College of Mathematics and Systems Science, Shandong University of Science and Technology, Qingdao 266590, China sdust.edu.cn
State Key Laboratory of Mining Disaster Prevention and Control Cofounded by Shandong Province and the Ministry of Science and Technology, Shandong University of Science and Technology, Qingdao 266590, China sdust.edu.cn
College of Mathematics and Systems Science, Shandong University of Science and Technology, Qingdao 266590, China sdust.edu.cn
State Key Laboratory of Mining Disaster Prevention and Control Cofounded by Shandong Province and the Ministry of Science and Technology, Shandong University of Science and Technology, Qingdao 266590, China sdust.edu.cn
College of Mathematics and Systems Science, Shandong University of Science and Technology, Qingdao 266590, China sdust.edu.cn
State Key Laboratory of Mining Disaster Prevention and Control Cofounded by Shandong Province and the Ministry of Science and Technology, Shandong University of Science and Technology, Qingdao 266590, China sdust.edu.cn
College of Mathematics and Systems Science, Shandong University of Science and Technology, Qingdao 266590, China sdust.edu.cn
State Key Laboratory of Mining Disaster Prevention and Control Cofounded by Shandong Province and the Ministry of Science and Technology, Shandong University of Science and Technology, Qingdao 266590, China sdust.edu.cn
SIR epidemic model with nonlinear pulse vaccination and lifelong immunity is proposed. Due to the limited medical resources, vaccine immunization rate is considered as a nonlinear saturation function. Firstly, by using stroboscopic map and fixed point theory of difference equations, the existence of disease-free periodic solution is discussed, and the globally asymptotical stability of disease-free periodic solution is proven by using Floquet multiplier theory and differential impulsive comparison theorem. Moreover, by using the bifurcation theorem, sufficient condition for the existence of positive periodic solution is obtained by choosing impulsive vaccination period as a bifurcation parameter. Lastly, some simulations are given to validate the theoretical results.
1. Introduction and Model Formulation
Infectious disease is one of the greatest enemies of human health. According to the World Health Statistics Report 2013 [1], nearly 2.5 million persons are infected by HIV each year. Although the number of infectious patients has dropped compared with 20 years ago, the absolute number of people with AIDS is still increasing, due to the fact that there are about 80,000 more infection cases than deaths. At the same time, AIDS has an important impact on adult mortality in high-prevalence countries. For example, the life expectancy in South Africa has fallen from 63 years old (in 1990) to 58 years old (in 2011). In Zimbabwe, the drop is six years during the same period. In recent years, due to the emergence of H7N9 avian influenza and other infectious diseases, the prevention and control situation is extremely grim all over the world. In order to prevent and control infectious diseases, vaccination is widely accepted. Generally, there are two types of strategies: continuous vaccination strategy (CVS) and pulse vaccination strategy (PVS) [2]. For certain kinds of infectious diseases, PVS is more affordable and easier to implement than CVS. Theoretical study about PVS was started by Agur and coworkers in [2]. In Central and South America [3, 4] and UK [5], PVS has a positive effect on the prevention of measles. With the encouragement of successful applications of PVS, many models are established to study the PVS [6–30]. Pang and Chen [29] studied a class of SIRS model with pulse vaccination and saturated contact rate as follows:
(1)
In model (1), S(t), I(t), and R(t) represent the number of susceptible, infected, and removed individuals at the time t, respectively. Constant p is the vaccination rate. However, for some emerging infectious diseases, vaccination is often restricted by limited medical resources. The vaccination success rate always has some saturation effect; that is, vaccination rate can be expressed as a saturation function as follows [31]:
(2)
Here, p is the maximum pulse immunization rate and θ is the half-saturation constant; that is, the number of susceptible when the vaccination rate is half to the largest vaccination rate. Thus we have that
(3)
Then, we establish a SIR epidemic model with the nonlinear impulsive vaccination and life-immunity as follows:
(4)
Here, constant A represents the total number of input population and a (0 < a ≤ 1) is the proportion of input population without immunity. μ is natural mortality, λ is the death rate due to illness, and γ is the recovery rate. The definitions of other symbols are shown in literature [29]. Note that variable R just appears in the third and the sixth equations of model (4), so we only need to consider the subsystem of (4) as follows:
(5)
This paper is organized as follows. In Section 2, we will firstly discuss the existence of the disease-free periodic solution by constructing stroboscopic map and using fixed point theory of difference equations. Then we will discuss the stability of the disease-free periodic solution by using the Floquet multipliers theory and the differential equations comparison theorem. In Section 3, we will discuss the existence of positive periodic solution and bifurcation by using the bifurcation theorem. Finally, we will give some numerical simulations and a brief decision in Section 4.
