Research article Special Issues

CAPTCHA recognition based on deep convolutional neural network

  • Aiming at the problems of low efficiency and poor accuracy of traditional CAPTCHA recognition methods, we have proposed a more efficient way based on deep convolutional neural network (CNN). The Dense Convolutional Network (DenseNet) has shown excellent classification performance which adopts cross-layer connection. Not only it effectively alleviates the vanishing-gradient problem, but also dramatically reduce the number of parameters. However, it also has caused great memory consumption. So we improve and construct a new DenseNet for CAPTCHA recognition (DFCR). Firstly, we reduce the number of convolutional blocks and build corresponding classifiers for different types of CAPTCHA images. Secondly, we input the CAPTCHA images of TFrecords format into the DFCR for model training. Finally, we test the Chinese or English CAPTCHAs experimentally with different numbers of characters. Experiments show that the new network not only keeps the primary performance advantages of the DenseNets but also effectively reduces the memory consumption. Furthermore, the recognition accuracy of CAPTCHA with the background noise and character adhesion is above 99.9%.

    Citation: Jing Wang, Jiaohua Qin, Xuyu Xiang, Yun Tan, Nan Pan. CAPTCHA recognition based on deep convolutional neural network[J]. Mathematical Biosciences and Engineering, 2019, 16(5): 5851-5861. doi: 10.3934/mbe.2019292

    Related Papers:

    [1] Jing Cai, Jianfeng Yang, Yongjin Zhang . Reliability analysis of s-out-of-k multicomponent stress-strength system with dependent strength elements based on copula function. Mathematical Biosciences and Engineering, 2023, 20(5): 9470-9488. doi: 10.3934/mbe.2023416
    [2] Amal S. Hassan, Najwan Alsadat, Christophe Chesneau, Ahmed W. Shawki . A novel weighted family of probability distributions with applications to world natural gas, oil, and gold reserves. Mathematical Biosciences and Engineering, 2023, 20(11): 19871-19911. doi: 10.3934/mbe.2023880
    [3] Ghada Mohammed Mansour, Haroon Mohamed Barakat, Islam Abdullah Husseiny, Magdy Nagy, Ahmed Hamdi Mansi, Metwally Alsayed Alawady . Measures of cumulative residual Tsallis entropy for concomitants of generalized order statistics based on the Morgenstern family with application to medical data. Mathematical Biosciences and Engineering, 2025, 22(6): 1572-1597. doi: 10.3934/mbe.2025058
    [4] Yarong Yu, Liang Wang, Sanku Dey, Jia Liu . Estimation of stress-strength reliability from unit-Burr Ⅲ distribution under records data. Mathematical Biosciences and Engineering, 2023, 20(7): 12360-12379. doi: 10.3934/mbe.2023550
    [5] Mahmoud El-Morshedy, Zubair Ahmad, Elsayed tag-Eldin, Zahra Almaspoor, Mohamed S. Eliwa, Zahoor Iqbal . A new statistical approach for modeling the bladder cancer and leukemia patients data sets: Case studies in the medical sector. Mathematical Biosciences and Engineering, 2022, 19(10): 10474-10492. doi: 10.3934/mbe.2022490
    [6] Thomas Hillen, Kevin J. Painter, Amanda C. Swan, Albert D. Murtha . Moments of von mises and fisher distributions and applications. Mathematical Biosciences and Engineering, 2017, 14(3): 673-694. doi: 10.3934/mbe.2017038
    [7] Mohamed Kayid . Some new results on bathtub-shaped hazard rate models. Mathematical Biosciences and Engineering, 2022, 19(2): 1239-1250. doi: 10.3934/mbe.2022057
    [8] Mohamed S. Eliwa, Buthaynah T. Alhumaidan, Raghad N. Alqefari . A discrete mixed distribution: Statistical and reliability properties with applications to model COVID-19 data in various countries. Mathematical Biosciences and Engineering, 2023, 20(5): 7859-7881. doi: 10.3934/mbe.2023340
    [9] Ibrahim Alkhairy . Classical and Bayesian inference for the discrete Poisson Ramos-Louzada distribution with application to COVID-19 data. Mathematical Biosciences and Engineering, 2023, 20(8): 14061-14080. doi: 10.3934/mbe.2023628
    [10] Ariel Cintrón-Arias, Carlos Castillo-Chávez, Luís M. A. Bettencourt, Alun L. Lloyd, H. T. Banks . The estimation of the effective reproductive number from disease outbreak data. Mathematical Biosciences and Engineering, 2009, 6(2): 261-282. doi: 10.3934/mbe.2009.6.261
  • Aiming at the problems of low efficiency and poor accuracy of traditional CAPTCHA recognition methods, we have proposed a more efficient way based on deep convolutional neural network (CNN). The Dense Convolutional Network (DenseNet) has shown excellent classification performance which adopts cross-layer connection. Not only it effectively alleviates the vanishing-gradient problem, but also dramatically reduce the number of parameters. However, it also has caused great memory consumption. So we improve and construct a new DenseNet for CAPTCHA recognition (DFCR). Firstly, we reduce the number of convolutional blocks and build corresponding classifiers for different types of CAPTCHA images. Secondly, we input the CAPTCHA images of TFrecords format into the DFCR for model training. Finally, we test the Chinese or English CAPTCHAs experimentally with different numbers of characters. Experiments show that the new network not only keeps the primary performance advantages of the DenseNets but also effectively reduces the memory consumption. Furthermore, the recognition accuracy of CAPTCHA with the background noise and character adhesion is above 99.9%.


    Tuberculosis (TB) is a common and fatal infectious disease. It has become a chronic infectious disease that threatens human health worldwide. Globally, in 2016 there were an estimated 10.4 million incident cases of TB, equivalent to 140 cases per 100000 population. Meanwhile, the proportion of people who develop TB and die from the disease (the case fatality ratio) was 16 [1]. Therefore, TB has become a global concern for social and public health issues.

    Many scholars have carried out a lot of excellent researches on the transmission mechanism and prevention strategies of TB [2,3,4,5,6]. Silva et al. [4] introduced delays in a TB model, and studied optimal control of TB with state and control delays. Huo et al. [5] presented a two-strain TB model with general contact rate which allows TB patients with the drug sensitive of strain Mycobacterium tuberculosis to be treated and gave a detailed qualitative analysis about positivity, boundedness, existence, uniqueness and global stability of the equilibria of the model. Huo and Zou [6] studied a TB model with two kinds of treatment, that is, treatment at home and treatment in hospital and showed that the treatment at home has a negative influence on the spread of TB.

