Computational and Mathematical Methods in Medicine
Volume 2019 (2019), Article ID 3859815, 10 pages
https://doi.org/10.1155/2019/3859815

Research Article

State-Dependent Pulse Vaccination and Therapeutic Strategy in an SI Epidemic Model with Nonlinear Incidence Rate

Kaiyuan Liu,¹ Tongqian Zhang,2,3 and Lansun Chen⁴

¹College of Mathematics and Information Science, Anshan Normal University, Anshan 114016, China
²College of Mathematics and Systems Science, Shandong University of Science and Technology, Qingdao 266590, China
³State Key Laboratory of Mining Disaster Prevention and Control Co-founded by Shandong Province and the Ministry of Science and Technology, Shandong University of Science and Technology, Qingdao 266590, China
⁴Academy of Mathematics and Systems Science, Chinese Academy of Sciences, Beijing 100190, China

Correspondence should be addressed to Tongqian Zhang; zhangtongqian@sdust.edu.cn

Received 26 July 2018; Revised 18 November 2018; Accepted 22 November 2018; Published 6 February 2019

Guest Editor: Ke-jun Dong

Copyright © 2019 Kaiyuan Liu et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract.

In this paper, the state-dependent pulse vaccination and therapeutic strategy are considered in the control of the disease. A pulse system is built to model this process based on an SI ordinary differential equation model. At first, for the system neglecting the impulse effect, we give the classification of singular points. Then for the pulse system, by using the theory of the semicontinuous dynamic system, the dynamics is analyzed. Our analysis shows that the pulse system exhibits rich dynamics and the system has a unique order-1 homoclinic cycle, and by choosing p as the control parameter, the order-1 homoclinic cycle disappears and bifurcates an orbitally asymptotical stable order-1 periodic solution when p changes. Numerical simulations by maple 18 are carried out to illustrate the theoretical results.

1. Introduction

Infectious diseases are caused by various pathogens that can be transmitted from person to person, animal to animal, or human to animal. The ever-changing changes in the pathogens of ancient infectious diseases and the emergence of new pathogens have brought new challenges to the discovery, diagnosis, and prevention of infectious diseases. According to the 2016 report of the World Health Organization [1], about 36.7 million people have been infected with HIV/AIDS, 1.0 million people died of HIV/AIDS, and more than 18 million people worldwide living with HIV are receiving antiviral drugs. And tuberculosis is currently the biggest “killer” caused by a single infectious pathogen after AIDS in the world. As of the end of 2016, there were 10.4 million new tuberculosis cases [2]. Therefore, the control and elimination of infectious diseases has attracted wide attention of people. Various dynamic models have been proposed by mathematicians to investigate the spread and evolution of infectious diseases [3–14]. In particular, mathematical models of differential equations have been extensively investigated, and among them, the most classical well-known model is SIR model [15] or SIS model [16], which have been widely investigated [17–23].

It is well known that vaccination is mostly a medical behavior that can evoke the individual’s natural defense mechanism to prevent possible future diseases. This kind of vaccination is known as prophylactic vaccination. Diphtheria, whooping cough, polio, tetanus, herpes, rubella, and mumps are the most common types of vaccines. There are many types of vaccination, the two common types are continuous vaccination and pulsed vaccination. Continuous vaccination is when people are vaccinated at birth to protect themselves from illness, while pulsed vaccination is when people are vaccinated at a fixed period of time in all age groups which was firstly investigated by Agur et al in [24]. Pulse vaccination strategy (PVS) has been studied by many scholars [25, 26]. For example, Lu et al. [27] studied the pulse epidemic model with bilinear incidence and compared the effectiveness of the continuous and pulsed vaccination strategies. Liu et al. [28] investigated the SIR epidemic model with the saturated transmission rate. However, the strategy is taken at certain fixed times and does not depend on the status of infectious diseases. In general, taking into account the limited medical resources and costs, vaccines to susceptible people according to the number of susceptible people or infected people are more reasonable than continuous vaccination and fixed time pulse vaccination. This control strategy relies on the individual (or susceptible individuals) of the infection state and is called a state-dependent pulse vaccination strategy. Based on this idea, Tang et al. [29], Nie et al. [30], Guo et al. [31], and Qin et al. [32] have considered a state-dependent pulse strategy in SIR model and SIRS model. In fact, using state-dependent feedback control strategies to simulate real-world problems is more reasonable. Therefore, the impulsive state feedback control is also widely applied to the population dynamics model [33–46], chemostat model [47], and turbidostat model [48].

