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

He, Zhi-Long; Nie, Lin-Fei

doi:https://doi.org/10.1155/2015/532494

Discrete Dynamics in Nature and Society

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Abstract Introduction Model Formulation and Preliminaries Acknowledgments References Copyright Related Articles

Research Article | Open Access

Volume 2015 | Article ID 532494 | https://doi.org/10.1155/2015/532494

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

Zhi-Long He¹and Lin-Fei Nie¹

Academic Editor: Gian I. Bischi

Received15 Apr 2015

Revised03 Jun 2015

Accepted04 Jun 2015

Published25 Jun 2015

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.

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 Section 2, 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:withwhere . , , and represent the densities or quantities of susceptible, infected, and recovered populations, respectively. All model parameters are positive constants, where is the natural birth/death rate, denotes the basic transmission rate, and represents the removed/recovered rate. When increases and reaches a certain level , 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 , to reflect the impact of media coverage to the effective contact rate.

From Proposition 2 in [20], authors showed that, for model (1) with , the disease-free equilibrium is globally asymptotically stable if , and the unique endemic equilibrium is globally asymptotically stable if , wherewhere the function is defined to be a multivalued inverse of the function satisfying .

We assume, throughout this paper, that and . That is to say, model (1) with has a unique endemic equilibrium , 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 reaches the higher threshold value. That is, when the density of the infected individuals reaches the higher hazardous threshold value () at time at the th 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 , , and , respectively, where and are the vaccination intensity and medication intensification effort, respectively. However, when the density of the susceptible reaches the relatively small threshold value () at time at the th 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 , , and , respectively, where is the medication intensification effort.

Under the above assumptions, it follows from model (1) that we propose the following multiple state dependent pulse control differential equations:(i), or , or : (ii) and :(iii) and :where and are the abscissa of intersection of the the horizontal isocline and the lines and , respectively, and and .

Let and . 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 any , (), and each component of the solution of systems (4)–(6) with initial value is positive and ultimately bounded for all .

The proof of Lemma 1 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 , therefore, systems (4)–(6) are equivalent to the following system:

Based on the biological background of model (7), we only consider dynamical behavior of model (7) in region .

Generally, a semidynamical system is denoted by . For any , the function defined as is continuous, and we call the trajectory passing through point . Consider the following general state dependent pulse differential equation:where . , , , and are continuous functions mapping into and is the set of impulses. According to the denotations in [22], we denote , for any , , where is the set of impulses, is the pulse function, and is the set of phase after impulses. Obviously, the solution mapping of system (8) is a semicontinuous dynamical system, which is denoted by . 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 is said to be a semidynamical system if is a metric space, is the set of all nonnegative real, and is a continuous map such that (a) for all ,(b) for all and .

Definition 3 (order-1 periodic solution [23]). A trajectory π is called order-1 periodic solution with period if there exists a point and such that and .

Definition 4 (orbitally asymptotically stable [23]). Suppose is an order-1 periodic solution of model (8). If, for any , there must exist and , such that for any point , one has for , where denotes a -neighborhood of point and is the distance from to . Then one calls the order-1 periodic solution orbitally asymptotically stable.

Definition 5 (successor function [23]). Suppose is a map. For any , if there exists a such that , then is called the successor function of point , and the point is called the successor point of .

The following lemma and remarks are on the properties of successor function .

Lemma 6. The successor function is continuous.

The proof of Lemma 6 is obvious; hence we omit it here.

Remark 7. From Lemma 6, it is obvious that model (8) exists as positive order-1 periodic solution if there exist two points satisfying .

Remark 8. In Lemma 6, if , then trajectory with initial point is an order-1 periodic solution of model (8).

The following Lemma 9 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]). The -periodic solution of system (8) is orbitally asymptotically stable if the Floquet multiplier satisfies the condition , whereand , , , , , , , and are calculated at the point , , , and is the time of the th jump.

To discuss the dynamical behaviors of model (7), we denote two pulse setsand two phase setswhere and are continuous functions.

