Research article Special Issues

Managing extreme cryptocurrency volatility in algorithmic trading: EGARCH via genetic algorithms and neural networks

  • The blockchain ecosystem has seen a huge growth since 2009, with the introduction of Bitcoin, driven by conceptual and algorithmic innovations, along with the emergence of numerous new cryptocurrencies. While significant attention has been devoted to established cryptocurrencies like Bitcoin and Ethereum, the continuous introduction of new tokens requires a nuanced examination. In this article, we contribute a comparative analysis encompassing deep learning and quantum methods within neural networks and genetic algorithms, incorporating the innovative integration of EGARCH (Exponential Generalized Autoregressive Conditional Heteroscedasticity) into these methodologies. In this study, we evaluated how well Neural Networks and Genetic Algorithms predict "buy" or "sell" decisions for different cryptocurrencies, using F1 score, Precision, and Recall as key metrics. Our findings underscored the Adaptive Genetic Algorithm with Fuzzy Logic as the most accurate and precise within genetic algorithms. Furthermore, neural network methods, particularly the Quantum Neural Network, demonstrated noteworthy accuracy. Importantly, the X2Y2 cryptocurrency consistently attained the highest accuracy levels in both methodologies, emphasizing its predictive strength. Beyond aiding in the selection of optimal trading methodologies, we introduced the potential of EGARCH integration to enhance predictive capabilities, offering valuable insights for reducing risks associated with investing in nascent cryptocurrencies amidst limited historical market data. This research provides insights for investors, regulators, and developers in the cryptocurrency market. Investors can utilize accurate predictions to optimize investment decisions, regulators may consider implementing guidelines to ensure fairness, and developers play a pivotal role in refining neural network models for enhanced analysis.

    Citation: David Alaminos, M. Belén Salas, Ángela M. Callejón-Gil. Managing extreme cryptocurrency volatility in algorithmic trading: EGARCH via genetic algorithms and neural networks[J]. Quantitative Finance and Economics, 2024, 8(1): 153-209. doi: 10.3934/QFE.2024007

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  • The blockchain ecosystem has seen a huge growth since 2009, with the introduction of Bitcoin, driven by conceptual and algorithmic innovations, along with the emergence of numerous new cryptocurrencies. While significant attention has been devoted to established cryptocurrencies like Bitcoin and Ethereum, the continuous introduction of new tokens requires a nuanced examination. In this article, we contribute a comparative analysis encompassing deep learning and quantum methods within neural networks and genetic algorithms, incorporating the innovative integration of EGARCH (Exponential Generalized Autoregressive Conditional Heteroscedasticity) into these methodologies. In this study, we evaluated how well Neural Networks and Genetic Algorithms predict "buy" or "sell" decisions for different cryptocurrencies, using F1 score, Precision, and Recall as key metrics. Our findings underscored the Adaptive Genetic Algorithm with Fuzzy Logic as the most accurate and precise within genetic algorithms. Furthermore, neural network methods, particularly the Quantum Neural Network, demonstrated noteworthy accuracy. Importantly, the X2Y2 cryptocurrency consistently attained the highest accuracy levels in both methodologies, emphasizing its predictive strength. Beyond aiding in the selection of optimal trading methodologies, we introduced the potential of EGARCH integration to enhance predictive capabilities, offering valuable insights for reducing risks associated with investing in nascent cryptocurrencies amidst limited historical market data. This research provides insights for investors, regulators, and developers in the cryptocurrency market. Investors can utilize accurate predictions to optimize investment decisions, regulators may consider implementing guidelines to ensure fairness, and developers play a pivotal role in refining neural network models for enhanced analysis.



    The World Health Organization defines sexually transmitted diseases (STDs) as various diseases that are transmitted through sexual contact, similar sexual behaviors and indirect contact. The common reasons of STDs are bacteria, yeast and viruses[1]. STDs such as Trichomoniasis, Gonorrhea, Syphilis, Genital Warts and Herpes have become a serious public health problem. Recently, mathematical models of epidemic or population dynamics have been widely used [2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20].

    The mathematical models in the early researches of STDs generally assume that both the males and females are evenly mixed, that is, the contacts of all individuals are equal. These models ignore the social and contact structures of the real population. For instance, the number of sexual partners in different individuals may vary[21]. One method described in [22] that includes contact heterogeneity is the core group model. Although this model divides the population into two categories with a large number of sexual contacts and less sexual contact, it is still considered that the individuals are well mixed. Thus this model is not suitable for the spread of disease among the general public and is more suitable for sex workers.

    People gradually have realized the importance of heterogeneous social networks in recent years. The spread of sexually transmitted diseases occurs in social networks based on real human contact. The so-called network contains many nodes representing different individuals in the real system and the edges of the connected nodes representing relationships between individuals. Nodes are often separated into two categories by sexual contacts, and only nodes of the opposite type can be connected. The contacts between the opposite sex are represented by a bipartite network[23].

    Most of the previous models on complex networks assume that disease transmission is a Poisson process, and every individual randomly selects an individual from the population. This assumption implies that the duration of the partnership is very short. The focus of many researches gradually begins to understand the role of some individuals those with many connections in two ways. One is assuming a short-lived partnership (the time of disease transmission is much longer than the duration of the partnership)[24], another is assuming that the network is static (the time of disease transmission is much shorter than the duration of the partnership)[25,26,27,28]. The edge-based compartmental model (EBCM) has the potential to unify these two approaches recently, and allows partnership durations to last from zero to infinity[29,30,31].

    The above models assumed that the initial infection ratio was infinitesimally small. The inapplicability of this assumption was R0<1 or the initial infection rate was not negligible[25]. Miller[32] extended the edge-based compartmental model to arbitrary initial conditions and gave a detailed explanation of the part of the initial proportion infected could not be ignored. This helps to resolve an obvious paradox in early work, that is, if there are too many people initially infected, the number of susceptible people may increase. This also helps to explain a significant small deviation observed between the simulation and theory in the previous paper[25]. This modification makes sense for us to consider vaccination or previous infections. Yan et al.[33] considered the spread of STDs SIR sexually transmitted diseases on bipartite networks representing heterosexual individuals.