Firstly, we consider an epidemic model with nonlinear incidence rate described by the ordinary differential equations as follows:

which is a special case in the study of Liu et al. [49] and and represents the number of susceptible and infected individuals at time respectively. θ is the birth rate, β is the contact rate, and γ is the natural death rate.

Motivated by the studies of Tang et al. [29], Nie et al. [30], and Zhang et al. [50], we consider state-dependent pulse vaccination and treatment strategy in model (1) and get the following model:

where When the amount of infected reaches the hazardous threshold value vaccination and treatment are taken into account, and the number of susceptible and infected suddenly turn to and , respectively, where denote the vaccination rate of susceptible individuals and treatment rate of infected individuals, respectively. By the scaling,

then, model (2) transforms into the following form:

where In the following, according to the actual condition, we always suppose that , and based on practical significance, our research scope is limited to the first quadrant, i.e., .

The purpose of this paper is to study the dynamic behavior under the effect of state-dependent pulse vaccination and treatment strategy. This article is organized as follows. In Section 2, we introduce some definitions and notations of the geometric theory of semicontinuous dynamic systems, which will be useful for the latter discussion. In Section 3, we qualitatively analyze the dynamics of model (3). In Section 4, the existence of the homoclinic cycle is studied by using the geometrical theory of semicontinuous dynamical systems. At last, we present some numerical simulations.

2. Preliminaries

In this section, we introduce some notations, definitions, and lemmas of the geometric theory of semicontinuous dynamic system, which will be useful for the following discussions. The following definitions and lemmas of semicontinuous dynamic system come from the studies of Chen et al. [51] and Wei and Chen [36].

Definition 1. Consider the following state-dependent impulsive differential system

The solution mapping of system (4) is called the semicontinuous dynamical system denoted by where is the semicontinuous dynamical system mapping with initial point the sets M and N are called the impulse set and phase set, which are lines or curves on The continuous function is called impulse mapping.

Remark 1. System (4) constitutes a semicontinuous dynamic system where

Definition 2. If there exists a point and such that and , then is called order-1 periodic solution.

Definition 3. The trajectory combining with impulse line is called the order-1 cycle. If the order-1 cycle has a singularity, then the order-1 cycle is called the order-1 singular cycle. Furthermore, if the order-1 cycle only has a saddle, then the order-1 singular cycle is called the order-1 homoclinic cycle.

Definition 4. We assume that G is a bounded closed simple connected region, which has the following properties:

(i)Impulse set M is a simple connected bounded closed straight line segments or curve segments which do not contain closed branch(ii)The boundaries , , and of region G are nontangent arcs of semicontinuous dynamical system (4). The boundary is the impulse set of system (4), and its phase set satisfies ;(iii)The orientation of the vector fields of semicontinuous dynamical system (4) on the , , and points of the internal of region G. There are no equilibriums on the boundaries and also in the internal of region G of semicontinuous dynamical system (4).

Then region G is called Bendixson’s region of semicontinuous dynamical system (4).

Lemma 1. (Bendixson theorem of semicontinuous dynamical system.) If region G is Bendixson’s region of semicontinuous dynamical system (4), then there exists at least an order-1 periodic solution in the internal of region G (Figure 1).

Figure 1: Bendixon region of the semicontinuous dynamical system. This figure is reproduced from the study of Wei and Chen [36] (under the Creative Commons Attribution License/public domain).

Next, we will give the definition of successor function of system (4). Firstly, we define a new number axis in set N. On straight line , take the origin at point of coordinate axis y and define the positive direction and unit length to be consistent with coordinate axis x, then we obtain a number axis l. For any , let be the coordinate of point A which is defined as the distance between point A and the y-axis, i.e.,

Definition 5. Suppose be a map. Let be the initial mapping point, for any , there exists a such that . Then, function is the successor function of point P, and the point is called the successor point of P (Figure 2).

Figure 2: Successor function.