3. Main Results

Since the endemic equilibrium of model (7) without pulse effect is globally asymptotically stable, then any positive solutions of model (7) without pulse effect will eventually tend to . Therefore, region is divided into four different domains with the vertical isocline and the horizontal isocline of model (7), where

For convenience, we denote the -axis intersect line at point . Suppose that the horizontal isocline line intersects lines , , and at points , , and , respectively. Suppose that the vertical isocline line intersects lines and at points and , respectively.

According to the uniqueness of solution to initial value, we know there exists a unique trajectory starting from the initial point and tangent to line at point . Letand let be a bounded domain by the phase trajectory and segments and . Obviously, if , then any trajectory with initial value will reach pulse set , and any trajectory with initial value will reach pulse set . So, in this section, we discuss the existence and stability of positive order-1 periodic solutions of model (7) in cases of initial values and , respectively.

Firstly, the following result is on the existence and stability of positive order-1 periodic solution for model (7).

Theorem 10. For any , , and , model (7) has always an orbitally asymptotically stable positive order-1 periodic solution and which starts from pulse set .

Proof. Obviously, any trajectory with initial value will reach pulse set and intersects pulse set infinite times due to pulse treatment . The trajectory passing through point which is tangent to phase set at point intersects with pulse set at point and then jumps to point due to pulse treatment , where . By the geometrical construction of phase region , we have that point is right point . That is, . Hence the successor function of point is .
On the other hand, trajectory from the initial point intersects pulse set at point and next jumps to point () on phase set . It follows from the geometrical construction of region that we have that point is right point . That is, . Therefore, we have .
By Lemma 6, we know that there exists a positive order-1 periodic solution of model (7), which starts from pulse set (for more details, see Figure 2(a)).
Next, we discuss the orbital stability of positive order-1 periodic solution which starts from pulse set . Since trajectories starting from any point will enter the set after several times pulse effects at most, then the initial point of the order-1 periodic solution only lies in . The set is mapped to set by the geometrical construction of the phase regions and . Subsequently, set is mapped to set due to the pulse vaccination and pulse treatment. From the geometrical construction of regions and , it is easy to know that . Repeating the abovementioned process, we can get two sequences and () on set and satisfyTherefore, the sequence is convergent monotonously and . It implies that there exists a unique point such that . Furthermore, we have . For any point , whereWithout loss of generality, let , the trajectory starting from the initial point intersects pulse set at point () and next jumps to point due to pulse vaccination and pulse treatment, where . Repeating the process, we obtain a sequence of set , where , . Since , then . That is, . Similarly, if , we also can get . Thus trajectory starts from any point of set , which ultimately tends to the positive order-1 periodic solution .
Given the above, we obtain that model (7) has a positive order-1 periodic solution, which starts from the initial value and is orbitally asymptotically stable. This completes the proof.

(a)

(b)

Theorem 11. If , for any , , and , then model (7) has two positive order-1 periodic solutions in region : one starts from pulse set , and the other starts from . Furthermore, let be an order-1 periodic solution which starts from the pulse set of model (7) with period ; ifthen is orbitally asymptotically stable. Thus, trajectory of model (7) with initial value will tend to the stable positive order-1 periodic solution which starts from pulse set , and trajectory of model (7) with initial value will tend to the stable positive order-1 periodic solution which starts from .