    The susceptible individuals who carry the pathogen developing into infectious individuals are different from the progression of TB transmission. Some people may become symptomatic infectious individuals after a few days or months, and some people may occur after several years or even decades. For the former, it is considered that the susceptible individuals directly develop symptomatic infectious individuals without going through the latency period after infection, which is called the fast progression of TB transmission. For the latter, it is considered that the susceptible individuals become the latent individuals carrying the pathogen after infection with Mycobacterium tuberculosis, and they can become infected by exogenous reinfection or endogenous infection, which is called a slow progression of TB transmission. Huo and Feng [7] constructed an HIV/AIDS epidemic model with different latent stages and treatment. The model allowed for the latent individuals to have the fast and slow latent compartments. Mccluskey [8] introduced the spread of TB through two models which included fast and slow progression to the infected class. Berge et al. [9] considered a two patch cholera model with the aim of investigating the impact of human population movements between two cities(patches). Song et al. [10] studied TB models with fast and slow dynamics. Many scholars have studied infectious diseases related to the fast and slow progression (see e.g. [11] and references cited therein).

    Media coverage is changing the way that we communicate with each other in our daily life, work and study. The media may be the most important source of public health information. At the same time, it also plays an important role in the spread and control of epidemics by providing some health information. Cui et al. [12] proposed a general contact rate β(I)=c1c2f(I) to reflect some intrinsic characters of media coverage. Huo and Zhang [13] introduced a novel alcoholism model which involves impact of Twitter, and showed that Twitter can serve as a good indicator of alcoholism model and affect the spread of the drinking. Huo et al. [14] presented a SEIS epidemic model with the impact of media coverage. Their results manifested that media can be regarded as a good indicator in controlling the emergence and spread of the epidemic disease. Many scholars have done a lot of researches on infectious diseases with or without media coverage [15,16,17,18,19,20,21].

    Motivated by the above, we construct a new TB model which not only involves fast and slow progression but also incorporates the impact of media coverage in this paper. We study the stability of all the equilibria. Furthermore, we also investigate the occurrence of backward and forward bifurcation. Our results show that media coverage can encourage people to take countermeasures to avoid potential infections.

    The rest of this paper is organized as follows. In Section 2, a new tuberculosis model with fast and slow progression and media coverage is constructed. In Section 3, we discuss the existence and stability of all the equilibria, then we analyze a forward and backward bifurcation. Some numerical simulations are presented in Section 4. Sensitivity analysis and some discussions are given in the last section.

    The total population N(t) is divided into four compartments: S(t), E(t), I(t) and R(t). S(t) denotes susceptible individuals. E(t) is referred to as undetected non-symptomatic (latent) carriers. I(t) is symptomatic infectious individuals. R(t) represents recovered individuals. M(t) represents the number of message that all of them provide about TB at time t. The total population N(t) is given by

    N(t)=S(t)+E(t)+I(t)+R(t).

    The transfer diagram of the model is shown in Figure 1. The transfer diagram leads to the following system of ordinary differential equations:

    {dS(t)dt=δβSIeαMμS,dE(t)dt=(1q)βSIeαM(μ+ε)E,dI(t)dt=qβSIeαM+εE(d+μ+γ)I,dR(t)dt=γIμR,dM(t)dt=μ1S+μ2E+μ3I+μ4RτM. (2.1)
    Figure 1.  The transfer diagram of system (2.1).

    All the parameters are positive constants. δ is the constant recruitment rate of the population. β is the transmission coefficient of TB. α is the coefficient that determines how effective the disease-related messages can influence the transmission rate and the transmission rate β is reduced by a factor eαM (see [13,14]). μ is the natural death rate. q is the proportion of disease by fast progression. ε is the progression rate from the exposed individuals to the infected individuals. d is the disease-related death rate of TB. γ is the recovery rate of TB. τ is the rate that message become outdated. μ1, μ2, μ3 and μ4 are the rates that susceptible individuals, exposed individuals, infectious individuals, recovered individuals may send messages about TB, respectively.

    In this section, we will show positivity and boundedness for system (2.1).

    Lemma 1. If S(0)0,E(0)0,I(0)0,R(0)0,M(0)0, the solutions S(t), E(t), I(t), R(t), M(t) of system (2.1) are positive for all t>0.

    Proof. If S(0)0, according to the first equation of system (2.1), we have

    dS(t)dt=δ[βI(t)eαM(t)+μ]S(t).

    It can be rewritten as:

    dS(t)dtexp{t0[βI(u)eαM(u)+μ]du}+S(t)[βI(t)eαM(t)+μ]exp{t0[βI(u)eαM(u)+μ]du}=δexp{t0[βI(u)eαM(u)+μ]du}.

    Therefore,

    ddt(S(t)exp{t0[βI(u)eαM(u)+μ]du})=δexp{t0[βI(u)eαM(u)+μ]du}.

    Hence,

    S(t)exp{t0[βI(u)eαM(u)+μ]du}S(0)=t0(δexp{u0[βI(v)eαM(v)+μ]dv})du.

    So,

    S(t)=S(0)exp{t0[βI(u)eαM(u)+μ]du}+exp{t0[βI(u)eαM(u)+μ]du}{t0(δexp{u0[βI(v)eαM(v)+μ]dv})du}>0.

    Similarly, we can show that E(t)>0,I(t)>0,R(t)>0,M(t)>0. So the solutions S(t), E(t), I(t), R(t), M(t) of system (2.1) with initial conditions S(0)0,E(0)0,I(0)0,R(0)0,M(0)0 are positive for all t>0. This completes the proof of Lemma 1.

    Lemma 2. The feasible region Ω defined by

    Ω={(S,E,I,R,M)R5+:0S+E+I+Rδμ,0Mδ(μ1+μ2+μ3+μ4)μτ}

    with initial conditions S(0)0,E(0)0,I(0)0,R(0)0,M(0)0 is positively invariant for system (2.1).

    Proof. Adding the former four equations of system (2.1), we obtain

    dN(t)dt=δμN(t)dI(t)δμN(t).

    It follows that

    0N(t)δμ+N(0)eμt,

    where N(0) is the initial value of total number of people. Thus,

    limtsupN(t)δμ.

    Then

    0S(t)+E(t)+I(t)+R(t)δμ.

    Further, from the last equation of system (2.1), we have

    dM(t)dt=μ1S(t)+μ2E(t)+μ3I(t)+μ4R(t)τM(t)δμ(μ1+μ2+μ3+μ4)τM(t).

    It follows that

    0M(t)δ(μ1+μ2+μ3+μ4)μτ+M(0)eτt,

    where M(0) represents the initial value of cumulative density media coverage. Thus,

    limtsupM(t)δ(μ1+μ2+μ3+μ4)μτ.

