H . So the pulse vaccination and medication time depend obviously on the solution, which makes our model “state dependent.” Remark 2 The control parameters m, p, and hazardous threshold H rely heavily on the characteristics of the disease. The choices of these values are very important for different diseases, which are closely related to the spread or prevention of the disease. Let R = (−∞, ∞) and R3+ = {(x, y, z)|x > 0, y > 0, z > 0}. The global existence and uniqueness of solution for system (2) are guaranteed by the smoothness of the right-hand sides of system (2). For more details, we refer to Lakshmikantham et al. (1989). Lemma 1 Suppose that (S, I, R) is a solution of system (2) with the initial value (S(t0 ), I (t0 ), R(t0 )) ∈ R3+ . Then S(t) > 0, I (t) > 0, and R(t) > 0 for all t ≥ t0 . Proof For any initial value (S(t0 ), I (t0 ), R(t0 )) ∈ R3+ , we discuss all possible cases by the relation of the solution (S, I, R) to the line L : I = H as follows. Case 1: The solution (S, I, R) intersects with L finitely many times. In this case, since the endemic equilibrium (S ∗ , I ∗ , R ∗ ) is globally asymptotically stable, S(t) > 0, I (t) > 0, and R(t) > 0 for all t ≥ t0 . Case 2: The solution (S, I, R) intersects with L : I = H infinitely many times. In this case, suppose that solution (S, I, R) intersects with L : I = H at times tk , k = 1, 2, . . . , limk→∞ tk = ∞. If the conclusion of Lemma 1 is invalid, then there is a t ∗ > t0 such that min{S(t ∗ ), I (t ∗ ), R(t ∗ )} = 0, and S(t) > 0, I (t) > 0, R(t) > 0 for all t0 ≤ t < t ∗ . For this t ∗ , there is a positive integer n such that tn−1 ≤ t ∗ < tn . There are three possible cases. (i) I (t ∗ ) = 0, S(t ∗ ) ≥ 0, and R(t ∗ ) ≥ 0. For this case, it follows from the second and fifth equations of system (2) that t ∗ (βS(τ )−b−γ ) dτ I t ∗ = (1 − p)n−1 I t0+ e t0 > 0, which leads to a contradiction with I (t ∗ ) = 0. (ii) R(t ∗ ) = 0, S(t ∗ ) ≥ 0, and I (t ∗ ) ≥ 0. It follows from the third and sixth equations of system (2) that − t ∗ (b+γ ) dt R t ∗ ≥ R t0+ e t0 ∗ t∗ − t ∗ (b+γ ) dt − t (b+γ ) dt − (b+γ ) dt > 0, + pH e t1 + e t2 + · · · + e tn−1
which leads to a contradiction with R(t ∗ ) = 0. (iii) S(t ∗ ) = 0, I (t ∗ ) ≥ 0, and R(t ∗ ) ≥ 0. It then follows from the first and fourth equations of system (2) that − t ∗ (βI (τ )+b) dτ S t ∗ ≥ (1 − m)n−1 S t0+ e t0 > 0,
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Fig. 1 The dynamic behavior of system (3) without pulse vaccination and medication (a), and with state dependent pulse vaccination and medication (b) (Color figure online)
which leads to a contradiction with S(t ∗ ) = 0. These contradictions show that such a t ∗ does not exist. Therefore, S(t) > 0, I (t) > 0, and R(t) > 0 for all t ≥ t0 . This completes the proof. Since from (2), the total population is normalized to unity that S(t) + I (t) + R(t) = 1, therefore system (2) is equivalent to ⎫ ⎧ dS(t) ⎪ ⎪ ⎪ = b + α − βS(t)I (t) − (b + α)S(t) − αI (t), ⎪ ⎬ ⎪ ⎪ dt ⎪ ⎪ I < H, ⎪ ⎪ ⎪ ⎨ dI (t) ⎪ ⎭ = βS(t)I (t) − bI (t) − γ I (t), dt ⎪ ⎪ ⎪ ⎪ S(t) = S t + − S(t) = −mS(t), ⎪ ⎪ ⎪ I = H. ⎪ ⎩ I (t) = I t + − I (t) = −pH,
(3)