Definition 6. Suppose is an order-1 periodic solution of system (4). If for any , there must exist and such that, for any point , we have for ; then we call the order-1 periodic solution is orbitally asymptotically stable.

3. Qualitative Analysis of System Neglecting the Impulse Effect

First, we consider the classification of singular points of the system neglecting the impulse effect. Neglecting the impulse effects, system (3) reduces to

system (5) always has one equilibrium . Denote if the system (5) has two positive equilibria where and .

Next, we will analysis of the stability of the equilibria of system (5). For equilibrium , we have

obviously is a stable node.

For , we have

and the characteristic equation is Let and be the two characteristic roots of the characteristic equation, then we have

obviously is a saddle.

For , we have

and by calculations, we get

obviously is a stable node (Figure 3).

Figure 3: Phase diagram of system (3) with .

Lemma 2. System (5) is uniformly bounded.

Proof. Firstly, if we have the isoclines (Figure 4) and the straight line then we get thus, according to the qualitative theory of ordinary differential equations, the trajectory of the system (5) passes through and goes from the right side of to the left side of Consider the straight line where M is large enough and Then, we obtain , and thus, the straight line is nontangent; then, according to the qualitative theory of ordinary differential equations, the trajectory of the system (5) passes through and goes from the upper right side of to the lower left side of Let us denote the intersections of and be and consider the straight line obviously, we have and then, the trajectory of model (5) passes through and goes from the top side of to the bottom side of Thus, the model (5) is uniformly upper bounded. This completes the proof.

Figure 4: System (3) is the uniformly bounded.

4. Homoclinic Cycle of Model about Parameter p

In this section, we will discuss the existence of order-1 homoclinic cycle of model (3) by choosing p as the control parameter.

Theorem 1. If then there exists such that model (3) has an order-1 homoclinic cycle.

Proof. In model (3), since the point is a saddle point, then there exist two manifolds which will enter or leave the saddle point one is the unstable manifold () and another is the stable manifold According to Lemma 2 and the qualitative theory of ordinary differential equations, and impulse set M must intersect, and the intersection is denoted as If we denote the intersection of impulse set M and the isocline as point the intersection of image set N and isoclines as point , and the intersection of image set N and as point by the qualitative theory of ordinary differential equations, the unstable manifold is above of the isoclines , and the stable manifold is below the isoclines (Figure 5). Because the monotonicity of the impulse function with respect to x and there must exist such that and then the stable manifold starting form the point the unstable manifold starting form the point , and the impulse line formed a homoclinic cycle.

Figure 5: The existence of the order-1 homoclinic cycle.

Remark 2. If , according to the theory of differential equations, the trajectory tends to , and in a biological sense, the disease eventually extincts. However, the relatively high vaccination rate will waste medical resources. So, we always assume that in the following theorem.

Theorem 2. If , , and , then the homoclinic cycle of model (3) disappears and bifurcates an unique order-1 periodic solution.

Proof. By Theorem 1, if then there exists such that model (3) has an order-1 homoclinic cycle, i.e., the stable manifold starting form the point the unstable manifold starting form the point , and the impulse line formed a homoclinic cycle. Now, we consider whether there will be a periodic solution that bifurcates out of the homoclinic cycle when p changes. In fact, consider the unstable manifold starting form the point when touches the impulsive set M (the intersection is denoted as A), then a pulse happens and then the impulsive function transfers the point A into and the point C into and according the definition of impulsive function, we have . If we obtain Since then we get and then by the definition of Bendixon region of semicontinuous dynamics system, (part of isoclinal ), (), (part of the ), and (part of the ) constitute the Bendixon region G of the system (3). According to Lemma 1 and Lemma 2 in [36], system (3) must exist an order-1 periodic solution, initial point of which is between and in image set N (Figure 6).
Next, we show the order-1 periodic solution of system (3) is unique if it exists. The idea of the proof comes from the study of Wei and Chen [36]. Select two points I and J in phase set arbitrarily, where . Let , after that due to the impulsive effects, points and jump to For , we have and , ; hence, we have . Using Definition 5 and Definition 6 in [36], we obtain and . Hence, we have ; that is, the successor function is monotonic in Therefore, there is an unique point H such that ; thus, system (3) has an unique order-1 periodic solution (Figure 7).