    It implies that the region

    Ω={(S,E,I,R,M)R5+:0S+E+I+Rδμ,0Mδ(μ1+μ2+μ3+μ4)μτ}

    is a positively invariant set for system (2.1). So we consider dynamics of system (2.1) on the set Ω in this paper. This completes the proof of Lemma 2.

    It is easy to see system (2.1) always has a disease-free equilibrium

    P0=(S0,E0,I0,R0,M0)=(δμ,0,0,0,μ1δμτ). (3.1)

    We can obtain the basic reproductive number R0 by using the next-generation method [22]. Let x=(E,I,R,S,M)T, then system (2.1) can be written as

    dxdt=F(x)V(x),

    where,

    F(x)=((1q)βSIeαMqβSIeαM000)andV(x)=((μ+ε)EεE+(d+μ+γ)IγI+μRδ+βSIeαM+μSμ1Sμ2Eμ3Iμ4R+τM).

    The Jacobian matrices of F(x) and V(x) at the disease-free equilibrium P0 are, respectively,

    DF(P0)=(F3×300000000),DV(P0)=(V3×3000βδμeαμ1δμτ0μ0μ2μ3μ4μ1τ),

    where

    F=(0(1q)βδμeαμ1δμτ00qβδμeαμ1δμτ0000),V=(μ+ε00εd+μ+γ00γμ).

    The basic reproductive number, denoted by R0 is thus given by

    R0=ρ(FV1)=(ε+μq)βδeαμ1δμτμ(μ+ε)(d+μ+γ). (3.2)

    Theorem 1. The disease-free equilibrium P0=(δμ,0,0,0,μ1δμτ) of system (2.1) is locally asymptotically stable if R0<1, and is unstable if R0>1.

    Proof. The Jacobian matrix corresponding to system (2.1) about P0=(δμ,0,0,0,μ1δμτ) is obtained as follows:

    J(P0)=(μ0βδμeαμ1δμτ000(μ+ε)(1q)βδμeαμ1δμτ000εqβδμeαμ1δμτ(d+μ+γ)0000γμ0μ1μ2μ3μ4τ).

    The characteristic equation corresponding to the Jacobian matrix J(P0) is given by |λEJ(P0)|=0, where λ is the eigenvalue and E is the unit matrix. Thus, we get

    (λ+τ)(λ+μ)2[λ2+(2μ+ε+d+γqβδμeαμ1δμτ)λ+(μ+ε)(d+μ+γ)βδ(ε+μq)μeαμ1δμτ]=0. (3.3)

    Obviously, Eq. (3.3) has three negative roots λ1=τ, λ2=λ3=μ, and the other two roots λ4 and λ5 are determined by

    λ2+(2μ+ε+d+γqβδμeαμ1δμτ)λ+(μ+ε)(d+μ+γ)βδ(ε+μq)μeαμ1δμτ=0. (3.4)

    According to the above calculation and analysis, we can obtain

    λ4λ5=(μ+ε)(d+μ+γ)βδ(ε+μq)μeαμ1δμτ=(μ+ε)(d+μ+γ)(1R0),
    λ4+λ5=qβδμeαμ1δμτ(2μ+ε+d+γ)=(μ+ε)(d+μ+γ)qε+μq[(ε+μq)βδeαμ1δμτμ(μ+ε)(d+μ+γ)ε+μq(μ+ε)qε+μq(d+μ+γ)q]=(μ+ε)(d+μ+γ)qε+μq[R0ε+μqεq+μqε+μq(d+μ+γ)q]<(μ+ε)(d+μ+γ)qε+μq[R01ε+μq(d+μ+γ)q].

    If R0<1, we have λ4λ5>0, λ4+λ5<0, hence λ4<0,λ5<0. Therefore, P0=(δμ,0,0,0,μ1δμτ) is locally asymptotically stable. If R0>1, Eq. (3.4) has two real roots that one is positive and another is negative. In this case, P0=(δμ,0,0,0,μ1δμτ) is unstable. This completes the proof of Theorem 1.

    Theorem 2. The disease-free equilibrium P0=(δμ,0,0,0,μ1δμτ) of the system (2.1) is globally asymptotically stable if R0<1 and M(t)μ1δμτ.

    Proof. Motivated by Huo and Zhang [13], we define the Lyapunov function

    V(t)=εE(t)+(μ+ε)I(t).

    It is clear that V(t)0 and the equality holds if and only if E(t)=I(t)=0.

    From the first equation of the system (2.1), we have

    dSdt=δβSIeαMμSδμS,

    and then we can obtain S(t)δμ.

    Differentiating V(t) with respect to time t yields:

    dV(t)dt=εdE(t)dt+(μ+ε)dI(t)dt=βSIeαM(ε+μq)(μ+ε)(d+μ+γ)Iβδ(ε+μq)eαμ1δμτμI(μ+ε)(d+μ+γ)I=(μ+ε)(d+μ+γ)I[βδ(ε+μq)eαμ1δμτμ(μ+ε)(d+μ+γ)1]=(μ+ε)(d+μ+γ)I(R01).

    It follows that V(t) is bounded and non-increasing. Therefore, limtV(t) exists. Note that dV(t)dt=0 if and only if E=I=R=0,S=S0=δμ,M=M0=μ1δμτ. The maximum invariant set of the system (2.1) on the set {(S,E,I,R,M):dV(t)dt=0} is the singleton P0=(δμ,0,0,0,μ1δμτ). And note that R0<1 guarantees that dV(t)dt0 for all t0. By LaSalle's Invariance Principle [23], the disease-free equilibrium P0=(δμ,0,0,0,μ1δμτ) is globally asymptotically stable when R0<1 and M(t)μ1δμτ. This completes the proof of Theorem 2.

    First, we introduce:

    Φ=αμτ(μq+ε){(μq+ε)(γμ4+μμ3)+(d+μ+γ)[μμ2(1q)μ1(μ+ε)]}, (3.5)
    R01=δ(μq+ε)Φ(μ+ε)(d+μ+γ), (3.6)
    Rc=R01e1R01. (3.7)

    Remark 1. It is clear to check that: R01>0 if and only if Φ>0; R01=0 if and only if Φ=0; R01<0 if and only if Φ<0.

    Theorem 3. For system (2.1),

    (ⅰ) If R0>max{1,R01}, there is a unique endemic equilibrium P1.

    (ⅱ) If Rc=R0<min{1,R01} and R01>0, there is a unique endemic equilibrium P2.

    (ⅲ) If Rc<R0<min{1,R01} and R01>0, there are two distinct endemic equilibria P3 and P4.

