An SIRS Model for Assessing Impact of Media Coverage

and Applied Analysis 3 a 3 = Aασ + Bμ 2 + Bμσ + Aμ (μ + σ + α + λ) > 0, a 1 a 2 − a 3 = σ(A + B) 2 + Aλσ + 5Aμσ + Aμλ + 4Bμσ + 5ABμ + Aμα + 4Bμ 2 + 6Aμ 2 + 2 ⋅ B 2 μ + 3A 2 μ + μσ 2 + 3σμ 2 + Bσ 2 + Aσ 2 + 2μ 3 + ABα + ABλ + Φ 2 α + A 2 λ > 0. (19) Thus, using Routh-Hurwitz criterion, all eigenvalues of J(E) have negative real parts which means E(S, I, R) is locally asymptotically stable. Theorem 3. If R 0 > 1, E(S, I, R) is globally asymptotically stable, provided that inequalities μ > σ and μ > λ hold true. In order to study the global stability of E(S, I, R), we use the geometrical approach which is developed in the papers of Smith [10] and Li and Muldowney [11]. We obtain simple sufficient conditions that E(S, I, R) is globally asymptotically stable when R 0 > 1. At first, we give a brief outline of this geometrical approach. Let x 󳨃→ f(x) ∈ R be a C function for x in an open set D ∈ R . Consider the differential equation x 󸀠 = f (x) . (20) Denote by x(t, x 0 ) the solution to (20) such that x(0, x 0 ). We make the following two assumptions. (i) There exists a compact absorbing set K ⊂ D. (ii) Equation (20) has a unique equilibrium x inD. The equilibrium x is said to be globally stable in D if it is locally stable and all trajectories inD converge to x. The following general global stability principle is established in [11]. Let x 󳨃→ P(x) be an ( n2 )×( n2 )matrix-valued function that is C for x ∈ D. Assume that P(x) exists and is continuous for x ∈ K, the compact absorbing set. A quantity q is defined as q = lim sup t→∞ sup x∈K 1


Introduction
Media coverage has an enormous impact on the spread and control of infectious diseases [1][2][3][4][5][6].The paper [7] considered that the evidence shows that, faced with lethal or novel pathogens, people will change their behavior to try to reduce their risk.
In [8], the authors studied the effect of media coverage on the spreading of disease by using the following model: where the authors proposed an  model with the general nonlinear contact function () =  1 −  2 () and  1 and  2 are positive constants.Here,  1 is the usual contact rate without considering the infective individuals and  2 is the maximum reduced contact rate due to the presence of the infected individuals.Everyone cannot avoid contact with others in every case so it is assumed  1 >  2 .When infective individuals appear in a region, people reduce their contact with others to avoid being infected when they are aware of the potential danger of being infected, and the more infective individuals being reported, the less contact the susceptible will make with others.Therefore, it is assumed that   () ≥ 0. The limited power of the infection due to contact is reflected by the saturating function lim  → ∞ () = 1.In summary, the functional () satisfies (0) = 0,   () ≥ 0, lim  → ∞ () = 1.
In this paper, using the same contact function as [8], we study an  model with media coverage.Let (), (), and () denote the number of susceptible individuals, infected individuals, and recovered individuals at time , respectively.The ordinary differential equation with nonnegative initial conditions is as follows: Here, all the variables and parameters of the model are nonnegative.Λ is the recruitment rate,  represents the natural death rate,  is the loss of constant immunity rate,  is the diseases induced constant death rate, and  is constant recovery rate.

