Computational and Mathematical Methods in Medicine

Volume 2014 (2014), Article ID 680743, 10 pages

http://dx.doi.org/10.1155/2014/680743

Research Article

An SIRS Epidemic Model Incorporating Media Coverage with Time Delay

Huitao Zhao,1,2 Yiping Lin,¹ and Yunxian Dai¹

¹Department of Applied Mathematics, Kunming University of Science and Technology, Kunming, Yunnan 650093, China

²Department of Mathematics and Information Science, Zhoukou Normal University, Zhoukou, Henan 466001, China

Received 24 December 2013; Accepted 17 January 2014; Published 3 March 2014

Academic Editor: Chengjun Sun

Copyright © 2014 Huitao Zhao 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.

An SIRS epidemic model incorporating media coverage with time delay is proposed. The positivity and boundedness are studied firstly. The locally asymptotical stability of the disease-free equilibrium and endemic equilibrium is studied in succession. And then, the conditions on which periodic orbits bifurcate are given. Furthermore, we show that the local Hopf bifurcation implies the global Hopf bifurcation after the second critical value of the delay. The obtained results show that the time delay in media coverage can not affect the stability of the disease-free equilibrium when the basic reproduction number . However, when , the stability of the endemic equilibrium will be affected by the time delay; there will be a family of periodic orbits bifurcating from the endemic equilibrium when the time delay increases through a critical value. Finally, some examples for numerical simulations are also included.

1. Introduction

Since Kermack and Mckendrick proposed the classical SIR epidemic model in 1927, mathematical modeling has become important tools in analyzing the spread and control of infectious diseases. Attempts have been made to develop realistic mathematical models for the transmission dynamics of infectious diseases. In recent years, epidemic models described by ordinary differential equations have been studied by many authors (see, e.g., [1–9] and the references cited therein).

One of the most fundamental compartment models based on differential equations is the SIRS model described by (1) below [10–15]. Let be the number of susceptible individuals, the number of infective individuals, and the number of removed individuals at time , respectively. A general SIRS epidemic model can be formulated as

where is the recruitment rate of the population, is the natural death rate of the population, is the natural recovery rate of the infective individuals, is the rate at which recovered individuals lose immunity and return to the susceptible class, and is the disease-induced death rate. The transmission of the infection is governed by the incidence rate , and is called the infection force.

In modelling of communicable diseases, the incidence rate may be affected by some factors, such as media coverage, density of population, and life style [16–22]. It is worthy to note that media coverage plays an important role in helping both the government authority make interventions to contain the disease and people respond to the disease [16, 19]. And a number of mathematical models have been formulated to describe the impact of media coverage on the transmission dynamics of infectious diseases. In particular, Cui et al. [16], Tchuenche et al. [18], and Sun et al. [20] incorporated a nonlinear function of the number of infective individuals (2) in their transmission term to investigate the effects of media coverage on the transmission dynamics:

where is the maximal effective contact rate between the susceptible and infective individuals and is the maximal reduced effective contact rate due to mass media alert in the presence of infective individuals; the terms measure the effect of reduction of the contact rate when infectious individuals are reported in the media. Because the coverage report cannot prevent disease from spreading completely we have . The half-saturation constant reflects the impact of media coverage on the contact transmission. The function is a continuous bounded function which takes into account disease saturation or psychological effects [20–22]. Then model (1) becomes

On the other hand, delays are ubiquitous in life, so it is in media coverage. Media coverage of an infectious outbreak can be seen as following two major routes [20, 23]. The first route is when the media report directly to the public on facts that they (the media) observe; the second has public health authorities using mass media or the Internet to communicate about the outbreak. For the second route, the number of infections and the number of suspected infections reported by media today are often the statistical result of yesterday or the day before. So the effects of media coverage on the transmission dynamics can be modified as follows:

where is a time delay representing the latent period of media coverage. Then model (3) can be modified as

In the following, we will investigate the effect of time delay on the dynamics of system (5). We suppose that the initial condition for system (5) takes the form

where , which is the Banach space of continuous functions mapping the interval into , where .

By the fundamental theory of functional differential equations [24], system (5) has a unique solution satisfying the initial condition (6).