    Motivated by [32,33], some sexually transmitted diseases have the latent period but the individuals are less infectivity during the latent period, so we introduce latent compartment in our model and assume that transmission rate of the latent period is zero in this paper. We first assume that the proportion of initial infections is infinitesimally small. Based on this, the qualitative and stability of our model is further considered when the proportion of initial infections is arbitrarily large.

    This paper consists of 6 parts. We derive the edge-based SEIR model for sexually transmitted diseases in section 2. We computer the reproduction number R0 and the final epidemic size of the disease and analyze the local dynamics of the model in consideration of the infinitesimal initial infection rates in section 3. In section 4, we further analyze the dynamical behavior of our model considering a large number of infected individuals at the initial moment. In section 5, we perform some simulations with different initial values on different networks and some sensitivity analysis. The final section of the paper gives some concluding remarks.

    In the section, the network during the epidemic is assumed to be fixed and be of configuration type[34], and disease deaths are ignored. In the network, (PM(k)) and (PF(k)) represent the distribution of male and female individuals, respectively. Following[29,33], We distribute stubs to every men and women according to (PM(k)) and (PF(k)) at random. Further we pick out two stubs attached to individuals of different genders and connect them. We keep repeating the process until no new edge appear. Multiple edges, degree correlation, self loops, and clustering are negligible on the network constructed by this method[33,35].

    In Table 1, we list some variables and parameters. UM(UF) is a male (female) individual being tested which is randomly selected at the initial moment. The proportion of individuals in a certain state in the population is equal to the probability that UM(UF) in this state. We modify UM(UF) so that he does not be transmitted to any of its partners when infected. More discussion about the tested male UM and female UF is in [29,36]. SM,EM,IM,RM and SF,EF,IF,RF are the proportion of the susceptible, exposed, infected, recovered individuals in male and female individuals, respectively. They also are the probabilities of the tested male UM and the tested female UF are susceptible, exposed, infected or recovered, respectively. Other variables and parameters can be found in Table 1.

    Table 1.  The variables and parameters description of the SEIR model.
    Variable/Parameter Definition
     θM/θF The probability that tested male/female individual has not been infected
    by his/her randomly chosen partner yet. Initially, θM(0)=θF(0)=1.
     ϕSFM/ϕSMF The probability that randomly chosen partner of  UM/UF is susceptible,
    and  UM/UF was not infected by the partner before.
     ϕEFM/ϕEMF The probability that randomly chosen partner of  UM/UF is exposed,
    and  UM/UF was not infected by the partner before.
     ϕIFM/ϕIMF The probability that randomly chosen partner of  UM/UF is infectious,
    and  UM/UF was not infected by the partner before.
     ϕRFM/ϕRMF The probability that randomly chosen partner of  UM/UF is recovered,
    and  UM/UF was not infected by the partner before.
     PM(k)/PF(k) The probability of randomly selected male/female having k partners.
     SM(k,0)/SF(k,0) The fraction of males/females have degree k and are susceptible initially.
     ΨM(x)=k=0SM(k,0)PM(k)xk The probability of generating function for the network degree
    distribution PM(k) with considering initial conditions.
     ΨF(x)=k=0SF(k,0)PF(k)xk The probability of generating function for the network degree
    distribution PF(k) with considering initial conditions.
     (1/vM)/(1/vF) Length of the latent period for male/female groups.
     βFM/βMF The transmission rate from a infected female individual to male/from
    a infected male individual to female.
     γM/γF The recovery rate of male/female infected individuals.

     | Show Table
    DownLoad: CSV

    In this section, we assume that individuals in exposed compartment are not infectivity, then deduce a system portraying the spread of sexually transmitted diseases including above variables and parameters. The fraction of the male individuals who are susceptible at time t SM(t) is derived at first. SM(k,0) is the probability which the tested male UM is susceptible and has degree k at t = 0, and θM(t) is the probability that he has not been infected by his partner for a period of time. So we have

    SM(t)=k=0PM(k)θkM(t)SM(k,0)=ΨM(θM(t)).

    So, the proportion of the male individuals who are susceptible at time t is

    SF(t)=k=0PF(k)θkF(t)SF(k,0)=ΨF(θF(t)).

    We can get SM(t) and SF(t) by the equations of θM and θF.

    We can obtain ˙IM=υM(1SMIMRM)γMIM by combining equations ˙IM=υMEMγMIM and SM+EM+IM+RM=1. We also know that RM satisfy ˙RM=γMIM and SM(t)=ΨM(θM(t)), so we can completely define SM,EM,IM and RM assuming θM(t) and initial conditions for RM and IM are known. Similarly, we can completely define SF,EF,IF and RF assuming θF(t) and initial conditions for RF and IF are known.

    If we get the equations of θM and θF, we can close the system. We already know θM(t) is the probability that a tested male individual has not been infected by his randomly chosen partner yet. These partners are made up of susceptible, exposed, infected and recovered females, so we have θM=ϕSFM+ϕEFM+ϕIFM+ϕRFM. Because ϕIFM is the probability of a randomly chosen partner of UM who does not transmit the disease to UM before is infectious at time t, we have

    ddtθM=βFMϕIFM. (2.1)

    Similarly, we have

    ddtθF=βMFϕIMF.

    Now we need to get the equations of ϕIFM and ϕIMF. Because we have the equation θM=ϕSFM+ϕEFM+ϕIFM+ϕRFM. We can find the ϕSFM class, noticing the probability that UM has a female partner who is susceptible at the initial moment is ϕSFM(0) and the probability of the susceptible female has degree k is kPF(k)SF(k,0)jjPF(j)SF(j,0). So her probability of being a susceptible individual after time t is kkPF(k)SF(k,0)θk1FjjPF(j)SF(j,0). Thus we have

    ϕSFM=ϕSFM(0)kkPF(k)SF(k,0)θk1FjjPF(j)SF(j,0)=ϕSFM(0)ΨF(θF)ΨF(1).

    Next we start to calculate the ϕRFM class, From (Figure 1) we notice that only one edge enters the ϕRFM class. We have

    ddtϕRFM=γMϕIFM. (2.2)
    Figure 1.  Flow diagrams of our model.