Stability of the Disease-Free Equilibrium
Theorem 1.The disease-free equilibrium  0 is locally asymptotically stable for R 0 < 1 and unstable for R 0 > 1.
The Jacobian matrix at  * ( * ,  * ,  * ) is The characteristic polynomial of the matrix ( * ) is given by where Abstract and Applied Analysis 3 Thus, using Routh-Hurwitz criterion, all eigenvalues of ( * ) have negative real parts which means  * ( * ,  * ,  * ) is locally asymptotically stable.
) is globally asymptotically stable, provided that inequalities  >  and  >  hold true.
In order to study the global stability of  * ( * ,  * ,  * ), we use the geometrical approach which is developed in the papers of Smith [10] and Li and Muldowney [11].We obtain simple sufficient conditions that  * ( * ,  * ,  * ) is globally asymptotically stable when R 0 > 1.At first, we give a brief outline of this geometrical approach.
Let   → () ∈   be a  1 function for  in an open set  ∈   .Consider the differential equation Denote by (,  0 ) the solution to (20) such that (0,  0 ).We make the following two assumptions.
(i) There exists a compact absorbing set  ⊂ .
(ii) Equation (20) has a unique equilibrium  in .
The equilibrium  is said to be globally stable in  if it is locally stable and all trajectories in  converge to .
The following general global stability principle is established in [11].
Let   → () be an (  2 )×(  2 ) matrix-valued function that is  1 for  ∈ .Assume that  −1 () exists and is continuous for  ∈ , the compact absorbing set.A quantity  is defined as where and  [2] is the second additive compound matrix of the Jacobian matrix .The matrix   is obtained by replacing each entry   of  by its derivative in the direction of ,   , and () is the Lozinskiȋ measure of  with respect to a vector norm | ⋅ | in   (where  = (  2 )) defined by [12]  () = lim It is shown in [11] that, if  is simply connected, the condition  < 0 rules out the presence of any orbit that gives rise to a simple closed rectifiable curve that is invariant for (20), such as periodic orbits, homoclinic orbits, and heteroclinic cycles.As a consequence, the following global stability result is proved in Theorem 3.5 of [11].
Lemma 4. Assume that  is simply connected and that the assumptions (i) and (ii) hold.Then, the unique equilibrium  of (20) is globally asymptotically stable in  if  < 0.
We now apply Lemma 4 to prove Theorem 3.

Simulation Study and Discussion
To complement the mathematical analysis carried out in the previous section, using the Runge-Kutta method, we now investigate some numerical properties of (2).Choose () = /( + ),  > 0, and  reflects the reactive velocity of people and media coverage to the disease.Related parameter values are listed in Table 1. Figure 1(a) shows that, when R 0 = 2.941 > 1, the number of infected individuals is asymptotically stable, and the media coverage is beneficial to decrease the number of infected individuals.Figure 1(b) shows that, when R 0 = 0.029 < 1, the number of infected individuals tends to zero point, and the media coverage can quicken the extinction of infectious disease.
Furthermore, the analysis of the impact of related parameters on the infectious disease progression is fairly important.From the definition of R 0 , it can be seen that Hence, R 0 is an increasing function of Λ and is a decreasing function of .The mathematical results show that the basic reproduction number R 0 satisfies a threshold property.When R 0 < 1, it has been proved that the diseasefree equilibrium  0 is locally asymptotically stable, and the diseases will be eliminated from the community.And, when R 0 > 1, the unique endemic equilibrium  * is globally asymptotically stable, and the diseases persist.This shows that R 0 reduces to a value less than unity by reducing Λ or increasing , so as to control the spread of infectious diseases.
From Figure 2, we can find that the number of infected individuals decreases as the recruitment rate (Λ) decreases.Organized measures such as limitation of travel, closure of public places, or isolation are beneficial to lessen the recruitment rate to control the spreading of infectious diseases.Figure 3 reveals that the number of infected individuals decreases as the recovery rate () increases.So timely and effective treatment is regarded as one good method in managing infectious diseases.
Based on the obtained results, we can get that media coverage has an effective impact on the control and spread of infectious diseases.It is hoped that these control strategies we considered may offer some useful suggestions for authorities.In addition, we can generalize the current model by incorporating some control methods, such as isolation and treatment strategies.A more realistic model deserves to be considered.

Figure 1 :Figure 2 :
Figure 1: The tendency of the infected population varies.The solid line represents the case when  2 = 0.0018, and the dashed line represents the case when  2 = 0.

Figure 3 :
Figure 3: Variation of the number of infected under different .The solid line represents the case when  = 0.05, and the dashed line represents the case when  = 0.5.
122| and | 21 | being the matrix norm with respect to the  1 vector norm.More specifically,

Table 1 :
Parameters for the simulation.