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Research article

How can carbon markets drive the development of renewable energy sector? Empirical evidence from China

  • Received: 06 March 2024 Revised: 22 March 2024 Accepted: 10 April 2024 Published: 08 May 2024
  • JEL Codes: C1, C12, C15, C16, G1, G18

  • The reduction of carbon emissions has attracted significant global attention. This paper empirically analyzes the dynamic nonlinear linkages among carbon markets, green bonds, clean energy, and electricity markets by constructing DCC-GARCH and TVP-VAR-SV models, and places the four markets under a unified framework to analyze the volatility risk from a time-varying perspective, thereby enriching the research on China's carbon market and renewable energy sector. We found that extreme events have a significant impact on the dynamic connectivity among the four markets. The analysis of the shock impact indicates that the carbon market has a positive effect on the power market in the short and medium terms, but has a mitigating impact in the long term. Especially, when the other markets are hit, the carbon market has evident fluctuation in 2020. The green bond market has a positive influence on the carbon market, whereas the power market demonstrates adverse effects in the short and medium terms. The New Energy Index negatively impacts the power market in the short and medium terms, but is expected to have a positive effect after 2020, highlighting the growing need for renewable energy in the power system transformation. According to the findings mentioned above, we put forward appropriate recommendations.

    Citation: Jiamin Cheng, Yuanying Jiang. How can carbon markets drive the development of renewable energy sector? Empirical evidence from China[J]. Data Science in Finance and Economics, 2024, 4(2): 249-269. doi: 10.3934/DSFE.2024010

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  • The reduction of carbon emissions has attracted significant global attention. This paper empirically analyzes the dynamic nonlinear linkages among carbon markets, green bonds, clean energy, and electricity markets by constructing DCC-GARCH and TVP-VAR-SV models, and places the four markets under a unified framework to analyze the volatility risk from a time-varying perspective, thereby enriching the research on China's carbon market and renewable energy sector. We found that extreme events have a significant impact on the dynamic connectivity among the four markets. The analysis of the shock impact indicates that the carbon market has a positive effect on the power market in the short and medium terms, but has a mitigating impact in the long term. Especially, when the other markets are hit, the carbon market has evident fluctuation in 2020. The green bond market has a positive influence on the carbon market, whereas the power market demonstrates adverse effects in the short and medium terms. The New Energy Index negatively impacts the power market in the short and medium terms, but is expected to have a positive effect after 2020, highlighting the growing need for renewable energy in the power system transformation. According to the findings mentioned above, we put forward appropriate recommendations.



    Human immunodeficiency virus (HIV) takes the most important CD4+T lymphocytes in the human immune system as the main target, destroys people's CD4+T and makes the body lose its immune function [1]. HIV is a major problem facing the human race and poses a serious health threat to human society. While there has been remarkable advancement in the development of antiretroviral therapy (ART) and prevention strategies, currently there are still many people living with HIV [2]. From the National Health Commission of the People's Republic of China, to the end of 2020, a total of 1.053 million people in China were infected with HIV, 351,000 deaths were reported, and the number of infections is expected to increase to 1.6 million in 2022. Therefore, it is beneficial to study the effective control measures and find the optimal control by using mathematical models.

    Early HIV models were primarily devoted to the virus-to-cell infection [3,4,5,6,7,8,9,10]. With the development of science and the improvement of medical standards, some studies show that virus can also spread by direct cell-to-cell transmission [11,12,13,14,15]. Thus, many researchers have begun working on an HIV model incorporating virus-to-cell and cell-to-cell transmission [16,17,18,19,20]. For example, by an HIV model, Wang et al. [16] showed the existence, positivity and boundedness of the model solution. Lai et al. [19] demonstrated global threshold dynamics by the basic reproduction number. For the control of acquired immune deficiency syndrome (AIDS), the main drug treatment is to prevent new HIV infections by blocking the transformation of viral RNA into DNA in T cells and reducing the number of viral particles [21]. With regard to AIDS control, there have already been some results [22,23,24]. Liu et al. [22], Akbari et al.[23] and Guo et al. [24] proposed an optimal control problem for an HIV infection model with cell-to-cell spread. However, these studies were focused on optimal control given over the entire infection period T, which will increase the cost of control measures. If threshold level Nt can be introduced, such that the control measures are not implemented when the total number of infected cells in an infected person is at a relatively low level (threshold level Nt), while otherwise effective measures are taken to suppress progression of viruses and infected cells, the time and cost will be reduced. The idea has been widely used in engineering [25,26], but little research has been done in epidemic diseases.

    In addition, for the case that the number of infected cells in the body exceeds the threshold level (Nt), from the perspectives of epidemiology and economics, how to control the spread of virus is a valuable question. As indicated by discussion above, in this paper, we analyze the sliding mode dynamics and optimal control of the HIV model with virus-to-cell and cell-to-cell transmission. Here, because the timing of controlling infected cells and viruses is uncertain, dynamic programming is considered, which demonstrates that not only can infected cells and viruses be controlled in time, but also the goal of minimizing the concentration of infected cells and viruses with a low cost of application control can be achieved. The main novelties are summarized as follows:

    ● About the HIV model with cell-to-cell transmission, the majority of the existing results only discussed dynamics, this article proposes a piecewise control function concerning threshold policy and discusses sliding mode dynamics.

    ● Differently from previous works on optimal control over the time period [0,T] for the HIV model, in this paper, optimal control strategies for infected cells and viruses are achieved when the total number of infected cells in the body exceeds the certain tolerance threshold level (Nt). Moreover, our results generalize and improve some published results in the literature, such as [23,24].