    Integrating Eq (2.1) and Eq (2.2), we have

    ddtϕRFM=γFβFMddtθM. (2.3)

    Integrating Eq (2.3) from 0 to t yields

    ϕRFM=γF(1θM)βFM+ϕRFM(0).

    So we have

    ϕEFM=θMγF(1θM)βFMϕRFM(0)ϕSFM(0)ΨF(θF)ΨF(1)ϕIFM. (2.4)

    In addition, in the Figure 1 we also notice that there are two edges leaving the class ϕIFM at rates βFM and γF, respectively. And an edge enters the class ϕIFM. We have

    ddtϕIFM=υFϕEFM(γF+βFM)ϕIFM. (2.5)

    At last, the following equation can be obtained,

    ddtϕEFM=ddtϕSFMvFMϕEFM=ϕSFM(0)βMFϕIMFΨF(θF)ΨF(1)vFMϕEFM.

    In summary, our model can be derived to be

    {ddtθM=βFMϕIFM,ddtθF=βMFϕIMF,ddtϕEFM=ϕSFM(0)βMFϕIMFΨF(θF)ΨF(1)υFMϕEFM,ddtϕEMF=ϕSMF(0)βFMϕIFMΨM(θM)ΨM(1)υMFϕEMF,ddtϕIFM=υFϕEFM(γF+βFM)ϕIFM,ddtϕIMF=υMϕEMF(γM+βMF)ϕIMF,SM(t)=k=0PM(k)θkM(t)SM(k,0)=ΨM(θM(t)),SF(t)=k=0PF(k)θkF(t)SF(k,0)=ΨF(θF(t)),dIMdt=υMEMγMIM,dIFdt=υFEFγFIF,EM=1SMRMIM,EF=1SFRFIF. (2.6)

    We can further simplify the model by substituting Eq (2.4) into Eq (2.5), we have

    ddtϕIFM=υF[θMγF(1θM)βFMϕRFM(0)ϕSFM(0)ΨF(θF)ΨF(1)](υF+γF+βFM)ϕIFM.

    Similarly, we have

    ddtϕIMF=υM[θFγM(1θF)βMFϕRMF(0)ϕSMF(0)ΨM(θM)ΨM(1)](υM+γM+βMF)ϕIMF.

    Now we can rewrite the model (2.6) as

    {ddtθM=βFMϕIFM,ddtθF=βMFϕIMF,ddtϕIFM=υF[θMγF(1θM)βFMϕRFM(0)ϕSFM(0)ΨF(θF)ΨF(1)](υF+γF+βFM)ϕIFM,ddtϕIMF=υM[θFγM(1θF)βMFϕRMF(0)ϕSMF(0)ΨM(θM)ΨM(1)](υM+γM+βMF)ϕIMF,SM(t)=k=0PM(k)θkM(t)SM(k,0)=ΨM(θM(t)),SF(t)=k=0PF(k)θkF(t)SF(k,0)=ΨF(θF(t)),dIMdt=υMEMγMIM,dIFdt=υFEFγFIF,EM=1SMRMIM,EF=1SFRFIF. (2.7)

    Considering the following equations of model (2.6) :

    {ddtθM=βFMϕIFM,ddtθF=βMFϕIMF,ddtϕEFM=ϕSFM(0)βMFϕIMFΨF(θF)ΨF(1)υFϕEFM,ddtϕEMF=ϕSMF(0)βFMϕIFMΨM(θM)ΨM(1)υMϕEMF,ddtϕIFM=υFϕEFM(γF+βFM)ϕIFM,ddtϕIMF=υMϕEMF(γM+βMF)ϕIMF. (3.1)

    It is easy to know that (ϕEFM,ϕEMF,ϕIFM,ϕIMF,θM,θF)=(0,0,0,0,1,1) is the disease free equilibrium of system (3.1). We calculate the basic reproduction number R0 by applying the method of the second generation matrix in [37]. In our model (3.1), the classes ϕEFM and ϕEMF act as "exposed" types and the classes ϕIFM and ϕIMF act as "infected" types. Variables θM and θF act as "susceptible" types since they can enter the ϕEFM and ϕEMF classes when the disease breaks out. So we only need to linearize these equations about ϕEFM, ϕEMF, ϕIFM and ϕIMF in (3.1) at the disease free equilibrium (ϕEFM=ϕEMF=ϕIFM=ϕIMF=0), we have

    {ddtϕEFM=ϕSFM(0)βMFϕIMFΨF(1)ΨF(1)υFϕEFM,ddtϕEMF=ϕSMF(0)βFMϕIFMΨM(1)ΨM(1)υMϕEMF,ddtϕIFM=υFϕEFM(γF+βFM)ϕIFM,ddtϕIMF=υMϕEMF(γM+βMF)ϕIMF.

    Applying the method of the second generation matrix, we obtain

    ddt(ϕEFMϕEMFϕIFMϕIMF)=(FV)(ϕEFMϕEMFϕIFMϕIMF),

    where

    F=(000ϕSFM(0)βMFΨF(1)ΨF(1)00ϕSMF(0)βFMΨM(1)ΨM(1)000000000),

    and

    V=(υF0000υM00υF0γF+βFM00υM0γM+βMF).

    Thus,

    FV1=(0ϕSFM(0)βMFΨF(1)(βMF+γM)ΨF(1)0ϕSFM(0)βMFΨF(1)(βMF+γM)ΨF(1)ϕSMF(0)βFMΨM(1)(βFM+γF)ΨM(1)0ϕSMF(0)βFMΨM(1)(βFM+γF)ΨM(1)000000000).

    Hence,

    R0=ρ(FV1)=βFMβMFϕSMF(0)ϕSFM(0)ΨM(1)ΨF(1)(βFM+γF)(βMF+γM)ΨM(1)ΨF(1), (3.2)

    where ρ represents the spectral radius and the R0 is the basic reproduction number. The biological interpretation of ΨM(1)ΨM(1) comes from observing the situation of a random individual's partner VM in the early stages of the epidemic. If VM is infected by that randomly infected female, then ΨM(1)ΨM(1) is the expectant number of other partners VM has (his excess degree). Then βMFβMF+γM is the possibility that an infected male individual transmit the disease to his partner. So we can get ϕSMF(0)βMFΨM(1)(βMF+γM)ΨM(1) is the number of individuals who may be infected by VM. We have a similar result for VF infected with male individual who are randomly infected, that is, ϕSFM(0)βFMΨF(1)(βFM+γF)ΨF(1) is the number of individuals who may be infected by VF. So R0 is the geometric mean of the number of individuals infected with VM and the number of individuals infected with VF, which is consistent with the result we calculated.

