Abstract

This paper is concerned with exploring the global dynamics of SEIR epidemic model with media impact, which incorporates latency and relapse delays. The permanence of the model is carefully discussed. By suitable Lyapunov functionals, we establish the global stability of the equilibria. It is found that the basic reproduction number completely determines the threshold dynamics of the SEIR model. Finally, the impact of media on the epidemic spread is studied, which reveals that timely response of media and individuals may play a more key role in disease control.

1. Introduction

The investigation of dynamics of epidemiological models has been of importance in improving our understanding of disease control [1–6]. Media education has been an important control strategy for the emerging and reemerging epidemics, such as HIV/AIDS [7], SARS [2], Ebola virus disease (EVD), Middle East Respiratory Syndrome (MERS), which can not only alert the general public to the hazard from the infectious diseases but also educate the people about the requisite preventive measures such as wearing protective masks, vaccination, voluntary quarantine, and avoidance of congregated places. The extensive media education will bring about reducing the frequency and probability of potentially contagious contacts among the well-informed people [2–4]. In order to describe the impact of media on the diseases, Cui and his coauthors [3] used the transmission rate of the form in SEI model with logistic growth, where is the transmission rate before media alert and denotes the number of infected individuals. This work provided a theoretical basis that a Hopf bifurcation can occur for weak media impact (small values of ) while the model may have up to three endemic equilibria for strong media impact (large values of ). Liu and Cui [4] proposed the transmission rate taking the form to capture the impact of media on disease spread, where represents the reduced maximum value of the transmission rate when approaches infinite and reflects the reactive velocity of media coverage and individuals to the epidemic disease. For more details concerning the application of this transmission rate, we refer the reader to recent works [5, 6].

On the other hand, the individuals infected by infectious disease may develop symptoms after an incubation period [8], such as Hepatitis B virus (HBV), Hepatitis C virus (HCV), the human tuberculosis (TB), and Herpes simplex virus type 2 (HSV-2). The average latency period after the genital acquisition of HSV-2 is approximately 4 days [9], and latent tuberculosis may take months, years, or even decades to become infectious. Moreover, it has been found clinically that numerous diseases may make the recovered individuals suffer from a relapse of symptoms, including HBV [10], HCV [11], the majority of TB due to incomplete treatment [12], and genital HSV-2 [13, 14]. Recently, many epidemiological models incorporating both latency and relapse have been extensively investigated and many good results have been obtained (e.g., [15–19]). However, there are few investigation on both latency and relapse delays in the epidemiological models with media impact.

Suppose that the total population at time is divided into four disjoint epidemic subclasses: susceptible , latent/exposed , infectious , and temporarily recovered , respectively. And denotes recruitment rate of susceptible class , is natural death rate, indicates the death rate due to the disease, and represents the recovered rate for infectious class due to natural recovery or treatment. As pointed by Cui et al. in [5], media can effectively reduce the contact rates among the population to a limited level. Hence, it may be more realistic to use the transmission rate compared to . By incorporating media impact into the bilinear incidence rate, we now consider the incidence rate function as follows: In this work, we assume that the latency and relapse periods are constants, denoted by and , respectively. Hence, the probabilities and of remaining in the latent class and the temporarily recovered class are the step-functions taking the forms This suggests that all individuals remain in latent class for a constant period and in temporarily recovered class for a constant period . One further assumes that the disease has been in the population for at least a time of .

Following closely the ideas of [5, 8, 15, 18] and incorporating media impact, we consider the following integro-differential epidemic model with latency and relapse delays: Differentiating the second and the fourth equations of (3), we derive the delay model where the term indicates the individuals surviving in the latent period and entering into infectious class at time and the term represents the individuals surviving in temporarily recovered period and entering into infectious class at time . The initial conditions for model (4) are given by Here, let , which denotes the Banach space of continuous functions mapping the interval into , equipped with the uniform norm defined by , where . In consideration of the continuity of the initial conditions, one requires

Our main aim of this study is concerned with investigating the global dynamics of model (4) and the impact of media on the disease spread. The basic structure of this paper is as follows. In the next section, we study the existence and the local stability of equilibria of (4). Section 3 carefully addresses the permanence of (4). In Section 4, global stability analysis of (4) is carried out. Finally, a discussion section ends this paper.

2. The Equilibria

2.1. The Existence of Equilibria

Throughout this paper, denote .

Lemma 1. Any solution of model (4) with the initial conditions (5) and (6) is unique, positive, and bounded on . Moreover, the biologically feasible region is a positive invariant with respect to (4).

