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Research article

The impact of digital technology on enterprise green innovation: quality or quantity?

  • Digital technology promotes the dual transformation of enterprise digitization and greenization, thereby promoting the synergistic efficiency between the digital economy and the green economy. This paper collected financial data from 2010 to 2021 from Chinese listed companies on the Shanghai and Shenzhen stock exchanges. Through an in-depth semantic analysis of textual data, the study constructed an index to measure the level of enterprise digitization. Utilizing panel data models, the paper explored the impact of digital technology on enterprise green innovation and its mechanisms from the perspectives of quality and quantity. The research findings are as follows: (1) Digital technology significantly enhances the capability of enterprises for green innovation, with an emphasis on quality rather than quantity; (2) digital technology effectively alleviates financing constraints and information constraints, thereby enhancing the level of enterprise green innovation, but the former's effect is limited to small and medium-sized enterprises; (3) the "quality over quantity" effect of digital technology on enterprise green innovation is more pronounced in state-owned enterprises, non-heavy polluting industries, and enterprises located in regions with moderate to low levels of economic development.

    Citation: Xinyu Fu, Yanting Xu. The impact of digital technology on enterprise green innovation: quality or quantity?[J]. Green Finance, 2024, 6(3): 484-517. doi: 10.3934/GF.2024019

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  • Digital technology promotes the dual transformation of enterprise digitization and greenization, thereby promoting the synergistic efficiency between the digital economy and the green economy. This paper collected financial data from 2010 to 2021 from Chinese listed companies on the Shanghai and Shenzhen stock exchanges. Through an in-depth semantic analysis of textual data, the study constructed an index to measure the level of enterprise digitization. Utilizing panel data models, the paper explored the impact of digital technology on enterprise green innovation and its mechanisms from the perspectives of quality and quantity. The research findings are as follows: (1) Digital technology significantly enhances the capability of enterprises for green innovation, with an emphasis on quality rather than quantity; (2) digital technology effectively alleviates financing constraints and information constraints, thereby enhancing the level of enterprise green innovation, but the former's effect is limited to small and medium-sized enterprises; (3) the "quality over quantity" effect of digital technology on enterprise green innovation is more pronounced in state-owned enterprises, non-heavy polluting industries, and enterprises located in regions with moderate to low levels of economic development.



    Human immunodeficiency virus (HIV) infection was regarded as a major global public health issue. The World Health Organization (WHO) has claimed more than 40.4 million (32.9–51.3 million) infection cases as of 13 July 2023, in which 0.63 million (0.48–0.88 million) individuals died from HIV-related causes during the period from January 2022 to December 2022 [1]. Moreover, China admitted 117.9 thousand HIV/AIDS infection cases with 30.0 thousand (25.48%) deaths from January 2007 to December 2011 [2], in which 3266 HIV/AIDS infection cases were diagnosed with 725 deaths (22.20%), among which, 2320 local HIV/AIDS infection cases with 530 deaths (22.84%) and, 946 HIV/AIDS infection cases with 195 deaths (20.61%) from other provinces and overseas were reported during the period from January 2004 to December 2011. Additionally, the symptom characteristics were well recorded and classified by the surveillance data. Infection cases with mild symptoms may have an influenza-like illness including fever, headache, rash, sore throat and other symptoms such as swollen lymph nodes, weight loss, diarrhea and a cough, as reported by the WHO. Without treatment, infection cases with mild symptoms can develop to include severe symptoms such as tuberculosis, cryptococcal meningitis, severe bacterial infections and cancers due to the weakness of the immune system [1]. Essentially, the severity of symptoms was characterized by CD4+ T cell counts [3,4,5], and the starting and ending points of structured treatment interruptions were determined in [6]. The transmission mechanism of symptom-dependent HIV/AIDS was usually described by mathematical models.

    Local governments and policy makers controlled and prevented infectious diseases by adopting mathematical models, which played irreplaceable roles and provided new perspectives when studying the complex dynamics of infectious diseases, as in [7]. Specifically, the compartment models were often governed to investigate the transmission mechanisms of infectious diseases, in which the compartment structures were determined by specific infectious diseases. In fact, various types of deterministic dynamical models were included such as ordinary differential equation models, difference equation models, delay-differential equation models, age-structured partial differential equations (PDE) models and diffusion models in [7,8,9,10,11,12,13,14,15,16]. Meanwhile, recent contributions also governed the stochastic differential equation models and the fractional differential equation models to investigate the basic reproduction number and dynamical properties in [17,18,19,20,21,22,23,24,25]. More precisely, HIV/AIDS incorporated with tuberculosis (TB) models were taken into account, where the effect of Antiretroviral Therapy (ART) treatment and both local and global stabilities of HIV-only model were studied in [19,20] by adopting the standard incidence rates, in which the transmission mechanism to the susceptible compartment in [19] presented a more complex incidence mechanism. After these two contributions, the parameter fluctuation was introduced into the HIV/AIDS model with the bilinear incidence rate in [21], and the systematic fluctuation was investigated in [22] based on the model in [20]; therein, the related survival analysis and dynamical properties were extensively studied. The research results of the stochastic HIV/AIDS models in [21,22,25,26] showed that the intensities of white noises diminished the scale and threshold of HIV/AIDS. The investigations of HIV/AIDS models in [23,24] reflected that protection awareness of the susceptible also diminished the number of HIV/AIDS infection cases.

    In this paper, we proposed symptom-dependent HIV/AIDS models to describe the epidemiological characteristics and dynamical properties of HIV/AIDS in the Fujian Province. Based on the features of real surveillance data from the Fujian Provincial Center for Disease Control and Prevention (Fujian CDC), the threshold R0 of the deterministic HIV/AIDS model was derived, and the local and global stabilities for the disease-free point and endemic equilibrium point were extensively investigated in Section 3. Meanwhile, we introduced environmental fluctuations into the deterministic HIV/AIDS model; the conditions for the stationary distribution and stochastic extinction were investigated in Section 4. Moreover, we conducted numerical simulations and made predictions on the scale of HIV/AIDS infections in the Fujian Province in Section 5. The results of this study provided new perspectives to control the scale of HIV infection for local governments and policy makers including, but not limited to, the Fujian Province.