We assume, throughout this paper, that R0 = β/(b + γ ) > 1. That is to say, system (3) without pulse effects has a unique globally asymptotically stable endemic equilibrium E ∗ (S ∗ , I ∗ ) (see Fig. 1(a)). Based on the biological background of system (3) we only consider system (3) in the region R2+ = {(S, I ) : S > 0, I > 0} where the biology makes sense. Let Γ be an arbitrary set in R2 and Y be an arbitrary point in R2 . The distance between Y and Γ is defined by d(Y, Γ ) = infY0 ∈Γ |Y − Y0 |. Let X = (S, I ) be any solution of system (3). We define the positive orbit starting from initial point X0 ∈ R2+ for t ≥ t0 as
O + (X0 , t0 ) = X(t) : X(t) = S(t), I (t) ∈ R2+ , t ≥ t0 , X(t0 ) = X0 . Definition 1 (Orbital Stability, Hale and Kocak 1991) The trajectory O + (X0 , t0 ) is said to be orbitally stable if for any given ε > 0, there exists a constant δ = δ(ε) > 0
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such that for any other solution X ∗ of system (3), d(X ∗ (t), O + (X0 , t0 )) < ε for all t > t0 when d(X ∗ (t0 ), O + (X0 , t0 )) < δ. Definition 2 (Orbital Asymptotical Stability, Hale and Kocak 1991) The trajectory O + (X0 , t0 ) is said to be orbitally asymptotically stable if it is orbitally stable, and there exists a constant η > 0 such that for any other solution X ∗ of system (3), limt→∞ d(X ∗ (t), O + (X0 , t0 )) = 0 when d(X ∗ (t0 ), O + (X0 , t0 )) < η. To discuss the dynamic behavior of system (3), we define two sections to the vector field of system (3) by
Γp := (S, I ) : 0 < S < 1 − (1 − p)H, I = (1 − p)H and
ΓH := (S, I ) : 0 < S < 1 − H, I = H .
For any point Pn (Sn , H ) ∈ ΓH , suppose that the trajectory O + (Pn , tn ) starting from the initial point Pn intersects section ΓH infinitely many times. That is, trajectory O + (Pn , tn ) jumps to point Pn+ ((1 − m)Sn , (1 − p)H ) on section Γp due to impulsive effects S(t) = −mS(t) and I (t) = −pH . Furthermore, trajectory O + (Pn , tn ) intersects section ΓH at point Pn+1 (Sn+1 , H ), and then jumps to point + ((1 − m)Sn+1 , (1 − p)H ) on section Γp again (see Fig. 1(b)). Repeating this Pn+1 process, we have two impulsive point sequences {Pn+ ((1 − m)Sn , (1 − p)H )} and {Pn (Sn , H )}, where Sn+1 is determined by Sn , m, p, H , and n = 1, 2, . . . . We then define the Poincaré map of section ΓH : Sn+1 = F (Sn , m, p, H ).
(4)
Definition 3 A trajectory O + (Pn , tn ) of system (3) is said to be order-k periodic if there exists a positive integer k ≥ 1 such that k is the smallest integer for Sn+k = Sn . Next, we consider the autonomous system with pulse effects of the following: ⎧ dy ⎨ dx = f (x, y), = g(x, y), ϕ(x, y) = 0, (5) dt dt ⎩ x = ξ(x, y), y = η(x, y), ϕ(x, y) = 0, where f and g are continuous differentiable functions defined on R2 and ϕ is a sufficiently smooth function with ∇ϕ = 0. Let (μ, ν) be a positive T -periodic solution of system (5). The following result comes from Corollary 2 of Theorem 1 of Simeonov and Bainov (1988). Lemma 2 (Analogue of Poincaré Criterion) If the Floquet multiplier μ satisfies |μ| < 1, where μ=
n j =1
κj exp 0
T ∂f (μ(t), ν(t))
∂x
+
∂g(μ(t), ν(t)) dt ∂y
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with κj =
∂ϕ ( ∂η ∂y ∂x −
∂η ∂ϕ ∂x ∂y
+
∂ϕ ∂x )f+ ∂ϕ ∂x f
∂ξ ∂ϕ + ( ∂x ∂y −
+
∂ξ ∂ϕ ∂y ∂x
+
∂ϕ ∂y )g+
∂ϕ ∂y g
