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Research article

Computing the conjugacy classes and character table of a non-split extension 26·(25:S6) from a split extension 26:(25:S6)

  • In this paper, we will demonstrate how the character table of a sub-maximal subgroup 26:(25:S6) of the sporadic simple group Fi22 can be used to obtain the conjugacy classes and character table of a non-split extension of the form 26(25:S6), which sits maximal in the unique non-split extension 26Sp6(2).

    Citation: Abraham Love Prins. Computing the conjugacy classes and character table of a non-split extension 26·(25:S6) from a split extension 26:(25:S6)[J]. AIMS Mathematics, 2020, 5(3): 2113-2125. doi: 10.3934/math.2020140

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  • In this paper, we will demonstrate how the character table of a sub-maximal subgroup 26:(25:S6) of the sporadic simple group Fi22 can be used to obtain the conjugacy classes and character table of a non-split extension of the form 26(25:S6), which sits maximal in the unique non-split extension 26Sp6(2).


    Zika virus, a mosquito-borne flavivirus, was first isolated in monkeys a rhesus in Uganda in 1947. Later, it was detected in humans in Uganda and the United Republic of Tanzania in 1952 [1]. From the 1960s to 1980s, rare sporadic cases of human infections were found throughout Africa and Asia. The first recorded outbreak was reported from the Island of Yap in 2007 [2]. In March 2015, Brazil reported a large outbreak of rash illness, soon identified as Zika virus infection, and later found to be associated with Guillain-Barr ˊe syndrome and microcephaly [3]. On February 1, 2016, WHO declared Zika as a "Public Health Emergency of International Concern" [4]. Since the outbreak of Zika in Brazil, the expansion of the Zika outbreak has seemed unstoppable. It spread rapidly from Brazil to northern Europe [5], Australia [6], through Canada [7], the USA [8], subsequently, arrived to reach Japan [9], China [10]. Zika cases have been reported in 90 countries and territories by November 4, 2019 [11]. Recently, November 6, 2021, 13 new cases were reported in Uttar Pradesh's Kanpur district, which took the case tally to 79 in the state during the past two weeks. Authorities in the Indian capital region said they were on alert in the wake of a spike in Zika virus cases in the neighbouring state of Uttar Pradesh. It is natural to ask how the previously unknown Zika virus spreads rapidly in the short term.

    Since the outbreak of Zika in Brazil, many models have been proposed to study spread, impact, and control of Zika disease and dynamic behaviors. Zhang et al. [12] employed a SEIR (Susceptible-Exposed-Infected-Removed)-SEI (Susceptible-Exposed-Infected) human-vector model to estimate the time of first introduction of Zika to Brazil. Zhao et al. [13] considered the limited medical resources in Zika model and obtained rich bifurcation phenomena, such as, backward bifurcation, Hopf bifurcation, Bogdanov-Takens bifurcation of codimension 2 and discontinuous bifurcation. Various ordinary differential equations (ODEs) models and dynamics analyses had been applied to study Zika outbreak [14,15,16,17,18,19,20]. However, the above models ignore the effects of the spatial factors. In fact, the spread of the disease concerns not only the time, but also the spatial location. For this purpose, some researchers began to describe spatio-temporal transmission of Zika disease through partial differential equations (PDEs) [21,22]. Miyaoka et al. [21] considered spatial movement of humans and vectors and formulated a reaction diffusion model to research the effect of vaccination on the transmission and control of Zika disease. Yamazaki [22] added diffusive terms in Zika model in order to capture the movement of human hosts and mosquitoes, considering the unique threat of the sexual transmission of Zika disease. In the above PDEs models, all the coefficients are positive constants. That is, the dynamics of humans and vectors are described in spatially homogeneous environments. However, the diffusion dynamics of the disease is affected by the natural landscapes, the urban and rural distribution, even cultural geographical factors [23]. To make the model more consistent with the spread laws of the disease, the spatial heterogeneity must be considered. Hence, it is necessary to understand the transmission dynamics of the Zika disease influenced by the spatial heterogeneity [24]. However, in the above studies, the contaminated aquatic environments, an important transmission route for Zika virus, seem to have received little attention.

    Recently, experimental result [25] demonstrated that Aedes mosquitoes can acquire and transmit Zika virus by breeding in contaminated aquatic environments. It implies that Aedes mosquitoes are infected by Zika virus not only through biting infectious hosts but also through urine excreted by Zika patients. This new transmission route makes the transmission cycle of Zika virus much shorter. It may be one of the major causes of rapid spread of Zika virus in nature. Therefore, it is more reasonable to introduce environment transmission route (That is, human-environment-mosquito-human transmission route) into Zika model [26]. However, few Zika models incorporate environment transmission route and spatial heterogeneity simultaneously.

    The paper is organized as follows. In Section 2, we propose a novel reaction-diffusion Zika model with environment transmission route in a spatial heterogeneous environment, which is different from all Zika models mentioned earlier. In Section 3, the well-posedness and some properties of the model are also discussed. The basic offspring number Rm0 and basic reproduction number R0 for our spatial model are established in Section 4. In Section 5, by using comparison arguments and the theory of uniform persistence, the threshold dynamics for the model in terms of Rm0 and R0 are analysed. A brief conclusion is given in Section 7.

    In this section, we propose a reaction-diffusion Zika model with environment transmission route. Considering the effects of individual mobility, we assume that a host population lives in a spatial heterogeneous environment, which is represented with a bounded domain Ω with smooth boundary Ω. Mosquitoes are classified in aquatic and adult mosquitoes. Here we combine the egg, larval and pupal stages as one aquatic stage. Aquatic mosquitoes are divided into susceptible and infectious compartments, and their spatial densities at location x and time t are represented by Sa(x,t) and Ia(x,t), respectively. Adult mosquitoes are divided into susceptible and infectious compartments with spatial densities Sm(x,t) and Im(x,t), respectively. We divide the density of total human population at location x and time t, denoted by Nh(x,t), into three categories: susceptible humans Sh(x,t), infectious humans Ih(x,t) and recovered humans Rh(x,t). So Nh(x,t)=Sh(x,t)+Ih(x,t)+Rh(x,t). V(x,t) represents the density at location x and time t of Zika virus within the contaminated aquatic environments. Implication of V(x,t) is similar to avian influenza virus concentration in water [27,28].