    HIV/AIDS infection cases from surveillance data (2004–2011) in the Fujian CDC were recorded as either mild cases or severe cases. Based on the transmission mechanism and symptoms of HIV/AIDS, in this paper, we proposed symptom-oriented HIV/AIDS models in a local population. Here, the total population was separated into four the following mutually-exclusive compartments: S, susceptible with no HIV/AIDS infection; E, HIV infected with no clinical symptoms but were able to transmit the HIV virus to others (HIV virus lived in the hosts but did not produce clinical symptoms, the hosts with an HIV virus lacked awareness of going to hospital and getting checked); Im, HIV infected with mild symptoms; and Is, HIV infected with severe symptoms. Hence, the total population at time t for a designated region was given by N(t)=S(t)+E(t)+Im(t)+Is(t). Especially, for the key men who have sex with men (MSM) population with HIV/AIDS, the transmission mechanism of HIV/AIDS was usually described by the bilinear incidence between S and E, as well as S and Im in recent contributions such as equations-oriented descriptions in [21,23,25,27] and fluctuation-oriented descriptions in [28,29]. The aforementioned HIV/AIDS models with extensive discussions motivated us to govern the bilinear incidence for investigating symptom-dependent HIV/AIDS models in this paper:

    {˙S(t)=ΛβeS(t)E(t)βmS(t)Im(t)μS(t),˙E(t)=βeS(t)E(t)+βmS(t)Im(t)(γe+μ)E(t),˙Im(t)=γeE(t)(γm+μ)Im(t),˙Is(t)=γmIm(t)(δ+μ)Is(t), (2.1)

    where Λ was the constant recruitment rate, βe depicted the effective contact coefficient between S and E, βm was the effective contact coefficient between S and Im, 1/γe represented the average time from the date of being infected by the HIV virus to the date with mild symptoms, 1/γm was the average time required from the date of being detected with mild symptoms to the date with severe symptoms, 1/δ was the average time from the date that they had severe symptoms to the date that they died, and μ denoted the nature death rate. All parameters were assumed to be positive by their biological meanings.

    Let X=(S,E,Im,Is) T. Adding up the four equations in (2.1), and using the standard comparison theorem, we obtained that N(t)λ/μ; furthermore, we considered the dynamical properties in the positive invariant sets Ω={XR4+|0S+E+Im+IsΛ/μ} and Ω0={XΩ|E=Im=Is=0} in this paper. It was easy to check that P0=(S0,E0,Im0,Is0) T=(Λ/μ,0,0,0) T was the disease-free equilibrium point. It was known that the basic reproduction number was an important threshold for describing the average number of new HIV/AIDS infection cases produced by one HIV infected individual, which was computed by the next generation matrix in [30]. Let F be the new infections in compartments E,Im,Is, and let V=VV+, in which V was the transfer rate for the individuals moving out of three compartments, V+ was the transfer rate for the individuals entering into the compartments by all other means. Then, we wrote down the following:

    F:=(βeSE+βmSIm00),V:=(γeE+μEγmImγeE+μImγmIm+δIs+μIs).

    The Jacobian matrices for F and V were computed respectively as follows:

    F=(βeSβmS0000000),V=(γe+μ00γeγm+μ00γmδ+μ).

    Substituting the disease-free equilibrium point P0 into matrix FV1 gave the following basic reproduction number:

    R0:=ρ(FV1)=k4Λk1k3μ, (3.1)

    where ρ() denoted the spectral radius, and k1=γm+μ,k2=δ+μ,k3=γe+μ,k4=γeβm+k1βe. R0 reflected the average number that an HIV/AIDS infection case transmitted the HIV virus to the susceptible individuals without interventions.

    Alternatively, the endemic equilibrium point P=(S,E,Im,Is) T existed for model (2.1) when R0>1, where

    S=k1Λk4E+k1μ,E=k4Λk1k3μk3k4,Im=γeEk1,Is=γeγmEk1k2.

    In fact, let fR denote the right-hand side of the second equation in (2.1) and substitute S,Im,Is into the equation. Then, the following expression was obtained:

    f(E):=βeSE+βmSImk3E=k3k4E2+(γeβmΛ+k1βeΛk1k3μ)Ek4E+k1μ=AE2+BEk4E+k1μ=ˆf(E)g(E),

    in which A=k3k4>0,B=k4Λk1k3μ=k1k3μ(R01)>0, ˆf(E)=AE2+BE, g(E)=k4E+k1μ. It was easy to check that ˆf(E) is a quadratic function with a negative quadratic coefficient and f(0)=0; therefore, when R0>1, then ˆf(E)=0 had a unique positive real root, which implied that f(E)=0 also admitted a unique positive real root since g(E)>0 held for ER+.

    Furthermore, in this part, the locally and globally asymptotic stabilities for two equilibrium points, P0 and P, to model (2.1) were concerned using the Routh-Hurwitz Theorem and LaSalle's Invariant Principle, respectively.

    Theorem 3.1. (i). If R0<1, then the disease-free equilibrium point P0 was locally asymptotically stable in domain Ω.

    (ii). If R0>1, then P0 was unstable, and the endemic equilibrium point P was locally asymptotically stable in ΩΩ0.

    Proof. (ⅰ) The Jacobian matrix of model (2.1) was expressed as follows:

    J(X)=(βeEβmImμβeSβmS0βeE+βmImβeSk3βmS00γek1000γmk2).