and f , g, ∂ξ/∂x, ∂ξ/∂y, ∂η/∂x, ∂η/∂y, ∂ϕ/∂x, and ∂ϕ/∂y have been calculated at the point (μ(τj ), ν(τj )), f+ = f (μ(τj+ ), ν(τj+ )), g+ = g(μ(τj+ ), ν(τj+ )), and τj (j ∈ N) is the time of the j th jump, then (μ, ν) is orbitally asymptotically stable. 3 Main Results Since endemic equilibrium E ∗ (S ∗ , I ∗ ) is globally asymptotically stable, any positive solution of system (3) without state dependent pulse will eventually tend to E ∗ (S ∗ , I ∗ ). From the geometrical structure of the phase space of system (3), it is easily to obtain that any trajectory of (3) without state dependent pulse starting from domain I := {(S, I ) ∈ R2 | S < 0, I < 0} will enter into domains II := {(S, I ) ∈ R2 | S > 0, I < 0}, III := {(S, I ) ∈ R2 | S > 0, I > 0}, or IV := {(S, I ) ∈ R2 | S < 0, I > 0} by order, and eventually tend to E ∗ (S ∗ , I ∗ ) (see Fig. 1(a)). Therefore, if H ≤ I ∗ = (b + α)(β − b − γ )/β(b + α + γ ), then the trajectory with given initial value (S0 , I0 ), I0 = (1 − p)H intersects section ΓH infinitely many times. However, if H > I ∗ , the trajectory starting from initial point (S0 , I0 ) with I0 = (1 − p)H may intersect section ΓH finitely many times. In this section, we give some sufficient conditions for the existence and stability of positive periodic solutions in the cases of H ≤ I ∗ and H > I ∗ , respectively. 3.1 The Case of H ≤ I ∗ The first result is on the existence of positive order-1 periodic solution. Theorem 2 For any m, p ∈ (0, 1), system (3) admits a positive order-1 periodic solution. Proof Let point E1 (ε, (1 − p)H ) ∈ Γp for sufficiently small ε with ε ≤ (1 − m)S ∗ . In view of the geometrical structure of the phase space of system (3), the trajectory O + (E1 , t0 ) of system (3) starting from the initial point E1 in domain II will enter into domains III or IV and intersect section Γp at point E1p (ε1p , (1 − p)H ), and then intersect section ΓH at point F1 (S1 , H ), where S ∗ < S1 . At point F1 , the trajectory O + (E1 , t0 ) jumps to the point E2 ((1 − m)S1 , (1 − p)H ) on section Γp due to impulsive effects S(t) = −mS(t) and I (t) = −pH . Furthermore, trajectory O + (E1 , t0 ) intersects section ΓH at point F2 (S2 , H ). Since ε ≤ (1 − m)S ∗ , E1 is on the left of E2 . We claim that F2 is on the left of F1 . In fact, if F2 is on the right of F1 , then the orbits E 1 F1 and E 2 F2 intersect at some point D(S0 , I0 ). This shows that there are two different solutions which start from the same initial point D(S0 , I0 ). This contradicts the uniqueness of solutions for system (3). Therefore, it follows from (4) that S2 = F (S1 , m, p, H ) and F (S1 , m, p, H ) − S1 = S2 − S1 < 0.
(6)
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On the other hand, suppose that the curve L : βSI − bI − γ I = 0 intersects section Γp at point A0 ((b + γ )/β, (1 − p)H ). Then the trajectory O + (A0 , t0 ) starting from the initial point A0 intersects section ΓH at point B1 (S1 , H ) and then jumps to point A1 ((1 − m)S1 , (1 − p)H ) on section Γp and finally reaches point B2 (S2 , H ) in section ΓH again. If there is a positive constant m∗ such that (1 − m∗ )S1 = (b + γ )/β, then A1 coincides with A0 for m = m∗ ∈ (0, 1), that is, B1 coincides with B2 . Otherwise, A1 is on the left of A0 for (1 − m)S1 < (b + γ )/β and A1 is on the right of A0 for (1 − m)S1 > (b + γ )/β. However, from the geometrical structure of phase space of system (3), B2 is on the right of B1 for any m ∈ (0, m∗ ) ∪ (m∗ , 1) (see Fig. 1(b)). To sum up, we get, from the above discussions, that (i) When S1 = S2 , system (3) has positive order-1 periodic solution. (ii) When S1 < S2 , F (S1 , m, p, H ) − S1 = S2 − S1 > 0.