    The whole organization of this work is as follows. Section 2 describes the different components of the HIV model, and it then further extends a new three-dimensional Filippov model with two control measures. Section 3 investigates the sliding mode dynamics of the model and shows the existence of a unique positive pseudo-equilibrium. In Section 4, the optimal control problem is discussed. We first give the objective function and prove the uniqueness and existence of the viscosity solution of the HJB equation, and we then obtain the optimal control through the Hamiltonian function. The theoretical results are verified by numerical simulations in Section 5. Finally, conclusions and outlook for further work are given in Section 6.

    Inspired by [17,19], the classical HIV model with virus-to-cell and cell-to-cell transmission is

    {dx(t)dt=Λβx(t)v(t)γx(t)y(t)ax(t),dy(t)dt=βx(t)v(t)+γx(t)y(t)by(t),dv(t)dt=ky(t)cv(t), (2.1)

    where x(t) denotes the concentration of uninfected target cells at time t, y(t) is the concentration of infected cells at time t, and v(t) denotes the concentration of virus particles at time t. Λ is the recruitment rate of healthy target cells, β is the rate at which an uninfected cell becomes infected by an infectious virus, γ represents the infection rate of productively infected cells, k shows the generation rate of virus particles, a expresses the loss rate of infected cells, b represents the natural death rate of uninfected cells, and c indicates the clearance rate of virions.

    Based on the existing HIV models, we give the sliding control system to maintain the number of viruses and infected cells below the threshold level. Research [27] has found that current drug treatment consisting of five antiretroviral drugs can suppress viral replication to a low level or increase the CD4+T cell, the two main types of HIV resistance: reverse transcriptase inhibitors (RTIs) and protease inhibitors (PIs) [28,29]. RTIs prevent new HIV infections by blocking the transformation of viral RNA into DNA in T cells, and PIs reduce the number of viral particles produced by actively infected T cells [21]. We represent by μ1 the RTIs control variable and by μ2 the PIs control variable. The control system is given as follows:

    {dx(t)dt=Λβx(t)v(t)γx(t)y(t)ax(t),dy(t)dt=βx(t)v(t)+γx(t)y(t)by(t)εμ1y(t),dv(t)dt=ky(t)cv(t)εμ2v(t), (2.2)

    with

    ε={0,y(t)Nt<0,1,y(t)Nt>0. (2.3)

    μ1 is the culling rate of infected cells, μ2 is the clearance rate of virus particles, and they are constants. The critical level of the total number of infected cells is represented by Nt. For convenience, y(t)Nt is defined as ω(M)=y(t)Nt with vector M=(x(t),y(t),v(t))R3+ and R3+={M=(x,y,v)|x0,y0,v0}.

    Remark 2.1. When ε=0, model (2.2) becomes model (2.1) and the control measures are taken when ε=1.

    In this section, according to system (2.2) with (2.3), we study the dynamics of the system. First, we define G1={MR3+|ω(M)<0}, G2={MR3+|ω(M)>0}. Furthermore, we describe the manifold Gs as Gs={MR3+|ω(M)=0} and the normal vector perpendicular to Gs is shown as n=(0,1,0). Then, we consider the following Filippov system:

    ˙M={f1(M),MG1,f2(M),MG2, (3.1)

    where

    f1(M)=(Λax(t)βx(t)v(t)γx(t)y(t)βx(t)v(t)+γx(t)y(t)by(t)ky(t)cv(t)),
    f2(M)=(Λax(t)βx(t)v(t)γx(t)y(t)βx(t)v(t)+γx(t)y(t)by(t)μ1y(t)ky(t)cv(t)μ2v(t)).

    Based on this, we present the following definitions of various equilibriums and sliding domain.

    Definition 3.1. [30] If f1(M)=0 with ω(M)<0, or f2(M)=0 with ω(M)>0, the point M is called a real equilibrium of system (3.1).

    Definition 3.2. [30] If f1(M)=0 with ω(M)>0, or f2(M)=0 with ω(M)<0, the point M is called a virtual equilibrium of system (3.1).

    Definition 3.3. [30] If it is an equilibrium of the sliding mode of system (3.1), the point M is called a pseudo-equilibrium.

    Definition 3.4. [30] S is the sliding domain, if n,f1>0 and n,f2<0 on SGs.

    In this subsection, we calculate the basic reproduction number and analyze the stability of equilibrium. The dynamics of system (3.1) in region G1 are indicated by

    (x(t)y(t)v(t))=(Λax(t)βx(t)v(t)γx(t)y(t)βx(t)v(t)+γx(t)y(t)by(t)ky(t)cv(t)). (3.2)

    In subsystem (3.2), the disease-free equilibrium E01=(Λa,0,0) and the endemic equilibrium E1=(x1,y1,v1) are given, where

    x1=bckβ+cγ,y1=Λ(kβ+cγ)acbb(kβ+cγ),v1=Λk(kβ+cγ)ackbbc(kβ+cγ),R01=Λ(βk+cγ)acb.

    Next, the following theorem about the local asymptotic stability of equilibria E01 and E1 are given.

    Theorem 3.5. The disease-free equilibrium E01 is locally asymptotically stable if R01<1; the endemic equilibrium E1 exists and is locally asymptotically stable if R01>1.

    Proof. Using the next generation matrix, the basic reproduction number R01 of system (3.2) can be deduced, and system (3.2) can be represented as

    M=F(M)V(M),

    where

    F(M)=[βx(t)v(t)+γx(t)y(t)00],V(M)=[by(t)cv(t)ky(t)ax(t)+βx(t)v(t)+γx(t)y(t)Λ].