    Substituting P0 into J(X), then the eigenvalues of the characteristic equation det(λIJ(P0))=0 were given by λ1=μ<0,λ2=k2<0 and

    λ3,4=12(βeS0γeγm)μ±12β2eS20+(2βe(γmγe)+4γeβm)S0+(γmγe)2.

    Here,

    λ3<12(βeS0γeγm)μ12|βeS0γe+γm|.

    If βeS0γe+γm0, then λ3<γmμ<0; if βeS0γe+γm<0, then we obtained that λ3<k3(R01)γeβmS0k1<0 by R0<1. The product of λ3 and λ4 gave the following expression:

    λ3λ4=μ(βeS0γeγm)+μ2(γmβe+γeβm)S0+γeγm=k1k3(R01)>0,

    together with λ3<0, which implied that λ4<0. By the Routh-Hurwitz Theorem, the disease-free equilibrium point P0 was local asymptotically stable.

    (ⅱ). If R0>1, then λ3λ4=k1k3(R01)<0. Thus, there existed one positive eigenvalue, and the disease-free equilibrium point P0 was unstable.

    Then, we substituted P into J(X) to investigate the local asymptotic stability of the endemic equilibrium point P as follows:

    J(P)=(βeEβmImμβeSβmS0βeE+βmImβeSk3βmS00γek1000γmk2):=(J10k2).

    Obviously, λ4=k2<0. We denoted k5=k3(k4E+k1μ)=Λk4. Let the characteristic equation of J(P) be λ3+a2λ2+a1λ+a0=0, where

    a2=k1+k3+μ+βeE+βmImβeS=Λk5[(k1+k3+μ)k4k1k3βe]+βeE+βmIm=k1+μ+Λk3γeβmk5+βeE+βmIm>0,a1=k1k3+(k1+k3)μ+(k1+k3)(βeE+βmIm)(k1βe+μβe+γeβm)S=Λk5(μk1k4+k2μγeβm)+(k1+k3)(βeE+βmIm)=μk1+Λk2μγeβmk5+(k1+k3)(βeE+βmIm)>0,a0=k1k3μ+k1k3(βeE+βmIm)μk4S=Λk5(k1k3k4μμk1k3k4)+k1k3(βeE+βmIm)=k1k3(βeE+βmIm)>0.

    Then,

    a2a1a0=[μ+1k5(Λk3γeβm)+βeE+βmIm]×[μk1+1k5(Λk2μγeβm)+(k1+k3)(βeE+βmIm)]+k1[μk1+1k5(Λk2μγeβm)+k1(βeE+βmIm)]>0.

    By the Routh-Hurwitz Theorem, the endemic equilibrium point P was locally asymptotically stable in ΩΩ0.

    Theorem 3.2. (i). If R01, then the disease-free equilibrium point P0 was globally asymptotically stable in domain Ω.

    (ii). If R0>1, then the endemic equilibrium point P was globally asymptotically stable in ΩΩ0.

    Proof. Noting that the compartment Is did not appear in the first three equations of model (2.1), we considered the following equivalent model:

    {˙S(t)=ΛβeS(t)E(t)βmS(t)Im(t)μS(t),˙E(t)=βeS(t)E(t)+βmS(t)Im(t)(γe+μ)E(t),˙Im(t)=γeE(t)(γm+μ)Im(t), (3.2)

    with the following positive invariant sets:

    ¯Ω={(S,E,Im)R3+|0S+E+ImΛ/μ},¯Ω0={(S,E,Im)¯Ω|E=Im=0}.

    Then, the dynamics of model (2.1) was the same as model (3.2).

    (i). We defined a C2-function V1:R3+R+ by the following:

    V1(S,E,Im)=SS0S0lnSS0+E+Λβmk1μIm.

    The time derivation of V1 along the positive solution was given by the following:

    ˙V1=(1S0S)(ΛβeSEβmSImμS)+βeSE+βmSImk3E+Λβmk1μ(γeEk1Im).

    Using relation S0=Λμ, we derived the following:

    ˙V1=(1S0S)(ΛΛSS0)+βeS0E+βmS0Imk3E+Λβmk1μ(γeEk1Im)=Λ(2SS0S0S)+(Λβeμ+Λβeγek1μk3)E=Λ(2SS0S0S)+k3(R01)E.

    Thus, ˙V10 when R01. The equality held if and only if S=S0,E=0, which corresponded to ¯Ω1={(S,E,Im)¯Ω:S=Λ/μ,E=0}¯Ω. If R0=1, then ˙V10; the equality was valid if and only if S=S0, which corresponded to ¯Ω2={(S,E,Im)¯Ω:S=Λ/μ}¯Ω. Recalling the boundedness of ¯Ω, the maximal compact invariant set of model (3.2) on ¯Ω1 and ¯Ω2 contained only one element, P0. According to LaSalle's Invariant Principle, the disease-free equilibrium point P0 was globally asymptotically stable.

    (ii). For convenience, we denoted x=SS,y=EE,z=ImIm. By multiplying 1S,1E and 1Im on both sides of model (3.2), together with the identities

    ΛβeSEβmSImμS=0,βeSE+βmSIm(γe+μ)E=0,γeE(γm+μ)Im=0,

    model (3.2) was transformed into the following form:

    {˙x=x[ΛS(1x1)βeE(y1)βmIm(z1)],˙y=y(βeS+βmSImyEk3),˙z=z(γeEyImzk1). (3.3)

    We constructed a C2-function V2:R3+R+ by

    V2(x,y,z)=S(x1lnx)+E(y1lny)+βmSI2mγeE(z1lnz)

    to verify the global stability of the endemic equilibrium point ˉP for model (3.3). Differentiating V2 with respect to t along the positive solution of model (3.3), we obtained the following:

    ˙V2=S(x1)˙xx+E(y1)˙yy+βmSI2mγeE(z1)˙zz=S(x1)[ΛS(1x1)βeE(y1)βmIm(z1)]+E(y1)(βeS+βmSImyEk3)+βmSI2mγeE(z1)(γeEyImzk1)=(2ΛβeSEβmSIm)Λ(x+1x)βeSE(xyxy)βmSIm(xzxz)+βeSE+βmSIm+βeSE(xyxy)+βmSIm(xzxzyy)+βmSIm+βmSIm(yzyz)=(ΛβmSIm)(2x1x)+βmSIm(31xxzyyz).