(7)
By (6) and (7), it follows that the Poincaré map (4) has a fixed point. This amounts to saying that system (3) has a positive order-1 periodic solution. The proof is complete. Remark 3 From Theorem 2, we see that when H ≤ I ∗ , system (3) always admits a positive order-1 periodic solution. In addition, from the geometrical structure of the phase space of system (3), it is easily to get that the disease could be controlled below the threshold value H due to the state dependent pulse strategy. The next result is on the orbital stability of positive order-1 periodic solution of model (3). Theorem 3 Let (φ, ψ) be a positive order-1 periodic solution of model (3) with period T . If T |μ| = |κ| exp − βψ(t) + b + α dt < 1, (8) 0
where κ=
(1 − m)[β(1 − m)φ(T ) − b − γ ] , βφ(T ) − b − γ
then (φ, ψ) is orbitally asymptotically stable. Proof Suppose that (φ, ψ) intersects the sections Γp and ΓH at points E + ((1 − m)φ(T ), (1 − p)H ) and E(φ(T ), H ), respectively. Comparing with system (5), we have f (S, I ) = b + α − βSI − (b + α)S − αI,
g(S, I ) = βSI − bI − γ I,
ξ(S, I ) = −mS, η(S, I ) = −pI , ϕ(S, I ) = I − H , (φ(T ), ψ(T )) = (φ(T ), H ), and (φ(T + ), ψ(T + )) = ((1 − m)φ(T ), (1 − p)H ). Thus, ∂f = −βI − (b + α), ∂S
∂g = βS − b − γ , ∂I
(9)
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and ∂ξ = −m, ∂S
∂η = −p, ∂I
∂ϕ = 1, ∂I
∂ξ ∂η ∂ϕ = = = 0. ∂I ∂S ∂S
(10)
Furthermore, it follows from (9) and (10) that κ=
and
∂ϕ ( ∂η ∂I ∂S −
∂η ∂ϕ ∂S ∂I
+
∂ϕ ∂S )f+ ∂ϕ ∂S f
∂ξ ∂ϕ + ( ∂S ∂I −
+
∂ξ ∂ϕ ∂I ∂S
∂ϕ ∂I )g+
∂ϕ ∂I g
=
(1 − m)g+ (φ(T + ), ψ(T + )) g(φ(T ), ψ(T ))
=
(1 − m)(1 − p)[β(1 − m)φ(T ) − b − γ ] βφ(T ) − b − γ
μ = κ exp
+
βφ(t) − b − γ − βψ(t) + b + α dt .
T
(11)
(12)
0
On the other hand, we integrate the both sides of the second equation of system + E to give (3) along the orbit E ln
1 = 1−p
H
(1−p)H
dI = I
T
T
βφ(t) − b − γ dt.
[βS − b − γ ] dt =
0
(13)
0
From (11)–(13), we obtain (1 − m)(1 − p)[β(1 − m)φ(T ) − b − γ ] 1 |μ| = 1 − p βφ(T ) − b − γ T × exp − βψ(t) + b + α dt 0
T (1 − m)[β(1 − m)φ(T ) − b − γ ] βψ(t) + b + α dt . = exp − βφ(T ) − b − γ 0 By condition (8), we see that system (3) satisfies all conditions of Lemma 2. It then follows from Lemma 2 that the order-1 periodic solution (φ, ψ) of system (3) is orbitally asymptotically stable and has asymptotic phase property. This completes the proof. From the proof of Theorem 3, integrating both sides of the first equation of system + E, we obtain (3) along the orbit E ln
1 − (1 − m)φ(T ) = 1 − φ(T )
φ(T )
(1−m)φ(T )
dS ≤ 1−S
T 0
(b + α) dt = (b + α)T .
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This shows that T
βψ(t) + b + α dt < exp −(b + α)T ≤ exp − 0
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1 − φ(T ) . 1 − (1 − m)φ(T )
The succeeding Corollary 1 is a direct consequence of Theorem 3. Corollary 1 Let (φ, ψ) be a positive order-1 periodic solution of system (3) with periodic T . If (1 − m)[β(1 − m)φ(T ) − b − γ ] 1 − φ(T ) |μ| = < 1, βφ(T ) − b − γ 1 − (1 − m)φ(T ) then (φ, ψ) is orbitally asymptotically stable. Remark 4 From the numerical simulation in next section, we come to a conclusion that system (3) has a positive order-1 periodic (ϕ, ψ) for any m, p ∈ (0, 1) and H ≤ I ∗ , and (ϕ, ψ) is orbitally asymptotically stable. We thus raise a conjecture as follows. Conjecture 1 For any m, p ∈ (0, 1) and H ≤ I ∗ , system (3) has a positive order-1 periodic solution, which is orbitally asymptotically stable. 