    The Jacobian matrix of F(M) and V(M) at the equilibrium point E01 is

    DF(E01)=[γΛaβΛa0000000],DV(E01)=[b00kc0γΛaβΛaa],

    where

    F=[γΛaβΛa00],V=[b0kc].

    Then, the generation matrix of system (3.2) is

    FV1=[Λ(βk+cγ)acbβΛac00].

    So, the spectral radius of FV1 is

    ρ(FV1)=Λ(βk+cγ)acb,

    where ρ is the spectral radius of a matrix. Therefore, the basic reproduction number of system (3.2) is

    R01=Λ(βk+cγ)acb.

    For subsystem (3.2), The Jacobian matrix is

    J1(x(t),y(t),v(t))=(aβv(t)γy(t)γx(t)βx(t)βv(t)+γy(t)b+γx(t)βx(t)0kc).

    If R01<1, cbcγΛ+kβΛa>0, c+bγΛa>0 can be obtained, and (c+bγΛa)24(cbcγΛ+kβΛa)>0. The characteristic equation at E01 is

    (λ+a)[(λ+bγΛa)(λ+c)kβΛa]=0,

    which indicates that

    λ1=a<0,λ2=(c+bγΛa)+(c+bγΛa)24(cbcγΛ+kβΛa)2<0,λ3=(c+bγΛa)(c+bγΛa)24(cbcγΛ+kβΛa)2<0.

    Thus, the eigenvalues of J(E01) are negative when R01<1, so we can obtain that E01 is locally asymptotically stable.

    If R01>1, then γcΛ+kβΛacb>1, and there exists the endemic equilibrium E1. Thus, the characteristic equation at E1 is

    λ3+a1λ2+a2λ+a3=0,

    where

    a1=a+c+b+βv1+γy1γx1=a+c+b+βΛk(kβ+cγ)ackbbc(kβ+cγ)+γΛ(kβ+cγ)acbb(kβ+cγ)γbckβ+cγ=c+a+bkβkβ+cγ>0,a2=bc+ac+ab(γc+kβ+aγ)x1+(cβ+bβ)v1+(cγ+bγ)y1=bc+ac+ab(γc+kβ+aγ)bckβ+cγ+(cβ+bβ)Λk(kβ+cγ)ackbbc(kβ+cγ)+(cγ+bγ)Λ(kβ+cγ)acbb(kβ+cγ)=ca+akbβkβ+cγ>0,a3=acb(acγ+akβ)x1+bcβv1+bcγy1=acb(acγ+akβ)bckβ+cγ+bcβΛk(kβ+cγ)ackbbc(kβ+cγ)+bcγΛ(kβ+cγ)acbb(kβ+cγ)=Λ(kβ+cγ)acb>0.

    Moreover, we can get

    a1a2a3=[a+c+b+βv1+γy1γx1][bc+ac+ab(γc+kβ+aγ)x1+(cβ+bβ)v1+(cγ+bγ)y1][acb(acγ+akβ)x1+bcβv1+bcγy1]=Λ(kβ+cγ)acbbc(kβ+cγ)[bbkβ+akbβ+accγ+bckβ+cckβ+cccγ+cakβ+cacγ]+(cakβ+cacγ+abkβ)bkβ+ckβ+ccγ+akβ+acγ(kβ+cγ)2+2kβγ(b+c)Λ(kβ+cγ)acbbc(kβ+cγ)Λ(kβ+cγ)acbb(kβ+cγ)+Λ(kβ+cγ)acbbc(kβ+cγ)(ackβ+cacγ+akbβ)>0.

    According to the Routh-Hurwitz Criterion, all eigenvalues of J(E1) have negative real parts. Hence, this represents that E1 is locally asymptotic stable.

    In region G2, we give the dynamics of system (3.1):

    (x(t)y(t)v(t))=(Λax(t)βx(t)v(t)γx(t)y(t)βx(t)v(t)+γx(t)y(t)by(t)μ1y(t)ky(t)cv(t)μ2v(t)). (3.3)

    In subsystem (3.3), the disease-free equilibrium E02=(Λa,0,0) and the endemic equilibrium E2=(x2,y2,v2) are given, where

    x2=(c+μ2)(b+μ1)kβ+γ(c+μ2),y2=Λb+μ1a(c+μ2)kβ+γ(c+μ2),v2=kΛ(c+μ2)(b+μ1)akkβ+γ(c+μ2),R02=Λ(kβ+γ(c+μ2))a(c+μ2)(b+μ1).

    Next, the following theorem about the local asymptotic stability of equilibria E02 and E2 are given.

    Theorem 3.6. The disease-free equilibrium E02 is locally asymptotically stable if R02<1. The endemic equilibrium E2 exists and is locally asymptotically stable if R02>1.

    Proof. Using the next generation matrix, we deduce the basic reproduction number R02 of system (3.3):

    R02=ρ([γΛaβΛa00][b+μ10kc+μ2]1)=Λ(kβ+γ(c+μ2))a(c+μ2)(b+μ1).

    For subsystem (3.3), the Jacobian matrix is

    J2(x(t),y(t),v(t))=(aβv(t)γy(t)γx(t)βx(t)βv(t)+γy(t)b+γx(t)μ1βx(t)0kcμ2).

    Let c+μ2=c1, b+μ1=b1. Similar to the proof of Theorem 3.5, if R02<1, then all eigenvalues of J(E02) are negative, and thus the local asymptotic stability of E02 can also be concluded. Furthermore, all eigenvalues of J(E2) have negative real parts, if R02>1, which shows that E2 is locally asymptotically stable.