    The following relationships ΛβmSIm>ΛβeSEβmSImμS=0,βmSIm>0 and 2x1x0,31xxzyyz0 yielded that ˙V20. The equality was valid if and only if x=1 and y=z, which implied that S=S and EE=ImIm. Therefore, the maximal compact invariant set of model (3.2) on

    ¯Ω3={(S,E,Im)R3+:S=S,EE=ImIm}¯Ω¯Ω0,

    was a singleton {P}. Hence, by LaSalle's Invariant Principle, the unique endemic equilibrium point P was globally asymptotically stable for R0>1.

    Epidemic models with environmental fluctuations were profitable to provide additional perspectives, such as the survival analysis, as compared with their deterministic counterparts [31]. In general, environmental fluctuations contained linear fluctuations in [32,33] and nonlinear fluctuations in [34]. Some studies have shown that the evolving process of HIV was naturally subject to environmental fluctuations [21,22,25] such as policies, medical systems, climate and so on. Motivated by the previous contributions, in this paper, we assumed that X=(S,E,Im,Is) T was a Markov process [35] and assumed that the environmental fluctuations were proportional to S,E,Im, and Is. Taking the environmental fluctuations into account, model (4.1) was described by the following form:

    {dS(t)=[ΛβeS(t)E(t)βmS(t)Im(t)μS(t)]dt+σ1S(t)dB1(t),dE(t)=[βeS(t)E(t)+βmS(t)Im(t)(γe+μ)E(t)]dt+σ2E(t)dB2(t),dIm(t)=[γeE(t)(γm+μ)Im(t)]dt+σ3Im(t)dB3(t),dIs(t)=[γmIm(t)(δ+μ)Is(t)]dt+σ4Is(t)dB4(t), (4.1)

    where Bi(t) were four independent standard Brownian motions defined on a complete filtered probability space (Ω,F,{Ft}t0,P) with a filtration {Ft}t0, which was increasing and right continuous while F0 contained all P-null sets [35] with the initial values Bi(0)=0; σi reflected the intensities of white noises for i=1,2,3,4.

    Then, we showed that there existed a unique global positive solution to model (4.1) for any given initial values, which was described by the undermentioned Theorem 4.1.

    Theorem 4.1. Model (4.1) had a unique global positive solution X(t)R4+ with the initial value X0R4+ for any t0.

    Proof. Since the coefficients of model (4.1) were locally Lipschitz continuous, there existed a unique local solution X(t) on t[0,τe) for any initial value X0R4+, where τe denoted the explosion time. In order to prove that X(t) was global, we needed to verify that τe= held almost surely. Let n0>1 be sufficiently large such that each component of X(0) stayed in [1n0,n0]. Let the infimum of an empty set equals . Obviously, {τn}nn0 was monotonically increasing as n. Set τ=limnτn; then, we obtained ττe by the definition of stopping time. The proof was given by a contradiction. We assumed that there existed a pair of positive constants T>0 and ε(0,1) such that the probability that τT was larger than ε for any nn0 (i.e., P{τnT}ε). We defined a C2-function V1:R4+R+ by the following:

    V1(X)=(S1lnS)+(E1lnE)+(Im1lnIm)+(Is1lnIs).

    Applying the Itô's formula, we obtained the following:

    LV1(X)<Λ+γe+γm+δ+4μ+12(σ21+σ22+σ23+σ24).

    The remaining parts of the proof followed the approaches in [36,37,38] and we omitted them here.

    Next, we provided two useful results, Lemmas 4.1 and 4.2, and the corresponding proofs were quite similar with Lemmas 2.1 and 2.2 in [39]; therefore, we omitted them here. We denoted X0=(S(0),E(0),Im(0),Is(0)) T.

    Lemma 4.1. Let X(t) be a solution of model (4.1) initiated with X0R4+; then,

    limtS(t)t=0,limtE(t)t=0,limtIm(t)t=0,limtIs(t)t=0a.s..

    Lemma 4.2. Suppose that μ>(σ21σ22σ23σ24)/2. Let X(t) be a solution of model (4.1) initiated with X0R4+; then,

    limt1tt0S(s)dB1(s)=0,limt1tt0E(s)dB2(s)=0a.s.,limt1tt0Im(s)dB3(s)=0,limt1tt0Is(s)dB4(s)=0a.s..

    Theorem 4.2. For any initial value X0R4+, the solution of model (4.1) was stochastically ultimately bounded.

    Proof. We defined V2(X)=Sθ+Eθ+Iθm+Iθs, where θ(0,1). By using Itô's formula, we obtained dV2(X)=LV2(X)dt+G(X)dB(t), in which

    LV2(X)=θSθ1(ΛβeSEβmSImμS)+θEθ1[βeSE+βmSIm(γe+μ)E]+θIθ1m[γeE(γm+μ)Im]+θIθ1s[γmIm(δ+μ)Is]+θ(θ1)2[σ21Sθ+σ22Eθ+σ23Iθm+σ24Iθs],

    and

    G(X)dB(t)=θσ1SθdB1(t)+θσ2EθdB2(t)+θσ3IθmdB3(t)+θσ4IθsdB4(t).