    Proof. The endemic equilibrium P(S,E,I,R,M) of system (2.1) is determined by equations

    {δβSIeαMμS=0,(1q)βSIeαM(μ+ε)E=0,qβSIeαM+εE(d+μ+γ)I=0,γIμR=0,μ1S+μ2E+μ3I+μ4RτM=0. (3.8)

    Further, we obtain

    S=δμ(μ+ε)(d+μ+γ)μ(μq+ε)I, (3.9)
    E=(1q)(d+μ+γ)μq+εI, (3.10)
    R=γμI, (3.11)
    M=μ1δμτΦαI, (3.12)

    where Φ is given by (3.5). Substituting S,M into the first equation of (3.8) yields

    R0[1(μ+ε)(d+μ+γ)δ(μq+ε)I]=eΦI. (3.13)

    According to (3.6) and (3.13), we have

    R0(1ΦR01I)eΦI=0.

    We consider a function F(I) defined by

    F(I)=R0(1ΦR01I)eΦI. (3.14)

    Then, we have

    F(0)=R01,F(+)=,
    F(I)=R0R01Φ+ΦeΦI,F(0)=R0R01Φ+Φ,
    F(I)=Φ2eΦI.

    Case 1. When Φ=0, according to (3.13), we have

    I=δ(μq+ε)(μ+ε)(d+μ+γ)(11R0).

    Therefore, there is a unique endemic equilibrium if Φ=0 and R0>1.

    Case 2. When Φ0, we have

    F(I)=Φ2eΦI<0.

    Thus, we get F(I)<F(0), which means ΦeΦI<Φ.

    (1) If R0>1, we have F(0)=R01>0, F(+)=<0, and

    F(I)=R0R01Φ+ΦeΦI<R0R01Φ+Φ=(1R0R01)Φ.

    When R0>R01>0, we have Φ>0 and 1R0R01<0, which means F(I)<0.

    When R0>0>R01, we have Φ<0 and 1R0R01>0, which means F(I)<0.

    Therefore, there is a unique endemic equilibrium if Φ0 and R0>max{1,R01}.

    In conclusion, there is a unique endemic equilibrium P1 if R0>max{1,R01}.

    (2) If R0<1, we have F(0)=R01<0, F(+)=<0, Let's suppose

    F(I)=R0R01Φ+ΦeΦI=0.

    Then we obtain

    Ic=1ΦlnR01R0.

    When R0<R01, we have Φ>0 and lnR01R0>0, which means Ic>0.

    Substituting Ic into (3.14), we get

    F(Ic)=R0+R0R01(lnR0R011).

    (a) When F(Ic)=0, we can obtain I2=Ic and R0=Rc, where Rc is given by (3.7).

    Therefore, there is a unique endemic equilibrium P2 if Rc=R0<min{1,R01} and R01>0.

    (b) When F(Ic)>0, we can obtain R0>Rc.

    Since F(0)=R01<0, F(+)=<0 and F(Ic)>0, we know that F(I)=0 has two different positive solutions I3 and I4. Let I3 and I4 satisfy I3<I2<I4. Therefore, there are two distinct endemic equilibria P3 and P4 if Rc<R0<min{1,R01} and R01>0. This completes the proof of Theorem 3.

    Theorem 4. When q=0, the endemic equilibria Pi(i=1,2,3,4) of system (2.1) have the following qualities:

    (ⅰ) If R0>max{1,R01}, a1(I1)a2(I1)a3(I1)>0, a3(I1)[a1(I1)a2(I1)a3(I1)][a1(I1)]2a4(I1)>0 and a4(I1)>0, the endemic equilibrium P1 is locally asymptotically stable.

    (ⅱ) If Rc=R0<min{1,R01} and R01>0, the endemic equilibrium P2 is unstable.

    (ⅲ) If Rc<R0<min{1,R01} and R01>0, the endemic equilibrium P3 is unstable.

    (ⅳ) If Rc<R0<min{1,R01} and R01>0, the stability of the endemic equilibrium P4 is uncertain.

    Proof. When q=0, the Jacobian matrix corresponding to system (2.1) about Pi(i=1,2,3,4) are obtained as follows:

    J(Pi)=(βIieαMiμ0βSieαMi0αβSiIieαMiβIieαMi(μ+ε)βSieαMi0αβSiIieαMi0ε(d+μ+γ)0000γμ0μ1μ2μ3μ4τ).

    The characteristic equation corresponding to the Jacobian matrix J(Pi) is given by |λEJ(Pi)|=0, where λ is the eigenvalue and E is the unit matrix. Thus, we get

    |λ+βIieαMi+μ0βSieαMi0αβSiIieαMiβIieαMiλ+μ+εβSieαMi0αβSiIieαMi0ελ+d+μ+γ0000γλ+μ0μ1μ2μ3μ4λ+τ|=0.

    We set Θ=βeαMi, then

    Θ=βeαμ1δμτeΦIi=μ(μ+ε)(d+μ+γ)eΦIiR0εδ=μΦeΦIiR0R01.

    From the second equation of (3.8), we have

    βSiIieαMi=(μ+ε)Ei=(μ+ε)(d+μ+γ)εIi,

    then

    ΘSi=βSieαMi=(μ+ε)(d+μ+γ)ε=δΦR01,

    and

    ΘSiIi=βSiIieαMi=(μ+ε)(d+μ+γ)εIi=δΦR01Ii.

    Therefore, the characteristic equation can be rewritten as:

    (λ+μ)F(λ)=0, (3.15)

    where

    F(λ)=λ4+a1(Ii)λ3+a2(Ii)λ2+a3(Ii)λ+a4(Ii), (3.16)

    where

    a1(Ii)=d+γ+3μ+ε+τ+ΘIi, (3.17)
    a2(Ii)=(d+γ+2μ+ε+τ)(μ+ΘIi)+τ(d+γ+2μ+ε)+αδΦR01Ii(μ2μ1), (3.18)
    a3(Ii)=μΘIi(d+γ+2τ+ε+μ)+τ(ε+d+γ)(μ+ΘIi)+ε(d+γ)ΘIi
    +αδΦR01Ii[(d+γ+2μ)(μ2μ1)+ε(μ3μ1)], (3.19)
    a4(Ii)=τ(μ+ε)(d+γ+μ)[(μ+ΘIi)μ(1+ΦIi)]. (3.20)

    (ⅰ) According to (3.17)-(3.20), we have

    a1(I1)=d+γ+3μ+ε+τ+ΘI1,a2(I1)=(d+γ+2μ+ε+τ)(μ+ΘI1)+τ(d+γ+2μ+ε)+αδΦR01I1(μ2μ1),a3(I1)=μΘI1(d+γ+2τ+ε+μ)+τ(ε+d+γ)(μ+ΘI1)+ε(d+γ)ΘI1+αδΦR01I1[(d+γ+2μ)(μ2μ1)+ε(μ3μ1)],a4(I1)=τ(μ+ε)(d+γ+μ)[(μ+ΘI1)μ(1+ΦI1)].

