Modelling the Impact of Media in Controlling the Diseases with a Piecewise Transmission Rate

An epidemic model with media is proposed to describe the spread of infectious diseases in a given region. A piecewise continuous transmission rate is introduced to describe that the media has its effect when the number of the infected exceeds a certain critical level. Furthermore, it is assumed that the impact of the media on the contact transmission is described by an exponential function. Stability analysis of the model shows that the disease-free equilibrium is globally asymptotically stable if the basic reproduction number is less than unity. On the other hand, when the basic reproduction number is greater than unity, a unique endemic equilibrium exists, which is also globally asymptotically stable. Our analysis implies that media coverage plays an important role in controlling the spread of the disease.


Introduction
When an infectious disease outbreaks, the spread and the control of the disease will be reported by the media including television programs, newspapers, and online social networks.Many examples are the massive reports and daily updates in the public media on the number of the infections and deaths, which had important impacts on the diseases control [1,2].It is shown that media coverage plays an important role in the spread and control of the infectious disease, such as the SARS in 2003 [3], the H1N1 in 2009 influenza epidemic [4,5], and the Ebola in 2014 in Africa [6].
Recently, such impact on disease spreading and controlling has been investigated by mathematical modeling approach [5,[7][8][9][10][11][12][13][14][15][16][17][18][19][20].In these research works some focused their attention on the incidence rate, and the recent survey identified three typical terms in [5].Liu et al. [7] described the impact of media coverage using the transmission coefficient  − 1 − 2 − 3  , and this impact leads to the change of avoidance and contact patterns at both individual and community levels.Cui et al. [8][9][10][11] developed a compartment model using incidence rate  −  with  > 0 to investigate the impact of media coverage on the transmission, and stability analysis of the models has shown that Hopf bifurcation can occur.Tchuenche et al. [14] used an exponentially decreasing function  −() to reveal the force of media and showed the potential short-term beneficial effect of awareness programs.
More recently nonsmooth media functions [21][22][23][24][25] have been studied.Xiao et al. [21] introduced a segmented function to describe the media impact  −  , here  is also the strength of the media effect, and   is a threshold that people take the controlling measures or not.A Filippov epidemic model was proposed to describe the real characteristics of media impact in the spread of infectious diseases by incorporating a piecewise continuous transmission rate  (−)  in [24], and mathematical analysis with regard to the local and global stability of equilibria and local sliding bifurcations are performed.
In fact, during infectious disease outbreaks, individuals may reduce their activities after receiving information about the risk of infection.For example, people will reduce the time that they go out, students will not attend school, and so on, and such information on the ongoing epidemics may impact the dynamics itself.In fact, at the initial stages of the prevalence of disease, most people and public mass media are unaware of the disease; thus, the individuals will not do any protective measures.Only when the number of infectious individuals reaches and exceeds a certain level, the individuals will take precautionary measures against the diseases.Based on the above facts, in this paper we also focus on the incidence rate.Here we introduce () =  −   to show this fact, where   is the threshold value.It is assumed that the impact of media is described by an exponential decreasing factor and the population obeys the logistic growth.
The rest of this paper is organized as follows: in the next section, a mathematical model is proposed in order to reveal the effect of media; then the existence of equilibria is given in Section 3. In Section 4 the local and global stability of the disease-free and the unique endemic equilibria are analyzed.Furthermore, in Section 5 some numerical simulations and discussions are given in the last section.

Mathematical Model with Media
In this model the population is divided into three types: the susceptible, the infective and the recovered.Let (), (), and () denote the number of susceptible, infective, and recovered individuals at time , respectively.It is assumed that the growth of the susceptible population obeys the logistic growth, the intrinsic growth rate of the population is , and the carrying capacity for the population is .The interactions between susceptible and infective individuals are assumed to be bilinear, and  is the contact rate of susceptible with infective individuals,  is the recovery rate of the infective individuals, and  is the natural death rate of the population.Consider that the cumulative density of media about the disease is ().The growth rate of cumulative density of media, , is assumed to be proportional to the number of infective individuals in the population, and ] is the depletion rate of cumulative density of media.Thus, we have the following model: In this model we assume that the impact of media is described by an exponential decreasing factor as  =  0  − , where the parameter  > 0 is to reflect the impact of media.When  = 0, the transmission rate is the constant  0 .When  = 1, as one can see if  is relatively small, the transmission rate  is close to the constant  0 .Thus, the model can be rewritten as follows: with where the function () =  −   means that when infective individuals reach and exceed the certain level   , the media has its effects.
For the solutions of system ( 2) with (3), the region of attraction is given by the set: and it attracts all solutions initiating in the interior of the positive orthant.The (, , , ) phase space is split into two parts: In region  1 , there is no effect of the media and the transmission rate  =  0 , while in region  2 , the transmission rate declines to  0  − .
For convenience, let the vector  = (, , )  , and denote Then system (2) with (3) can be rewritten as the following switching system: In the following, system (6) defined in region  1 is the subsystem  1 and that defined in region  2 is the subsystem  2 , and we consider the two subsystems in isolation firstly.