    We defined F(X):=V2(X)+LV2(X), which was bounded in R4+ to model (4.1). Furthermore, there existed a constant H1 such that F(X)H1<; hence, LV2(X)H1V2(X). Again, by Itô's formula, for etV2(X), we obtained the following:

    d[etV2(X)]=et[V2(X)+LV2(X)]dt+etG(X)dB(t)H1etdt+etG(X)dB(t). (4.2)

    Integrating (4.2) and taking the expectation yielded the following:

    E[eτntV2(X(τnt))]V2(X0)+H1Eτnt0esds.

    Let n, we obtained the following:

    etEV2(X(t))V2(X0)+H1(et1)V2(X0)+H1et.

    Noticing that

    |X|θ4θ/2max1i4Xθi4θ/2V2(X),

    we obtained

    E|X(t)|θ2θ(etV2(X0)+H1),

    which revealed that

    lim suptE|X(t)|θ2θH1.

    Now, for any ε>0, let H=(2θH1/ε)2. By Chebyshev's Inequality, we obtained the following:

    P{|X(t)|>H}E|X(t)|1/2H1/2.

    Let θ=1/2. Then, we derived

    lim suptP{|X(t)|>H}21/2H1H1/2=ε,

    which was rewritten as follows:

    lim suptP{|X(t)|H}1ε.

    The proof was complete.

    In the undermentioned Theorem 4.3, we established the sufficient conditions to guarantee the existence of a stationary distribution and the ergodicity of model (4.1). This main result showed that HIV exhibited a sustainable behavior in a long run. We denoted

    Rs0=Λk4(k1+14σ23)(k3+12σ22)(μ+12σ21),

    which degenerated to R0 when σ1=σ2=σ3=0. Here, the expression of Rs0 was independent of the fluctuation of Is. In order to show the main result for Theorem 4.3, we used Lemma 3.1 of [25] to check the details.

    Theorem 4.3. Model (4.1) admitted a unique stationary distribution, and it was ergodic when Rs0>1.

    Proof. The positive definite diffusion matrix of model (4.1) was A(X)=diag{σ21S2,σ22E2,σ23I2m,σ24I2s}; therefore, condition (ⅰ) of Lemma 3.1 of [25] was satisfied. To prove condition (ⅱ), we constructed a C2-function J(X):=MV3+V4+V5+V6, with V3=2c1lnSc2lnEc3lnIm, V4=(S+E+Im+Is)m+1, V5=lnS, V6=lnIs, ciR+ (i=1,2,3); M was a sufficiently large positive constant, while m was a sufficiently small positive constant. Meanwhile, M and m satisfied the following relations:

    3M(3Rs01)+(2c1βeM+βe)ε1+Fe11,3M(3Rs01)+(2c1βmM+βm)ε1+Fe21,ˆM:=μ12m(σ21σ22σ23σ24)>0, (4.3)

    where B, F, e1, e2 were defined in (4.6) and (4.12), respectively. Then, we obtained the following:

    limk infXR4+DkJ(X)=+,

    where Dk=(1k,k)×(1k,k)×(1k,k)×(1k,k). Obviously, J(X) was a continuous function. We assumed that the minimum value of J(X) was ˜J. We defined a non-negative C2-function Q(X)=J(X)˜J. Itô's formula acted on V3, which gave the following:

    LV3=2c1ΛS+2c1βeE+2c1βmIm+2c1(μ+12σ21)c2βeSc2βmSImE+c2(k3+12σ22)c3γeEIm+c3(k1+12σ23)=(c1ΛS+c2βmSImE+c3γeEIm)(c1ΛS+c2βeS+c3k1)+2c1βeE+2c1βmIm+2c1(μ+12σ21)+c2(k3+12σ22)+2c3(k1+14σ23). (4.4)

    By applying a+b+c33abc and 3a+3b3a+b for positive a,b and c, expression (4.4) turned to the following:

    LV333c1c2c3Λ(βmγe+βek1)+2c1βeE+2c1βmIm+2c1(μ+12σ21)+c2(k3+12σ22)+2c3(k1+14σ23).

    We set

    c1=1μ+12σ21,c2=1k3+12σ22,c3=1k1+14σ23,

    then,

    LV33(3Rs01)+2c1βeE+2c1βmIm+2. (4.5)

    Similarly,

    LV4=(m+1)Nm(ΛμNδIs)+12(m+1)mNm1(σ21S2+σ22E2+σ23I2m+σ24I2s)(m+1)ΛNm(m+1)ˆMNm+1B12(m+1)ˆM(Sm+1+Em+1+Im+1m+Im+1s), (4.6)

    with

    B=supNR+(m+1)(ΛNm12ˆMNm+1)<.

    Additionally, we derived the following:

    LV5=ΛS+βeE+βmIm+μ+12σ21,LV6=γmImIs+δ+μ+12σ24. (4.7)

    From (4.5)–(4.7), we obtained the following:

    LQ3M(3Rs01)+(2c1βeM+βe)E+(2c1βmM+βm)Im12(m+1)ˆM(Sm+1+Em+1+Im+1m+Im+1s)ΛSγmImIs+δ+2μ+12σ21+12σ24+B+2M. (4.8)

    We defined a bounded region as follows:

    H={XR4+:ε1S1ε1,ε1E1ε1,ε1Im1ε1,ε21Is1ε21},

    where ε1>0 was sufficiently small and satisfied the following relations:

    min{Λ,γm}ε1+F1, (4.9)
    ˆM4εm+11(m+1)+F1, (4.10)
    ˆM2ε2m+21(m+1)+F1, (4.11)

    with

    F:=2M+B+e1+e2+δ+2μ+12σ21+12σ24,e1:=supER+{ˆMEm+14(m+1)+(2c1βeM+βe)E}<,e2:=supImR+{ˆMIm+1m4(m+1)+(2c1βmM+βm)Im}<. (4.12)

    Obviously, R4+H=D1D2D8, where

    D1={XR4+:0<S<ε1},D2={XR4+:0<E<ε1},D3={XR4+:0<Im<ε1},D4={XR4+:0<Is<ε21,Imε1},D5={XR4+:S1/ε1},D6={XR4+:E1/ε1},D7={XR4+:Im1/ε1},D8={XR4+:Is1/ε21}.