    In this paper, we extend our previous model [26] to consider mosquitoes and humans in spatially heterogeneous environments. So, the transmission path of Zika virus is similar to literature [26]. In order to incorporate the multiple factors of diffusion and spatial heterogeneity in the spatial domain Ω, we assume that the parameters K(x), ϑ(x), ω(x), μa(x), μm(x), Λ(x), μh(x), r(x), a(x), β1(x), β2(x), βv(x), θ(x), δ(x) are functions of the spatial location x where the contact occurs, and these space dependent parameters are continuous and strictly positive. Mathematically, we assume that all aquatic mosquitoes do not diffuse, and all adult mosquitoes have the same diffusion rate, denoted by dm>0, while all humans have the same diffusion rate, denoted by dh>0. The biological meanings of all parameters are shown in Table 1.

    Table 1.  Parameters description.
    Parameter Description
    K(x) The environment carrying capacity of aquatic mosquitoes at location x
    l The fraction of hatched female mosquitoes from all eggs
    s Probability of mosquito egg-to-adult survival
    ϑ(x) Intrinsic oviposition rate at location x
    ω(x) Maturation rate from aquatic stages to adult mosquitoes at location x
    μa(x) Death rate of aquatic mosquitoes at location x
    μm(x) Death rate of adult mosquitoes at location x
    Λ(x) Recruitment rate of humans at location x
    μh(x) Natural death rate of humans at location x
    r(x) Recovery rate of humans at location x
    a(x) Biting rate of mosquitoes at location x
    β1(x) Transmission probability from infectious humans to susceptible mosquitoes at location x
    β2(x) Transmission probability from infectious mosquitoes to susceptible humans at location x
    βv(x) Transmission rate from contaminated aquatic environments to aquatic mosquitoes at location x
    θ(x) Excretion rate for each infected individual at location x
    δ(x) Clearance rate of Zika virus in contaminated aquatic environments at location x
    dm Adult mosquito diffusion rate
    dh Human diffusion rate

     | Show Table
    DownLoad: CSV

    On the basis of above assumptions, following the flow diagram in Figure 1, we will focus on the spatiotemporal reaction-diffusion Zika model with environment transmission route as follows:

    Figure 1.  Flow diagram of Zika transmission. The parameters are given in Table 1.
    {Sat=lsϑ(x)(Sm+Im)(1Sa+IaK(x))βv(x)SaVω(x)Saμa(x)Sa,xΩ,t>0,Iat=βv(x)SaVω(x)Iaμa(x)Ia,xΩ,t>0,Smt=dmSm+ω(x)Saa(x)β1(x)IhSmNhμm(x)Sm,xΩ,t>0,Imt=dmIm+ω(x)Ia+a(x)β1(x)IhSmNhμm(x)Im,xΩ,t>0,Sht=dhSh+Λ(x)a(x)β2(x)ImShNhμh(x)Sh,xΩ,t>0,Iht=dhIh+a(x)β2(x)ImShNhr(x)Ihμh(x)Ih,xΩ,t>0,Rht=dhRh+r(x)Ihμh(x)Rh,xΩ,t>0,Vt=θ(x)Ihδ(x)V,xΩ,t>0,Smn=Imn=Shn=Ihn=Rhn=0,xΩ,t>0,Sa(x,0)=Sa0(x)0,Ia(x,0)=Ia0(x)0,Sm(x,0)=Sm0(x)0,xΩ,Im(x,0)=Im0(x)0,Sh(x,0)=Sh0(x)0,Ih(x,0)=Ih0(x)0,xΩ,Rh(x,0)=Rh0(x)0,V(x,0)=V0(x)0,xΩ, (2.1)

    where represents the Laplacian operator. The density of total human population Nh(x,t) can be determined by the following equation

    {Nht=dhNh+Λ(x)μh(x)Nh,xΩ,t>0,Nhn=0,xΩ,t>0. (2.2)

    From Lemma 1 in [29], system (2.2) admits a globally attractive positive steady state H(x) in C(ˉΩ,R+). For simplicity, we assume that the density of total human population at location x and time t stabilizes at H(x). That is, Nh(x,t)H(x), t0,xΩ. Therefore, it suffices to consider the following reduced system:

    {Sat=ρ(x)(Sm+Im)(1Sa+IaK(x))βv(x)SaVω(x)Saμa(x)Sa,xΩ,t>0,Iat=βv(x)SaVω(x)Iaμa(x)Ia,xΩ,t>0,Smt=dmSm+ω(x)Saa(x)β1(x)H(x)IhSmμm(x)Sm,xΩ,t>0,Imt=dmIm+ω(x)Ia+a(x)β1(x)H(x)IhSmμm(x)Im,xΩ,t>0,Sht=dhSh+Λ(x)a(x)β2(x)H(x)ImShμh(x)Sh,xΩ,t>0,Iht=dhIh+a(x)β2(x)H(x)ImShr(x)Ihμh(x)Ih,xΩ,t>0,Vt=θ(x)Ihδ(x)V,xΩ,t>0,Smn=Imn=Shn=Ihn=0,xΩ,t>0,Sa(x,0)=Sa0(x)0,Ia(x,0)=Ia0(x)0,xΩ,Sm(x,0)=Sm0(x)0,Im(x,0)=Im0(x)0,xΩ,Sh(x,0)=Sh0(x)0,Ih(x,0)=Ih0(x)0,V(x,0)=V0(x)0,xΩ, (2.3)

    where ρ(x)=lsϑ(x).