The rest of the paper is organized as follows. In Section 2, we show the positivity and the boundedness of solutions of system (5) with initial condition (6). In Section 3, we study the local stability of the equilibria and the existence of the Hopf bifurcation at the positive equilibrium. In Section 4, we consider the global existence of bifurcating periodic solutions. In Section 5, we will give some numerical simulations to support the theoretical prediction. In Section 6, a brief discussion is given.

2. Positivity and Boundedness

In this section, we study the positivity and boundedness of solutions of system (5) with initial condition (6).

Theorem 1. Solutions of system (5) with initial condition (6) are positive for all .

Proof. Assume is a solution of system (5) with initial condition (6). Let us consider for . It follows from the second equation of system (5) that

From the initial condition (6), we have , for . Then, from the third equation of system (5), we have

A comparison argument shows that

From the initial condition (6), we have , for .

Next, we prove that is positive. Assume the contrary; then, let be the first time such that . By the first equation of (5) we have

This means for , where is an arbitrarily small positive constant. This leads to a contradiction. It follows that is always positive for . This ends the proof.

Theorem 2. Solutions of system (5) with initial condition (6) are ultimately bounded.

Proof. From Theorem 1, solutions of system (5) with initial condition (6) are positive for all . Let . From (5), we have

Therefore, for all large , where is an arbitrarily small positive constant. Thus, , , and are ultimately bounded.

3. Local Stability and Hopf Bifurcation Analysis

3.1. Previous Results

We now state some key results from [17], which provide the context for the main results of this paper. The basic reproduction number [17, 21] for the model is

From [21], when , system (5) has a disease-free equilibrium , which exists for all parameter values. When , system (5) has a unique endemic equilibrium , where

Denoting , the following results in [17, 21] are here just recalled.

Lemma 3. For , we have the following.

(i)The disease-free equilibrium is globally asymptotically stable if and unstable if in the set .(ii)The endemic equilibrium is globally asymptotically stable if in the set .

3.2. Local Stability at

The characteristic equation of system (5) at is

which is equivalent to

It is easy to see that, when , (15) has three negative roots and that, when , (15) has one positive root and two negative roots. Thus, we have the following.

Theorem 4. For any time delay , we have the following:

(i)the disease-free equilibrium is locally asymptotically stable if .(ii)the disease-free equilibrium is unstable if .

3.3. Local Stability and Hopf Bifurcation at

In this subsection, we suppose that . In what follows, using time delay as the bifurcation parameter, we investigate the Hopf bifurcation for system (5) and the stability of by using the method in [25, 26].

The characteristic equation of system (5) at is

where , , , , , and . Equation (16) is equivalent to

where , , , , , and .

Obviously, is a root of (17) if and only if satisfies

Separating the real and imaginary parts, we have

which is equivalent to

Let and denote , , and . Then (20) becomes

Next, we need to seek the conditions under which (21) has at least one positive root. Denote

Since , we conclude that if , then (21) has at least one positive root.

From (22), we have

Clearly, if , then the function is monotone increasing in . Thus, when and , (21) has no positive real root. On the other hand, when and , the following equation

has two real roots

It is easy to see that and . It follows that and are the local minimum and the local maximum of , respectively. Hence, we have the following simple property.

Lemma 5. Suppose that and . Then (21) has positive root if and only if and .

From Lemma 5 and the discussion above, we have the following.

Lemma 6. For the polynomial equation (21), we have the following results.

(i)If , then (21) has at least one positive root.(ii)If and , then (21) has no positive root.(iii)If and , then (21) has positive roots if and only if and .

Suppose that (21) has positive root. Without loss of generality, we assume that it has three positive roots, defined by , , and , respectively. Then (20) has three positive roots

From (19), we have

Thus, if we denote

where and , then is a pair of purely imaginary roots of (17) with . Define

Note that, from Lemma 3, when , the endemic equilibrium is stable if . Till now, we can employ a result from Ruan and Wei [25] to analyze (17), which is stated as follows.

Lemma 7. Consider the exponential polynomial

where and are constants. As vary, the sum of the order of the zeros of on the open right half plane can change only if a zero appears on or crosses the imaginary axis.