    With (4.3) and (4.8), we discussed each case as follows:

    Case 1. When XD1, by (4.9), we obtained the following:

    LQΛS+FΛε1+F1.

    Case 2. When XD2, due to (4.8), we derived the following:

    LQ3M(3Rs01)+(2c1βeM+βe)E+Fe13M(3Rs01)+(2c1βeM+βe)ε1+Fe11.

    Case 3. When XD3, in view of (4.8), we obtained the following:

    LQ3M(3Rs01)+(2c1βmM+βm)Im+Fe23M(3Rs01)+(2c1βmM+βm)ε1+Fe21.

    Case 4. When XD4, according to (4.9), we derived the following:

    LQγmImIs+Fγmε1+F1.

    Case 5. When XD5, by (4.10), we obtained the following:

    LQˆMSm+12(m+1)+FˆM2εm+11(m+1)+F1.

    Case 6. When XD6, due to (4.10), we derived the following:

    LQˆMEm+12(m+1)+(2c1βeM+βe)E+Fe1ˆMEm+14(m+1)+FˆM4εm+11(m+1)+F1.

    Case 7. When XD7, in view of (4.10), we obtained the following:

    LQˆMIm+1m2(m+1)+(2c1βmM+βm)Im+Fe2ˆMIm+1m4(m+1)+FˆM4εm+11(m+1)+F1.

    Case 8. When XD8, according to (4.11), we derived the following:

    LQ12(m+1)ˆMIm+1s+F12(m+1)ˆMε2m+21+F1.

    Therefore, LQ1 when XR4+H. Consequently, condition (ii) of Lemma 3.1 was satisfied.

    The sufficient conditions for the extinction were established with notation S(t)=1tt0S(s)ds.

    Theorem 4.4. If the following conditions held,

    Re0=Λ(βe+βm)μ(μ+13ˆσ)<1,μ>12(σ21σ22σ23σ24),ˆσ=12σ2212σ23(δ+12σ24),

    then the solution of model (4.1) satisfies the following:

    limt1tln(E(t)+Im(t)+Is(t))<0a.s.,

    which implied that HIV/AIDS became extinct with an exponential rate.

    Proof. Integrating the first equation of model (4.1), then, by Lemmas 4.1 and 4.2, we obtained the following:

    limt1t(S(t)S(0))limt(ΛμS(t)+σ1tt0S(s)dB1(s)),

    which indicated

    limtS(t)Λμa.s.. (4.13)

    By Itô's formula, we defined W(X)=E+Im+Is and obtained the following:

    dlnW(X)=LlnW(X)dt+σ2EWdB2(t)+σ3ImWdB3(t)+σ4IsWdB4(t), (4.14)

    where

    LlnW(X)=βeSEW+βmSImWμδIsW12W2(σ22E2+σ23I2m+σ24I2s)(βe+βm)Sμ1W2[σ222E2+σ232I2m+(δ+σ242)I2s](βe+βm)SμE2+I2m+I2sW2ˆσ(βe+βm)Sμ13ˆσ.

    The integration on (4.14) provided the following:

    1t(lnW(X(t))lnW(X(0)))(βe+βm)S(t)μˆσ3+σ2tB2(t)+σ3tB3(t)+σ4tB4(t). (4.15)

    Applying the strong law of large numbers [35], we obtained the following:

    limtBi(t)t=0(i=2,3,4). (4.16)

    Recalling (4.13), (4.16), Re0<1, and letting t, we derived from (4.15) that

    lim suptlnW(X(t))tΛ(βe+βm)μμˆσ3=(Re01)(μ+ˆσ3)<0.

    Thus, the number of infected individuals decreased to zero with an exponential rate in the long run.

    Remark 4.1. The following relationships held:

    R0>Rs0andRe0>Rs0.

    Moreover, R0<Re0 when min{γe,γm}13ˆσ. Precisely, model (4.1) had a unique stationary distribution when Rs0>1. Furthermore, R0>Rs0>1, which implied that the solution of model (2.1) eventually approached the endemic equilibrium point P. However, R0>1 revealed that HIV/AIDS became an endemic disease for model (2.1), but not for model (4.1) , since Rs0>1 was not guaranteed.

    In this section, we mainly analyzed the features of the surveillance data from the Fujian CDC in 2004–2011. Figure 1 presents the yearly incidence of HIV/AIDS infection cases with gender groups and age groups, and the maximum age, the minimum age and the average age for all HIV/AIDS infection cases in the Fujian Province are clearly presented within the left panel. Meanwhile, the yearly incidence of deaths with age groups are presented within the right panel of Figure 1, in which the cumulative incidence of deaths over eight years were 2.05/100,000 and 0.20/100,000 for all ages and for 60 years old and over, respectively. Precisely, the monthly incidence of infection cases and severe cases by genders were investigated in Figure 2.

    Figure 1.  The maximum age, the minimum age, the average age of HIV/AIDS infection cases (left), and the yearly incidence (1/100,000) of deaths at all ages as well as 60 years old and over (right).
    Figure 2.  Monthly incidence (1/100,000) for infection cases (left) and severe cases (right) by genders in 2004–2011 from the Fujian CDC.

    According to the report in [40], the total population scale and the life expectancy of the Fujian Province in 2004 were 35,290,000 and 73.834, respectively, which then gave the following initial values of this study:

    N(0)=35,290,000,Λ=35,290,00073.834,μ=173.834.