    In this section, we will study the well-posedness of system (2.3). Let X:=C(ˉΩ,R7) be the Banach space with the supremum norm . Define X+:=C(ˉΩ,R7+), then (X,X+) is a strongly ordered Banach space. Let XK be the subset in X defined by

    XK:={ϕ=(ϕ1,ϕ2,ϕ3,ϕ4,ϕ5,ϕ6,ϕ7)TX+:0ϕ1(x)+ϕ2(x)K(x),xˉΩ}.

    In order to simplify notations, we set u=(u1,u2,u3,u4,u5,u6,u7)T=(Sa,Ia,Sm,Im,Sh,Ih,V)T, and the initial data satisfies u0=(u01,u02,u03,u04,u05,u06,u07)T=(Sa0,Ia0,Sm0,Im0,Sh0,Ih0,V0)T. Throughout, for any wC(ˉΩ,R), we denote ¯w:=maxxˉΩw(x), w_:=minxˉΩw.

    We define Ti(t), Tj(t), T5(t), T6(t), T7(t) : C(ˉΩ,R)C(ˉΩ,R) as the C0 semigroups associated with

    (ω()+μa()),dmμm(),dhμh(),dh(r()+μh()),σ()

    subject to the Neumann boundary condition, respectively, i=1,2, j=3,4. Then it follows that for any ψC(ˉΩ,R),t0,

    (Tk(t)ψ)(x)=ΩΓk(x,y,t)ψ(y)dy,k=1,2,3,4,5,6,7, (3.1)

    in which Γi, Γj, Γ6, Γ7 and Γ7 are the Green functions associated with

    (ω()+μa()),dmμm(),dhμh(),dh(r()+μh()),σ()

    subject to the Neumann boundary condition, respectively, i=1,2, j=3,4.

    It is well-known that for all t>0 and k=1,2,3,4,5,6,7, Tk is compact and strongly positive (see [30], Section 7.1 and Corollary 7.2.3). Moreover, T(t)=(T1(t),T2(t),T3(t),T4(t),T5(t),T6(t),T7(t))T, t0, is a C0 semigroup. For xΩ and u=(u1,u2,u3,u4,u5,u6,u7)TXK, the nonlinear operator F=(F1,F2,F3,F4,F5,F6,F7)T : XKX is defined by

    {F1(u)(x)=ρ(x)(u3+u4)(1u1+u2K(x))βv(x)u1u7,F2(u)(x)=βv(x)u1u7,F3(u)(x)=ω(x)u1a(x)β1(x)H(x)u6u3,F4(u)(x)=ω(x)u2+a(x)β1(x)H(x)u6u3,F5(u)(x)=Λ(x)a(x)β2(x)H(x)u4u5,F6(u)(x)=a(x)β2(x)H(x)u4u5,F7(u)(x)=θ(x)u6. (3.2)

    Then system (2.3) can be rewritten as the following integral equation

    uk(t)=Ti(t)u0k+t0Γi(ts)Fk(u(,s))ds,k=1,2,3,4,5,6,7. (3.3)

    For any ϕXK and h0, then we have

    ϕ+hF(ϕ)=[ϕ1+h(ρ(x)(ϕ3+ϕ4)(1ϕ1+ϕ2K(x))βv(x)ϕ1ϕ7)ϕ2+hβv(x)ϕ1ϕ7ϕ3+h(ω(x)ϕ1a(x)β1(x)H(x)ϕ6ϕ3)ϕ4+h(ω(x)ϕ2+a(x)β1(x)H(x)ϕ6ϕ3)ϕ5+h(Λ(x)a(x)β2(x)H(x)ϕ4ϕ5)ϕ6+ha(x)β2(x)H(x)ϕ4ϕ5ϕ7+hθ(x)ϕ6][ϕ1(1h¯βvϕ7)ϕ2ϕ3(1h¯a¯β1H_ϕ6)ϕ4ϕ5(1h¯a¯β2H_ϕ4)ϕ6ϕ7],

    and

    K(x)(ϕ1+hF1(ϕ)+ϕ2+hF2(ϕ))=K(x)((ϕ1+ϕ2)+hρ(x)(ϕ3+ϕ4)(1ϕ1+ϕ2K(x)))=(K(x)(ϕ1+ϕ2))[1hρ(x)K(x)(ϕ3+ϕ4)]. (3.4)

    This means that

    limh0+1hdist(ϕ+hF(ϕ),XK)=0,ϕXK.

    It then follows from Corollary 8.1.3 in [31] (see also Corollary 4 in [32]), we have the following result.

    Lemma 3.1. For every initial value function ϕXK, system (2.3) admits a unique mild solution, denoted by

    u(,t,ϕ)=(Sa(,t,ϕ),Ia(,t,ϕ),Sm(,t,ϕ),Im(,t,ϕ),Sh(,t,ϕ),Ih(,t,ϕ),V(,t,ϕ))T

    on its maximal existence interval [0,bϕ) with u0=ϕ, where bϕ.Moreover, u(,t,ϕ)XK for t(0,bϕ) and u(,t,ϕ) is a classical solution of system (2.3).

    Next, we will show that solutions of system (2.3) exist globally on [0,), and admit a global compact attractor on XK.

    Lemma 3.2. For every initial value function ϕXK, system (2.3) has a unique solution, denoted by

    u(,t,ϕ)=(Sa(,t,ϕ),Ia(,t,ϕ),Sm(,t,ϕ),Im(,t,ϕ),Sh(,t,ϕ),Ih(,t,ϕ),V(,t,ϕ))T

    on [0,) with u0=ϕ. Moreover, define the semiflow Φ(t):XKXK associated withsystem (2.3) by

    Φ(t)ϕ=u(,t,ϕ),ϕ=u0XK,t0.

