Abstract
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:In model (1), , , and represent the number of susceptible, infected, and removed individuals at the time , respectively. Constant 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]:Here, is the maximum pulse immunization rate and 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 thatThen, we establish a SIR epidemic model with the nonlinear impulsive vaccination and life-immunity as follows:Here, constant represents the total number of input population and is the proportion of input population without immunity. is natural mortality, is the death rate due to illness, and is the recovery rate. The definitions of other symbols are shown in literature [29]. Note that variable just appears in the third and the sixth equations of model (4), so we only need to consider the subsystem of (4) as follows:
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 , which satisfies Clearly, we have , and then system (4) is ultimately bounded. Next, we will discuss the existence of the disease-free periodic solution of model (5). Let in system (5); then we get the subsystem of system (5) as follows:We have the following lemma for the property of subsystem (7).
Lemma 1. System (7) has a unique globally asymptotically stable periodic solution .
Proof. Solving the first equation of system (7), we getFrom the second equation of system (7), we get the pulse condition Then, we haveLet , and construct a stroboscopic map as follows:then system (11) has a unique fixed point. In fact, assume that is the fixed point of (11); then , and it satisfies where Since , then the equation has a unique positive root therefore system (11) has a unique fixed point .
From system (11), we haveClearly, ; then ; thus, is a stable fixed point of (11). Substituting the expression of into (8), we have Since is the unique stable fixed point of the difference equations, then 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 .
Next we will discuss the stability of the periodic solution. Suppose that is any positive solution of the system (5); let then the linearized system of the system (5) for the disease-free periodic solution ishere . Let be the fundamental solution matrix of the system; thus and , where is the unit matrix. Thenwhere . Here, is not required in the following analysis. Then we get For , the pulse condition isLet then the single-valued matrix of the system (18) is
The eigenvalues of the matrix are and . By the Floquet multiplier theory [32], the disease-free periodic solution is locally asymptotically stable if ; that is, Denote we can draw a conclusion as follows.
Theorem 3. The disease-free periodic solution of system (5) is locally asymptotically stable if .
Next we will prove the global attractivity of the disease-free periodic solution of system (5).
Theorem 4. The disease-free periodic solution of system (5) is globally attractive if .
Proof. Let be any solution of the system (5). Since , one can choose small enough such that From the first and third equations of the system (5), we have Consider the following impulsive comparison system: By the comparison theorem of impulsive differential equation, we have and as . Hence there exists such thatfor all large enough. For simplification we may assume (30) holds for all . From the second equation of system (5), we havewhich leads to Hence and as . Therefore as since for . Without loss of generality, we may assume that for all . From the first equation of system (5), we have Then, we have and , , where and are solutions ofrespectively. Consider , . Therefore, for any , there exists a such that
Letting , we havefor large enough, which implies as . So the disease-free periodic solution 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 of the system (5) is globally asymptotically stable if .
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 is proportional to the pulse vaccination period , and for large enough. In this case, the disease-free periodic solution of the system (5) is unstable. Assume that as . We choose the pulse vaccination period as a bifurcation parameter. Denote , and then the system (5) can be rewritten asWe assume that is the solution of the system (37) through the initial point . 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:Obviously, is equivalent to . ConsiderThen, we haveFrom the bifurcation theorem, the system (37) could produce nontrivial periodic solutions by the boundary solution with the condition , and the bifurcation is a supercritical bifurcation if , or a subcritical bifurcation if . Lettingthen the following theorem is gotten.
Theorem 6. If the condition holds, then system (37) has a supercritical branch at the point ; that is, a nontrivial periodic solution could be produced by the boundary periodic solution . Here, satisfies .
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 . If , then the disease-free periodic solution of the system (5) is globally asymptotically stable. Otherwise, it is unstable and will show a supercritical branch for . The threshold is related to all parameters of the model (5).
Next, we focus on the relations of the with the parameters and . The model (5) adopts the saturated vaccination rate . Here, represents the degree of restriction about medical resources. The relations of with and can be seen in Figure 1.
From Figure 1, if we fix , the threshold is an increasing function of the vaccination period . And if we fix the vaccination period , is an increasing function of the parameter . By the meaning of and , if we enrich the medical resources (i.e., decrease ) or reduce the vaccination period (i.e., decrease ), 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 , , , , , , , , and . We have an example as follows:
Case 1. Let , the initial point is . By calculation, we obtain . 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 , which is globally asymptotically stable.
Case 2. Letting , the initial point is . By calculation, we obtain . 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.
Conflict of Interests
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.