Global dynamics for a Filippov system with media e ﬀ ects

: In the process of spreading infectious diseases, the media accelerates the dissemination of information, and people have a deeper understanding of the disease, which will signiﬁcantly change their behavior and reduce the disease transmission; it is very beneﬁcial for people to prevent and control diseases e ﬀ ectively. We propose a Filippov epidemic model with nonlinear incidence to describe media’s inﬂuence in the epidemic transmission process. Our proposed model extends existing models by introducing a threshold strategy to describe the e ﬀ ects of media coverage once the number of infected individuals exceeds a threshold. Meanwhile, we perform the stability of the equilibriua, boundary equilibrium bifurcation, and global dynamics. The system shows complex dynamical behaviors and eventually stabilizes at the equilibrium points of the subsystem or pseudo equilibrium. In addition, numerical simulation results show that choosing appropriate thresholds and control intensity can stop infectious disease outbreaks, and media coverage can reduce the burden of disease outbreaks and shorten the duration of disease eruptions.


Introduction
With the development of human civilization, infectious diseases have emerged gradually. For instance, in the mid-14th century, the Black Death killed 25 million Europeans, or a third of the total population of Europe at the time [1]. In 2003, the SARS epidemic swept through 32 countries and regions around the world, and infected 8422 people [2,3]. Regarding COVID-19 at the end of 2019, it took a great toll on humanity. As of August 20, 2021, according to the latest real-time statistics from the World Health Organization, the cumulative number of confirmed COVID-19 cases worldwide was approximately 200 million and the cumulative death toll was approximately 4 million [4][5][6]. Therefore, infectious diseases are an important research topic, that have attracted the attention of many scholars, who have been trying to establish and improve realistic mathematical models of infectious disease transmission dynamics to increase their knowledge and understanding of infectious diseases.
Based on epidemic transmission dynamics, researchers are increasingly focusing on the effect of social factors on disease transmission, such as vaccination, lifestyle, and media coverage. The impact of vaccination on disease control has been studied in the literature [7][8][9]. At the same time, mass media (internet, books, newspapers and others) can be effective as an important way to obtain information and deliver preventive health messages at the beginning of a disease outbreak. A large number of people can learn the disease, media coverage will have a profound psychological impact on people, which will greatly change their personal behaviors, and reduce the spread of the disease. It is very beneficial for people to actively prevent the disease and control it effectively [10,11]. Accordingly, the establishment of mathematical models related to media coverage and in-depth research are also of great practical importance for the prevention and control of infectious diseases. The timeliness of the media impact is also a very important issue in the control of epidemics. Therefore, to further identify the potential effects of media coverage on the spread of infectious diseases, with the aid of mathematical modeling methods, we explore and analyze the predicted spread of diseases to provide a rational qualitative description of disease transmission dynamics [12][13][14].
The media is regarded as an important factor in disease transmission, and a great number of scholars have conducted thorough research on this issue. In 2008, Cui and Sun [12] proposed a nonlinear transmission rate as β (I) = µe −mI , where µ is a positive number. When parameter m = 0, the transmission rate is linear, and when m > 0, this reflects the effect of media coverage on contact transmission. In 2008, Liu and Cui [13] proposed the SIR model with the transmission rate β (I) = β 1 − β 2 (mI+I) , which is a good response to the media's effect on the disease transmission process. Furthermore , the Filippov system provides a natural and reasonable framework for many realistic problems, and has been widely used in the process of various epidemics and the predator-prey relationship, particularly in controlling epidemics. When the number of infected individuals is less than a certain level, no measures are taken, and when the number of infected individuals is greater than a certain level, the media begins to report the disease. Thus, threshold strategies provide a natural description of such systems. For example, in 2012, Xiao et al., [14] proposed the SIR model with a threshold strategy using β (1 − f) S I to reflect the effect of the media. In 2015, Xiao et al., [15] proposed a Filippov model with the transmission rate β(I) = exp −M 1 (t) β 0 to reflect the impact of media coverage. Additionally, the impact of the media on infectious diseases has also been thoroughly explored [16,17].