Applying Lemmas 6 and 7 and the discussion above, we obtain the following lemma.

Lemma 8. For the third degree transcendental equation (17), we have the following:

(i)if and , then all roots of (17) have negative real parts for all ;(ii)if either or , , , and , then all roots of (17) have negative real parts for .

Let

be the root of (17) near satisfying and . Then, from Lemma  8 in [26], we have the following transversality condition.

Lemma 9. Suppose that and , where is defined by (22). Then

and has the same sign with .

The proof of Lemma 9 is similar to that in the proof of Lemma  8 in [26], and here we omit it.

Then, from the above discussion and Lemmas 8 and 9, we have the following theorem.

Theorem 10. Suppose holds, and , , and are defined by (28) and (29), respectively. Then

(i)if and , the endemic equilibrium of system (5) is locally asymptotically stable for all ;(ii)if either or , , , and , the endemic equilibrium of system (5) is locally asymptotically stable for ;(iii)if the conditions of (ii) are satisfied and , then system (5) exhibits Hopf bifurcation at the endemic equilibrium when pass through .

4. Global Continuation of Local Hopf Bifurcations

In this section, we study the global continuation of periodic solutions bifurcating from the positive equilibrium of system (5).

Throughout this section, we follow closely the notations in [27]. For simplification of notations, setting , we may rewrite system (5) as the following functional differential equation:

where = . It is obvious that if holds, then system (5) has a semitrivial equilibrium and a positive equilibrium . Following the work of [27], we need to define

Let denote the connected component passing through in , where and are defined by (28) and (29). From Theorem 10, we know that is nonempty.

We first state the global Hopf bifurcation theory due to Wu [27] for functional differential equations.

Lemma 11. Assume that is an isolated center satisfying the hypotheses in [27]. Denote by the connected component of in . Then either

(i) is unbounded, or(ii) is bounded, is finite, and

for all , where is the th crossing number of if , or it is zero if otherwise.

Clearly, if (ii) in Lemma 11 is not true, then is unbounded. Thus, if the projections of onto -space and onto -space are bounded, then the projection onto -space is unbounded. Further, if we can show that the projection of onto -space is away from zero, then the projection of onto -space must include interval . Following this ideal, we can prove our results on the global continuation of local Hopf bifurcation.

From Theorems 1 and 2, it is easy to have the following.

Lemma 12. If the condition holds, then all nonconstant periodic solutions of (5) with initial condition (6) are uniformly bounded.

From [21], we know the following lemma.

Lemma 13. If the condition holds, then when , the positive equilibrium is globally stable in .

Lemma 14. If , then system (5) has no nonconstant periodic solution with period .

Proof. Suppose for a contradiction that system (5) has nonconstant periodic solution with period . Then the following system of ordinary differential equations has nonconstant periodic solution:

System (36) has the same equilibria as system (5), that is, and a positive equilibrium . Note that -axis and -axis are the invariable manifold of system (36) and the orbits of system (36) do not intersect each other. Thus, there is no solution that crosses the coordinate axis. On the other hand, note the fact that if system (36) has a periodic solution, then there must be the equilibrium in its interior and are located on the coordinate axis. Thus, we conclude that the periodic orbit of system (36) must lie in the first quadrant. From Lemma 13, the positive equilibrium is asymptotically stable and globally stable in ; thus, there is no periodic orbit in the first quadrant. This ends the proof.

Theorem 15. Let and be defined in (28) and (29). If , then system (5) has at least periodic solutions for every .

Proof. It is sufficient to prove that the projection of onto -space is for each , where .

The characteristic matrix of (33) at an equilibrium takes the following form:

where is called a center if and . A center is said to be isolated if it is the only center in some neighborhood of . It follows from (37) that

where , , , , , and   are defined as in Section 3. From the discussion in Section 3, each of (38) has no purely imaginary root provided that . Thus, we conclude that (33) has no center of the form as and . On the other hand, from the discussion in Section 3 about the local Hopf bifurcation, it is easy to verify that is an isolated center, and there exist , , and a smooth curve such that , for all and

Let

It is easy to verify that on ,

Therefore, the hypotheses in [27] are satisfied. Moreover, if we define

then we have the crossing number of isolated center as follows:

Thus, we have

where has all or parts of the form .