3.2 The Case of H > I ∗ Theorem 4 For any m, p ∈ (0, 1) and H > I ∗ , one of the following statements is valid. (i) If there is a positive constant S˜ = ρ(H ) ∈ (0, S ∗ ) such that the trajectory ˜ (1 − p)H ) is tanO + (A0 , t0 ) of system (3) starting from the initial point A0 (S, ˜ gent to the line L : I = H at point B((b + β)/γ , H ) and (1 − m)(1 − H ) < S, then system (3) has a positive order-1 or order-2 periodic solution, which is orbitally asymptotically stable. (ii) If for any S˜ ∈ (0, S ∗ ), the trajectory O + (A0 , t0 ) of system (3) starting from the ˜ (1 − p)H ) cuts the line L : I = H at point B(S0 , H ), where initial point A0 (S, ∗ S0 > S , then system (3) has a positive order-1 or order-2 periodic solution, which is orbitally asymptotically stable. (iii) If for any S˜ ∈ (0, S ∗ ), the trajectory O + (A0 , t0 ) of system (3) starting from ˜ (1 − p)H ) does not intersect the line L : I = H , then the initial point A0 (S, system (3) has no positive order-k (k ≥ 1) periodic solution and the endemic equilibrium E ∗ (S ∗ , I ∗ ) is globally asymptotically stable. Proof We first prove (i). If there is a positive constant S˜ = ρ(H ) ∈ (0, S ∗ ) such ˜ (1 − that the trajectory O + (A0 , t0 ) of system (3) starting from the initial point A0 (S, ˜ p)H ) crosses section Γp at point A0p (S1 , (1 − p)H ) and is tangent to the line L : I = H at point B(S ∗ , H ), then the trajectory starting from the initial point (S, (1 − p)H )
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˜ S˜1 ) will tend to endemic equilibrium E ∗ (S ∗ , I ∗ ) and not intersect with with S ∈ (S, ˜ then (1 − p)S < S˜ for any point section ΓH . Moreover, if (1 − m)(1 − H ) < S, (S, I ) ∈ ΓH . Therefore, for any E(S, H ) ∈ ΓH , trajectory O + (E, t0 ) starting from the initial point E(S, H ) will intersect with ΓH infinitely many times due to the impulsive effects S(t) = −mS(t) and I (t) = −pH . On the other hand, for any two points Ei (Si , H ) and Ej (Sj , H ) on section ΓH , where Si , Sj ∈ (S ∗ , 1 − H ) and Si < Sj , in view of impulsive effects, Ei+ ((1 − m)Si , (1 − p)H ) is on the left of Ej+ ((1 − m)Sj , (1 − p)H ). We claim that 0 < Sj +1 < Si+1 < 1 − H.
(14)
In fact, if (14) is false, that is, Sj +1 ≥ Si+1 , then Ej +1 (Sj +1 , H ) is on the right of Ei+1 (Si+1 , H ), or the two points coincide. So we have trajectories O + (Ei+ , t0 ) and O + (Ej+ , t0 ) intersect at some point (S0∗ , I0∗ ). This implies that there are two different solutions which start from the same initial point (S0∗ , I0∗ ), which contradicts the uniqueness of solution. Inequality (14) is thus valid. Suppose that the trajectory O + (E0 , t0 ) of system (3) starting from the initial point E0 (S0 , H ) on section ΓH jumps to point E0+ (Sˆ0 , (1 − p)H ) on section Γp due to impulsive effects S(t) = −mS(t) and I (t) = −pH and reaches section ΓH at point E1 (S1 , H ) again, where S1 ∈ (S ∗ , 1 − H ), and then jumps to point E1+ (Sˆ1 , h) at section Γp . At state E1+ , trajectory O + (E0 , t0 ) intersects section ΓH at point E2 (S2 , H ), where S2 ∈ (S ∗ , 1 − H ). By the Poincaré map (4) of section ΓH , it follows that S1 = F (S0 , m, p, H ) and S2 = F (S1 , m, p, H ). Repeating the above process, we have Sn+1 = F (Sn , m, p, H ) (n = 0, 1, . . .). In particular, system (3) has a positive order-1 periodic solution when S0 = S1 and a positive order-2 periodic solution when S0 = S1 and S0 = S2 . Next, we look at the general situation where S0 = S1 = S2 = · · · = Sk (k > 2). We discuss the problem in different cases. Case 1. S0 < S1 . In this case, it follows from (14) that S2 < S1 . This results in the relation of S0 , S1 , and S2 to be one of the following cases. (1) S2 < S0 < S1 . In this case, S3 > S1 > S2 by (14). Repeating the above process, we have S ∗ < · · · < S2k < · · · < S2 < S0 < S1 < · · · < S2k+1 < · · · < 1 − H. (2) S0 < S2 < S1 . Similar to 1), we have S0 < S2 < · · · < S2k < · · · < S2k+1 < · · · < S3 < S1 < 1 − H. Case 2. S0 > S1 . In this case, it follows from (14) that S1 < S2 . This results in the relation of S0 , S1 , and S2 to be one of the following cases.