    Remark 3.7. The basic reproduction number R02 is related to control variables μ1 and μ2, and when μ1=μ2=0, that is, the control measures are not implemented, we have R01=R02.

    In order to study the dynamics of sliding mode of system (3.1) in this subsection, we initially examine the existence of the sliding mode. The manifold Gs is defined as y(t)=Nt, and we have

    n,f1=(010),(Λax(t)βx(t)v(t)γx(t)y(t)βx(t)v(t)+γx(t)y(t)by(t)ky(t)cv(t))=βx(t)v(t)bNt+γx(t)Nt=σ1(x(t),v(t)), (3.4)

    and

    n,f2=(010),(Λax(t)βx(t)v(t)γx(t)y(t)βx(t)v(t)+γx(t)y(t)by(t)μ1y(t)ky(t)cv(t)μ2v(t))=βx(t)v(t)bNt+γx(t)Ntμ1Nt=σ2(x(t),v(t)), (3.5)

    which shows that σ2(x(t),v(t))<σ1(x(t),v(t)). Because n,f1>0 and n,f2<0, we can obtain

    n,f1>0,ifx(t)>bNtβv(t)+γNt=H1(v(t)),n,f2<0,ifx(t)<bNt+μ1Ntβv(t)+γNt=H2(v(t)),

    and

    H2(v(t))>H1(v(t)).

    It is obvious that H1(v(t)) is positive on the interval (0,Nt), and so is H2(v(t)). Then, the sliding domain SGs is defined as follows:

    S{(x(t),y(t),v(t))Gs:H1(v(t))<x(t)<H2(v(t)),y(t)=Nt}. (3.6)

    Moreover, in order to obtain sliding mode equations in the region S, we use the Utkin equivalent control method, as used in [31]. First, according to ω(y(t))=y(t)Nt=0 and dω/dt=0,

    dωdt=ωy(t)dy(t)dt=βx(t)v(t)by(t)+γx(t)y(t)εμ1y(t)=0. (3.7)

    By solving Eq (3.7), the function about ε is obtained:

    ε(v(t))=βx(t)v(t)by(t)+γx(t)y(t)μ1y(t)=βx(t)v(t)bNt+γx(t)Ntμ1Nt. (3.8)

    Substitute Eq (3.8) into system (2.2), and the dynamics of sliding mode in S can be given by differential equations as follows:

    {dx(t)dt=Λax(t)βx(t)v(t)γx(t)Nt,dy(t)dt=0,dv(t)dt=kμ1N2tcμ1Ntv(t)+bμ2Ntv(t)γμ2Ntx(t)v(t)μ2βx(t)v2(t)μ1Nt. (3.9)

    Next, we exhibit the following proposition about the existence of equilibrium of sliding mode (3.9).

    Proposition 3.8. The existence of a unique positive pseudo-equilibrium Es=(Λa+βv+γNt,Nt,v) of sliding mode (3.9) is obtained if ckb<μ2μ1<cb and EsS.

    Proof. Let dx(t)dt=0 and dv(t)dt=0. It obtains that

    x(t)=Λa+βv(t)+γNt, (3.10)

    and

    kμ1N2tcμ1Ntv(t)+bμ2Ntv(t)γμ2Ntx(t)v(t)μ2βx(t)v2(t)=0. (3.11)

    After substituting Eq (3.10) into (3.11), we get the following equation of v(t):

    τ1v2(t)+τ2v(t)+τ3=0, (3.12)

    where

    τ1=βNt(bμ2cμ1)μ2βΛ,τ2=abμ2Ntcaμ1NtγΛμ2Nt+bγμ2N2tcγμ1N2t+kβμ1N2t,τ3=akμ1N2t+kγμ1N3t,Δ=τ224τ1τ3=N2t[abμ2caμ1γΛμ2+bγμ2Ntcγμ1Nt+kβμ1Nt]24(βNt(bμ2cμ1)μ2βΛ)[akμ1N2t+kγμ1N3t].

    If ckb<μ2μ1<cb, then τ1<0,τ3>0 and Δ>0 can be received. The Vieta Theorem shows

    v1v2=τ3τ1<0. (3.13)

    Due to Δ>0, the existence of the root is proven, and there are two roots, one positive root and one negative root from (3.13). We can conclude that Eq (3.12) has a unique positive root denoted by v. Thus, the equilibrium point Es can be obtained from (3.10):

    Es=(x,y,v)=(Λa+βv+γNt,Nt,v).

    Furthermore, Es is a unique pseudo-equilibrium if EsSGs holds.

    Similar to the previous theorem, it is worth considering the local asymptotical stability of Es, and then the following theorem is given.

    Theorem 3.9. Under the same conditions of Proposition 3.8, Es is locally asymptotically stable on the sliding domain S.