Proof. The proof of the existence and stability of the positive order-1 periodic solution starting from pulse set 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 .
Suppose that the trajectory passing through point intersects with pulse set at point . Since point , then pulse occurs at ; supposing point is subject to pulse effects to point , where , the position of has the following three cases (for more details, see Figure 2(b)):(i)If point coincides with , that is, , then the successor function of is that . Thus, model (7) has a positive order-1 periodic solution which starts from pulse set .(ii)If point is left point , that is, , thus the successor function of is that . On the other hand, choose a point which is left point , where is small enough. Suppose that trajectory from the initial point intersects pulse set at point and next jumps to point on phase set due to pulse effects. According to the existence and uniqueness of solution for pulse differential equation, point is right point and point is right point . Therefore, the successor function of point is . By Lemma 6, we know that there exists a positive order-1 periodic solution of model (7), which starts from pulse set .(iii)If point is right point , that is, , thus, the successor function of is that . In the meanwhile, suppose that trajectory from the initial point intersects pulse set at point and next jumps to point on phase set again, where . From the geometrical construction of phase region of model (7), we have . Therefore, point is left point . That is, . Thus, the successor function of is . By Lemma 6, we know that model (7) admits a positive order-1 periodic solution, which starts from pulse set .
To sum up the above discussion, model (7) exists as an order-1 periodic solution which starts from pulse set .
Next, we show the orbitally asymptotical stability of this order-1 periodic solution . According to Lemma 9, suppose that intersects phase set and pulse set at points and , respectively. Comparing with model (8), we have, , and . Thus,Furthermore, it follows thatOn the other hand, we integrate both sides of the second equation of model (7) along the orbit , and we haveThen, we haveBy condition (16), we note that model (7) satisfies all conditions of Lemma 9. Therefore, the order-1 periodic solution starting from the pulse set is orbitally asymptotically stable. This completes the proof.

From the proof of Theorem 11, integrating both sides of the first equation of model (7) along the orbit , we obtain thatThis shows that

The following Corollary 12 is a direct consequence of Theorem 11.

Corollary 12. Let be the order-1 periodic solution starting from pulse set of model (7) with period . Ifthen is orbitally asymptotically stable.

For , in this case, suppose that orbit starting from the initial point will intersect lines and at points and , respectively.

Integrating both sides of the second equation of model (7) along orbit , we haveThat is,

Further, integrating both sides of the first equation of model (7) along orbit , we have

From the geometrical construction of phase region , we have . Since the point in region , it yields thatThis together with (26) and (27) gives

Theorem 13. Assuming that , for any , , and , model (7) has a positive order-1 periodic solution starting from pulse set which is orbitally asymptotically stable. Furthermore, ifthen model (7) has a positive order-1 periodic solution starting from pulse set . Let be the order-1 periodic solution of model (7) with period , which starts from pulse set . Further, if (16) holds, then is orbitally asymptotically stable. Thus, model (7) has two positive order-1 periodic solutions in region , which are orbitally asymptotically stable.

The proof of the existence and stability of positive order-1 periodic solution starting from pulse set 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 , , , and . That is, we consider the following SIR epidemic model with state-dependent pulse vaccination and medication:By directly calculating, we have . It is easy to known that model (31) without pulse control has a unique globally asymptotically stable endemic equilibrium .

Firstly, we choose control parameters , , and , respectively. By calculating, we get and . From numerical simulation, we know model (31) has a unique trajectory starting from the initial point which is tangent to line at point , as shown in Figure 3(a). Obviously, . Now, we take ; it is easy to calculate thatTherefore, 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 , and the other starts from point , which are orbitally asymptotically stable.

(a)

(b)

(c)

(d)

Nextly, we choose control parameter and other parameters fixed as above. It is easy to calculate that and . By numerical simulations, model (31) has a trajectory which starts from the initial point and is tangent to line . This is what show in Figure 4(d). Obviously, . It can be easily calculated thatFurther, if we choose , it is easy to calculate thatTherefore, 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 , and the other starts from point .

(a)

(b)

(c)

(d)

We choose, finally, the vaccination intensity . 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 . We choose, however, control parameters to be , , , and . From numerical simulation and calculation, we have that and . By Theorem 13, we can know that model (31) has two positive order-1 periodic solutions satisfying . In particular, take 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.

(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 , which is orbitally asymptotically stable for the initial value (see Theorem 10), and exists two positive order-1 periodic solutions in region with condition or . 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.

Conflict of Interests

The authors declare that there is no conflict of interests regarding the publication of this paper.

Acknowledgments

This research has been partly supported by the National Natural Science Foundation of China (Grants nos. 11461067, 11402223, and 11271312) and the China Postdoctoral Science Foundation (Grants nos. 2012T50836 and 20110491750).

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Copyright © 2015 Zhi-Long He and Lin-Fei Nie. 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.

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