    We usually calculate basic reproduction number of the disease at infinitesimal initial values (ϕSFM(0)=ϕSMF(0)=1, ϕRFM(0)=ϕRMF(0)=0), then R0 in Eq (3.2) becomes

    ^R0=βFMβMFΨM(1)ΨF(1)(βFM+γF)(βMF+γM)ΨM(1)ΨF(1).

    Yan et al.[33] derived the basic reproduction number of a new edge-based SIR model of sexually transmitted diseases on bipartite networks. Comparing to that of [33], we know that the R0 of our model is the same. We also note that R0 is symmetric in the parameters describing of male and female properties being consistent with [38].

    The final size relation has been done for various models with small or large initial conditions [38,36]. Motivated by [38,36], the final epidemic size of our model be derived in what follows.

    We set ddtθM=ddtθF=ddtϕIFM=ddtϕIMF=ddtIF=ddtIM=ddtEF=ddtEM=0, so ϕEFM()=ϕEMF()=EF()=EM()=0 and ϕIFM()=ϕIMF()=IF()=IM()=0. From (2.7), we have

    θM=βFMβFM+γF(γFβFM+ϕRFM(0)+ϕSFM(0)ΨF(θF)ΨF(1)),

    and

    θF=βMFβMF+γM(γMβMF+ϕRMF(0)+ϕSMF(0)ΨM(θM)ΨM(1)).

    Since ΨF(θF)ΨF(1)=kkPF(k)SF(k,0)θk1FjjPF(j)SF(j,0) and ΨM(θM)ΨM(1)=kkPM(k)SM(k,0)θk1MjjPM(j)SM(j,0), we have

    θM=βFMβFM+γF(γFβFM+ϕRFM(0)+ϕSFM(0)kkPF(k)SF(k,0)θk1FjjPF(j)SF(j,0))=βFMβFM+γF[γFβFM+ϕRFM(0)+ϕSFM(0)kkPF(k)SF(k,0)jjPF(j)SF(j,0)(βMFβMF+γM)k1(γMβMF+ϕRMF(0)+ϕSMF(0)ΨM(θM)ΨM(1))k1],

    and

    θF=βMFβMF+γM[γMβMF+ϕRMF(0)+ϕSMF(0)kkPM(k)SM(k,0)jjPM(j)SM(j,0)(βFMβFM+γF)k1(γFβFM+ϕRFM(0)+ϕSFM(0)ΨF(θF)ΨF(1))k1].

    Then

    θM()=βFMβFM+γF[γFβFM+ϕRFM(0)+ϕSFM(0)kkPF(k)SF(k,0)jjPF(j)SF(j,0)(βMFβMF+γM)k1(γMβMF+ϕRMF(0)+ϕSMF(0)ΨM(θM())ΨM(1))k1],

    and

    θF()=βMFβMF+γM[γMβMF+ϕRMF(0)+ϕSMF(0)kkPM(k)SM(k,0)jjPM(j)SM(j,0)(βFMβFM+γF)k1(γFβFM+ϕRFM(0)+ϕSFM(0)ΨF(θF())ΨF(1))k1].

    Since we have SM()=ΨM(θM()) and SF()=ΨF(θF()), we can get the final epidemic size with arbitrary initial conditions are

    RM()=1SM()RM(0)=1ΨM(θM())RM(0), (3.3)

    and

    RF()=1SF()RF(0)=1ΨF(θF())RF(0). (3.4)

    We calculate the final epidemic size of the disease at infinitesimal initial values, that is, ϕRFM(0)=ϕRMF(0)=0,ϕSFM(0)=ϕSMF(0)=1,RM(0)=RF(0)=0. Then,

    θM()=βFMβFM+γF[γFβFM+kkPF(k)SF(k,0)jjPF(j)SF(j,0)(βMFβMF+γM)k1(γMβMF+ΨM(θM())ΨM(1))k1],

    and

    θF()=βMFβMF+γM[γMβMF+kkPM(k)SM(k,0)jjPM(j)SM(j,0)(βFMβFM+γF)k1(γFβFM+ΨF(θF())ΨF(1))k1].

    Further we get the final epidemic size at infinitesimal initial values

    RM()=1SM()RM(0)=1ΨM(θM()),

    and

    RF()=1SF()RF(0)=1ΨF(θF()).

    From Eqs (3.2), (3.3) and (3.4), we know that the basic reproduction number and the final size of an epidemic are not related to the length of latent period in the SEIR model without infectivity during the latent period.

    In the section, we investigate the disease equilibrium of our model at the infinitesimal initial conditions, i.e. ϕEFM(0)=ϕIFM(0)=0 and ϕEMF(0)=ϕIMF(0)=0 or ϕSFM(0)+ϕRFM(0)=1 and ϕRMF(0)+ϕSMF(0)=1. We only need to study the equation group consisting of the equations of ϕIFM,ϕIMF,θM and θF in the model (2.7), i.e.

    {ddtϕIFM=υF[θMγF(1θM)βFMϕRFM(0)ϕSFM(0)ΨF(θF)ΨF(1)](υF+γF+βFM)ϕIFM,ddtϕIMF=υM[θFγM(1θF)βMFϕRMF(0)ϕSMF(0)ΨM(θM)ΨM(1)](υM+γM+βMF)ϕIMF,ddtθM=βFMϕIFM,ddtθF=βMFϕIMF. (3.5)

    Noting that ϕIFM,ϕIMF,θM and θF are all probabilities, we only need to consider this system in Ω={(ϕIFM,ϕIMF,θM,θF)|0ϕIFM,ϕIMF,θM,θF1}. It is easy to verify that Ω is a positive invariant set of system (3.5). We have these results in what follows.