    Then the semiflow Φ(t):XKXK admits a global compact attractor on XK,t0.

    Proof. Clearly, for ϕXK, we have 0Sh(,t,ϕ),Ih(,t,ϕ)H() for all t0. The comparison principle ([30], Theorem 7.3.4) implies that Sh(x,t,ϕ) and Ih(x,t,ϕ) are uniformly bounded and ultimately bounded. It then follows from the seventh equation of (2.3) that

    V(x,t)tθ(x)H(x)δ(x)V(x,t),¯θ¯Hδ_V(x,t)t>0. (3.5)

    Thus, the comparison principle shows that V(,t,ϕ) is uniformly bounded on [0,bϕ), and

    limtV(x,t)¯θ¯Hδ_,uniformlyinx¯Ω. (3.6)

    More precisely, there exists a t1>0 such that

    V(,t)2¯θ¯Hδ_,tt1.

    Letting N1=max{maxt[0,t1],xˉΩV(x,t),2¯θ¯Hδ_}<, we deduce

    V(x,t)N1forallxΩ,t>0. (3.7)

    Let A(x,t)=Sa(x,t)+Ia(x,t), M(x,t)=Sm(x,t)+Im(x,t). Then it follows from the first four equations of (2.3) that (A(x,t),M(x,t)) satisfies

    {At=ρ(x)(1AK(x))M(ω(x)+μa(x))A,xΩ,t>0,Mt=dmM+ω(x)Aμm(x)M,xΩ,t>0,Mn=0,xΩ,t>0,A(x,0)=A0(x),M(x,0)=M0(x),xΩ, (3.8)

    where A0(x)=ϕ1(x)+ϕ2(x), M0(x)=ϕ3(x)+ϕ4(x). It is easy to see that there exists a positive vector ν=(ν1,ν2):=(¯K,f¯ω¯Kμm_) such that

    ρ(x)(1ν1K(x))ν2(ω(x)+μa(x))ν10,ω(x)ν1μm(x)ν20.

    Thus, ν is an upper solution of (3.8). The comparison principle implies that solutions of (3.8) are uniformly bounded on [0,bϕ). Hence, so are Sa(x,t),Ia(x,t), Sm(x,t) and Im(x,t). Then, we can extend the local unique solution from Lemma 3.1 to global in time via a standard a priori estimates and continuation of local theory. That is, solutions of (2.3) exist on [0,). Next, we show that Sa(x,t),Ia(x,t), Sm(x,t) and Im(x,t) are ultimately bounded.

    From Lemma 3.1, we have Sa(x,t)+Ia(x,t)¯K for all xˉΩ,t0. This implies that Sa(x,t) and Ia(x,t) are ultimately bounded and

    Sa(x,t)¯K,Ia(x,t)¯K,xˉΩ,t0. (3.9)

    It then follows from the third equation of (2.3) that

    {SmtdmSm+ω(x)¯Kμm(x)SmdmSm+¯ω¯Kμm_Sm,xΩ,t>0,Smn=0,xΩ,t>0,Sm(x,0)=ϕ3(x),xΩ. (3.10)

    Consider

    {Wt=dmW+¯ω¯Kμm_W,xΩ,t>0,Wn=0,xΩ,t>0,W(x,0)=ϕ3(x),xΩ. (3.11)

    From Lemma 1 in [29], ¯ω¯Kμm_ is a unique positive steady state that is globally attractive in C(ˉΩ,R+). Hence there exists t20 such that W(x,t)2¯ω¯Kμm_. By comparison principle, Sm(,t)2¯ω¯Kμm_ when t>t2.

    Then,

    {ImtdmIm+¯ω¯K+2¯a¯β1¯ω¯Kμm_μm_Im,xΩ,t>t2,Imn=0,xΩ,t>t2,Im(x,t2):=Im2(x),xΩ. (3.12)

    Similarly, Im(x,t) is ultimately bounded. More precisely, there exists a t3>0 such that Im(,t)2¯ω¯Kμm_(1+2¯a¯β1μm_), for t>t3. Thus, we can obtain

    Im(,t)N2,fort0, (3.13)

    where N2=max{maxt[0,t3],xˉΩIm(x,t),2¯ω¯Kμm_(1+2¯a¯β1μm_)}<.

    In addition, since the first two equations and the last equation of (2.3) have no diffusion term, the solution semiflow Φ(t) is not compact. However, due to ω(x)μa(x)<0,δ(x)<0,xˉΩ, using similar arguments from Theorem 4.1 in [33] (also Lemma 4.1 in [34] and Theorem 2.6 in [35]) that the semiflow Φ(t):XKXK has a global compact attractor on XK,t0. This completes the proof of Lemma 3.2.

    The following result shows that the solution of system (2.3) is strictly positive.

    Lemma 3.3. Let (Sa(,t,ϕ),Ia(,t,ϕ),Sm(,t,ϕ),Im(,t,ϕ),Sh(,t,ϕ),Ih(,t,ϕ),V(,t,ϕ))T be the solution of system (2.3) with the initial value ϕXK. If there exists some t00 such that Ia(,t0,ϕ)0, Im(,t0,ϕ)0, Ih(,t0,ϕ)0, V(,t0,ϕ)0, then the solution of system (2.3)satisfies

    Sa(x,t,ϕ)>0,Ia(x,t,ϕ)>0,Sm(x,t,ϕ)>0,Im(x,t,ϕ)>0,Sh(x,t,ϕ)>0,Ih(x,t,ϕ)>0,V(x,t,ϕ)>0,t>t0,xˉΩ.

