
We consider a volume constraint problem for the nonlocal doubly nonlinear parabolic equation, called the nonlocal p-Sobolev flow, and introduce a nonlinear intrinsic scaling, converting a prototype nonlocal doubly nonlinear parabolic equation into the nonlocal p-Sobolev flow. This paper is dedicated to Giuseppe Mingione on the occasion of his 50th birthday, who is a maestro in the regularity theory of PDEs.
Citation: Masashi Misawa, Kenta Nakamura, Yoshihiko Yamaura. A volume constraint problem for the nonlocal doubly nonlinear parabolic equation[J]. Mathematics in Engineering, 2023, 5(6): 1-26. doi: 10.3934/mine.2023098
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We consider a volume constraint problem for the nonlocal doubly nonlinear parabolic equation, called the nonlocal p-Sobolev flow, and introduce a nonlinear intrinsic scaling, converting a prototype nonlocal doubly nonlinear parabolic equation into the nonlocal p-Sobolev flow. This paper is dedicated to Giuseppe Mingione on the occasion of his 50th birthday, who is a maestro in the regularity theory of PDEs.
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)=β1−(β2(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)SI to reflect the effect of the media. In 2015, Xiao et al., [15] proposed a Filippov model with the transmission rate β(I)=exp−M1(t)β0 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 S1 and control system S2. 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.
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 βSI1+αI2 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 I0, 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:
{dSdt=Λ−βSI1+εαI2−μS,dIdt=βSI1+εαI2−μI−γI,dRdt=γI−μR, | (2.1) |
with
ε={0,I<I0,1,I>I0. | (2.2) |
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)∈R2+|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−I0 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−I0. For convenience, let the vector Z=(S,I)T. Then the discontinuous switching surface Σ can be defined as
Σ={Z∈R2+|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 (S1) and the control system (S2).
G1={Z∈R2+|H(Z)<0}andG2={Z∈R2+|H(Z)>0}. |
Let
F1(Z)=[F11F12]=[Λ−βSI−μSβSI−μI−γI], | (2.3) |
F2(Z)=[F21F22]=[Λ−βSI1+αI2−μSβSI1+αI2−μI−γI]. | (2.4) |
Hence, system (2.1) can be rewritten as the Filippov system using the following form:
˙Z={F1(Z),Z∈G1,F2(Z),Z∈G2. | (2.5) |
For convenience, we present the following definitions of all the types of equilibria of the Filippov system [21,22].
Definition 2.1. If the equilibrium point Z∗ of the sliding line region Σs of system 2.1 satisfies λF1(Z∗)+(1−λ)F2(Z∗)=0,H(Z∗)=0 with 0<λ<1, where
λ=⟨HZ(Z∗),F2(Z∗)⟩⟨HZ(Z∗),F2(Z∗)−F1(Z∗)⟩, |
then Z∗ is called the pseudo-equilibrium point of the system.
Definition 2.2. If Z∈G1 and F1(Z)=0 or Z∈G2 and F2(Z)=0, then Z is the real equilibrium point of system 2.1. If Z∈G1 and F2(Z)=0 or Z∈G2 and F1(Z)=0, then Z is the virtual equilibrium point of system 2.1.
Definition 2.3. If Z∈Σ and F1(Z)=0 or F2(Z)=0, then Z is called the boundary point of system 2.1.
Moreover, if Fi(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].
Definition 2.4. If Z∈Σ and HF1(Z)=0 or HF2(Z)=0, then Z is called the tangent point of system 2.1.
Before analyzing the complete switching system, it is first necessary to determine the dynamical behaviors of the two subsystems. According to the method in [26,27], the basic reproduction number is considered as R0=βΛ(μ+γ)μ in system S1. Meanwhile, according to this equation μα(μ+γ)I2+(μ+γ)βI+μ(μ+γ)−Λβ=0, we obtain the basic reproduction number Rc=βΛ(μ+γ)μ in system S2. Therefore, it is obvious that R0=Rc=βΛ(μ+γ)μ.
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 E0=(Λμ,0) and the endemic equilibrium point is E1=(μ+γβ,Λβ−(μ2+μγ)β(μ+γ)).
Lemma 1. The disease-free equilibrium point E0=(Λμ,0) in system S1 is globally asymptotically stable if R0<1 and unstable if R0>1, whereas the endemic equilibrium point E1=(S1,I1) is globally asymptotically stable if R0>1.