    Theorem 1. There is a disease free equilibrium E0(0,0,1,1) in the system (3.5) with infinitesimal infected initial values. Moreover,

    (Ⅰ) if R0<1, the disease free equilibrium E0 is locally asymptotically stable,

    (Ⅱ) if R0>1, there exists only one endemic equilibrium E=(0,0,θM,θF) in which 0<θM,θF<1, and it is locally asymptotically stable.

    Proof. We know that the equilibria need to satisfy the following.

    {υF[θMγF(1θM)βFMϕRFM(0)ϕSFM(0)ΨF(θF)ΨF(1)](υF+γF+βFM)ϕIFM=0,υM[θFγM(1θF)βMFϕRMF(0)ϕSMF(0)ΨM(θM)ΨM(1)](υM+γM+βMF)ϕIMF=0,βFMϕIFM=0,βMFϕIMF=0.

    Now we study system (3.5) in Ω. It is easy to know that E0=(0,0,1,1) is always a disease free equilibrium in the system (3.5). Below we use three steps to complete our proof.

    (1) Noting that for system (3.5) whose initial infection is infinitesimal, the characteristic equation at the disease free equilibrium E0 is

    |υF+βFM+γF+λ0υFυFγFβFMυFϕSFM(0)ΨF(1)ΨF(1)0υM+βMF+γM+λυMϕSMF(0)ΨM(1)ΨM(1)υMυMγMβMFβFM0λ00βMF0λ|=0. (3.6)

    Therefore, Eq (3.6) can be written as

    λ4+a1λ3+a2λ2+a3λ+a4=0,

    where

    a1=βFM+βMF+γF+γM+υF+υM,a2=(βFM+γF)(βMF+γM)+(βFM+βMF+γF+γM)(υF+υM)+υFυM,a3=(βFM+γF)(βMF+γM)(υF+υM)+υFυM(βFM+βMF+γF+γM),a4=[(βFM+γF)(βMF+γM)ϕSMF(0)ϕSFM(0)ΨM(1)ΨM(1)ΨF(1)ΨF(1)βFMβMF]υFυM.

    Applying Routh-Hurwitz criteria, if R0<1, we have βFMβMFϕSMF(0)ϕSFM(0)ΨM(1)ΨF(1)(βFM+γF)(βMF+γM)ΨM(1)ΨF(1)<1, further we can get

    (βFM+γF)(βMF+γM)>ϕSMF(0)ϕSFM(0)ΨM(1)ΨM(1)ΨF(1)ΨF(1)βFMβMF.

    So a4>0, and we have

    b1=a1>0,b2=a1a2a3>0,b3=|a1a301a2a40a1a3|>0,b4=a4b3>0.

    For the sake of clarity, we put the formulae of bi(i=1,2,3,4) in the Appendix A. We can clearly know that the disease free equilibrium E0 is locally asymptotically stable if R0<1.

    (2) We show that there is a unique endemic equilibrium E=(0,0,θM,θF) in which 0<θM,θF<1 when R0>1. Inspired by [33], we know that ϕIFM and ϕIMF are always equal to 0 at the equilibrium and construct the auxiliary function from system (2.7):

    f(θM)=θMβFMϕSFM(0)βFM+γFΨF(θF)ΨF(1)γF+βFMϕRFM(0)βFM+γF,

    where

    θF=βMFϕSMF(0)βMF+γMΨM(θM)ΨM(1)+γM+βMFϕRMF(0)βMF+γM.

    We only need to prove that f(θM) has a unique solution between 0 and 1. It is easy for us to compute that f(0)<0 and f(1)=0, we have

    df(θM)dθM=1βFMϕSFM(0)βFM+γFΨF(θF)ΨF(1)dθFdθM=1βFMϕSFM(0)βFM+γFΨF(θF)ΨF(1)βMFϕSMF(0)βMF+γMΨM(θM)ΨM(1), (3.7)

    and

    d2f(θM)dθ2M=βFMϕSFM(0)βFM+γFΨ(3)F(θF)ΨF(1)(βMFϕSMF(0)βMF+γMΨM(θM)ΨM(1))2βFMϕSFM(0)βFM+γFΨF(θF)ΨF(1)βMFϕSMF(0)βMF+γMΨ(3)M(θM)ΨM(1). (3.8)

    According to d2f(θM)dθ2M0, we find that f(θM) is concave. Therefore df(θM)dθM is monotonically decreasing in the interval (0, 1). We have βFMϕSFM(0)βFM+γFΨF(1)ΨF(1)βMFϕSMF(0)βMF+γMΨM(1)ΨM(1)>1 according R0>1, we also find df(0)dθM=1>0 and df(1)dθM<0 from Eq (3.7). Therefore, we can derive a θM in the interval (0, 1) when R0>1 so that df(θM)dθM=0 according to the intermediate value theorem. Since f(θM) is a concave function in the interval (0, 1), we can get θM to make f(θM) the largest in the interval, so θM(0,1) and f(θM)>f(1)=0. Considering the monotonicity of f(θM), we use the intermediate value theorem to find a θM(0,θM) makes f(θM)=0 and θM is unique on the basis of the increment of f(θM) in (0,θM) and the decrement of f(θM) in (θM,1) (see Figure 2).

    Figure 2.  The simple graph of f(θM) in interval [0, 1] if R0>1.

    (3) We prove that the endemic equilibrium E is locally asymptotically stable in Ω. The characteristic equation at the endemic equilibrium E is

    |υF+βFM+γF+λ0υFυFγFβFMυFϕSFM(0)ΨF(θF)ΨF(1)0υM+βMF+γM+λυMϕSMF(0)ΨM(θM)ΨM(1)υMυMγMβMFβFM0λ00βMF0λ|=0. (3.9)

    Therefore, Eq (3.9) can be written as

    λ4+c1λ3+c2λ2+c3λ+c4=0,

    where

    c1=υF+υM+βFM+βMF+γF+γM,c2=(βFM+γF)(βMF+γM)+(βFM+βMF+γF+γM)(υF+υM)+υFυM,c3=(βFM+γF)(βMF+γM)(υF+υM)+υFυM(βFM+βMF+γF+γM),c4=[(βFM+γF)(βMF+γM)ϕSFM(0)ϕSMF(0)ΨM(θM)ΨM(1)ΨF(θF)ΨF(1)βFMβMF]υFυM.