    It is clear that a1(I1)>0, according to RouthHurwitz criteria [24], the proof (ⅰ) of Theorem 4 is obtained.

    (ⅱ) According to the proof of (ⅱ) of Theorem 3, we have I2=1ΦlnR01R0. Then, we can get ΘI2=μΦI2. Therefore, based on (3.17)-(3.20), we can obtain

    a1(I2)=d+γ+3μ+ε+τ+ΘI2>0,a4(I2)=τμ(μ+ε)(d+γ+μ)[(1+ΦI2)(1+ΦI2)]=0.

    It is easy to know that a3(I2)0, and a1(I2)a2(I2)a3(I2)<0. Therefore, we know that Eq. (3.15) has negative, positive and zero eigenvalues. So the endemic equilibrium P2 of system (2.1) is unstable.

    (ⅲ) Due to I3<I2=1ΦlnR01R0, we can get ΘI3<μΦI3. Therefore, based on (3.17)-(3.20), we can obtain

    a1(I3)=d+γ+3μ+ε+τ+ΘI3>0,

    and

    a4(I3)<τμ(μ+ε)(d+γ+μ)[(1+ΦI3)(1+ΦI3)]=0.

    Let gj(I3)(j=1,2,3,4) be the solutions of F(λ)=0, and we assume that the real parts satisfy Re(g1(I3))Re(g2(I3))Re(g3(I3))Re(g4(I3)), where Re means the real part of a complex number. Then we can obtain gj(I3)(j=1,2,3,4) satisfying

    g1(I3)+g2(I3)+g3(I3)+g4(I3)=a1(I3)<0,

    and

    g1(I3)g2(I3)g3(I3)g4(I3)=a4(I3)<0.

    So, we have Re(g1(I3))<0 and Re(g4(I3))>0. Then, we know that the endemic equilibrium P3 of system (2.1) is unstable.

    (ⅳ) Due to I4>I2=1ΦlnR01R0, we have ΘI4>μΦI4. Therefore, based on (3.17)-(3.20), we can obtain

    a1(I4)=d+γ+3μ+ε+τ+ΘI4>0

    and

    a4(I4)>τμ(μ+ε)(d+γ+μ)[(1+ΦI4)(1+ΦI4)]=0.

    Let gj(I4)(j=1,2,3,4) be the solutions of F(λ)=0, and we assume that the real parts satisfy Re(g1(I4))Re(g2(I4))Re(g3(I4))Re(g4(I4)). Then we can obtain gj(I4)(j=1,2,3,4) satisfying

    g1(I4)+g2(I4)+g3(I4)+g4(I4)=a1(I4)<0 (3.21)

    and

    g1(I4)g2(I4)g3(I4)g4(I4)=a4(I4)>0.

    Therefore, if Re(gj(I4))<0(j=1,2,3,4), the endemic equilibrium P4 of system (2.1) is stable. However, if Re(g1(I4))Re(g2(I4))<0<Re(g3(I4))Re(g4(I4)) and |Re(g1(I4))|+|Re(g2(I4))|>|Re(g3(I4))|+|Re(g4(I4))|, the endemic equilibrium P4 of system (2.1) is unstable. Thus, the stability of the endemic equilibrium P4 is uncertain. This completes the proof of Theorem 4.

    Theorem 5. (ⅰ) If R01<1, system (2.1) exhibits a forward bifurcation at R0=1.

    (ⅱ)If R01>1, system (2.1) exhibits a backward bifurcation at R0=1.

    Proof. We suppose x1=S,x2=E,x3=I,x4=R,x5=M, system (2.1) becomes

    {dx1dt=δβx1x3eαx5μx1:=f1,dx2dt=(1q)βx1x3eαx5(μ+ε)x2:=f2,dx3dt=qβx1x3eαx5+εx2(d+μ+γ)x3:=f3,dx4dt=γx3μx4:=f4,dx5dt=μ1x1+μ2x2+μ3x3+μ4x4τx5:=f5.

    When R0=1, we obtain β=βc=μ(μ+ε)(d+μ+γ)δ(ε+μq)eαμ1δμτ. When β=βc, the Jacobian matrix corresponding to system (2.1) about the disease-free equilibrium P0=x0=(δμ,0,0,0,μ1δμτ) is given by

    J(x0)=(μ0(μ+ε)(d+μ+γ)ε+μq000(μ+ε)(1q)(μ+ε)(d+μ+γ)ε+μq000εq(μ+ε)(d+μ+γ)ε+μq(μ+d+γ)0000γμ0μ1μ2μ3μ4τ).

    It is clear that 0 is a simple eigenvalue of J(x0). A right eigenvector ω corresponding to the 0 eigenvalue is ω=(ω1,ω2,ω3,ω4,ω5)T, where

    ω1=(μ+ε)(d+μ+γ)μ,ω2=(1q)(d+μ+γ),ω3=ε+μq,ω4=γ(ε+μq)μ,ω5=(ε+μq)Φα.

    The left eigenvector υ corresponding to the 0 eigenvalue satisfying υJ=0 and υω=1 is υ=(υ1,υ2,υ3,υ4,υ5), where

    υ1=υ4=υ5=0,υ2=1(1q)(d+μ+γ)+(ε+μq)(μ+ε),
    υ3=μ+εε[(1q)(d+μ+γ)+(ε+μq)(μ+ε)].

    Furthermore, we have a=5k,i,j=1υkωiωj2fk(x0)xixj and b=5k,i=1υkωi2fk(x0)xiβ. Substituting the values of all second order derivatives evaluated at the disease-free equilibrium x0=(δμ,0,0,0,μ1δμτ), we obtain

    a=2υ2ω1ω32f2(x0)x1x3+2υ2ω3ω52f2(x0)x3x5+2υ3ω1ω32f3(x0)x1x3+2υ3ω3ω52f3(x0)x3x5=2βeαμ1δμτ(ω1ω3αδμω3ω5)(υ2(1q)+υ3q)=2β(ε+μq)2(μ+ε)(d+μ+γ)(R011)eαμ1δμτμε[(1q)(d+μ+γ)+(ε+μq)(μ+ε)],

    and

    b=υ2ω32f2(x0)x3β+υ3ω32f3(x0)x3β=ω3δμeαμ1δμτ[(1q)υ2+qυ3]=δ(ε+μq)2eαμ1δμτμε[(1q)(d+μ+γ)+(ε+μq)(μ+ε)].