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(3) S1 < S0 < S2 . In this case, S2 > S1 > S3 by (14). Repeating the above process, we have S ∗ < · · · < S2k+1 < · · · < S1 < S0 < S2 < · · · < S2k < · · · < 1 − H. (4) S1 < S2 < S0 . Similar to (3), we have S ∗ < S1 < · · · < S2k+1 < · · · < S2k < · · · < S2 < S0 < 1 − H. Moreover, in (1) of Case 1, we have limk→∞ S2k = λ2 and limk→∞ S2k+1 = λ1 , where S ∗ < λ2 < λ1 < 1 − h. Hence, λ1 = F (λ2 , m, p, H ) and λ2 = F (λ1 , m, p, H ). So system (3) has an orbit of asymptotically stable positive order-2 periodic solution. Similarly, in (2) of Case 1 and (4) of Case 2, system (3) has an orbital asymptotically stable positive order-1 periodic solution. In (3) of Case 2, system (3) has an orbit of asymptotically stable positive order-2 periodic solution. This proves (i). Now we turn to (ii). If for any S˜ ∈ (0, S ∗ ), the trajectory O + (A0 , t0 ) of system ˜ (1 − p)H ) cuts the line L : I = H at point (3) starting from the initial point A0 (S, B(S0 , H ), where S0 > S ∗ , then for any E(S, H ) ∈ ΓH , trajectory O + (E, t0 ) will intersect with section ΓH infinitely many times due to the impulsive effects S(t) = −mS(t) and I (t) = −pH . Similar to (i), we can also obtain that system (3) has a positive order-1 or order-2 periodic solution, which is orbitally asymptotically stable. (ii) thus follows. Finally, we show (iii). If for any S˜ ∈ (0, (b + β)/γ ), the trajectory O + (A0 , t0 ) ˜ (1 − p)H ) does not intersect the of system (3) starting from the initial point A0 (S, line L : I = H , then the trajectory starting from the point (S, (1 − p)H ) of section Γp with S ∈ (0, (b + β)/γ ) will tend to endemic equilibrium E ∗ (S ∗ , I ∗ ) and not intersect with section ΓH . Furthermore, any other trajectory intersects section ΓH at most finitely many times, and then tends to endemic equilibrium E ∗ (S ∗ , I ∗ ). In this case, system (3) has no positive order-k (k ≥ 1) periodic solution and endemic equilibrium E ∗ (S ∗ , I ∗ ) is globally asymptotically stable. This is (iii). The proof is complete. Remark 5 From part (i) of Theorem 4, we note that (1 − m)(1 − H ) < S˜ is a sufficient condition for system (3) to has a positive order-1 or order-2 periodic solution. Remark 6 Part (iii) of Theorem 4 shows that the state dependent pulse effects are invalid when the vaccination m and medication intensification effort p remain at a relatively low level and threshold value H is greater than I ∗ .
4 Numerical Simulation and Discussion To illustrate the results and the feasibility of the state dependent pulse feedback control strategy, we consider the following SIRS epidemic system with the state depen-
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Fig. 2 The orbitally asymptotically stable period solution of system (15) with m = 0.3, p = 0.6, and H = 0.3 < I ∗ (Color figure online)
dent pulse vaccination and medication: ⎫ ⎧ dS(t) ⎪ ⎪ = 0.15(S(t) + I (t) + R(t)) − 0.8S(t)I (t) − 0.15S(t) + 0.1R(t), ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ dt ⎪ ⎪ ⎪ ⎪ ⎬ ⎪ ⎪ dI (t) ⎪ ⎪ I < H, ⎪ = 0.8S(t)I (t) − 0.15I (t) − 0.2I (t), ⎪ ⎪ ⎪ dt ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ dR(t) ⎪ ⎭ = 0.2I (t) − 0.15R(t) − 0.1R(t), ⎪ dt ⎪ ⎪ ⎫ ⎪ ⎪ S(t) = S t + − S(t) = −mS(t), ⎪ ⎪ ⎪ ⎪ ⎪ ⎬ ⎪ + ⎪ ⎪ ⎪ − I (t) = −pH, I (t) = I t I = H. ⎪ ⎪ ⎪ ⎪ ⎪ + ⎩ ⎭ R(t) = R t − R(t) = pH + mS(t), (15) It is obvious that system (15) without the pulse effects has a unique globally asymptotically stable endemic equilibrium (S ∗ , I ∗ , R ∗ ) = (0.4375, 0.3125, 0.25). Firstly, we choose the control parameters to be m = 0.3, p = 0.6, and H = 0.3 < I ∗ = 0.3125, respectively. By Theorem 2, we know that system (15) has a positive order-1 periodic solution (φ, ψ, ϕ), which is shown in Figs. 2(a)–(c). At the same time, simulation results also show that the periodic solution starting from initial point (0.4753, 0.06, 0.4647).