    Applying Routh-Hurwitz, we have

    d1=c1>0,d2=c1c2c3>0,d3=|c1c301c2c40c1c3|>0,d4=c4d3=[(βFM+γF)(βMF+γM)ϕSFM(0)ϕSMF(0)ΨM(θM)ΨM(1)ΨF(θF)ΨF(1)βFMβMF]υFυMd3=(βFM+γF)(βMF+γM)(1βFMϕSFM(0)βFM+γFΨF(θF)ΨF(1)βMFϕSMF(0)βMF+γMΨM(θM)ΨM(1))d3=(βFM+γF)(βMF+γM)df(θM)dθMd3>(βFM+γF)(βMF+γM)df(θM)dθMd3=0.

    For the sake of clarity, we put formulae of di (i=1,2,3,4) in the Appendix B. We can clearly know that the endemic equilibrium E is locally asymptotically stable. The proof is completed.

    We analyze the local dynamics of system (2.7) with larger initial value of infection in this section. The null lines of θM and θF in the model (2.7) are in what follows:

    LM:γF(1θM)βFM+ϕRFM(0)+ϕSFM(0)ΨF(θF)ΨF(1)=0,
    LF:γM(1θF)βMFϕRMF(0)+ϕSMF(0)ΨM(θM)ΨM(1)=0.

    Where ϕRMF(0) and ϕRFM(0) are fixed, as the values of ϕIMF(0) and ϕIFM(0) increase from 0, LM and LF move to the left and down, respectively. We have the following two claims:

    (1) If R0<1, system (2.7) for infinitesimal initial conditions have only one locally asymptotically stable disease free equilibrium E0(0,0,1,1) in Ω. As ϕIMF(0) and ϕIFM(0) gradually increase from 0, the disease free equilibrium E0 moves from (0, 0, 1, 1) to the lower left to an internal point E0=(0,0,θ(0)M,θ(0)F) of Ω in which 0<θ(0)M,θ(0)F<1, and E0 and E0 are consistent in local stability.

    (2) If R0>1, system (2.7) for infinitesimal initial conditions has one locally asymptotically stable endemic equilibrium E=(0,0,θM,θF) and an unstable disease free equilibrium E0(0,0,1,1) in Ω. As ϕIMF(0) and ϕIFM(0) gradually increase from 0, disease free equilibrium E0 moves from (0, 0, 1, 1) to the upper right to an external point E0=(0,0,θ(0)M,θ(0)F) with min{θ(0)M,θ(0)F}>1 (we do not analyze its dynamics since this point is not in Ω), and the endemic equilibrium E moves from (0,0,θM,θF) slightly to the lower left to another internal point E=(0,0,θM,θF) of Ω in which 0<θ()M<θM and 0<θ()F<θF. In addition. The stability of E0 and E is consistent with E0 and E respectively.

    In summary, we give the following theorem for system (2.7) with max{ϕIMF(0),ϕIFM(0)}>0.

    Theorem 2. For our system (2.7) with large initial values of infection.

    (Ⅰ) if R0<1, system has only one disease free equilibrium E0=(0,0,θ(0)M,θ(0)F) of Ω with 0<θ(0)M,θ(0)F<1, and the solution gradually approaches (0,0,θ(0)M,θ(0)F) from (0, 0, 1, 1),

    (Ⅱ) if R0>1, system has only one endemic equilibrium E=(0,0,θM,θF) in Ω, and the solution gradually approaches (0,0,θM,θF) from (0, 0, 1, 1).

    Proof. Similar to the proof of Theorem 1, we show that there is a unique equilibrium in Ω. Since ϕIFM and ϕIMF are always equal to zero at the equilibrium point, we construct the following auxiliary function:

    g(θM)=θMβFMϕSFM(0)βFM+γFΨF(θF)ΨF(1)γF+βFMϕRFM(0)βFM+γF,

    where

    θF=βMFϕSMF(0)βMF+γMΨM(θM)ΨM(1)+γM+βMFϕRMF(0)βMF+γM.

    We only need to prove that there is a unique solution for g(θM) between 0 and 1. Getting g(0)<0 is easy for us, and we have

    dg(θM)dθM=1βFMϕSFM(0)βFM+γFΨF(θF)ΨF(1)dθFdθM=1βFMϕSFM(0)βFM+γFΨF(θF)ΨF(1)βMFϕSMF(0)βMF+γMΨM(θM)ΨM(1),

    and

    d2g(θM)dθ2M=βFMϕSFM(0)βFM+γFΨ(3)F(θF)ΨF(1)(βMFϕSMF(0)βMF+γMΨM(θM)ΨM(1))2βFMϕSFM(0)βFM+γFΨF(θF)ΨF(1)βMFϕSMF(0)βMF+γMΨ(3)M(θM)ΨM(1). (4.1)

    According to d2g(θM)dθ2M0 for all θM0 in Eq (4.1), we find that g(θM) is concave. When the initial infection is arbitrary large, we have

    0<ϕSFM(0)+ϕRFM(0)<1,0<ϕSMF(0)+ϕRMF(0)<1,θM(0)=1,θF(0)=1.

    According to 0<θF<1, we have

    g(1)=1βFMϕSFM(0)βFM+γFΨF(θF)ΨF(1)γF+βFMϕRFM(0)βFM+γF>1βFMϕSFM(0)βFM+γFγF+βFMϕRFM(0)βFM+γF=1βFM(ϕSFM(0)+ϕRFM(0))+γFβFM+γF>0.

    Considering g(θM) is concave when θM0, we use the intermediate value theorem to find that there is a unique ~θM in (0, 1) to make g(~θM)=0. And we have

    ~θF=βMFϕSMF(0)βMF+γMΨM(~θM)ΨM(1)+γM+βMFϕRMF(0)βMF+γM.

    We conclude that our system (2.7) has only one equilibrium ˜E=(0,0,~θM,~θF) in Ω. It is easy to conclude that ˜E is locally asymptotically stable which is similar to the proof process of Theorem 1. And the solutions of our system gradually approach ˜E from (0, 0, 1, 1). The proof is completed.