Based on the aforementioned research, in this study, we present a Filippov epidemic model that relies on the number of infected individuals using a threshold strategy, and include a nonlinear incidence rate. Our proposed model extends the existing model by introducing a threshold strategy to describe the effect that is revealed by media coverage once the number of infected individuals exceeds a threshold. We implement an epidemiological model using adaptive switching behavior when the number of infected individuals exceeds a threshold. We also investigate which threshold levels can be used to guide the eradication of infectious diseases.
The remainder of the paper is organized as follows: In Section 2, we develop a Filippov epidemic model with nonliner incidence. In Section 3, we examine the dynamic behaviors of two subsystems: free system S 1 and control system S 2 . In Section 4, we present the sliding mode dynamics and identify the pseudo-equilibrium, and show that it exists under certain conditions and is asymptotically stable. In Section 5, we discuss the global dynamics analysis of the Filippov system. In Section 6, we perform boundary equilibrium bifurcation analysis. In Section 7, we investigate the effect of key parameters of the Filippov system. Finally, we present the conclusion and discussion in the last section.

Model establishment
In the classical model of infectious diseases, the transmission rate is bilinear. However, media reports, vaccination, and population density may directly or indirectly affect it, and the bilinear transmission rate function cannot adequately explain the complex phenomenon of epidemic transmission. Meanwhile, at the beginning of the disease outbreak, media can be effective as an important way to obtain information and deliver preventive health messages. A large number of people can learn about the disease that is prevalent, and simultaneously, media coverage will have a profound psychological impact on people, which will greatly change their personal behaviors, and reduce the spread and proliferation of the disease. It is very beneficial for people to actively prevent the disease and control it effectively; hence, we choose βS I 1+αI 2 as the transmission rate. It can be used to explain the "psychological" effect and show the effect of the media.
When the number of infected individuals reaches a critical level I 0 , the media makes relevant reports and people go out less to avoid being infected, which reduces the effective exposure and transmission rate. Meanwhile, the government implements control measures to reduce the spread of the disease. Based on the work in [14,15], we classify the population into three types, that is, susceptible, infected, and recovered, and establish a SIR model with a threshold strategy. Let S (t), I (t) and R (t) refer to the proportions of susceptible, infected, and removed individuals at the time t changes, respectively. Thus, the model is as follows: System (2.1) with (2.2) is a particular form of Filippov system, where a (constant) recruitment birth rate Λ is introduced in the susceptible population, β is the transmission rate, µ is the natural death rate, and γ is the recovered rate.
Because the recovered class R in system (2.1) does not affect the dynamics of the first and second equations, we only consider the first two equations of system (2.1) with (2.2) in the following, which are easily obtained as: (S , I) ∈ R 2 + |0 < S + I ≤ Λ/µ = Ω. For (S , I) ∈ Ω, dS /dt| S =0 > 0, dI/dt| I=0 = 0, and d (S + I)/dt| S +I=Λ/µ < 0. Thus, all solutions are in the Ω region; hence, Ω is an attractive domain. Let H (Z) = I − I 0 be a function with a threshold value that depends on the number of infected individuals, and ε be a segmentation function that depends on I − I 0 . For convenience, let the vector Z = (S , I) T . Then the discontinuous switching surface Σ can be defined as Σ = Z ∈ R 2 + |H (Z) = 0 , we call Σ the switching manifold, which is the separating boundary of the regions [18][19][20]. The two regions represent the free system (S 1 ) and the control system (S 2 ). (2.4) Hence, system (2.1) can be rewritten as the Filippov system using the following form: For convenience, we present the following definitions of all the types of equilibria of the Filippov system [21,22].
then Z * is called the pseudo-equilibrium point of the system. Definition 2.2. If Z ∈ G 1 and F 1 (Z) = 0 or Z ∈ G 2 and F 2 (Z) = 0, then Z is the real equilibrium point of system 2.1. If Z ∈ G 1 and F 2 (Z) = 0 or Z ∈ G 2 and F 1 (Z) = 0, then Z is the virtual equilibrium point of system 2.1.