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Fig. 3 The trajectory of system (15) with p = 0.6, H = 0.3242 > I ∗ = 0.3125, and m = 0.85 in (a) and m = 0.55 in (b), respectively (Color figure online)
In addition, it is easily to calculate that (1 − m)[β(1 − m)φ(T ) − b − γ ] 1 − φ(T ) |μ| = 1 − (1 − m)φ(T ) βφ(T ) − b − γ ≈
1 − 0.679 0.7(0.8 × 0.4753 − 0.15 − 0.2) × 0.8 × 0.679 − 0.15 − 0.2 1 − 0.4753
≈ 0.067 < 1. Therefore, the order-1 periodic solution of system (15) is orbitally asymptotically stable and has asymptotic phase property by Corollary 1, which is shown in Fig. 2(d). In addition, further more numerical simulations show that for any m, p ∈ (0, 1), system (15) with H < I ∗ has a positive orbitally asymptotically stable order-1 periodic solution. This is just Conjecture 1 in Sect. 3. This shows that the conditions of Theorem 3 or Corollary 1 are sufficient not necessary. Next, let m = 0.9, p = 0.6, H = 0.3242, and (1 − m)(1 − H ) = 0.06758 < Sˆ = 0.098. It is easy to see that system (15) has a positive orbitally asymptotically stable order-1 periodic solution by part (i) of Theorem 4, which is shown in Fig. 3(a). However, if we choose m = 0.55, p = 0.6, and H = 0.3242, numerical simulations show that the trajectories of system (15) intersect the line L : I = H at most finitely many times and then tend to endemic equilibrium (0.4375, 0.3125, 0.25), which is shown in Fig. 3(b). This is just part (iii) of Theorem 4. This implies that system (15) has no positive order-k (k ≥ 1) periodic solution in this case. Thirdly, we discuss how the state dependent pulse control strategy affects the prevention and control of infectious diseases and the existence and stability of the periodic solutions. Here, we choose H = 0.1 < I ∗ , p = 0.6, and m to be 0.4, 0.6, and 0.8, respectively. Numerical simulations show that the period T of order-1 periodic solution for system (15) increases with the increase of immune strength m, which is shown in Fig. 4(a). Furthermore, let H = 0.1, m = 0.55, and p to be 0.4, 0.6, and 0.8, respectively. The corresponding numerical results are presented in Fig. 4(b). These results indicate that the recurrence time for infectious diseases can be prolonged by increasing immune strength m or medication intensification effort p. That is, the infected density can be kept at a low level by adjusting immune or medication strength
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Fig. 4 The trajectory of system (15) with H = 0.1 and (a): p = 0.6, m = 0.4, 0.6, 0.8, respectively; (b) m = 0.55, p = 0.4, 0.6, 0.8, respectively (Color figure online)
Fig. 5 The trajectory of system (15) with H = 0.32 > I ∗ and (a): p = 0.4, m = 0.2, 0.4, 0.6, 0.8, 0.9, respectively; (b) m = 0.6, p = 0.2, 0.4, 0.6, 0.8, 0.9, respectively (Color figure online)
(see Fig. 6(a)). The strong consistency between theoretical result and real situation is obviously observed. However, if H = 0.32 > I ∗ , p = 0.4, and m to be taken 0.2, 0.4, 0.6, 0.8, and 0.9, respectively, then unique endemic equilibrium (S ∗ , I ∗ , R ∗ ) of system (15) is asymptotically stable when the immune strength m remains at a relatively low level. However, with the increase of immune strength m, (S ∗ , I ∗ , R ∗ ) loses its global asymptotic stability and system (15) has a positive order-1 orbitally asymptotically stable periodic solution. This is observed in Fig. 5(a). Similar results can also be obtained from H = 0.32 > I ∗ , m = 0.6, and p to be taken as 0.2, 0.4, 0.6, 0.8, and 0.9, respectively. This is presented in Fig. 5(b). Numerical simulations also indicate that we can control infectious disease at a relatively low level over a long period of time by adjusting immune or medication strength when H < I ∗ or H > I ∗ (see Figs. 6(a) and (b)). Finally, we compare the state dependent pulse control strategy and FTPS. First, we give a cost of control measures. Assume that the control cost is proportional to the densities of treated infected group and immunized susceptible group. For simplicity, we use the following total cost: C=
mS ti (H ) + pI ti (H ) .