For the control system S2, the disease-free equilibrium point is easily obtained as E2=E0=(Λμ,0) and the endemic equilibrium point is the solution of the following algebraic equation:
{Λ−βSI1+αI2−μS=0,βSI1+αI2−μI−γI=0, |
the equation for I is obtained as
AI2+BI+C=0, |
where
A=μα(μ+γ), |
B=(μ+γ)β, |
C=μ(μ+γ)[1−R0]. |
Then, we can solve the equation with respect to I to obtain
Ii=−B±√Δ2A(i=3,4), |
where
Δ=(μβ+γβ)2−4α(μγ+μ2)2[1−R0]. |
Because parameters μ,α,β,γ are non-negative, A>0 and B>0. When C<0, that is, R0>1, a unique positive solution
I3=−B+√Δ2A>0 |
exists; hence, we have the following summary:
ⅰ) If R0<1, then no positive equilibrium exists.
ⅱ) If R0>1, then a unique positive equilibrium E3=(S3,I3) exists, which is called the endemic equilibrium and given by
S3=(μ+γ)(1+αI23)β, |
I3=−(μβ+γβ)+√Δ2μα(μ+γ). |
Lemma 2. If R0<1, then the disease-free equilibrium point E2 in the system S2 is globally asymptotically stable.
Theorem 1. If R0>1, then the endemic equilibrium point E3 is locally asymptotically stable.
proof. The Jacobian matrix of the control system S2 at E3 is
M=(−βI31+αI23−μ−βS31+αI23+2αβS3I23(1+αI23)2βI31+αI23βS31+αI23−2αβS3I23(1+αI23)2−μ−γ), |
the characteristic equation, for this matrix, becomes
λ2−tr(M)λ+det(M)=0, |
where
det(M)=(βI31+αI23+μ)(μ+γ)+μβS3(αI23−1)(1+αI23)2, |
tr(M)=−βI31+αI23−2μ+βS31+αI23−2αβS3I23(1+αI23)2−γ. |
If det(M)>0, tr(M)<0, that is, R0>1, then the endemic equilibrium point E3 is a stable focus or node.
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
σ(Z)=⟨HZ(Z),F1(Z)⟩⟨HZ(Z),F2(Z)⟩, |
where ⟨⋅,⋅⟩ denotes the standard scalar product and HZ(Z) is the invariant gradient of the smooth scalar function H(Z) in Σ. The sliding domain is defined as Σs={Z∈Σ|σ(Z)<0}, and H(I)=I−I0, which means that HZ(Z)=(0,1)T. Since F1(Z)=(F11(Z),F12(Z))T, F2(Z)=(F21(Z),F22(Z))T. For more information about the Filippov system, readers can refer to [30,31,32].
When
σ(Z)=⟨HZ(Z),F1(Z)⟩⟨HZ(Z),F2(Z)⟩=F12(Z)F22(Z)≤0, |
then it's obvious that F12(Z)>F22(Z). Hence, we obtain
Σs={(S,I)∈Σ|F12(S,I)≥0,F22(S,I)≤0}. |
Let the endpoints of the slide segment be A(SL,I0),B(SR,I0), where SL=μ+γβ, SR=(μ+γ)(1+αI20)β; hence, the sliding domain of the Filippov system is as follows:
Σs={(S,I)∈R2+|SL≤S≤SR,I=I0}. |
Assuming that the second equation of the system (2.1) is equal to 0, we obtain βSI1+εαI2=μI+γI. Substituting I=I0 and βSI1+εαI2=μI+γI into the first equation of system (2.1), we obtain
dSdt=Λ−μI0−μS−γI0=f(S). |
Let f(S)=0. Thus, we obtain
Sp=Λ−μI0−γI0μ, |
therefore, the coordinates of the possible pseudo-equilibrium point are given by Ep=(Sp,I0).
When
Sp−SL=Λ−μI0−γI0μ−(μ+γ)β=Λβ−(μ+γ)βI0−μ(μ+γ)μβ>0, |
Sp−SR=Λ−μI0−γI0μ−(μ+γ)(1+αI20)β=Λβ−(μ+γ)βI0−μ(μ+γ)(1+αI20)μβ<0, |
(i.e., (μ+γ)(βI0+μ)<Λβ<(μ+γ)[βI0+μ(1+αI2)]). Thus, when SL≤Sp≤SR, the pseudo-equilibrium point Ep is in sliding mode. Additionally, because f(Sp)′≤0, the pseudo-equilibrium point Ep is locally asymptotically stable. When Sp>SR or Sp<SL, the pseudo-equilibrium point Ep is not in the sliding mode.