Definition 2.3. If Z ∈ Σ and F 1 (Z) = 0 or F 2 (Z) = 0, then Z is called the boundary point of system 2.1. Moreover, if F i (Z) = 0 is invertible, we say that a boundary equilibrium bifurcation occurs at Z. These bifurcations are classified as the boundary focus bifurcation and boundary node bifurcation in [23][24][25].

Dynamics for the free system S 1
In this section, we analyze the stability of the equilibrium point in the free system and define a suitable Lyapunov function to verify the global stability. The free system has two equilibrium points: disease-free equilibrium and endemic equilibrium. The disease-free equilibrium point is E 0 = Λ µ , 0 and the endemic equilibrium point is .
3.2. Dynamics for the control system S 2 For the control system S 2 , the disease-free equilibrium point is easily obtained as E 2 = E 0 = Λ µ , 0 and the endemic equilibrium point is the solution of the following algebraic equation: Then, we can solve the equation with respect to I to obtain Because parameters µ, α, β, γ are non-negative, A > 0 and B > 0. When C < 0, that is, R 0 > 1, a unique positive solution exists; hence, we have the following summary: i) If R 0 < 1, then no positive equilibrium exists. ii) If R 0 > 1, then a unique positive equilibrium E 3 = (S 3 , I 3 ) exists, which is called the endemic equilibrium and given by β , Lemma 2. If R 0 < 1, then the disease-free equilibrium point E 2 in the system S 2 is globally asymptotically stable. Theorem 1. If R 0 > 1, then the endemic equilibrium point E 3 is locally asymptotically stable. proof. The Jacobian matrix of the control system S 2 at E 3 is the characteristic equation, for this matrix, becomes If det(M) > 0, tr(M) < 0, that is, R 0 > 1, then the endemic equilibrium point E 3 is a stable focus or node.

Sliding mode and its dynamics
In this section, we consider the sliding mode and its dynamics. First, we review the definition of the sliding mode segment, calculate the pseudo-equilibrium point, and provide sufficient conditions for the existence of the pseudo-equilibrium point. In mathematics, there are two approaches to determine sufficient conditions for the appearance of sliding mode on discontinuous surfaces: the Filippov convex method [28] and Utkin's equivalent control method [29]. Then, according to Definition 2.1, we determine the existence of the sliding mode domain of system 2.1.
Let β ; hence, the sliding domain of the Filippov system is as follows: Assuming that the second equation of the system (2.1) is equal to 0, we obtain βS I 1+εαI 2 = µI + γI. Substituting I = I 0 and βS I 1+εαI 2 = µI + γI into the first equation of system (2.1), we obtain Let f (S ) = 0. Thus, we obtain therefore, the coordinates of the possible pseudo-equilibrium point are given by E p = S p , I 0 . When . Thus, when S L ≤ S p ≤ S R , the pseudoequilibrium point E p is in sliding mode. Additionally, because f S p ≤ 0, the pseudo-equilibrium point E p is locally asymptotically stable. When S p > S R or S p < S L , the pseudo-equilibrium point E p is not in the sliding mode.
To analyze the relationship between the sliding domain and the attraction domain, we solve the following equation: It is easy to derive Regarding the bifurcation set phase diagram of the control intensity α − I 0 shown in Figure 1, we obtain the following: If I 0 < I 3 (i.e., region Γ 1 in Figure 1), the equilibrium point E 1 of the free system S 1 is virtual, which is denoted by E V 1 , and the equilibrium point E 3 of the control system S 2 is real, which is denoted by E R 3 . If I 3 < I 0 < I 1 (i.e., region Γ 2 ∪ Γ 3 in Figure 1), the endemic equilibria E 1 and E 3 of both system S 1 and system S 2 are virtual, which are denoted by E V 1 and E V 3 , respectively. At this moment, E V 1 , E V 3 , and E p coexist. When I 1 < I 0 (i.e., region Γ 4 ∪ Γ 5 ∪ Γ 6 in Figure 1), the endemic equilibrium point E 1 of the free system S 1 is real, which is denoted by E R 1 . The endemic equilibrium point E 3 of the control system S 2 is virtual, which is denoted by E V 3 (see Definition 2.2 for details).