    Example 1. We assume that both males and females in system (2.7) obey the Poisson distributions given by PM(k)=PF(k)=λkeλk! with λ=5, ϕRFM(0)=ϕRMF(0)=0. We get R0=0.5556<1 and a unique locally asymptotically stable disease free equilibrium E0=(0,0,1,1) in Ω (see Figure 3(a)). Moreover, we also get R0=0.25<1 and a unique locally asymptotically stable disease free equilibrium E0=(0,0,0.9070,0.9071) in Ω (see Figure 4(a)). It is obvious that the solution gradually approaches (0,0,0.9070,0.9071) from (0, 0, 1, 1) as ϕIFM(0) and ϕIMF(0) increase from 0.

    Figure 3.  (a) Phase plane plot of model (2.7) with R0<1, where ϕSMF(0)=ϕSFM(0)=1,ϕIMF(0)=ϕIFM(0)=0,γM=γF=0.8,βFM=βMF=0.1. (b) Phase plane plot of model (2.7) with R0>1, where ϕSMF(0)=ϕSFM(0)=1,ϕIMF(0)=ϕIFM(0)=0,γM=γF=0.2,βFM=βMF=0.1.
    Figure 4.  (a) Phase plane plot of model (2.7) with R0<1, where ϕSMF(0)=ϕSFM(0)=0.4,ϕIMF(0)=ϕIFM(0)=0.4,γM=γF=0.8,βFM=βMF=0.01. (b) Phase plane plot of model (2.7) with R0>1, where ϕSMF(0)=ϕSFM(0)=0.4,ϕIMF(0)=ϕIFM(0)=0.4,γM=γF=0.2,βFM=βMF=0.01.

    We set parameter values and initial values, getting R0=2>1, a disease free equilibrium E0 and the locally asymptotically stable endemic equilibrium E=(0,0,0.6813,0.6813) in Ω (see Figure 3(b). Moreover, we also get R0=1.8>1 and a unique locally asymptotically stable endemic equilibrium E=(0,0,0.6763,0.6765) in Ω (see Figure 4(b)). It is obvious that the solution gradually approaches (0,0,0.6763,0.6765) from (0, 0, 1, 1) as ϕIFM(0) and ϕIMF(0) increase from 0.

    We implement the comparison between stochastic simulations and numerical predictions to our SEIR model on Poisson and scale-free networks in this section. The distributions of Poisson network and scale-free network are given by P(k)=λkeλk!(1k10) and P(k)=(r1)m(r1)kr(3k10), respectively. The average degree of these two networks is 5. For network degree distributions of male and female individuals, we use the configuration model described in Section 2 to generate random contact.

    Our model is defined in a random simulation as follows. The model has no infectivity during the latent period. Once a susceptible male is infected with a rate of βFMiF, where iF is the partner (female) node which he is exposed to, and he first passes through the latent period with an average length of 1/υM, then enters the infectious class. The same is true for female nodes. We define parameters are βFM=βMF=0.01,γF=γM=0.04,υF=υM=0.2. The initial values are ϕEFM(0)=ϕEMF(0)=0.01,ϕIFM(0)=ϕIMF(0)=0.04,ϕRFM(0)=ϕRMF(0)=0.06,EM(0)=EF(0)=500,IM(0)=IF(0)=200,RM(0)=RF(0)=300. If there are no exposed and infected individuals in the networks, the entire spread of epidemic will stop. As we can see from Figures 5 and 6, the prediction of model and average of stochastic simulation fit well on both types of networks with the same average degree. This shows that model we built can accurately simulate the spread of disease.

    Figure 5.  The comparison of system (2.7) prediction value (black lines) with the ensemble averages (red circles) of 100 runs of stochastic simulations (blue lines) on a Poisson bipartite network with NM=NF=5000 and P(k)=λkeλk!(1k10). Disease parameters are βFM=βMF=0.01,γF=γM=0.04,υF=υM=0.2. The initial values are ϕEFM(0)=ϕEMF(0)=0.1,ϕIFM(0)=ϕIMF(0)=0.04,ϕRFM(0)=ϕRMF(0)=0.06,EM(0)=EF(0)=500,IM(0)=IF(0)=200,RM(0)=RF(0)=300.
    Figure 6.  The comparison of system (2.7) prediction value (black lines) with the ensemble averages (red circles) of 100 runs of stochastic simulations (blue lines) on a scale-free bipartite network with P(k)=(r1)m(r1)kr(3k10), where m stands for the minimum number of partners for individuals and r is variable of power law exponent. Let m = 3, r = 3 and NM=NF=5000. Disease parameters are βFM=βMF=0.01,γF=γM=0.04,υF=υM=0.2. The initial values are ϕEFM(0)=ϕEMF(0)=0.1,ϕIFM(0)=ϕIMF(0)=0.04,ϕRFM(0)=ϕRMF(0)=0.06,EM(0)=EF(0)=500,IM(0)=IF(0)=200,RM(0)=RF(0)=300.

    Considering that many sexually transmitted diseases have different proportions of exposed and infectious individuals among male and female populations. We set the initial values of the male population different from the female population on two networks in Figures 7 and 8. We observe that when the initial infection of women is lower than that of men, the peak of female infectious individuals will be higher than that of male. This result is consistent on both Poisson and scale-free networks. We can find that this is consistent with the real situation. When the number of female individuals infected is small initially, the number of male individuals infected by female will be smaller, and finally the peak of male individuals infected is lower than that of female.

    Figure 7.  The comparison of SEIR dynamics by setting the difference between the male initial values and the female initial values on a Poisson bipartite network(P(k)=λkeλk!(1k10)) with 100 runs of stochastic simulations. We set initial values are ϕEFM(0)=0.01,ϕEMF(0)=0.1,ϕIFM(0)=0.004,ϕIMF(0)=0.04,ϕRFM(0)=0.006,ϕRMF(0)=0.06,EM(0)=500,IM(0)=200,RM(0)=300,EF(0)=50,IF(0)=20 and RF(0)=30 with NM=NF=5000. Disease parameters are βFM=βMF=0.01,γF=γM=0.04,υF=υM=0.2.
    Figure 8.  The comparison of SEIR dynamics by setting the difference between the male initial values and the female initial values on scale-free networks (P(k)=(r1)m(r1)kr(4k10)) with 100 runs of stochastic simulations. We set initial values are ϕEFM(0)=0.01,ϕEMF(0)=0.1,ϕIFM(0)=0.004,ϕIMF(0)=0.04,ϕRFM(0)=0.006,ϕRMF(0)=0.06,EM(0)=500,IM(0)=200,RM(0)=300,EF(0)=50,IF(0)=20 and RF(0)=30 with NM=NF=5000. Disease parameters are βFM=βMF=0.01,γF=γM=0.04,υF=υM=0.2.