Stability of Equilibria
In this section we present the locally and globally asymptotical stability of the equilibria of subsystem  1 and subsystem  2 .
Proof.The characteristic equation corresponding to the subsystem  1 at  2 0 is the form where  is the eigenvalue.We get Thus,  2 0 is locally asymptotically stable if  0 < 1 and unstable when  0 > 1.The proof of the disease-free equilibrium of the subsystem  2 is similar to  1 ; then this proof is omitted.
Proof.To establish the global stability of the disease-free equilibrium of subsystem  2 , we consider the positive definite function () = () by using Lyapunov's method.Now differentiating  with respect to , we get If  0 < 1, then / ≤ 0. Thus, LaSalle's invariance principle implies that  2 0 is globally asymptotically stable in Ω.We can use the same positive definite function when we prove the global stability of the disease-free equilibrium of subsystem  1 .
In fact, when  0 < 1 curves starting from  2 will converge to (, 0, 0) that belongs to  1 , and curves starting from  1 also converge to (, 0, 0); therefore,  2 0 = (, 0, 0) is globally asymptotically stable.In the following we consider the stability of the endemic equilibrium  * 0 of subsystem  1 and  * of subsystem  2 .According to Hurwitz criterion, it is easy to get the following proposition.Proposition 4. The unique endemic equilibrium  * 0 of subsystem  1 is locally asymptotically stable if  0 > 1.The unique endemic equilibrium  * of subsystem  2 is locally asymptotically stable if  0 > 1.When the endemic equilibrium  * 0 belongs to  1 , we get 1 <  0 < /( −  0   ) from the expression of  * 0 , so if  0 > /( −  0   ),  * 0 belongs to  2 , which lead to the existence of  * that belongs to  2 .Let   = /( −  0   ); then we have the following result.
and we make  1 = (1 − /) −  0  − ,  2 =  0  −  − ( + ), and  3 =  − ] for the subsystem  2 .Then Hence, we can exclude the existence of limit cycles in region  2 .The same method can be applied in discussion for region  1 .When 1 <  0 <   , it is easy to get the endemic equilibrium  * 0 that belongs to  1 and all curves starting from  1 will verge to  * 0 .We know that all curves starting from  2 will converge to  * , but  * also belongs to  1 if 1 <  0 <   , and there is a unique endemic equilibrium and no limit cycles in region  1 ; hence, curves starting from  2 will enter  1 and converge to  * 0 .Thus, when 1 <  0 <   the endemic equilibrium  * 0 is globally asymptotically stable.Similarly when  0 >   the endemic equilibrium  * 0 belongs to  2 ; hence, curves starting from  1 will enter  2 and converge to the unique endemic equilibrium  * in region  2 .So when  0 >   the endemic equilibrium  * is globally asymptotically stable.

Numerical Simulations and Results
To check the analysis in Section 4 about  * 0 and  * , we do some numerical simulations in this section.
In the following we consider the impact of media coverage on the spread of diseases by drawing the curves between (), (), and , respectively, as shown in Figure 2. It is shown that the number of () reduces with the increase of , which is opposite to ().This is common sense that media coverage influences people's behavior and reminds them to prevent themselves from the diseases through staying at home, reducing travel or quarantine, and so on.So it is clear that media coverage plays a key role in the preventing and controlling the diseases.

Discussion
In this paper, a nonlinear mathematical model has been proposed and analyzed to describe the impact of media coverage on the transmission dynamics of infectious diseases.Considering the fact that public mass media usually do not work when the number of the infected individuals is generally small at the initial stages of prevalence of the disease, and by incorporating a piecewise continuous transmission rate to represent that media coverage has its effects only when  the number of the infected individuals exceeds a certain level.The disease-free equilibrium has been shown to be globally asymptotically stable when the basic reproduction number  0 < 1.When  0 > 1 and 0 <  <  0 sufficiently small, it leads to the global stability of endemic equilibrium.But when 1 <  0 <   the trajectories all approach towards  * 0 whenever initial starts ∈  1 or  2 , and the trajectories approach towards  * if  0 >   .Numerical simulations prove the stability of equilibria and the impact of media coverage in the spread of diseases, which suggest that if we want to reduce the number of the infected individuals, we should increase media coverage.

Figure 1 :
Figure 1: The stability of the endemic equilibria  * 0 and  * .