Dynamics Analysis of an Epidemiological Model with Media Impact and Two Delays

This paper is concernedwith exploring the global dynamics of SEIR epidemicmodel withmedia 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.


Introduction
The investigation of dynamics of epidemiological models has been of importance in improving our understanding of disease control [1][2][3][4][5][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][3][4].In order to describe the impact of media on the diseases, Cui and his coauthors [3] used the transmission rate of the form  exp(−) 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  −  1 /( + ) to capture the impact of media on disease spread, where  1 ≤  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][16][17][18][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  −  1 ()/( + ()) compared to  exp(−()).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  1 and  2 , respectively.Hence, the probabilities P 1 () and P 2 () 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  1 and in temporarily recovered class for a constant period  2 .One further assumes that the disease has been in the population for at least a time of  >  fl max{ 1 ,  2 }.Following closely the ideas of [5,8,15,18] and incorporating media impact, we consider the following integrodifferential epidemic model with latency and relapse delays: Differentiating the second and the fourth equations of (3), we derive the delay model where the term  − 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.4) with the initial conditions ( 5) and ( 6) is unique, positive, and bounded on [0, +∞).Moreover, the biologically feasible region

The Equilibria
is a positive invariant with respect to (4).

Permanence
In order to study the permanence of model ( 4), we first discuss its uniform persistence when R 0 > 1 by the persistence theory for infinite dimensional systems [24].
Mathematical Problems in Engineering In the sequel, some notations and terminology are introduced.Denote Φ(),  ≥ 0, as the family of solution operators with respect to (4).Consider  ∈  with the uniform norm ‖‖.Let us define the -limit set as () fl { ∈  | there is a sequence   → ∞ as  → ∞ with lim  → ∞ Φ(  ) = }.The semigroup Φ() is referred to as being asymptotically smooth, if for any bounded subset U of , for which Φ()U ⊂  ∀ ≥ 0, there is a compact set A such that (Φ()U, A) → 0 as  → ∞.Set It can be seen that  0 = / 0 = , where  represents the boundary of .
In an epidemiological sense, uniform persistence of model ( 4) implies that there are always infectious individuals if the disease is initially present and R 0 > 1.

Global Stability
We are now in a position to study the global asymptotic behaviors of model ( 4).

Lemma 9.
The equilibrium  * of ( 33) is globally asymptotically stable in Proof.Construct the following Lyapunov function () =  1 () +  2 () +  3 (), where It follows from Theorem 6 that the variables  and  are sufficiently bounded and bounded away from 0. This ensures the boundedness of () for  ≥ 0, and thus () is well defined.Together with the properties of H(), we find that () ≥ 0 with global minimum 0 at  * .By  = ( * ,  * )+ * and  =  − 1 ( * ,  * )/ * + − 2 , differentiating  1 () along the solutions of (33) yields And the time derivatives of  2 () and  3 () along the solutions of (4), respectively, read The fact that  * is local stable when it exists, implies that it is also globally asymptotically stable in We finish the proof.
Additionally, since R 0 / 1 < 0 and R 0 / 2 < 0, it is an advantage for controlling the disease spread to increase both latent and relapse delays.In practice, the latent period may be hard to change, but the likelihood of symptomatic recurrence [14] and the frequency of subclinical (asymptomatic) viral shedding [13] can be substantially reduced under suppressive therapy rather than episodic treatment, such that the relapse period (delay) can be lengthened.
Note that media education does not change the basic reproduction number R 0 [4].However, the greater the reactive velocity of media coverage and individuals (i.e., the smaller the value of ), the endemic level  * will be controlled to a much lower level, seeing the blue lines shown in Figure 3 (where we change the values of  and  1 but keep the same initiate condition ((0), (0), (0), (0)) = (0.8, 0.05, 0.01, 0.05) and the reminding parameters values are the same with Figure 2).On the contrary, if media  departments and the public do not respond timely to the epidemic, the effect of media propaganda on the disease transmission is almost the same with the case with no media impact (i.e.,  1 = 0, seeing the red line in Figure 3).Hence, timely response of media coverage and individuals plays a more key role in controlling the epidemic.
and Key Scientic Research Project of Higher Education Institutions of Henan Province (16A110005).

Figure 2 :Figure 3 :
Figure 2: The EE  * of model (4) is globally asymptotically stable when R 0 = 1.5308,where  = 0.02,  1 = 0.01, and initiate conditions and the reminding parameters values are kept the same as Figure1.
indicates the individuals surviving in the latent period  1 and entering into infectious class at time  and the term  − 2 ( −  2 ) represents the individuals surviving in temporarily recovered period  2 and entering into infectious class at time .