Global behavior of the Filippov system
In the previous section, we studied the regular equilibrium, virtual equilibrium, and pseudoequilibrium. The pseudo-equilibrium point exists and is locally asymptotically stable under certain conditions. In this section, we discuss the dynamical behaviors of the Filippov system 2.1 with 2.2. We adjust the parameters and find that the system changed from stable to unstable because of the change of parameter β. The phase portrait of system 2.1 shows that S and I eventually tended to a stable value as time t changed, as shown in Figure 2. The time response of the states of system 2.1 for different initial values is shown in Figure 3 Next, we specifically describe the branching phase diagram in (Figure 1). If R 0 < 1, the disease is eradicated; hence, we only consider the dynamical behaviors of the system if R 0 > 1.
1) If I 0 < I 3 (region Γ 1 ), the endemic equilibrium point E 1 is virtual in the free system S 1 , which is denoted by E V 1 , and the endemic equilibrium point E 3 in the control system S 2 is real , which is denoted by E R 3 . The trajectories of different initial values eventually converge to the real equilibrium E R 3 , as shown in Figure 4(a),(b).
2) If I 3 < I 0 < I 1 (region Γ 2 ∪ Γ 3 ), the endemic equilibrium points E 1 and E 3 in the free system and control system are virtual denoted by E V 1 and E V 3 , respectively. In this case , the pseudo-equilibrium point exists and is locally asymptotically stable, as shown in Figure 4(c). The trajectories of different initial values eventually converge to the pseudo-equilibrium point E p , as shown in Figure 4(c),(d).
3) If I 1 < I 0 (region Γ 4 ∪ Γ 5 ∪ Γ 6 ), the endemic equilibrium point E 1 of the free system S 1 is the real equilibrium point, which is denoted by E R 1 , and the endemic equilibrium point E 3 of the control system S 2 is the virtual equilibrium point, which is denoted by E V 3 . The trajectories of different initial values eventually converge to the real equilibrium E R 1 , as shown in Figure 4(e),(f). To obtain the richer dynamical behaviors that the system may exhibit, we chose the parameters α = 0.6 and γ = 0.1, and all other parameters remained unchanged. The corresponding simulations are shown in Figure 5.
In Figure 5, for different initial values, our main results show that the system eventually stabilizes at the equilibrium points E 1 and E 3 , or the pseudo-equilibrium E p , which is strongly related to the threshold value I 0 . In particular, when we choose a sufficiently small threshold level I 0 (i.e., I 0 < I 3 ), the control policy is always triggered; hence, the solution of the system eventually tends to the endemic equilibrium or disease-free equilibrium (not labeled in the Figure), as shown in Figure 5(a),(b). Then, we choose the threshold I 0 as an intermediate value (i.e., I 3 < I 0 < I 1 ), the trajectories of different initial values eventually converge to the pseudo-equilibrium point E p , as shown in Figure 5(c). When we choose a sufficiently large threshold level I 0 (i.e., I 0 > I 1 ), the control strategy is not triggered, and the solution converges to the endemic equilibrium E R 1 of the subsystem at this moment, which depends on the individual parameter values, as shown in Figure 5

Boundary equilibrium bifurcation analysis
To demonstrate the boundary equilibrium bifurcation of system 2.1, we choose I 0 as the bifurcation parameter and fix the other parameters. The definitions of the boundary equilibrium point and tangent point are shown in the Definitions 2.3 and 2.4, respectively. The equations satisfied by the boundary equilibrium of system 2.1 are as follows: To ensure that the boundary equilibrium point exists, for both = 0 and = 1, it is necessary to satisfy which indicates that I 0 = I 1 (I 0 = I 3 ) for = 0 ( = 1). Thus, there are four possible boundary equilibria:    βS I 1 + αI 2 − µI − γI = 0, I = I 0 , by solving the above equation, we obtain the tangent point as When the threshold value I 0 passes a certain critical value, a boundary equilibrium bifurcation may occur if the real equilibrium, tangent point, and pseudo-equilibrium (or tangent point and real equilibrium) collide [14,23,30]. When the threshold value I 0 passes the first critical value I 0 = I 3 , the equilibrium E R 3 , pseudo-equilibrium point E p , and tangent point T 2 collide, which is denoted by E 1 B , as shown in Figure 6, where I 0 = 1.2122. In this case, the boundary equilibrium E 1 B is an attractor. When the threshold value I 0 passes the second critical value I 0 = I 1 , the equilibrium E R 1 , pseudo-equilibrium point E p , and tangent point T 1 collide, which is denoted by E 2 B , where I 0 = 1.8. In this case, the boundary equilibrium E 1 B is an attractor. Overall, there are two boundary equilibrium bifurcations in Figure 6. 