    According to Theorem 4.1 of [25], note that the coefficient b is always positive. If R01<1, the coefficient a is negative. In this case, the direction of the bifurcation of system (2.1) at R0=1 is forward. If R01>1, the coefficient a is positive. Under this circumstance, the direction of the bifurcation of system (2.1) at R0=1 is backward. This completes the proof of Theorem 5.

    In this section, we will give some simulations using the parameter values which are given in Table 1.

    Table 1.  The parameters description of the tuberculosis model.
    Parameter Description Estimated value Source
    δ Constant recruitment rate of the population 0.8day1 [14]
    β Transmission coefficient of TB 0.0099-0.8person1day1 Estimate
    α The coefficient that determines how effective TB 0.00091-0.8day1 [14]
    information can influence the transmission rate
    μ Nature death rate 0.009-0.6year1 Estimate
    q The proportion of disease by fast progression 0-0.5year1 Estimate
    ε The progression rate from E to I 0.02-0.99day1 Estimate
    d The disease-related death rate of TB 0.002-0.5day1 Estimate
    γ The recovery rate of TB 0.006-0.99day1 Estimate
    μ1 The rate that susceptible individuals may send 0.04-0.99day1 [26]
    message about TB
    μ2 The rate that exposed individuals may send 0.008-0.8day1 [26]
    message about TB
    μ3 The rate that infectious individuals may send 0.08-0.8day1 [26]
    message about TB
    μ4 The rate that recovered individuals may send 0-1day1 Estimate
    message about TB
    τ The rate that message become outdated 0.03-0.6year1 [26]

     | Show Table
    DownLoad: CSV

    We choose a set of the following parameters: δ=0.8, β=0.8, α=0.08, μ=0.6, q=0.5, ε=0.09, d=0.02, γ=0.7, μ1=0.99, μ2=0.4, μ3=0.8, μ4=0.8, τ=0.6. It is easy to check that the basic reproductive number R0=0.383<1. Then the unique disease-free equilibrium P0=(1.3333,0,0,0,2.2) of system (2.1) is globally asymptotically stable (see Figure 2).

    Figure 2.  The disease-free equilibrium of system (2.1) is globally asymptotically stable when R0<1.

    Next, we select a set of the following parameters: δ=0.8, β=0.8, α=0.08, μ=0.2, q=0.1, ε=0.4, d=0.02, γ=0.6, μ1=0.2, μ2=0.8, μ3=0.8, μ4=0.8, τ=0.6. It is easy to check that the basic reproductive number R0=2.4553>1. Then, from Theorem 4, the endemic equilibrium P1 of system (2.1) is locally asymptotically stable when R0>max(1,R01), where R01=0.0158 (see Figure 3).

    Figure 3.  The endemic equilibrium P1 of system (2.1) is locally asymptotically stable when R0>max{1,R01}.

    The backward and forward bifurcation diagram of system (2.1) is shown in Figure 4, and the direction of bifurcation depends upon the value of R01. As seen in the backward bifurcation diagram of Figure 4(a) when R01=4.4936>1, there is a threshold quantity Rc which is the value of R0. The disease-free equilibrium is globally asymptotically stable when R0<Rc, where Rc=0.1350. There are two endemic equilibria and a disease-free equilibrium when Rc<R0<1, the upper ones are stable, the middle ones are unstable and the lower ones is globally asymptotically stable. There is a stable endemic equilibrium and an unstable disease-free equilibrium when R0>1. As seen in the forward bifurcation diagram of Figure 4(b) when R01=0.5357<1, the disease-free equilibrium is globally asymptotically stable when R0<1. There are a stable endemic equilibrium and an unstable disease-free equilibrium when R0>1.

    Figure 4.  (a) Illustration of backward bifurcation when one parameter β in R0 is varied. (b) Illustration of forward bifurcation when one parameter β in R0 is varied.

    In this section, we discuss sensitivity analysis of the basic reproductive number R0 and the infectious individuals I at first. We study the influence of α, μ1 and β to R0. It is straightforward from (3.2) that R0 increases as β increases. This agrees with the intuition that higher transmission coefficient increases the basic reproduction number. In order to see the relationship of these parameters and R0, we regard R0 as a function about those parameters. Note that

    R0α=(ε+μq)βδ2μ1eαμ1δμτμ2τ(μ+ε)(d+μ+γ)<0,
    R0μ1=(ε+μq)αβδ2eαμ1δμτμ2τ(μ+ε)(d+μ+γ)<0,
    R0q=μβδeαμ1δμτμ(μ+ε)(d+μ+γ)>0.

    Therefore, we find that α and μ1 have a negative influence on the basic reproductive number R0. However, q has a positive influence on the basic reproductive number R0. The parameter values are δ=0.8, q=0.1, β=0.8, μ=0.2, ε=0.4, γ=0.6, d=0.02, μ2=0.8, μ3=0.8, μ4=0.8, τ=0.6. From Figure 5, we know that the basic reproductive number R0 will decrease when α and μ1 increase. However, the basic reproductive number R0 will increase when q increases.

    Figure 5.  The relationship among R0, α, μ1 and q.

    Next, in order to evaluate the effect of media coverage on the dynamics of tuberculosis, we choose different values of α and τ (see Figure 6). The parameters are δ=0.8, q=0.1, β=0.8,μ=0.2, ε=0.4, γ=0.6, d=0.02, μ1=0.2, μ2=0.8, μ3=0.8, μ4=0.8.

    Figure 6.  The effect of message-related parameters on the dynamics of infectious individuals.

    From Figure 6, we know that infected number will decrease when α increase, and increase when τ increases. Therefore, we find that media coverage has a great impact on the transmission of tuberculosis.

    Choosing β as a parameter, it is also observed that with β increasing, the positive equilibrium point P1 loses its stability and a Hopf bifurcation occurs when β passes a critical values β.

    We select a set of the following parameters: δ=0.8, β=0.0099, α=0.007, μ=0.009, q=0.1, ε=0.99, d=0.5, γ=0.99, μ1=0.08, μ2=0.8, μ3=0.8, μ4=0.8, τ=0.6. The endemic equilibrium P1 of system (2.1) is locally asymptotically stable when R0>max{1,R01} and β<β (see Figure 7).

    Figure 7.  Endemic equilibrium P1 of system (2.1) is locally asymptotically stable when β<β.

    To illustrate the existence of Hopf bifurcation, we choose a set of the following parameters: δ=0.8, α=0.007, μ=0.009, q=0.5, ε=0.99, d=0.02, γ=0.1, μ1=0.09, μ2=0.008, μ3=0.08, μ4=0.08, τ=0.03. When β passes through the critical value β, we find the positive endemic equilibrium P1 loses its stability and a Hopf bifurcation occurs (see Figure 8).

    Figure 8.  Endemic equilibrium P1 of system (2.1) occurs a Hopf bifurcation when R0>max{1,R01} and β>β.