    Moreover, for any initial value ϕXK, there exists some positive constant ζ0 such that

    lim inftSh(x,t,ϕ)ζ0,uniformlyforxˉΩ. (3.14)

    Proof. For a give ϕXK, it is easy to see that Ia(x,t,ϕ), Im(x,t,ϕ), Ih(x,t,ϕ) and V(x,t,ϕ) satisfy

    {Iat(¯ω+¯μa)Ia,xΩ,t>0,ImtdmIm¯μmIm,xΩ,t>0,IhtdhIh(¯r+¯μh)Ih,xΩ,t>0,Vt¯δV,xΩ,t>0,Ian=Imn=Ihn=Vn=0,xΩ.

    If there exists some t00 such that Ia(,t0,ϕ)0, Im(,t0,ϕ)0, Ih(,t0,ϕ)0, V(,t0,ϕ)0, it then follows from the strong maximum principle (see Proposition 13.1 in [33]) that Ia(x,t,ϕ)>0,Im(x,t,ϕ)>0,Ih(x,t,ϕ)>0,V(x,t,ϕ)>0,t>t0,xˉΩ.

    Next, we will prove Sa(x,t,ϕ)>0,t>t0,xˉΩ. To this end, we first show A(x,t)=Sa(x,t)+Ia(x,t)<K(x) for all xˉΩ,t0. If not, then there exists x1ˉΩ,t10, such that A(x1,t1)=K(x1). Since A(x,t) satisfies

    {At=ρ(x)(1AK(x))M(ω(x)+μa(x))A,xΩ,t>0,A(x,0)=A0(x),xΩ,
    0=A(x1,t1)t=ρ(x)(1A(x1,t1)K(x1))M(x1,t1)(ω(x1)+μa(x1))A(x1,t1).

    It implies that A(x1,t1)=0. Then K(x1)=0 which contradicts that K(x) is strictly positive. So, Sa(x,t),Ia(x,t)<K(x) for all xˉΩ,t0. Next, we show Sa(x,t,ϕ)>0,t>t0,xˉΩ. Suppose not, there exists x2ˉΩ,t2>t0, such that Sa(x2,t2)=0. From the first equation of system (2.3), we have

    0=Sa(x2,t2)t=ρ(x2)(Sm(x2,t2)+Im(x2,t2))(1Ia(x2,t2)K(x2))>0.

    It is contradictory. Thus, Sa(x,t,ϕ)>0,t>t0,xˉΩ. Similarly, we can show Sm(x,t,ϕ)>0,Sh(x,t,ϕ)>0,t>t0,xˉΩ.

    Moreover, for a give ϕXK, it follows from the fifth equation of system (2.3) and (3.13) that Sh(x,t,ϕ) satisfies

    {ShtdhSh+Λ_(¯a¯β2N2H_+¯μh)Sh,xΩ,t>0,Shn=0,xΩ.

    According to the comparison principle, we have

    lim inftSh(x,t,ϕ)Λ_¯a¯β2N2H_+¯μh,uniformlyforxˉΩ.

    This completes the proof of Lemma 3.3.

    One of the most important concepts in epidemiology is the basic reproduction number R0 which is a threshold index to determine the disease invasion. It is defined to be the average number of secondary cases produced in a completely susceptible population, by a typical infective individual, during its lifetime as infectious. By using the concept of next generation operators, Diekmann, Heesterbeek and Metz [36] presented a general approach to R0 for autonomous epidemic models. Van den Driessche and Watmough [37] gave a computation formula of R0 for compartmental models of ordinary differential equations with constant coefficients. For the reaction-diffusion system with spatially dependent coefficients, Wang and Zhao [38] defined the basic reproduction ratio as the spectral radius of the next infection operator. According to the above methods, in this section, in order to obtain the basic reproduction number, we should first find the disease free equilibrium (DFE) (infection-free steady state). System (2.3) admits two possible DFEs: E01(x)=(0,0,0,0,H(x),0,0) and E02(x)=(A(x),0,M(x),0,H(x),0,0). E01(x) is characterized by the absence of mosquitoes. E02(x) represents an eradication of Zika in the presence of mosquito population. Firstly, we give the basic offspring number which determines whether mosquito population persists, corresponds to the stability of E01(x).

    Since (A(x,t),M(x,t)) satisfies system (3.8), it suffices to consider system (3.8). Obviously, system (3.8) always admits a DFE (0,0). Linearizing system (3.8) at (0,0), we have

    {At=ρ(x)M(ω(x)+μa(x))A,xΩ,t>0,Mt=dmM+ω(x)Aμm(x)M,xΩ,t>0,Mn=0,xΩ,t>0,A(x,0)=A0(x),M(x,0)=M0(x),xΩ. (4.1)

    The eigenvalue problem associated with (4.1) is as follows

    {ρ(x)ψ2(ω(x)+μa(x))ψ1=λ1ψ1,xΩ,dmψ2+ω(x)ψ1μm(x)ψ2=λ1ψ2,xΩ,ψ2n=0,xΩ, (4.2)

    where ψ1 and ψ2 are both positive for xΩ. Let Bm be defined as follows

    Bm=[(ω(x)+μa(x))ρ(x)ω(x)dmμm(x)]. (4.3)

    Denote the basic offspring number as Rm0. According to Lemma 4.2 in [38], 1Rm0 is the unique positive eigenvalue of the eigenvalue problem

    {dmφ+μm(x)φ=λ2ω(x)ρ(x)ω(x)+μa(x)φ,xΩ,ϕn=0,xΩ. (4.4)

    By using the variational characterization of principal eigenvalue [39], we can obtain

    Rm0=supφH10(Ω),ϕ0{Ωω(x)ρ(x)ω(x)+μa(x)φ2Ωdm|φ|2+μm(x)φ2}. (4.5)

    By Lemma 2.2 and Lemma 2.3 in [40], we have the following observation.

    Lemma 4.1. Let λm=s(Bm) be the spectral bound of Bm.

    (A1) If λm0, then λm is the principal eigenvalue of (4.2) with a strongly positive eigenfunction.