    Proof. In system (3.9), the Jacobian matrix of the first two equations is

    Js(x,v)=(J11J12J21J22). (3.14)

    From Proposition 3.8, ckb<μ2μ1<cb, we set

    P=dx(t)dt=Λax(t)βx(t)v(t)γx(t)Nt,Q=dv(t)dt=kμ1N2tcμ1Ntv(t)+bμ2Ntv(t)γμ2Ntx(t)v(t)μ2βx(t)v2(t)μ1Nt, (3.15)

    and obtain

    J11=Px|(x,v)=aβvγNt<0,J12=Pv|(x,v)=βx<0,J21=Qx|(x,v)=γμ2vNtμ2β(v)2μ1Nt<0,J22=Qv|(x,v)=cμ1Nt+bμ2Ntγμ2Ntx2μ2βxvμ1Nt<0. (3.16)

    Thus, we get

    J11J22J12J21>0, (3.17)

    since J11+J22<0, and we can obtain that

    J11J22J12J21=[aβvγNt]cμ1Nt+bμ2Ntγμ2Ntx2μ2βxvμ1Nt(βx)γμ2vNtμ2β(v)2μ1Nt=(a+βv+γNt)(cμ1Ntbμ2Nt)+(a+γNt)γμ2Ntx+(a+βv+γNt)μ2βxvμ1Nt>0. (3.18)

    Therefore, all eigenvalues of (3.14) have negative real parts. That Es is locally asymptotically stable can be obtained easily.

    Remark 3.10. The sliding domain S represents the region between the two lines associated with x(t),v(t) belonging to the plane y=Nt, and the local asymptotical stability of pseudo-equilibrium Es in sliding domain S is related to the control variables, which needs to satisfy the condition of the parameters.

    In order to reduce the number of infected cells and viruses, while keeping the cost to apply the control at the minimum level at any time, we show an optimal control problem on the basis of system (2.2). In this section, any time refers to when the number of infected cells is greater than threshold level Nt, that is, when ε=1 in system (2.2). If the control variable in system (2.2) is a time dependent variable, how do we find the optimal control? Thus, we represent by u1(t) the RTIs control variable and by u2(t) the PIs control variable. First, we have u(t)=(u1(t),u2(t))V[s,T]={u():[s,T]U|u1(t)andu2(t)aremeasurable:0u1(t)1,0u2(t)1}, where U is a metric space and convex, T>0, and the control problem of the model is given by

    {dx(t)dt=Λβx(t)v(t)γx(t)y(t)ax(t),dy(t)dt=βx(t)v(t)+γx(t)y(t)by(t)u1(t)y(t),t>0,dv(t)dt=ky(t)cv(t)u2(t)v(t). (4.1)

    Let (s,M)[0,T)×R3, and we consider the following control system over [s,T]:

    {dM(t)dt=b(t,M(t),u(t)),t[s,T],M(s)=M0, (4.2)

    where M(t)=(x(t),y(t),v(t))R3. Next, we construct the following objective functional:

    J(u1(t),u2(t))=Ts(A1u21(t)+A2u22(t)+A3y(t)+A4v(t))dt+h(M(T)), (4.3)

    where Ai(i=1,2,3,4) are weights to make the terms of the integrand keep balance. The term Ts(A1u21(t)+A2u22(t))dt gives the total cost of using the control strategy, h(M(T)) is the penalty function corresponding to the terminal state, and Ts(A3y(t)+A4v(t))dt represents the total number of infected cells and viruses over the time period T. It is important to find an optimal control pair (u1(t),u2(t)),t[s,T] such that

    J(u1(t),u2(t))=minu1(t),u2(t)V[s,T]J(u1(t),u2(t)). (4.4)

    Further, the value function is as follows:

    {V(s,M0)=infu1(t),u2(t)V[s,T]J(s,M0;u1(t),u2(t)),(s,M0)[0,T)×R3,V(T,M0)=h(M0),M0R3. (4.5)

    Before further study, we give the following assumption.

    Assumption 1. (U,˜d) is a separable metric space.

    Then, we would like to study V(,) in great detail and present the following results called Bellman's principle of optimality by [32].

    Theorem 4.1. Let 1 hold, and U is convex. Then, for any (s,M0)[0,T)×R3, we have

    V(s,M0)=infu1(t),u2(t)V[s,T]{ˆss(A1u21(t)+A2u22(t)+A3y(t)+A4v(t))dt+V(ˆs,M(ˆs;s,M0,u()))},0sˆsT. (4.6)

    Proof. Let us define

    ˉV(s,M0)={ˆss(A1u21(t)+A2u22(t)+A3y(t)+A4v(t))dt+V(ˆs,M(ˆs;s,M0,u()))}. (4.7)

    By (4.6), we obtain

    ˉV(s,M0)J(s,M0;u(t))=ˆss(A1u21(t)+A2u22(t)+A3y(t)+A4v(t))dt+J(ˆs,M(ˆs);u())),u(t)V[s,T]. (4.8)

    Therefore, we take the infimum over u(t)V[s,T] and get

    V(s,M0)ˉV(s,M0), (4.9)

    and there exists a uε(t)V[s,T] for ε>0 such that

    V(s,M0)+εJ(s,M0;uε(t))ˆss(A1u21(t)+A2u22(t)+A3y(t)+A4v(t))dt+J(ˆs,Mε(ˆs))ˉV(s,M0), (4.10)

    where Mε()=M(;s,M0,uε()). Combining (4.9) and (4.10), (4.6) is obvious. This completes the proof.

    (4.6) is called the dynamic programming equation, and the equation is difficult to handle. Thus, we construct the Hamilton-Jacobi-Bellman (HJB) equation as follows.

    Theorem 4.2. Let 1 hold, and U is convex. Suppose VC1([0,T]×R3). Then, V(s,M0) is a solution to the following terminal value problem of a first-order partial differential equation:

    {0=Vt+supu(t)UH(t,M,u(t),VM),(t,M)[0,T]×R3,V|t=T=h(M),MR3, (4.11)

    where

    H(t,M,u(t),VM)=Vx(Λax(t)βx(t)v(t)γx(t)y(t))Vy(βx(t)v(t)+γx(t)y(t)by(t)u1(t)y(t))Vv(ky(t)cv(t)u2(t)v(t))A1u21(t)A2u22(t)A3y(t)A4v(t). (4.12)

    We call (4.11) the Hamilton-Jacobi-Bellman (HJB) equation associated with (4.5).