Proof. From the fundamental theory of functional differential equations [20], (4) admits a unique solution satisfying the initial conditions (5) and (6).
Firstly, one shows that the solution is positive, . If not, we assign to be the first time such that , which implies that for . Thus, we must have from the first equation of (4). Then there is a sufficiently small constant such that holds for . This leads to a contradiction. So for . Secondly, also holds . In fact, if would lose its positivity and were the first time such that , then for . Solving the third equation of (4) on gives Due to , the right hand side of the above equality is positive, which yields that , contracting to . Thirdly, since the second equation of (4) is equivalent to the second equation of (3) and , it follows from the second equation of (3) that . Similarly, we can obtain that . The positivity of solutions is proved.
Finally, the boundedness of the solutions is shown. Since , we get that . This suggests that , , , are bounded on . Hence, the feasible region is a positive invariance that attracts all solutions of (4) in . The proof is completed.

Theorem 2. Model (4) admits a unique endemic equilibrium (EE) if , and there always exists a disease-free equilibrium (DFE) .

Proof. Model (4) always has a disease-free equilibrium , where . Applying the theory of the next generation matrix in [21], we derive that is the basic reproduction number of (4), which stands for the average number of new infections brought out by a typical infectious individual during the whole infectious period [22].
Let be any positive equilibrium (if it exists), then Solving the second and the fourth equations of (10) yields that From the first and the third equations of (10), eliminating leads to Denote . Because of , one has . Now, one turns to studying the following equation: By , we get that , . Applying gives whence one obtains when and is sufficiently small. One thus deduces that (13) admits a positive real root, denoted by . This also suggests that model (4) at least admits positive equilibrium from (11).
In fact, is proved to be a unique EE. From (13), it follows that . Due to , we can examine that and which leads to which implies that strictly decreases at any positive equilibrium . Note that is continuously differentiable on . Assume that (13) has more than one positive root; then there must exist certain one positive equilibrium such that , resulting in a contraction. Thus, is unique and so is if .

2.2. The Local Stability of Equilibria

In the following sections, one uses the notations and , .

Theorem 3. For model (4), the DFE is locally asymptotically stable if but unstable if . Moreover, the EE is locally asymptotically stable if .

Proof. The characteristic equation of model (4) at some equilibrium is calculated as where .
(1) By , evaluating (17) at yields Clearly, an eigenvalue of (18) is , and the remaining ones satisfy Suppose that . From (19), we directly get which means that (19) has at least one positive root. That is, is unstable if . Suppose that . Assign Using (19) yields . Let be any root of (19). If , one has contradicting with (21), and thus . So is locally asymptotically stable if .
(2) From (17), the characteristic equation of model (4) at reads Apparently, is an eigenvalue of (23). Furthermore, assume that (23) has another root with , and then From (15) and , we can deduce that Recall that the left side of (24) satisfies . This contradicts with (24). It follows from Theorem  9.17.4 in [23] that (23) does not admit any root with a nonnegative real part. So is locally asymptotically stable if . This completes the proof.

3. Permanence

In order to study the permanence of model (4), we first discuss its uniform persistence when by the persistence theory for infinite dimensional systems [24].

Definition 4. Denote as the interior of . If there is a constant independent of initial values in , such that , , , and , then model (4) is uniformly persistent in .

In the sequel, some notations and terminology are introduced. Denote , , as the family of solution operators with respect to (4). Consider with the uniform norm . Let us define the -limit set as . The semigroup is referred to as being asymptotically smooth, if for any bounded subset of , for which , there is a compact set such that as . Set It can be seen that , where represents the boundary of .

Lemma 5 (see [24], Theorem  4.2). Let the following conditions be satisfied:(i) is open and dense in with and .(ii)The solution operators satisfy , .(iii) is point dissipative in .(iv) is bounded in if is bounded in .(v) is asymptotically smooth.(vi) is isolated and has an acyclic covering , where is the global attractor of restricted to and , .(vii), holds, where denotes the stable set.Then is a uniform repeller with respect to ; that is, there exists a constant such that, , .

Theorem 6. Model (4) is permanent provided that .

Proof. From (26), one examines that (i) and (ii) clearly hold. And (iii)-(iv) immediately follow from Lemma 1. It is seen that (here, ) is isolated, which implies that the covering is simply . Since no orbit connects to itself in , we have that is acyclic. Thus, (vi) is checked out.
Now, we prove that , where . Assume by contradiction that there is a solution , such that From Lemma 1, we know that for . Choose a Lyapunov function From (27), it follows that there exists such that . And the time derivative of along the solutions of (4) reads By (27), we can use L’Hospital’s rule, yielding If , by (9), then the function is not decreasing when is large enough. For the given above , one thus gets for . Note that prevents from converging to as , This contradicts to . For the dissipative system (4), uniform persistence is equivalent to permanence, completing the proof.