(16)
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Fig. 6 The trajectory of system (15) with (a): H = 0.2 < I ∗ , (b): H = 0.32 > I ∗ (Color figure online)
Fig. 7 The state dependent pulse vaccination and medication vs. the fixed time impulsive vaccination and medication, where m = 0.35, p = 0.4, and (a) I (0) = 0.04, (b) I (0) = 0.02 and 0.04, respectively (Color figure online)
The disease is controlled by the impulsive effects S(t) = −0.35S(t), I (t) = −0.4H , and R(t) = 0.4H + 0.35S(t). Numerical simulations show that infected group is controlled within a relatively low level by the state dependent pulse control when the number of infected population reaches the hazardous threshold. This is well explained in Fig. 7. Assume that the hazardous threshold is 0.05 and initial value is (0.6, 0.04, 0.36). The trajectory of this solution is (0.6, 0.04, 0.36) → (0.6825, 0.05, 0.2675) → (0.7226, 0.05, 0.1774) → (0.7232, 0.05, 0.2268) → (0.7232, 0.05, 0.2268) → · · · . By (16), we have the control cost as follows: C1 = 0.35 × 0.6 + 0.4 × 0.04 = 0.226. When the infected group is controlled within 0.05 at t ≈ 1.346, the solution tends quickly to a stable periodic solution by state feedback control strategy, in other words, the disease is completely controlled. However, for the same initial value (0.6, 0.04, 0.36), if we take control measures at a fixed time t = 4k (k = 1, 2, . . .), the density of infected population can not controlled under the hazardous threshold 0.05, and hence the disease will be spreading. Furthermore, if we take control for the infected population at a fixed time T = 3k (k = 1, 2, . . .), the trajectory of this solution with initial value (0.6, 0.04, 0.36) is (0.6, 0.04, 0.36) → (0.7347, 0.0719, 0.1934) →
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(0.6878, 0.0639, 0.2483) → (0.6834, 0.0545, 0.2621) → (0.69, 0.0467, 0.2633) → · · · . The infected population is controlled within 0.05 at t ≈ 11.998 and the control cost is C2 = 0.35(0.6 + 0.7347 + 0.6878 + 0.6834) + 0.4(0.04 + 0.0719 + 0.0639 + 0.0545) ≈ 1.039. Therefore, if we choose the same immune and medication strength m and p and the appropriate control parameters, the state dependent pulse control would be very economic. In Fig. 7(a), we also note that the fixed time control with T = 3k is slightly frequent than the state dependent control. However, if the initial value of infected population is changed, FTPS is difficult to achieve the expected purpose, which is observed in Fig. 7(b). We can therefore say that the state dependent pulse vaccination and medication is more effective and easier implementable than the fixed time pulse control strategy. 5 Concluding Remarks The dynamic behavior of a SIRS epidemic model with state dependent pulse control strategy is studied in this paper. The state dependent pulse control strategy causes the complexity for the dynamic behavior of system (2) such as frequent switching between states, irregular motion, and some uncertainties. This is the distinguished feature compared with other control strategies. By the Poincaré map, the analogue of Poincaré criterion, and qualitative analysis method, some sufficient conditions on the existence and orbital stability of positive order-1 or order-2 periodic solution of system (2) are presented. This amounts to that we can control the density of infectious disease at a low level over a long period of time by adjusting immune or medication strength. It is concluded that the state dependent pulse control strategy is more feasible, effective, and importantly, easier implementable than the fixed-time pulse control. Finally, we mention some future possible works along the present work: (a) the uniqueness and global stability of positive order-1 periodic solution for H < I ∗ ; (b) the optimal control strategy for state dependent pulse control problem; (c) application to more complicated epidemic models. Acknowledgements The authors would like to thank antonymous referees for their constructive suggestions and comments that improve substantially the original manuscript. This work was supported in part by the Natural Science Foundation of Xinjiang (Grant No. 2011211B08), the National Natural Science Foundation of China (Grant No. 11001235, 11271312, and 11261056), the China Postdoctoral Science Foundation (Grant No. 20110491750 and 2012T50836), the Scientific Research Programmes of Colleges in Xinjiang (Grant No. XJEDU2011S08), the National Basic Research Program of China (2011CB808002), and the National Research Foundation of South Africa.
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