    As of January 2004, nine HIV/AIDS infection cases were recorded by the Fujian CDC, in which seven mild cases (72.7%) and two severe cases (27.3%) were reported with no deaths. According to the research results in [41], the basic reproduction number of HIV/AIDS ranged from 1.3 to 6.0. Subsequently, we computed the total HIV/AIDS infection cases as follows: 7×6.0+2×1.3=44.645. By proportion, we derived Im(0)=33 and Is(0)=12 because we took potential HIV/AIDS infection cases before 2004 into account in this study. Furthermore, we assumed that the number of HIV/AIDS infection cases with no clinical symptoms was 1,000, that is, E(0)=1,000. By S(0)=N(0)E(0)Im(0)Is(0), we computed that S(0)=35,288,955. Therefore, all initial values were given herewith.

    With modified HIV policies from the Chinese government in February 2006 in [42], the targeted population with HIV test was extended, which produced an increase in the number of HIV/AIDS infection cases from 2006. Therefore, the values of βe were usually regarded as distinct before and after 2006. Utilizing the least squares method, βe was estimated by two parts: β1e=0.25835,290,000 for 2004–2005, β2e=0.41735,290,000 for 2006–2011. Moreover, βm was estimated by 0.12035,290,000. By comparing the population scales of Mexico and the Fujian Province, we took γe=0.1411 in [43]. The surveillance data from the Fujian CDC indicated that the probability that an HIV/AIDS infection case developed into a severe case was 0.2731. Meanwhile, the average time from Im to Is was 2.0162 years, which gave γm=0.27312.0162. The average time for a severe case to death was 1.1419 years, which indicated that δ=11.1419. We collected all parameters in Table 1.

    Table 1.  Parameters in Fujian province.
    Parameter Value Period Source
    Λ 35,290,00073.834 2004–2011 [40]
    μ 173.834 2004–2011 [40]
    β1e 0.25835,290,000 2004–2005 Estimated
    β2e 0.41735,290,000 2006–2011 Estimated
    βm 0.12035,290,000 2004–2011 Estimated
    γe 0.1411 2004-2011 [43]
    γm 0.27312.0162 2004–2011 Fujian CDC
    δ 11.1419 2004–2011 Fujian CDC

     | Show Table
    DownLoad: CSV

    By the surveillance data from the Fujian CDC, the gender and age distributions of HIV/AIDS infection cases in 2004–2011 were recorded, in which the proportions for male infection cases (69.9%) and female infection cases (30.1%) were investigated during the period 2004–2011, and the percentages for key infection cases (85.9%, 20–49 years old) and ordinary infection cases (14.1%, 1–19 years old as well as 50 years old and over) were discussed by clustering the data. The curves for cumulative incidences of infection cases by genders are plotted on left panel of Figure 3, in which the simulations for total incidence (in yellow), male incidence (in blue) and female incidence (in pink) were performed. The analysis for the cumulative incidences of infection cases by age are shown on right panel of Figure 3. By the same argument, similar investigations on the cumulative incidence of severe cases, the proportions of male and female cases, and the proportions of key and ordinary cases are presented in Figure 4.

    Figure 3.  Monthly and cumulative incidences (1/100,000) of HIV/AIDS infection cases in 2004–2011 by genders, 69.9% for male, 30.1% for female (left); 85.9% for key infection cases, 14.1% for ordinary infection cases (right).
    Figure 4.  Monthly and cumulative incidences (1/100,000) of HIV/AIDS severe cases in 2004–2011 by genders, 68.35% for male, 31.65% for female (left); 79.75% for key severe cases, 20.25% for ordinary severe cases (right).

    Furthermore, predictions of HIV/AIDS incidence for the period 2012–2014 were made. We chose the values of βe from 0.37735,290,000 to 0.41735,290,000, and kept other parameters the same as Table 1. Then, the results of the predictions were derived, which ranged from 22.97/100,000 to 23.84/100,000 at the end of 2014 as presented on left panel of Figure 5. The minimum value of βe was estimated by 2β1e+6β2e8, and the maximum value was same with β2e. By the proportion of male infection cases, the possible incidence rate ranged from 16.6/100,000 and 16.66/100,000, and by the percentage of key infection cases, the possible incidence rate ranged from 19.73/100,000 to 20.48/100,000 at the end of 2014. Especially, by the same discussion, the predictions of incidence for severe cases during the period 2012–2014 were operated, which produced the range from 7.12/100,000 to 7.23/100,000 at the end of 2014, and were presented on right panel of Figure 5.

    Figure 5.  Predictions of cumulative incidences of infection cases and severe cases in 2012–2014.

    The recent contributions in [44,45] established approaches regarding the positivity preserving truncated Euler-Maruyama (PPTEM) method, which supported the numerical simulations on the extinction and persistence of stochastic models. Moreover, the PPTEM method was governed on the extinction of stochastic models to avoid the negative values. We used the Milstein's higher order (MHO) method [46] to establish the discretization equations of model (4.1), due to its efficiency and performance of the persistence of stochastic models.

    As given by Table 1, the basic reproduction number R0 was 2.4032 for the period between 2004 and 2005 and 3.3990 for the period between 2006 and 2011. Our investigation showed that the solution of model (2.1) eventually approached the globally asymptotically stable P, but was repelled by the unstable P0 as claimed in Theorem 3, where P and P0 were calculated as P=(2.91×104,6.21×103,5.88×103,8.59×102) T, and P0=(105,0,0,0) T. Additionally, our investigation revealed that model (4.1) admitted the indicator Rs03.3849 when taking σi=0.01(i=1,2,3,4). By Theorem 4.3, we concluded that HIV/AIDS was stochastically persistent for a long run as demonstrated on left panel of Figures 68. When we repeated 3,000 runs, the corresponding curves for the histogram were respectively carried out as shown on right panel of Figures 6-8. Specifically, we noticed that R0>Rs0>1 held in Remark 4, which indicated that the stochastic persistence to model (4.1) implied the persistence to model (2.1) for HIV/AIDS.

    Figure 6.  Stochastic persistence (left) and histogram (right) of E(t).
    Figure 7.  Stochastic persistence (left) and histogram (right) of Im(t).
    Figure 8.  Stochastic persistence (left) and histogram (right) of Is(t).