    (A2) Rm01 and λm have the same sign.

    Let YK:={(A0,M0)TC(ˉΩ,R2+):0A0(x)K(x)),xˉΩ}. The following result is concerned with the global dynamics of system (3.8).

    Lemma 4.2. ([40], Lemma 2.5) Suppose that Rm0>1.System (3.8) admits a unique steady state (A(x),M(x)) which is globally asymptotically stable in YK{(0,0)}. Moreover, 0<A(x)<K(x),xˉΩ.

    Below we use the method proposed in Wang and Zhao [38] to introduce the basic reproduction number.

    This sub-section is devoted to formulate of the reproduction number for system (2.3) that determines invasion of Zika disease. So, it is the essential condition that guarantees the persistence of mosquito population. From Lemma 4.2, we assume that Rm0>1, and then E02(x) exists. Linearizing system (2.3) at E02(x), and then considering only the equations of infective compartments, we have

    {Iat=βv(x)A(x)V(ω(x)+μa(x))Ia,xΩ,t>0,Imt=dmIm+ω(x)Ia+a(x)β1(x)M(x)H(x)Ihμm(x)Im,xΩ,t>0,Iht=dhIh+a(x)β2(x)Im(r(x)+μh(x))Ih,xΩ,t>0,Vt=θ(x)Ihδ(x)V,xΩ,t>0,Imn=Ihn=0,xΩ,t>0,Ia(x,0)=Ia0(x),Im(x,0)=Im0(x),Ih(x,0)=Ih0(x),V(x,0)=V0(x),xΩ. (4.6)

    Let ˉT(t) : C(ˉΩ,R4)C(ˉΩ,R4) be the solution semigroup generated by system (4.6). It is easy to see that ˉT(t) is a positive C0 semigroup, and its generator B can be written as

    B=[(ω(x)+μa(x))00βv(x)A(x)ω(x)dmμm(x)a(x)β1(x)M(x)H(x)00a(x)β2(x)dh(r(x)+μh(x))000θ(x)δ(x)].

    Further, B is a closed and resolvent positive operator (see Theorem 3.12 in [41]). The eigenvalue problem associated with (4.6) is as follows

    {λ3φ1=βv(x)A(x)φ4(ω(x)+μa(x))φ1,xΩ,t>0,λ3φ2=dmφ2+ω(x)φ1+a(x)β1(x)M(x)H(x)φ3μm(x)φ2,xΩ,t>0,λ3φ3=dhφ3+a(x)β2(x)φ2(r(x)+μh(x))φ3,xΩ,t>0,λ3φ4=θ(x)φ3δ(x)φ4,xΩ,t>0,φ2n=φ3n=0,xΩ,t>0. (4.7)

    By a similar argument as Theorem 7.6.1 in [30], we have the following observation.

    Lemma 4.3. Let λ=s(B) be the spectral bound of B. If λ0, then λ is the principal eigenvalue of the eigenvalue problem (4.7) with a strongly positive eigenfunction.

    In the following, we will use the ideas in [38] to define the basic reproduction number. Let T(t): C(ˉΩ,R4)C(ˉΩ,R4) be the solution semigroup generated by the following linear system

    {Iat=(ω(x)+μa(x))Ia,xΩ,t>0,Imt=dmIm+ω(x)Iaμm(x)Im,xΩ,t>0,Iht=dhIh(r(x)+μh(x))Ih,xΩ,t>0,Vt=θ(x)Ihδ(x)V,xΩ,t>0,Imn=Ihn=0,xΩ,t>0,Ia(x,0)=Ia0(x),Im(x,0)=Im0(x),Ih(x,0)=Ih0(x),V(x,0)=V0(x),xΩ. (4.8)

    It is easy to see that T(t) is a C0 semigroup on C(ˉΩ,R4).

    We define

    F(x)=[000βv(x)A(x)00a(x)β1(x)M(x)H(x)00a(x)β2(x)000000],V(x)=[(ω(x)+μa(x))000ω(x)μm(x)0000(r(x)+μh(x))000θ(x)δ(x)].

    In order to define the basic reproduction number for system (2.3), we assume that the state variables are near DFE E02, and introduce the distribution of initial infective individuals described by ϕ(x)C(ˉΩ,R4). Thus, it is easy to see that T(t)ϕ(x) represents the distribution of those infective individuals at time t. Thus, F(x)T(t)ϕ(x) represents the distribution of new infective individuals at time t.

    Define L:C(ˉΩ,R4)C(ˉΩ,R4) as follows

    L(ϕ)(x):=0F(x)T(t)ϕ(x)dt. (4.9)

    It then follows that L(ϕ)(x) represents the distribution of the total new population generated by initial infective individuals ϕ(x) during their infection period. So, L is the next generation operator. We define the spectral radius of L as the basic reproduction number of system (2.3). That is,

    R0:=r(L). (4.10)

    From [38], we have the following observation.

    Lemma 4.4. R01 and λ have the same sign.

    The following result indicates that basic offspring number Rm0 is a threshold index for eradication or persistence of the Zika disease.

    We firstly focus on the global dynamic behaviors of the DFEs E01 and E02 of system (2.3).

    Theorem 5.1. If Rm0<1, then the DFE E01(x) is globally attractive in XK for system (2.3).

    Proof. Assume Rm0<1. It follows from Lemma 4.1 that λm>0. λm is the principal eigenvalue of eigenvalue problem (4.2) with a strongly positive eigenfunction (ψ1,ψ2). Since (A(x,t),M(x,t)) satisfies system (3.8), it follows that

    {Atρ(x)M(ω(x)+μa(x))A,xΩ,t>0,Mt=dmM+ω(x)Aμm(x)M,xΩ,t>0,Mn=0,xΩ,t>0. (5.1)

    For any given ϕ=(ϕ1,ϕ2)YK, there exists some q>0 such that

    (A(x,0,ϕ),M(x,0,ϕ))q(ψ1,ψ2),xΩ.