    In Figures 9-11, we change the initial conditions on the Poisson and the scale-free networks, and observe the dynamics of the model. In Figure 9(a), we change the value of the initial infectious individuals IM(0)(=IF(0)) with RM(0)(=RF(0)) is fixed, observing that the initial values change does not affect the peak arrival time of disease, but it affects the peak size of disease on different networks (i.e. the larger IM(0)(=IF(0)), the greater the peak of the disease). In Figure 9(b), we change the value of the initial recoverers RM(0)(=RF(0)) with IM(0)(=IF(0)) is fixed, observing that the change in the initial values does not affect the disease peak arrival time, but it affects the peak size on different networks (i.e. the larger RM(0)(=RF(0)), the greater the peak of the disease).

    Figure 9.  The comparison of SEIR dynamics by varying the initial infections IM(0)(=IF(0)) on different networks. 100 simulations were performed for each initial infections, and each curve represents the average of 100 random simulations. We set disease parameters are βFM=βMF=0.01,γF=γM=0.04,υF=υM=0.2.
    Figure 10.  Disease parameters are the same as in Figure 9. (a) Contour map of the initial infectious individuals IM(0)(=IF(0)) on a Poisson network. (b) Contour map of the initial infectious individuals IM(0)(=IF(0)) on a scale-free network.
    Figure 11.  Disease parameters are the same as in Figure 9. (a) Contour map of the initial recoverers RM(0)(=RF(0)) on a Poisson network. (b) Contour map of the initial recoverers RM(0)(=RF(0)) on a scale-free network.

    Figure 10(a) and (b) are contour maps of the initial infectious individuals on Poisson and scale-free networks respectively, then we observe that the change in IM(0)(=IF(0)) does not affect the peak arrival time of the infection but affects the peak value with that peak size is proportional to IM(0) and IF(0) on two types of networks. In addition, we also find that the scale-free network has a larger disease peak than the Poisson network in the same initial infections.

    Figure 11(a) and (b) are contour maps of the initial recoverers on Poisson and scale-free networks respectively, then we find that the initial recoverers RM(0)(=RF(0)) affect the peak arrival time on two types of networks. It is obvious that the larger the RM(0) and RF(0) are, the earlier the peak arrives. We also find that the scale-free network has a larger disease peak than the Poisson network in the same initial recoverers.

    From Figure 12, we know that changing the length of the latent period 1/υM(=1/υF) can affects both the peak size of the infection and its arrival time. The shorter the latent period is, the larger the peak value an the earlier the arrival time. So we can take some measures to regulate the length of the latent period to interfere with the spread of the disease.

    Figure 12.  Disease parameters are βFM=βMF=0.01,γF=γM=0.04. (a) Contour map of the length of the latent period 1/υM(=1/υF) on a Poisson network. (b) Contour map of the length of the latent period 1/υM(=1/υF) on a scale-free network.

    Figure 13(a) shows the final epidemic size RM(). Parameters ρ and k represent the ratio of the initial infected and exposed individuals and the average degree of the males and females respectively. We assume that the degree distribution in system (2.7) is subject to the Poisson network (P(k)=λkeλk!(1k20)). We have ΨM(x)=ΨF(x)=eλ(x1) and ΨM(1)=ΨF(1)=λ=k. We set initial values ϕRFM(0)=ϕRMF(0)=0 and parameters βFM=βMF=0.01,γF=γM=0.04,υM=υF=0.2. We have ϕSFM(0)=ϕSMF(0)=1ρ. We can see from the figure that the final epidemic size increases when the average degree k increases, where ρ remains unchanged. And when k remains unchanged, the final epidemic size increases with the increase of ρ. That is, the final epidemic size is proportional to ρ and k.

    Figure 13.  (a) Phase diagram of the SEIR model on a Poisson network (P(k)=λkeλk!(1k20)). The RM is shown as a function of the ratio of the initial infected and exposed individuals ρ and average degree k. (b) Sensitivity analysis. The partial rank correlation coefficients(PRCCs) results for the dependence of R0 on each parameter, and gray rectangles indicate sensitivity between 0.2(-0.4) and 0.4(-0.2).

    In Figure 13(b), we study the effect of each parameter on R0. The PRCCs are calculated with respect to βFM,βMF,γF,γM,ϕSFM(0),ϕSMF(0) and k with 2000 simulations. The input variables are subject to uniform distribution, and the positive and negative signs indicate that the effect is positive or negative, respectively. Sensitivity between 0 and 0.2 indicates that the parameter is weakly correlated, 0.2 to 0.4 is moderately correlated, and above 0.4 is highly correlated. As shown in Figure 13(b), we can see that βFM,βMF,ϕSFM(0) and ϕSMF(0) have a positive influence and are highly correlated on R0, γF and γM have a negative influence and are highly correlated on R0, and k is an insensitive parameter.

    In this paper, we extend an edge-based sexually transmitted SEIR model with no infectivity during the latent period, which the relationship between individuals is described by a bipartite network. We assume that the contact network is static and ignore the birth, death and migration of the population. We derive the basic reproduction number and the implicit formulas of the final epidemic size. We further analyze the dynamics of our model on different initial conditions, and observe the effects of different initial values on disease epidemics numerically. The basic reproduction number is consistent with that of [33] if we assume that transmission rate of the latent period is zero. Furthermore we also find that the length of the latent period is very much related to the arrival time and size of disease peak. How to construct and study the edge-based sexually transmitted models when the contact network is dynamic and the birth, death and migration of the population are considered are interesting. We leave this work in future.

    We are grateful to the anonymous referees and the editors for their valuable comments and suggestions which improved the quality of 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.



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