It follows from Lemma 11 that the connected component through in is unbounded. From (28), we can know that if holds, for ,

Now we prove that the projection of onto -space is , where . Clearly, it follows from the proof of Lemma 14 that system (5) with has no nontrivial periodic solution. Hence, the projection of onto -space is away from zero.

For a contradiction, we suppose that the projection of onto -space is bounded; this means that the projection of onto -space is included in an interval . Noticing and applying Lemma 14, we have for belonging to . Applying Lemma 12, we know that the projection of onto -space is bounded. So the component of is bounded. It contradicts our conclusion that is unbounded. The contradiction implies that the projection of onto -space is unbounded above.

Hence, system (5) has at least periodic solution for every , . This completes the proof.

5. Numerical Simulation

Example 1. In this case, we set , , , , , , , and . From (12), we compute . Furthermore, from (13), system (5) has only a disease-free equilibrium . From Theorem 4, we know that the disease-free equilibrium is locally asymptotically stable for any time delay .

Figure 1 shows that is locally asymptotically stable, and the trajectories of always converge to zero for taking some different values.

Figure 1: The trajectories of with , respectively. is always stable.

Example 2. In this case, we set , , , , , , , and . From (12), we compute . Furthermore, from (13), we get a disease-free equilibrium and an endemic equilibrium of system (5). From the algorithm of Section 3.3, we can compute and . Thus, from Theorems 4 and 10, we know that the disease-free equilibrium is unstable for all and that the endemic equilibrium is stable for . When crosses , a family of periodic orbits bifurcate from .

Figure 2 shows that the endemic equilibrium is stable with . Figure 3 shows that the endemic equilibrium is unstable and a periodic orbit bifurcate from with . Figure 4 shows that the endemic equilibrium is still unstable and a periodic orbit bifurcate from with . We can see from Figures 3 and 4 that the period and amplitude of the oscillation are increasing with the increasing of time delay. Furthermore, Figure 5 shows that the local Hopf bifurcation implies the global Hopf bifurcation after the second critical value of .

(a)

(b)

Figure 2: The trajectories and phase graphs of system (5) with . is stable.

(a)

(b)

Figure 3: The trajectories and phase graphs of system (5) with , is unstable and a periodic orbit bifurcate from .

(a)

(b)

Figure 4: The trajectories and phase graphs of system (5) with . is unstable and a periodic orbit bifurcate from .

(a)

(b)

Figure 5: The trajectories and phase graphs of system (5) with . is unstable and a periodic orbit bifurcate from .

6. Discussion

In this paper, we proposed an SIRS epidemic model incorporating media coverage with time delay. We first investigated the positivity and boundedness of the solution of system (5) and show that the solution of system (5) with the initial condition (6) is positive and bounded.

Secondly, we studied the stability of the disease-free equilibrium. Our results show that the disease-free equilibrium is globally stable for all when the basic reproduction number . This is to say, the time delay in media coverage cannot influence the stability of the disease-free equilibrium. In other words, we can ignore the effect of time delay for .

However, when , the stability of the endemic equilibrium will be affected by the time delay in media coverage. We found that there existed a critical value of time delay , such that the stability of the endemic equilibrium changed and periodic oscillations occurred when the time delay passes through this critical value. Furthermore, we show that the local Hopf bifurcation implies the global Hopf bifurcation after the second critical value of delay.

These results mean that, when and the time delay is small enough, the epidemic will eventually become endemic disease. However, if the delay of information about and appraisal of an epidemic on media coverage is too large, it will lead to repeated episodes of epidemic, and then it is unfavourable for the containment of the epidemic. We suggest that it is helpful for controlling epidemic to communicate about the outbreak of an epidemic as soon as possible.

Conflict of Interests

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

Acknowledgment

The authors would like to thank the anonymous referee for the very helpful suggestions and comments which led to improvements to their original paper. This work is supported by the National Natural Science Foundation of China (no.11061016), Science and Technology Department of Henan Province (no. 122300410417), and Education Department of Henan Province (no. 13A110108).

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