Figure 6: The existence of order-1 periodic solution of system (3).

Figure 7: The monotonicity of the successor function G in the segment

Theorem 3. If , , and , then the order-1 periodic solution of model (3) is orbitally asymptotically stable.

Proof. By Theorem 2, we have that the order-1 periodic solution in system (3) is unique. Let the initial point of the order-1 periodic solution is , where . Set , then due to the impulsive effects jumps to which is the successor point of (Figure 8). We have . Set . Owing to trajectories do not intersect, we have and , where F is the impulsive point of the order-1 periodic solution.
Similar to the above method, let , we have and . We can repeat the above steps and have a sequence of impulse set M and a sequence of image set N satisfying , , . That is, we have

Thus, series is monotonically increasing, and is monotonically decreasing; , as , and , as . Select a point that is different from point H. If (otherwise, , the discussion is similar to ), there must be an integer k such that . The orbit starting from point will also experience an infinite number of impulsive effects. Let the phase point be which is after the impulsive effect. Then for any l, we have and . Hence, is monotonically decreasing, and is monotonically increasing. Thus, after the pulse effects the successor points are attracted to the point H, which means that the order-1 periodic solution of the system (3) is orbitally asymptotically stable.

Figure 8: The stability of order-1 periodic solution of system (3).

5. Numerical Simulations

In this section, we give some numerical simulations to illustrate the theoretical results we previously obtained. First, we consider the system neglecting state-dependent pulse strategy, let and simple calculations show ; then system (3) has three equilibria, i.e., , and (Figure 3), and among them, is a stable node, is a saddle, and is a stable node.

Then, we consider the state-dependent pulse control strategy in system (3). First, we take more moderate preventive and therapeutic measures, and let then system (3) has a homoclinic cycle composed of the unstable manifold (), the stable manifold (), and the pulse straight line (Figure 9, the initial value is ). And from the time series diagrams, we can see that x and y show periodic oscillations over time (Figures 10 and 11). If we maintain a certain intensity of treatment (fix parameter ) and reduce the intensity of prevention, for example, let by Theorem 2, the order-1 homoclinic cycle disappears and bifurcates a order-1 periodic solution, which is shown in Figures 12–14. And if we take more stringent preventive measures which means a larger vaccination rate, for example, let then the disease will become extinct, which is shown in Figures 15–17.

Figure 9: Order-1 homoclinic cycle of system (3) with

Figure 10: Time series diagram of of system (3) with

Figure 11: Time series diagram of of system (3) with

Figure 12: Order-1 homoclinic bifurcation of system (3) with

Figure 13: Time series diagram of of system (3) with

Figure 14: Time series diagram of of system (3) with

Figure 15: Phase diagram of system (3) with

Figure 16: Time series diagram of of system (3) with

Figure 17: Time series diagram of of system (3) with

6. Conclusion

In this paper, a different strategy from tradition, i.e., the state-dependent pulse vaccination and therapeutic strategy, is considered in the control of the disease. A pulse system is built to model this process based on an SI ordinary differential equation model. By using the theory of semicontinuous dynamic system, the dynamics of the pulse system is analyzed. Our results show the pulse system exhibits rich dynamics; for example, the system has a unique order-1 homoclinic cycle, and by choosing p as the control parameter, we prove that when p changes, the order-1 homoclinic cycle disappears and bifurcates an orbitally asymptotical stable order-1 periodic solution. However, it should be pointed out here that, in this work, we focused on the theoretical framework and realistic parameters can be incorporated into our model. State-dependent impulsive vaccination strategy may be used a supplementary control measure besides routine vaccination, or it may be used in the situation when vaccine stockpile is limited (for example, the yellow fever outbreaks in Nigeria and Congo in 2017 [52]). The realistic approach in childhood infection and other infections will be conducted in future work.

Data Availability

All data are hypothetical to verify the theoretical results of this study.

Conflicts of Interest

The authors declare that there are no conflicts of interest regarding the publication of this paper.

Acknowledgments

This work was supported by the National Natural Science Foundation of China (No. 11671346 and 61751317), SDUST Research Fund (No. 2014TDJH102), and Scientific Research Foundation of Shandong University of Science and Technology for Recruited Talents.

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