    Note that the linear system (4.1) admits a solution qeλmt(ψ1,ψ2),t0. Then the comparison principle implies that

    (A(x,t,ϕ),M(x,t,ϕ))qeλmt(ψ1,ψ2),t0,xˉΩ.

    Hence, limt(A(x,t,ϕ),M(x,t,ϕ))=(0,0), uniformly for all xˉΩ. Then, from A(x,t)=Sa(x,t)+Ia(x,t) and M(x,t)=Sm(x,t)+Im(x,t), together with positivity of solutions, for system (2.3), we have for every initial value function ϕ=(ϕ1,ϕ2,ϕ3,ϕ4,ϕ5,ϕ6,ϕ7)TXK,

    limtSa(x,t,ϕ)=0,limtIa(x,t,ϕ)=0,limtSm(x,t,ϕ)=0,limtIm(x,t,ϕ)=0,uniformlyforallxˉΩ.

    Then, Sh(,t) in system (2.3) is asymptotic to the following system

    {Sht=dhSh+Λ(x)μh(x)Sh,xΩ,t>0,Shn=0,xΩ,t>0,Sh(x,0)=ϕ5,xΩ. (5.2)

    By the theory for asymptotically autonomous semiflows (see Corollary 4.3 in [42]), together with Lemma 1 in [29], it follows that

    limtSh(x,t,ϕ)=H(x),uniformlyforallxˉΩ.

    Similarly, Ih(,t) in system (2.3) is asymptotic to the following system

    {Iht=dhIh(r(x)+μh(x))Ih,xΩ,t>0,Ihn=0,xΩ,t>0,Ih(x,0)=ϕ6,xΩ. (5.3)

    Therefore,

    limtIh(x,t,ϕ)=0,uniformlyforallxˉΩ.

    Then V(,t) in system (2.3) is asymptotic to the following system

    {V(x,t)t=δ(x)V(x,t),xΩ,t>0,V(x,0)=ϕ7,xΩ. (5.4)

    Thus,

    limtV(x,t,ϕ)=0,uniformlyforallxˉΩ.

    This completes the proof of Theorem 5.1.

    Remark 1. Biologically, Theorem 5.1 shows that the basic offspring number Rm0 can be used as a control parameter which determines whether mosquito population is absent or not. It means that mosquito population can be vanished, and the Zika virus will eradicate in human population and contaminated aquatic environment by reducing Rm0 below 1.

    Theorem 5.2. Let u(x,t,ϕ) be the solution of system (2.3) with u(,0,ϕ)=ϕXK.If Rm0>1, R0<1, then the DFE E02(x) is globally attractive for system (2.3). That is, for any ϕXK, if (ϕ1,ϕ3)(0,0), then

    limtu(x,t,ϕ)=E02(x),uniformlyforallxˉΩ.

    Proof. Suppose R0<1. By Lemma 4.4, we have λ<0. So, there exists a sufficiently small positive number ϵ0 such that λϵ0<0, where λϵ0<0 is the principal eigenvalue of the following eigenvalue problem

    {λφϵ02=βv(x)(A(x)+ϵ0)φϵ07(ω(x)+μa(x))φϵ02,xΩ,t>0,λφϵ04=dmφϵ04+ω(x)φϵ02+a(x)β1(x)(M(x)+ϵ0)H(x)φϵ06μm(x)φϵ04,xΩ,t>0,λφϵ06=dhφϵ06+a(x)β2(x)φϵ04(r(x)+μh(x))φϵ06,xΩ,t>0,λφϵ07=θ(x)φϵ06δ(x)φϵ07,xΩ,t>0,φϵ04n=φϵ06n=0,xΩ,t>0, (5.5)

    with a strongly positive eigenfunction (φϵ02,φϵ04,φϵ06,φϵ07). It follows from the condition Rm0>1, Lemma 4.2 and the positivity of solutions that there exists a t0>0 such that

    Sa(x,t)A(x)+ϵ0,Sm(x,t)M(x)+ϵ0,

    for all xˉΩ, tt0. Hence, by the Ia, Im, Ih and V equations of system (2.3), it follows that

    {Iatβv(x)(A(x)+ϵ0)V(ω(x)+μa(x))Ia,xΩ,tt0,ImtdmIm+ω(x)Ia+a(x)β1(x)(M(x)+ϵ0)H(x)Ihμm(x)Im,xΩ,tt0,IhtdhIh+a(x)β2(x)Im(r(x)+μh(x))Ih,xΩ,tt0,Vt=θ(x)Ihδ(x)V,xΩ,tt0,Imn=Ihn=0,xΩ,tt0. (5.6)

    For any given ϕXK, there exists some q1>0 such that

    (Ia(x,t0,ϕ),Im(x,t0,ϕ),Ih(x,t0,ϕ),V(x,t0,ϕ))q1(φϵ02,φϵ04,φϵ06,φϵ07),xˉΩ.

    Note that the following linear system

    {Iat=βv(x)(A(x)+ϵ0)V(ω(x)+μa(x))Ia,xΩ,tt0,Imt=dmIm+ω(x)Ia+a(x)β1(x)(M(x)+ϵ0)H(x)Ihμm(x)Im,xΩ,tt0,Iht=dhIh+a(x)β2(x)Im(r(x)+μh(x))Ih,xΩ,tt0,Vt=θ(x)Ihδ(x)V,xΩ,tt0,Imn=Ihn=0,xΩ,tt0, (5.7)

    admits a solution q1eλϵ0(tt0)(φϵ02,φϵ04,φϵ06,φϵ07),tt0. Then the comparison principle implies that

    (Ia(x,t,ϕ),Im(x,t,ϕ),Ih(x,t,ϕ),V(x,t,ϕ))q1eλϵ0(tt0)(φϵ02,φϵ04,φϵ06,φϵ07),tt0,xˉΩ.