    Proof. Fix a uU. Let M(t) be the state trajectory corresponding to the control u(t)u. According to (4.6) with ˆss, we get

    0V(ˆs,M(ˆs))V(s,M0)ˆss1ˆssˆss(A1u21(t)+A2u22(t)+A3y(t)+A4v(t))dtVt(s,M0)(Vx(Λax(t)βx(t)v(t)γx(t)y(t))+Vy(βx(t)v(t)+γx(t)y(t)by(t)u1(t)y(t))+Vv(ky(t)cv(t)u2(t)v(t)))(A1u21(t)+A2u22(t)+A3y(t)+A4v(t)),uU, (4.13)

    which results in

    0Vt(s,M0)+supuU{Vx(Λax(t)βx(t)v(t)γx(t)y(t))Vy(βx(t)v(t)+γx(t)y(t)by(t)u1(t)y(t))Vv(ky(t)cv(t)u2(t)v(t))A1u21(t)A2u22(t)A3y(t)A4v(t)}. (4.14)

    For any ε>0,0sˆsT with ˆss>0 small enough, there exists a u=uε,ˆs()V[s,T] such that

    V(s,M0)+ε(ˆss)ˆss(A1u21(t)+A2u22(t)+A3y(t)+A4v(t))dt+V(ˆs,M(ˆs)), (4.15)

    so it follows that

    εV(ˆs,M(ˆs))V(s,M0)ˆss1ˆssˆss(A1u21(t)+A2u22(t)+A3y(t)+A4v(t))dt=1ˆssˆss{Vt(t,M)[Vx(Λax(t)βx(t)v(t)γx(t)y(t))+Vy(βx(t)v(t)+γx(t)y(t)by(t)u1(t)y(t))+Vv(ky(t)cv(t)u2(t)v(t))][A1u21(t)+A2u22(t)+A3y(t)+A4v(t)]}dt1ˆssˆss{Vt(t,M)+supuU{Vx(Λax(t)βx(t)v(t)γx(t)y(t))Vy(βx(t)v(t)+γx(t)y(t)by(t)u1(t)y(t))Vv(ky(t)cv(t)u2(t)v(t))A1u21(t)A2u22(t)A3y(t)A4v(t)}}dtVt(s,M0)+supuU{Vx(Λax(t)βx(t)v(t)γx(t)y(t))Vy(βx(t)v(t)+γx(t)y(t)by(t)u1(t)y(t))Vv(ky(t)cv(t)u2(t)v(t))A1u21(t)A2u22(t)A3y(t)A4v(t)}. (4.16)

    Combining (4.14) and (4.16), we complete the proof.

    For further study, the definition of viscosity solution of (4.11) by [32] is given.

    Definition 4.3. A function VC[0,T]×R3) is called a viscosity subsolution (or supersolution) of (4.11) if

    V(T,M)h(M),(V(T,M)h(M))MR3, (4.17)

    and for any φC1([0,T]×R), whenever Vφ attains a local maximum(or minimum) at (t,M)[0,T]×R3, we have

    φt(t,M)+supu(t)UH(t,M,u(t),φM(T,M))0,(φt(t,M)+supu(t)UH(t,M,u(t),φM(T,M))0). (4.18)

    In this case that V is both a viscosity subsolution and supersolution of (4.11), it is a viscosity solution of (4.11).

    If we get the value function V by solving the HJB equation, then an optimal pair could be constructed. Thus, the characterization of the value function of the viscosity solution to the HJB equation is given below.

    Theorem 4.4. Let 1 hold, and U is convex. Then, the value function V(,) satisfies

    |V(s,M0)V(ˉs,¯M0)|K|sˉs|,(s,M0),(ˉs,¯M0)[0,T]×R3, (4.19)

    for some K>0. Moreover, V is the only viscosity solution of (4.11) in the class C([0,T]×R3).

    Proof. We have

    |V(s,M0)V(ˉs,¯M0)|=|infu1,u2V[s,T]Ts[A1u21(t)+A2u22(t)+A3y(t)+A4v(t)]dtinfu1,u2V[s,T]Tˉs[A1u21(t)+A2u22(t)+A3y(t)+A4v(t)]dt||ˉss[A1u21(t)+A2u22(t)+A3y(t)+A4v(t)]dt|K|sˉs|, (4.20)

    where K=max{A1u21(t)+A2u22(t)+A3y(t)+A4v(t)}>0. Then, similar to [32, Theorem 2.5], the value function V clearly satisfies the condition and is the only viscosity solution of (4.11).

    Next, in order to discuss the existence of the optimal control pair, we define H(t,M,u(t),p) as the Hamiltonian as follows:

    H(t,M,u(t),p)=(Λax(t)βx(t)v(t)γx(t)y(t))p1+(βx(t)v(t)+γx(t)y(t)by(t)u1(t)y(t))p2+(ky(t)cv(t)u2(t)v(t))p3+A1u21(t)+A2u22(t)+A3y(t)+A4v(t). (4.21)

    Theorem 4.5. There exists an optimal control pair (u1(t),u2(t)) and a corresponding optimal state (x(t),y(t),v(t)) such that

    J(u1(t),u2(t))=minu1(t),u2(t)UJ(u1(t),u2(t)). (4.22)

    Proof. Note that both the control variables and state variables are nonnegative, and the objective functional (4.3) is convex with respect to the control variables. The convexity and closure of the control set U are obtained according to the definition of it. Furthermore, the optimal control is bounded. Therefore, we conclude that there exists an optimal control u1(t),u2(t) for J(u1(t),u2(t))=minu1(t),u2(t)UJ(u1(t),u2(t)).