7. The effect of the key parameters of the Filippov system For system 2.1, our aim is to find an effective strategy to adjust the infected population below a certain value or to be eliminated. It is important to implement the control strategy by setting an appropriate threshold I 0 . We choose control intensity α and recovery rate γ as key parameters and study how these parameters affect the dynamics of the system.
The red dashed line represents the free system and the blue solid line represents the control system affected by the media coverage. From Figure 7(a), it can be obtained that the media coverage makes the peak lower and delays the appearance of the peak. In Figure 7(b), we find that the peak reduces as the control intensity α increases. A similar trend appears in Figure 7(c), which means that media coverage and recovery treatment can be helpful in controlling infectious diseases.

Conclusions and discussion
In early epidemic models, incidence rates were bilinear, but bilinear incidence rates cannot explain the spread of epidemics very well under realistic conditions. Meanwhile, the Filippov system provides a natural and reasonable framework for many realistic problems and has been widely used in the process of various epidemics and the predator-prey relationship, particularly in controlling epidemics. Accordingly, we proposed a Filippov epidemic model with nonlinear incidence to describe the influence of the media on the epidemic transmission process. Our proposed model extends existing models by introducing a threshold strategy to describe the media effects.
When the number of infected individuals reaches or exceeds threshold I 0 , another adaptive system with a nonlinear incidence rate is used, at which time the disease is widely reported immediately and people become aware of the disease, and thus take certain protective measures to reduce the possibility of being infected. Thus, the number of new infections per day is reduced to a certain level. This shows that the typical threshold behavior is completely valid. First, we analyzed the stability of each equilibrium in system 2.1. Then, we used the theory of the Filippov system [28,31,32] to discuss the existence of sliding regions, the pseudo-equilibrium, and the real and virtual properties of each equilibrium of the system. Next, we studied the global stability of system 2.1 and boundary equilibrium bifurcation. Finally, we obtained biological conclusions from the theoretical and numerical simulation results of the system.
Our main results showed that the system eventually stabilized at the equilibrium points E 1 or E 3 or the pseudo-equilibrium E p , which was strongly related to the threshold value I 0 . Meanwhile, our findings indicate that by setting the appropriate threshold I 0 , the relevant health authorities can decide whether to intervene to effectively control the disease at a relatively low level. Simultaneously, the media's real-time coverage of the disease has had a psychological impact on humans that leads them to change their behavior, which results in a decrease in the number of infections. In Figure 7, we find that media coverage decreases the peak of the disease outbreak and delays its occurrence. Meanwhile, to a certain extent, the peak of the disease outbreak decreases with the increase of media coverage. This implies that media coverage is effective in controlling infectious diseases. We further demonstrated that media coverage is important for disease prevention and control.
Although we determined meaningful implications for disease control in this study, it still has some drawbacks. For example, we only considered the relationship between the number of infected individuals and a certain threshold to construct the switching condition. Actual disease-control strategies depend on more than the number of infected individuals. We did not take rapid growth rates into account, and this will be our next step in future work.