2. The Existence and Stability of the Disease-Free Periodic Solution
Let the total population number of model (4) be N(t) = S(t) + I(t) + R(t), which satisfies
(6)
Clearly, we have limsupt→∞N(t) < A/μ, and then system (4) is ultimately bounded. Next, we will discuss the existence of the disease-free periodic solution of model (5). Let I(t) = 0 in system (5); then we get the subsystem of system (5) as follows:
(7)
We have the following lemma for the property of subsystem (7).
Lemma 1. System (7) has a unique globally asymptotically stable periodic solution S*(t).
Proof. Solving the first equation of system (7), we get
(8)
From the second equation of system (7), we get the pulse condition
(9)
Then, we have
(10)
Let S(kT+) = Sk, and construct a stroboscopic map as follows:
(11)
then system (11) has a unique fixed point. In fact, assume that is the fixed point of (11); then , and it satisfies
(12)
where
(13)
Since , then the equation has a unique positive root
Clearly, 0 < f′(Sk) < 1; then ; thus, is a stable fixed point of (11). Substituting the expression of into (8), we have
(16)
Since is the unique stable fixed point of the difference equations, then S*(t) is the unique global asymptotically stable periodic solution of the system (7). The proof is completed.
According to Lemma 1, we have the following theorem.
Theorem 2. System (5) has a disease-free periodic solution (S*(t), 0).
Next we will discuss the stability of the periodic solution. Suppose that (S(t), I(t)) is any positive solution of the system (5); let
(17)
then the linearized system of the system (5) for the disease-free periodic solution (S*(t), 0) is
(18)
here t ≠ kT. Let Φ(t) be the fundamental solution matrix of the system; thus
(19)
and Φ(0) = E, where E is the unit matrix. Then
(20)
where dφ22(t)/dt = [βS*(t)/(αS*(t) + 1) − λ − γ − μ]φ22(t). Here, φ12(t) is not required in the following analysis. Then we get
(21)
For t = kT, the pulse condition is
(22)
Let
(23)
then the single-valued matrix of the system (18) is
(24)
The eigenvalues of the matrix M are λ1 = ((1 − p + pθ2/(S*(T) + θ) 2))e−μT < 1 and λ2 = φ22(T). By the Floquet multiplier theory [32], the disease-free periodic solution (S*(t), 0) is locally asymptotically stable if λ2 < 1; that is,
(25)
Denote
(26)
we can draw a conclusion as follows.
Theorem 3. The disease-free periodic solution (S*(t), 0) of system (5) is locally asymptotically stable if R1 < 1.
Next we will prove the global attractivity of the disease-free periodic solution (S*(t), 0) of system (5).
Theorem 4. The disease-free periodic solution (S*(t), 0) of system (5) is globally attractive if R1 < 1.
Proof. Let (S(t), I(t)) be any solution of the system (5). Since R1 < 1, one can choose ε > 0 small enough such that
(27)
From the first and third equations of the system (5), we have
(28)
Consider the following impulsive comparison system:
(29)
By the comparison theorem of impulsive differential equation, we have S(t) ≤ x(t) and x(t) → S*(t) as t → ∞. Hence there exists ε > 0 such that
(30)
for all t large enough. For simplification we may assume (30) holds for all t ≥ 0. From the second equation of system (5), we have
(31)
which leads to
(32)
Hence I(kT) ≤ I(0+)ekσ and I(kT) → 0 as k → ∞. Therefore I(t) → 0 as t → ∞ since 0 < I(t) ≤ I(kT)eβT for kT < t ≤ (k + 1)T. Without loss of generality, we may assume that 0 < I(t) < ε(ε < αaA/β) for all t ≥ 0. From the first equation of system (5), we have
(33)
Then, we have z1(t) ≤ S(t) ≤ z2(t) and , z2(t) → S*(t), where z1(t) and z2(t) are solutions of
(34)
respectively. Consider , kT < t ⩽ (k + 1)T. Therefore, for any ε1 > 0, there exists a T > 0 such that
(35)
Letting ε → 0, we have
(36)
for t large enough, which implies S(t) → S*(t) as t → ∞. So the disease-free periodic solution (S*(t), 0) of system (5) is global attractivity. The proof is completed.
Synthesizing Theorems 3 and 4, we have the following.
Theorem 5. The disease-free periodic solution (S*(t), 0) of the system (5) is globally asymptotically stable if R1 < 1.