    The HJB equation and the value function have been discussed. Further, we figure out the optimal control as follows.

    Theorem 4.6. Let u1(t),u2(t) be optimal control variables, and x(t),y(t),v(t) are corresponding optimal state variables. There exists adjoint process p(t)=(p1(t),p2(t),p3(t)), satisfying the following adjoint equation:

    {dp1(t)=[(aβv(t)γy(t))p1(t)+(βv(t)+γy(t))p2(t)]dtdp2(t)=[(γx(t))p1(t)+(γx(t)bu1(t))p2(t)+kp3(t)+A3]dtdp3(t)=[(βx)p1(t)+(βx)p2(t)+(cu2(t))p3(t)+A4]dtpi(T)=0,i=1,2,3. (4.23)

    Furthermore, the optimal control is given as follows:

    ui(t)=min{max{Bi,0},1},i=1,2, (4.24)

    where B1=y(t)p2(t)2A1,B2=v(t)p3(t)2A2.

    Proof. Because the Hamiltonian function H is given. Moreover, by using the optimal condition, and we obtain u1(t) and u2(t)

    Hu1=0,Hu2=0. (4.25)

    Hence,

    u1(t)=y(t)p2(t)2A1,u2(t)=v(t)p3(t)2A2. (4.26)

    So, B1=y(t)p2(t)2A1,B2=v(t)p3(t)2A2 are taken, and the optimal control u1(t),u2(t) follows:

    u1(t)={0,ify(t)p2(t)2A1<0,y(t)p2(t)2A1,if0y(t)p2(t)2A11,1,ify(t)p2(t)2A1>1,u2(t)={0,ifv(t)p3(t)2A2<0,v(t)p3(t)2A2,if0v(t)p3(t)2A21,1,ifv(t)p3(t)2A2>1. (4.27)

    Thus, the optimal value can be obtained.

    Remark 4.7. In practical problems, when the parameters are known, the optimal control u1(t) and u2(t) can be calculated by using computer programming for Eqs (4.1), (4.23) and (4.27), that is, the dosage intensity of RTIs and PIs at each time can be calculated. If patients are treated in such a proportion, an optimal control strategy to minimize the costs and the number of viruses and infected cells can be obtained.

    In this section, several numerical simulations are presented to prove and verify the above theoretical results by MATLAB software.

    In this subsection, numerical simulations are performed to illustrate the theoretical results. We discuss the dynamics behavior of system (3.1). Since the choice of the threshold level Nt is different, system (3.1) will show various dynamics. There are three cases to consider. First, we discretize system (3.1) as follows:

    {x(j+1)=x(j)+Δt(Λax(j)βx(j)v(j)γx(j)y(j)),y(j+1)=y(j)+Δt(βx(j)v(j)+γx(j)y(j)by(j)εμ1y(j)),v(j+1)=v(j)+Δt(ky(j)cv(j)εμ2v(j)). (5.1)

    In Table 1, all parameter values are exhibited. Some values are collected from different papers [6,17], and others are assumed.

    Table 1.  Parameter values of numerical experiments for model (2.2).
    Parameter Value Source of data
    Λ 10 [17]
    a 0.01 [17]
    β 0.00034 Assumed
    γ 0.0001 Assumed
    b 0.26 [6]
    k 11 [6]
    c 0.1 [6]
    μ1 0.3 Assumed
    μ2 0.1 Assumed

     | Show Table
    DownLoad: CSV

    Case 1: E1 is a real equilibrium, and E2 is a virtual equilibrium.

    This situation is established. Suppose that the following conditions are satisfied:

    y1<Ntandy2<Nt. (5.2)

    In this case, both equilibriums are located on the same side of the plane Nt, namely, the region G1. Thus, when (5.2) is satisfied, we obtain that E1 can achieve stability. In Figure 1 E1G1 achieves stability with Nt=60, and the possible trajectories of this figure are shown: A trajectory that starts in region G1 will converge to E1 as t+ with hitting and sliding down on the sliding domain SGs; a trajectory of the initial point in region G2 will cross the manifold Gs, then enter region G1 and finally converge to E1; a trajectory that begins in region G1 or G2 will hit and slide to the boundary of the sliding domain SGs before moving towards E1.

    Figure 1.  E1G1 achieves stability with Nt=60.

    Case 2: E2 is a real equilibrium, and E1 is a virtual equilibrium.

    This situation is established. Suppose that the following conditions are satisfied:

    y1>Ntandy2>Nt. (5.3)

    In this case, both equilibriums are located on the same side of the plane Nt, namely, the region G2. Thus, when (5.3) is satisfied, we obtain that E2 can achieve stability. In Figure 2 E2G2 achieves stability with Nt=15, and the possible trajectories of this figure are shown: A trajectory with initial condition in region G2 will converge to E2 as t+ with hitting and sliding up on the sliding domain SGs; a trajectory with initial point in region G1 will pass through the manifold Gs from G1 to G2; a trajectory that starts in region G1 or G2 will hit and slide up on the boundary of the sliding domain SGs before moving towards E2.

    Figure 2.  E2G2 achieves stability with Nt=15.