    Furthermore, we performed an operation for models (2.1) and (4.1) in 400 years. The research results showed that the solution for model (4.1) and the solution for model (2.1) matched well when we took σi=0.01(i=1,2,3,4). The deviations are shown in Figure 9. Then, we defined the proportion of deviation for the solutions of models (2.1) and (4.1) by the following form:

    ds(t)=solution of model (2.1)solution of model (4.1)solution of model (2.1),

    where the proportion of deviations varied with the time t. The research results showed that when we took σi=0.01(i=1,2,3,4), the maximum absolute values of the proportion of deviations were 5.35% for E, 5.93% for Im and 6.15% for Is.

    Figure 9.  Deviations for the solutions to models (2.1) and (4.1).

    Furthermore, the changes of the solutions and the probability density functions to model (4.1) were explored when the intensities of white noises σi=0.01,0.02,0.03,0.04,0.05 (i=1,2,3,4) were chosen and other parameters were kept the same as Table 1. Then, the corresponding numerical simulations of the solutions were derived on left panels of Figures 1012. Meanwhile, the changes of the probability density functions were demonstrated on right panels of Figures 1012 when the intensities of white noises increased.

    Figure 10.  Comparison of stochastic persistence for E.
    Figure 11.  Comparison of stochastic persistence for Im.
    Figure 12.  Comparison of stochastic persistence for Is.

    The research results showed that the tendencies of the numerical simulations for next five decades were very similar when we compared Figure 6.1 for Indonesia in [25] and Figures 78 for the Fujian Province in this study. The incidence (1/100,000) of HIV/AIDS infection cases for the Fujian Province reached a peak with about 2.86×104, then rapidly declined and gradually became stable around at 1.61×104, which revealed that the potential risk of medical burden raised near the infection peak. We suggested that the local government adopted more tough strategies before the medical burden appeared, and medical service centers were asked to prepare the future medical resources as early as possible.

    By surveillance data from 2004 to 2011 of the Fujian CDC, we proposed symptom-dependent HIV/AIDS models since the mild and severe symptom cases were recorded well, and further investigated the survival dynamics such as persistence, extinction and stationary distribution. The research results showed that the disease-free equilibrium P0 was locally and globally asymptotically stable for model (2.1) when R0<1; additionally, P0 was unstable, whereas the endemic equilibrium point P was locally and globally asymptotically stable when R0>1. When it came to the environmental fluctuations, the interaction mechanism among compartments of model (2.1) remained the same. Based on the existence and uniqueness of the global positive solution, the ergodic stationary distribution of stochastic model (4.1) was valid when Rs0>1, which indicated that HIV/AIDS prevailed for the long run. Moreover, we investigated the sufficient condition of extinction for HIV/AIDS for a short period of time.

    The surveillance data of HIV/AIDS from the Fujian CDC gave that R03.3990>1 to model (2.1), which indicated that HIV/AIDS became an endemic disease for a long time in the Fujian Province, and that HIV/AIDS displayed an exponential upward trend in Figure 5. Meanwhile, model (4.1) displayed small fluctuations (say σi=0.01) provided that Rs03.3849>1, which revealed that HIV/AIDS cases of the Fujian Province persisted almost surely, as shown in Figures 6-8. However, when σi=0.10 were taken in Theorem 4 such that Rs02.3876>1, the sample paths did not present the stationary properties very well. To perform the 90–90–90% plan of WHO in [47], the scale to the HIV infection mainly relied on the reductions of βe (the effective contact coefficient between S and E) and βm (the effective contact coefficient between S and Im). For the susceptible population, the reduction of HIV infection might come from the following: individual-oriented protection such as male or female condom use; medical-system-oriented protection such as HIV tests and voluntary medical male circumcision in [48]; and social-oriented protection such as harm reduction services for people who inject and use drugs. Alternatively, some medicines and medical devices such as antiretroviral drugs (ARVs), dapivirine vaginal rings and injectable long acting cabotegravir were extensively available to prevent HIV, which reduced the value of γm, and the numbers of severe cases and deaths were therefore declined.

    Figure 5 indicated that the HIV/AIDS incidence rate ranged from 22.97/100,000 to 23.84/100,000 in the Fujian Province at the end of 2014; in comparison, the research results in [25] showed that the HIV/AIDS incidence rate was 36.36/100,000. In 2020, the incidence rate of HIV/AIDS infection cases for Mainland of China admitted 76.47/100,000 in [25], while the prediction by model (2.1) showed that the lower and upper incidence rates for the Fujian Province were 120.16/100,000 and 149.04/100,000, respectively, at the end of 2020. In comparison with other previous contributions, our models consisted of four equations with seven parameters to handle the surveillance data from the Fujian CDC, and the optimal fittings were performed very well. Our model (2.1) modified model (2.1) in [25] by combining the equations for I and C as an equation of Im, and keeping the equation of A as an equation of Is.

    The authors declare that they have not used Artificial Intelligence (AI) tools in the creation of this article.

    The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.

    Authorship contribution statement: Wenshuang Li: Modelling, Data curation, Software, Visualization, Writing-original draft. Shaojian Cai: Data curation, Validation, Writing-review. Xuanpei Zhai: Writing-original draft. Jianming Ou: Writing-review. Kuicheng Zheng: Writing-review. Fengying Wei: Methodology, Validation, Writing-review & editing. Xuerong Mao: Supervision, Methodology, Writing-review.

    The research of F.Wei was supported in part by Natural Science Foundation of Fujian Province of China (2021J01621); the research of X. Mao was supported by the Royal Society, UK (WM160014, Royal Society Wolfson Research Merit Award), the Royal Society of Edinburgh (RSE1832), Engineering and Physical Sciences Research Council, UK (EP/W522521/1) for their financial support.



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