To analyze the relationship between the sliding domain and the attraction domain, we solve the following equation:
μ+γβ=Λμ−Iand(μ+γ)(1+αI2)β=Λμ−I. |
It is easy to derive
I10=Λμ−μ+γβ,I30=−μβ+√Δ2μα(μ+γ), |
where
Δ=(μβ)2−4μα(μ+γ)[μ(μ+γ)−Λβ]. |
Regarding the bifurcation set phase diagram of the control intensity α−I0 shown in Figure 1, we obtain the following: If I0<I3 (i.e., region Γ1 in Figure 1), the equilibrium point E1 of the free system S1 is virtual, which is denoted by EV1, and the equilibrium point E3 of the control system S2 is real, which is denoted by ER3. If I3<I0<I1 (i.e., region Γ2∪Γ3 in Figure 1), the endemic equilibria E1 and E3 of both system S1 and system S2 are virtual, which are denoted by EV1 and EV3, respectively. At this moment, EV1, EV3, and Ep coexist. When I1<I0 (i.e., region Γ4∪Γ5∪Γ6 in Figure 1), the endemic equilibrium point E1 of the free system S1 is real, which is denoted by ER1. The endemic equilibrium point E3 of the control system S2 is virtual, which is denoted by EV3 (see Definition 2.2 for details).
In the previous section, we studied the regular equilibrium, virtual equilibrium, and pseudo-equilibrium. 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(a), (b). The S-I phase portrait presents the periodic solution, as shown in Figure 3(c).
Next, we specifically describe the branching phase diagram in (Figure 1). If R0<1, the disease is eradicated; hence, we only consider the dynamical behaviors of the system if R0>1.
1) If I0<I3 (region Γ1), the endemic equilibrium point E1 is virtual in the free system S1, which is denoted by EV1, and the endemic equilibrium point E3 in the control system S2 is real, which is denoted by ER3. The trajectories of different initial values eventually converge to the real equilibrium ER3, as shown in Figure 4(a), (b).
2) If I3<I0<I1 (region Γ2∪Γ3), the endemic equilibrium points E1 and E3 in the free system and control system are virtual denoted by EV1 and EV3, 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 Ep, as shown in Figure 4(c), (d).
3) If I1<I0 (region Γ4∪Γ5∪Γ6), the endemic equilibrium point E1 of the free system S1 is the real equilibrium point, which is denoted by ER1, and the endemic equilibrium point E3 of the control system S2 is the virtual equilibrium point, which is denoted by EV3. The trajectories of different initial values eventually converge to the real equilibrium ER1, 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 E1 and E3, or the pseudo-equilibrium Ep, which is strongly related to the threshold value I0. In particular, when we choose a sufficiently small threshold level I0 (i.e., I0<I3), 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 I0 as an intermediate value (i.e., I3<I0<I1), the trajectories of different initial values eventually converge to the pseudo-equilibrium point Ep, as shown in Figure 5(c). When we choose a sufficiently large threshold level I0 (i.e., I0>I1), the control strategy is not triggered, and the solution converges to the endemic equilibrium ER1 of the subsystem at this moment, which depends on the individual parameter values, as shown in Figure 5(d). Clearly, the trajectories are influenced by the isoclines and the switching line in Figures 4 and 5.
To demonstrate the boundary equilibrium bifurcation of system 2.1, we choose I0 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:
∧−βSI1+ϵαI2−μS=0,βSI1+ϵαI2−μI−γI=0,I=I0. |
To ensure that the boundary equilibrium point exists, for both ϵ=0 and ϵ=1, it is necessary to satisfy
Λ(1+ϵαI20)βI0+μ(1+ϵαI20)=(μ+γ)(1+ϵαI20)β, |
which indicates that I0=I1 (I0=I3) for ϵ=0 (ϵ=1). Thus, there are four possible boundary equilibria:
E11b=(ΛβI1+μ,I1),E21b=(Λ(1+αI21)βI1+μ(1+αI21),I1), |
E12b=(ΛβI3+μ,I3),E22b=(Λ(1+αI23)βI3+μ(1+αI23),I3). |
The equation satisfied by the tangent point is as follows:
βSI1+ϵαI2−μI−γI=0,I=I0, |
by solving the above equation, we obtain the tangent point as
T1=(μ+γβ,I0),T2=((μ+γ)(1+αI20)β,I0). |
When the threshold value I0 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 I0 passes the first critical value I0=I3, the equilibrium ER3, pseudo-equilibrium point Ep, and tangent point T2 collide, which is denoted by E1B, as shown in Figure 6, where I0=1.2122. In this case, the boundary equilibrium E1B is an attractor. When the threshold value I0 passes the second critical value I0=I1, the equilibrium ER1, pseudo-equilibrium point Ep, and tangent point T1 collide, which is denoted by E2B, where I0=1.8. In this case, the boundary equilibrium E1B is an attractor. Overall, there are two boundary equilibrium bifurcations in Figure 6.
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 I0. 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.
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 I0, 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 E1 or E3 or the pseudo-equilibrium Ep, which was strongly related to the threshold value I0. Meanwhile, our findings indicate that by setting the appropriate threshold I0, 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.
This work was supported by The National Natural Science Foundation of China (Grant No.11961024).
The authors declare no conflict of interest.
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