    Case 3: Both E1 and E2 are virtual equilibria.

    This situation is established. Suppose that the following conditions are satisfied:

    y1>Ntandy2<Nt. (5.4)

    For this case, we can find that E1 and E2 belong to regions G2 and G1, respectively. If ESSGs, it is a pseudo-equilibrium. Thus, when (5.4) is satisfied, we obtain that Es is stable when it exists. Figure 3 EsS achieves stability with Nt=25, and the possible trajectories of this figure are shown: A trajectory with initial condition in region G1 will converge to Es as t+; a trajectory that begins in region G2 will cross the manifold Gs, then enter region G1 and finally converge to Es; a trajectory with initial condition in region G1 or G2 will hit and slide down on SGs before converging to Es.

    Figure 3.  EsS achieves stability with Nt=25.

    In this subsection, we also discretize the optimal control system (4.1) as

    {x(j+1)=x(j)+Δt(Λax(j)βx(j)v(j)γx(j)y(j)),y(j+1)=y(j)+Δt(βx(j)v(j)+γx(j)y(j)by(j)u1(j)y(j)),v(j+1)=v(j)+Δt(ky(j)cv(j)u2(j)v(j)). (5.5)

    The parameter values Λ, a, b, β, γ, k and c are chosen as shown in Table 1, and A1=20,A2=50,A3=0.1,A4=0.1. Then, we compare the effects of different control intensities by the following figures. The cost of control strategies must be considered, and the cost of each measure is different, so we want to know the change under only one control strategy compared with two control strategies. Thus, we make simulations under only u1(t) (see Figure 4) or u2(t) (see Figure 5) and optimal control (see Figure 6).

    Figure 4.  The states of x(t),y(t) and v(t) when u1=0,0.3,0.6 and u2=0,0,0.
    Figure 5.  The states of x(t),y(t) and v(t) when u1=0,0,0 and u2=0,0.3,0.6.
    Figure 6.  The paths of x(t), y(t) and v(t) with and without optimal control.

    When initial value (x0,y0,v0)=(50,150,800), Figure 4 illustrates that if the control strategy is only applied to infected cells (u2(t)=0), the concentration of infected cells and viruses will decrease. Meanwhile, the number of infected cells and viruses decreased with the increase of u1(t) intensity. Figure 5 shows that if the control strategy is only applied to viruses (u1(t)=0), the density of infected cells and viruses will also decrease, and the number of infected cells and viruses decreased with the increase of u2(t) intensity.

    The expressions of optimal control u1(t) and u2(t) are obtained through calculation (in Eq (4.27)), and then we obtain the optimal control states of uninfected cells, infected cells and viruses in Figure 6, which shows that the concentration of infected cells and viruses decreases to some extent after the control is applied, and the combination of multi-drug works better than a single-drug approach. Figure 7 shows the values of control variables u1(t) and u2(t) in each time. Eventually, the concentration of infected cells and viruses and control intensity gradually decrease and stabilize. Thus, the control strategies in our model have significant influence on the spread of HIV.

    Figure 7.  The path of the two controls.

    In this paper, we have proposed an HIV model with cell-to-cell transmission and analyzed the sliding mode dynamics of the model and optimal control problem. First, we extended a novel Filippov model (3.1), which indicates that corresponding control measures (i.e., antiretroviral drugs RTIs and PIs) are triggered once the total number of infected cells reaches the threshold level Nt, where RTIs prevent new HIV infections by blocking the transformation of viral RNA into DNA in T cells, and PIs reduce the number of viral particles produced by actively infected T cells [21]. Further, the sliding domain and sliding mode dynamics of system (2.2) have been examined. In addition, the simulation results show that the model solution is either near a real equilibrium point or near a pseudo-equilibrium point according to the different threshold levels. It is worth mentioning that some parameters can be simulated through actual data, and the basic reproduction number can be calculated to judge the stability. In addition, the number of viruses and infected cells can reduce to a previously desired level when the threshold level is chosen properly. Because different patients have different initial viral loads, an individualized therapy is suggested, which shows that the choice of a treatment strategy for a given patient should depend on HIV viruses and infected cells and proposed threshold level.

    Moreover, the cost of treatment is beyond the reach of many infected patients. Therefore, we have introduced an optimal therapy to minimize the cost of treatment and reduce the viral load and the number of infected cells. Then, the efficacies of RTIs and PIs and their combinations have been measured. In addition, we have discussed an efficient numerical method based on optimal control to determine the best treatment strategy for HIV infection. Our results indicate that with the increase of treatment intensity, the number of infected cells and viruses decreases, while the density of CD4+T increases. Due to the multiple transmission routes of HIV, the combined use of multiple drugs is better than the use of a single drug. From a biological point of view, it can be concluded that optimal control is adopted when the number of infected cells in the patient is higher than threshold level Nt, and at this time, the economic cost can be considered to select the optimal control measures. Control measures are not necessary when the concentration of infected cells is low. Optimal control path is shown in Figure 7.

    The results of this paper have practical implications for controlling HIV transmission. However, our work is only a preliminary exploration of the impact of some control measures on HIV transmission, which can be improved in many aspects. The number of viruses in the infected person varies with age, so it is interesting and challenging to consider the sliding mode control of HIV model with age structure. We can also refer to the methods in the literature [33,34] to study the fractional-order HIV model. These issues will be the focus of our future research.

    The research was supported by the Key Research and Development Program of Ningxia (2020BEB04007) and the Fundamental Research Funds for the Central Universities, North Minzu University (2020KYQD17).

    The authors declare there is no conflict of interest.



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