3. Existence of Positive Periodic Solution and Bifurcation
In this section, we will discuss the existence of the positive periodic solution and the branch of the system (5) by using the bifurcation theorem [33].
Obviously, the threshold value R1 is proportional to the pulse vaccination period T, and R1 ≥ 1 for T large enough. In this case, the disease-free periodic solution (S*(t), 0) of the system (5) is unstable. Assume that T = T0 as R1 = 1. We choose the pulse vaccination period T as a bifurcation parameter. Denote x1(t) = S(t), x2(t) = I(t), and then the system (5) can be rewritten as
(37)
We assume that X(t) = (x1(t), x2(t)) T = Φ(t, X0) = (Φ1(t, X0), Φ2(t, X0)) T is the solution of the system (37) through the initial point X(0) = X0 = (x10, x20) T. By Theorem 2, the system (37) has a boundary periodic solution , and . In order to apply the bifurcation theorem (see [33]), we make the calculations as follows:
(38)
Obviously, is equivalent to R1 = 1. Consider
(39)
Then, we have
(40)
From the bifurcation theorem, the system (37) could produce nontrivial periodic solutions by the boundary solution with the condition BC ≠ 0, and the bifurcation is a supercritical bifurcation if BC < 0, or a subcritical bifurcation if BC > 0. Letting
(41)
then the following theorem is gotten.
Theorem 6. If the condition (H) holds, then system (37) has a supercritical branch at the point T0; that is, a nontrivial periodic solution could be produced by the boundary periodic solution (S*(t), 0). Here, T0 satisfies R1(T0) = 1.
4. Discussion and Numerical Simulation
In this paper, we have considered a SIR epidemic model with the nonlinear impulsive vaccination and life-immunity. The whole dynamics of the model is investigated under nonlinear impulsive effect. Firstly, the existence of disease-free periodic solution is discussed by using stroboscopic map and fixed point theory of difference equations, the globally asymptotical stability of disease-free periodic solution is proven by using Floquet multiplier theory and differential impulsive comparison theorem. Then, by choosing impulsive vaccination period as a bifurcation parameter, sufficient condition for the existence of positive periodic solution was obtained by using the bifurcation theorem. We have found that the dynamics of the model (5) depends on the threshold R1. If R1 < 1, then the disease-free periodic solution (S*(t), 0) of the system (5) is globally asymptotically stable. Otherwise, it is unstable and will show a supercritical branch for R1(T0) = 1. The threshold R1 is related to all parameters of the model (5).
Next, we focus on the relations of the R1 with the parameters θ and T. The model (5) adopts the saturated vaccination rate p(t) = pS(t)/(S(t) + θ). Here, θ represents the degree of restriction about medical resources. The relations of R1 with θ and T can be seen in Figure 1.
From Figure 1, if we fix θ, the threshold R1 is an increasing function of the vaccination period T. And if we fix the vaccination period T, R1 is an increasing function of the parameter θ. By the meaning of θ and T, if we enrich the medical resources (i.e., decrease θ) or reduce the vaccination period (i.e., decrease T), then the disease will be extinction; otherwise, the disease will be permanent.
To show the influence of restriction of medical resources on the model dynamics, we give some numerical simulations. Let p = 0.7, a = 0.2, A = 2, β = 2, α = 0.7, μ = 0.4, λ = 0.2, γ = 0.2, and T = 2. We have an example as follows:
(42)
Case 1. Let θ = 0, the initial point is (0.8,0.06). By calculation, we obtain R1 = 0.8324. Figure 2 shows that the number of susceptible individuals produces periodic oscillation. Figure 3 shows that the disease will eventually be eliminated. Figure 4 shows that system (5) has a disease-free periodic solution (S*(t), 0), which is globally asymptotically stable.
Case 2. Letting θ = 0.9, the initial point is (0.8,0.06). By calculation, we obtain R1 = 1.1456. Figure 5 shows that the number of susceptible individuals produces periodic oscillation under the pulsed effect. Figure 6 shows that the disease will be persistent. Figure 7 shows that system (5) has a globally asymptotically stable positive periodic solution.
The authors declare that there is no conflict of interests regarding the publication of this paper.
Acknowledgments
This work is supported by the National Natural Science Foundation of China (no. 11371230), Natural Sciences Fund of Shandong Province of China (no. ZR2012AM012), a Project for Higher Educational Science and Technology Program of Shandong Province of China (no. J13LI05), SDUST Research Fund (2014TDJH102), and Joint Innovative Center for Safe and Effective Mining Technology and Equipment of Coal Resources, Shandong Province of China.
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