    In this paper, we propose and analyse a TB model with fast and slow progression and media coverage. By means of the next-generation matrix, we obtain the basic reproductive number R0, which plays a crucial role in our model. By constructing Lyapunov function, we prove the global stability of the disease-free equilibrium. In addition, we obtain the existence and the local stability of the endemic equilibrium. By using the center manifold theory, we get a backward and forward bifurcation. Furthermore, we give a numerical result about a Hopf bifurcation occurs when β passes through the critical value β. At last, we also use numerical method to simulate outcomes which we have been proved.

    The initially exposed individuals have a higher risk of developing active TB. They still have the possibility of progressing to infectious TB with time passing. The likelihood of becoming an active infectious case decreases with the age of the infection. Taking these factors into consideration, we set up a new tuberculosis with fast and slow progression and media coverage. Through simulations, we know that β plays an important role and induces Hopf bifucation in our model. Furthermore, we have done some simulations (not shown). We did not find other critical parameters (including q) for Hopf bifurcation. q is the proportion of disease by fast progression. Since R0=ρ(FV1)=(ε+μq)βδeαμ1δμτμ(μ+ε)(d+μ+γ), we can find the basic reproductive number R0 will increase when q increases. Tuberculosis may breakout due to the increase of q. The fast and slow progression can not induce Hopf bifurcation, but it still plays an important role in TB transmission and has a positive influence on the basic reproductive number R0.

    Our results show that media coverage has a substantial influence on the dynamics of tuberculosis and it can greatly influence the spread of the tuberculosis, thus, it is crucial to remind people to take countermeasures to avoid potential infections by media coverage.

    In our model (2.1), we only consider the form of ordinary equation. Note that all of the people have a time delay in releasing and receiving information, it is more realistic to explore a time delay in the rate that media coverage become outdated. On the other hand, as suggested by Styblo et al. [27], recovered individuals may only have partial immunity. Indeed, TB is one kind of chronic infectious diseases that has a certain relapse rate due to the drug-resistant tuberculosis and lack of combination drug regimen. Thus, it is a very interesting and more realistic to study our model with reinfection, that is some individuals in the recovered class can relapse back into the active TB state. We leave these interesting works for the future.

    The authors are very grateful to the Editor-in-Chief and the anonymous referees for their valuable comments and suggestions which helped us to improve the paper. This work is supported by the National Natural Science Foundation of China (11861044 and 11661050), and the HongLiu first-class disciplines Development Program of Lanzhou University of Technology.

    The authors declare there is no conflict of interest.



    [1] L. Wang, R. Zhang and D. Yin, Image verification code identification of hyphen, Comput. Eng. Appl., 28 (2011), 150–153.
    [2] J. Yan and A. S. E. Ahmad, A low-cost attack on a Microsoft CAPTCHA, Proceedings of the ACM Conference on Computer and Communications Security, (2008), 543–554.
    [3] L. Zhang, S. W. Huang, Z. X. Shi, et al., CAPTCHA recognition method based on LSTM RNN, Pattern Recogn., 1 (2011), 40–47.
    [4] L. Yin, D. Yin and R. Zhang, A recognition method of twisted and pasted character verification code, Pattern Recogn., 3 (2014), 235–241.
    [5] H. Li, J. H. Qin and X. Y. Xiang, An efficient image matching algorithm based on adaptive threshold and RANSAC, IEEE Access, 6 (2018), 66963–66971.
    [6] L.Y. Xiang, Y. Li and W. Hao, Reversible natural language watermarking using synonym substitution and arithmetic coding, Comput. Mat. Con., 3 (2018), 541–559.
    [7] L.Y. Xiang, X. B. Shen, J. H. Qin, et al., Discrete multi-graph hashing for large-scale visual search, Neur. Process. Lett., 49 (2019), 1055–1069.
    [8] Y. L. Liu, H. Peng and J. Wang, Verifiable diversity ranking search over encrypted outsourced Data, Comput. Mater. Con., 1 (2018), 37–57.
    [9] H. T. Tang, Verification code recognition model and algorithm of self-organizing incremental neural network, MA thesis, Guangdong University of technology, 2016.
    [10] Y. Wang, Y. Q. Xu and Y. B. Peng, Verification code identification of xiaonei network based on KNN technology, Comput. Moder., 2 (2017),93–97.
    [11] Y. S. Chen and Y. Zhang, Design and implementation of character-based image verification code recognition algorithm, Comput. K. T., 1 (2017),190–192.
    [12] Y. Wang and M. Lu, A self-adaptive algorithm to defeat text-based CAPTCHA, IEEE International Conference on Industrial Technology, (2016), 720–725.
    [13] W. T. Ma, J. H. Qin, X. Y. Xiang, et al., Adaptive median filtering algorithm based on divide and conquer and its application in CAPTCHA recognition, Comput. Mater. Con., 58 (2019), 665–677.
    [14] J. W. Wang, T. Li and X. Y. Luo, Identifying computer generated images based on quaternion central moments in color quaternion wavelet domain, IEEE T. Circ. Syst. Vid., 1 (2018), 1.
    [15] X. W. Liu, L. Wang, Jian Zhang, et al., Global and local structure preservation for feature selection, IEEE T. Neur. Net. Lear., 25 (2014), 1083–1095.
    [16] J. H. Qin, H. Li, X. Y. Xiang, et al., An encrypted image retrieval method based on Harris corner optimization and LSH in cloud computing, IEEE Access, 17 (2019), 24626–24633.
    [17] M. L. Wen, X. Zhao, M. Q. Cai, et al., End-to-end verification code recognition based on deep learning, Wireless Inter. technol., 14 (2017), 85–86.
    [18] Y. Peng, Research on verification code recognition based on deep convolutional neural network, Commu. world, 1 (2018), 66–67.
    [19] Z. Zhang, S. F. Wang and L. Dong, Verification code recognition based on deep learning, J. hubei univ. technol., 2 (2018), 5–11.
    [20] S. R. Zhou, W. L. Liang, J. G. Li, et al., Improved VGG model for road traffic sign recognition, Comput. Mat. Con., 1 (2018), 11–24.
    [21] W. Fang, F. H. Zhang and V. S. Sheng, A method for improving CNN-based image recognition using DCGAN, Comput. Mat. Con., 1 (2018), 167–178.
    [22] Y. P. Lv, F. P. Cai, D. Z. Lin, et al., Chinese character CAPTCHA recognition based on convolution-neural network, Proceedings of the IEEE Congress on Evolutionary Computation, (2016), 4854–4859.
    [23] G. Garg and C. Pollett, Neural network CAPTCHA crackers, Proceedings of the Future Technologies Conference, (2016), 853–861.
    [24] Y. H. Shen, R. G. Ji and D. L. Cao, Hacking Chinese touclick CAPTCHA by multiscale corner struc-ture model with fast pattern matching, Proceedings of the ACM International Conference on Multimedia, (2014), 853–856.
    [25] W. Fan, J. G. Han, Fan Gou, et al., Chinese character verification code recognition by convolutional neural network, Comput. Eng. Appl., 3 (2018), 160–165.
    [26] G. Huang, Z. Liu, L. V. D Maaten, et al., Densely connected convolutional networks, Proceedings of the IEEE Conference on Computer Vision and Pattern Recognition, (2017), 2261–2269.
    [27] N. Ma, X. Zhang, H. T. Zheng, et al., ShuffleNet V2: practical guidelines for efficient CNN architecture design, Computer Vision and Pattern Recognition, preprint, arXiv:1807.11164.
  • This article has been cited by:

    1. Muqrin A. Almuqrin, A new flexible distribution with applications to engineering data, 2023, 69, 11100168, 371, 10.1016/j.aej.2023.01.046
    2. Mahdi Rasekhi, Mohammad Mehdi Saber, G. G Hamedani, M. M.Abd El-Raouf, Ramy Aldallal, Ahmed M. Gemeay, Melike Kaplan, Approximate Maximum Likelihood Estimations for the Parameters of the Generalized Gudermannian Distribution and Its Characterizations, 2022, 2022, 2314-4785, 1, 10.1155/2022/4092576
    3. Yinghui Zhou, Zubair Ahmad, Zahra Almaspoor, Faridoon Khan, Elsayed tag-Eldin, Zahoor Iqbal, Mahmoud El-Morshedy, On the implementation of a new version of the Weibull distribution and machine learning approach to model the COVID-19 data, 2022, 20, 1551-0018, 337, 10.3934/mbe.2023016
    4. Abdulaziz S. Alghamdi, M. M. Abd El-Raouf, A New Alpha Power Cosine-Weibull Model with Applications to Hydrological and Engineering Data, 2023, 11, 2227-7390, 673, 10.3390/math11030673
    5. Najwan Alsadat, Aijaz Ahmad, Muzamil Jallal, Ahmed M. Gemeay, Mohammed A. Meraou, Eslam Hussam, Ehab M.Elmetwally, Md. Moyazzem Hossain, The novel Kumaraswamy power Frechet distribution with data analysis related to diverse scientific areas, 2023, 70, 11100168, 651, 10.1016/j.aej.2023.03.003
    6. Najwan Alsadat, A new modified model with application to engineering data sets, 2023, 72, 11100168, 1, 10.1016/j.aej.2023.03.050
    7. Xiangming Tang, Jin-Taek Seong, Randa Alharbi, Aned Al Mutairi, Said G. Nasr, A new probabilistic model: Theory, simulation and applications to sports and failure times data, 2024, 10, 24058440, e25651, 10.1016/j.heliyon.2024.e25651
    8. Ibrahim Alkhairy, A new approach of generalized Rayleigh distribution with analysis of asymmetric data sets, 2024, 100, 11100168, 1, 10.1016/j.aej.2024.04.070
    9. Shahid Mohammad, Isabel Mendoza, A New Hyperbolic Tangent Family of Distributions: Properties and Applications, 2024, 2198-5804, 10.1007/s40745-024-00516-5
    10. Amulya Kumar Mahto, Yogesh Mani Tripathi, Sanku Dey, M.M. Abd El-Raouf, Najwan Alsadat, Efficient estimation of the density and distribution functions of Weibull-Burr XII distribution, 2024, 104, 11100168, 576, 10.1016/j.aej.2024.07.118
    11. Faridoon Khan, Zubair Ahmad, Saima K. Khosa, Mohammed Ahmed Alomair, Abdullah Mohammed Alomair, Abdulaziz khalid Alsharidi, A new modification of the flexible Weibull distribution based on power transformation: Monte Carlo simulation and applications, 2023, 9, 24058440, e17238, 10.1016/j.heliyon.2023.e17238
    12. Laxmi Prasad Sapkota, Vijay Kumar, Ahmed M. Gemeay, M. E. Bakr, Oluwafemi Samson Balogun, Abdisalam Hassan Muse, New Lomax-G family of distributions: Statistical properties and applications, 2023, 13, 2158-3226, 10.1063/5.0171949
    13. Xiaochun Liu, Jian Ji, Afaf Alrashidi, Fatimah A. Almulhim, Etaf Alshawarbeh, Jin-Taek Seong, A new probabilistic model with mixed-state failure rates: Modeling time-to-event scenarios in reliability and music engineering, 2024, 96, 11100168, 99, 10.1016/j.aej.2024.03.103
    14. M.M. Abd El-Raouf, Mohammed AbaOud, A novel extension of generalized Rayleigh model with engineering applications, 2023, 73, 11100168, 269, 10.1016/j.aej.2023.04.063
    15. Aijaz Ahmad, Najwan Alsadat, Mintodê Nicodème Atchadé, S. Qurat ul Ain, Ahmed M. Gemeay, Mohammed Amine Meraou, Ehab M. Almetwally, Md. Moyazzem Hossain, Eslam Hussam, New hyperbolic sine-generator with an example of Rayleigh distribution: Simulation and data analysis in industry, 2023, 73, 11100168, 415, 10.1016/j.aej.2023.04.048
    16. Zubir Shah, Ehab M. Almetwally, Dost Muhammad Khan, Farrukh Jamal, A novel odd Type-X family of distributions: Model, theory, and applications to medical, insurance, and engineering data sets, 2025, 18, 16878507, 101451, 10.1016/j.jrras.2025.101451
    17. Suleman Nasiru, Christophe Chesneau, Selasi Kwaku Ocloo, Abdul Ghaniyyu Abubakari, The Log-Cosine-Power Generated Family of Distributions, 2025, 2730-9657, 10.1007/s44007-025-00162-0
    18. Badr Aloraini, Improved estimation of population parameter of in the existence of nonresponse using auxiliary information, 2025, 10, 2473-6988, 12312, 10.3934/math.2025558
  • Reader Comments
  • © 2019 the Author(s), licensee AIMS Press. This is an open access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0)
通讯作者: 陈斌, bchen63@163.com
  • 1. 

    沈阳化工大学材料科学与工程学院 沈阳 110142

  1. 本站搜索
  2. 百度学术搜索
  3. 万方数据库搜索
  4. CNKI搜索

Metrics

Article views(10531) PDF downloads(1189) Cited by(69)

Figures and Tables

Figures(4)  /  Tables(5)

/

DownLoad:  Full-Size Img  PowerPoint
Return
Return

Catalog