    Hence, limt(Ia(x,t,ϕ),Im(x,t,ϕ),Ih(x,t,ϕ),V(x,t,ϕ))=(0,0,0,0), uniformly for all xˉΩ. Then, (Sa(,t),Sm(,t)) in system (2.3) is asymptotic to system (3.8). By the theory for asymptotically autonomous semiflows, together with Lemma 1 in [29], it follows that

    limt(Sa(x,t,ϕ),Sm(x,t,ϕ))=(A(x),M(x)),uniformlyforallxˉΩ.

    Similarly, Sh(,t) in system (2.3) is asymptotic to system (5.2). That is,

    limtSh(x,t,ϕ)=H(x),uniformlyforallxˉΩ.

    This completes the proof of Theorem 5.2.

    Remark 2. Biologically, Theorem 5.2 shows that mosquito population is present when the basic offspring number Rm0>1. Under this premise, the basic reproduction number R0 can be used as a control parameter which determines whether the disease will eventually die out or not. It means that the disease can be eradicated by reducing R0 below 1.

    Before giving the disease persistence, we first give the following lemma.

    Lemma 5.1. Suppose that Rm>1, and ϕi0,i=2,4. If there exists some ζ1>0 such that

    lim inft+Ih(x,t,ϕ)ζ1,uniformlyforallxˉΩ,

    then there exists some ζ2>0 such that

    lim inft+Sa(x,t,ϕ)ζ2,lim inft+Ia(x,t,ϕ)ζ2,lim inft+Sm(x,t,ϕ)ζ2,lim inft+Im(x,t,ϕ)ζ2,lim inft+Sh(x,t,ϕ)ζ2,lim inft+Ih(x,t,ϕ)ζ2,lim inft+V(x,t,ϕ)ζ2,uniformlyforallxˉΩ. (5.8)

    Proof. From lim inft+Ih(x,t,ϕ)ζ1,uniformlyforallxˉΩ, we have that there exists t11>0 such that

    Ih(x,t)13ζ1,tt11,xˉΩ.

    It follows from the last equation of system (2.3) that V(x,t) satisfies

    {Vt13ζ1θ_¯δV,xΩ,tt11,Vn=0,xΩ.

    By comparison principle, we have

    lim inft+V(x,t)ζ1θ_3¯δ:=e1,uniformlyforallxˉΩ.

    Thus, there is a t12>t11 such that

    V(x,t)13e1,tt12,xˉΩ.

    Due to Rm>1 and ϕi0,i=2,4, and from Lemma 4.2, we can obtain that there exists t13>t12 such that

    Sa(x,t)+Ia(x,t)A(x)+13(K(x)A(x))=13K(x)+23A(x),
    Sm(x,t)+Im(x,t)13M(x).

    From the first equation of system (2.3) and (3.7), one has

    \begin{eqnarray*} \left\{ \begin{aligned} \frac{\partial S_{a}}{\partial t} &\geq \frac{1}{3}M^*(x) \rho(x) \left(1-\frac{ \frac{1}{3} K(x)+\frac{2}{3}A^*(x) }{K(x)}\right)-(\beta_{v}(x)N_{1} +\omega(x) +\mu_{a}(x)) S_{a}&\\ &\geq \frac{2}{9}\underline{M^*} \underline{\rho} \left(1-\frac{ \overline{ A^*}}{\underline{K}}\right) - (\overline{\beta_{v}} N_{1} +\overline{\omega} +\overline{\mu_{a}} ) S_{a}, \; &x\in\Omega, \; t > t_{13}, \\ & \frac{\partial S_{a}}{\partial n} = 0, \; &x\in\partial\Omega. \\ \end{aligned} \right. \end{eqnarray*}

    Then, we can obtain

    \liminf\limits_{t\rightarrow +\infty}S_a(x, t)\geq \frac{ \frac{2}{9}\underline{M^*} \underline{\rho} \left(1-\frac{ \overline{ A^*}}{\underline{K}}\right) }{\overline{\beta_{v}} N_{1} +\overline{\omega} +\overline{\mu_{a}} }: = e_2 , \; uniformly \; for \; all \; x\in \bar{\Omega}.

    Thus, there is a t_{14} > t_{13} such that

    S_a(x, t)\geq \frac{1}{3 } e_2 , \; \forall t\geq t_{14}, \; x\in \bar{\Omega}.

    From the second equation of system (2.3), we have

    \begin{eqnarray*} \left\{ \begin{aligned} &\frac{\partial I_{a}}{\partial t}\geq \underline{\beta_{v}}\times\frac{1}{3 } e_1 \times\frac{1}{3 }e_2 -(\overline{\omega}+\overline{\mu_{a}})I_{a}, \; &x\in\Omega, \; t\geq t_{14}, \\ & \frac{\partial I_{a}}{\partial n} = 0, \; &x\in\partial\Omega. \\ \end{aligned} \right. \end{eqnarray*}

    Then

    \liminf\limits_{t\rightarrow +\infty}I_a(x, t)\geq \frac{\underline{\beta_{v}} e_1 e_2 }{9(\overline{\omega}+\overline{\mu_{a}}) }: = e_3 , \; uniformly \; for \; all \; x\in \bar{\Omega},

    which implies that there is a t_{15} > t_{14} such that

    I_a(x, t)\geq \frac{1}{3 } e_3, \; \forall t\geq t_{15}, \; x\in \bar{\Omega}.

    Similarly, it follows from the third and fourth equations of system (2.3) that

    \liminf\limits_{t\rightarrow +\infty}S_m(x, t)\geq \frac{f\underline{ \omega} e_2 }{3(\overline{b}\overline{\beta_1}+\overline{\mu_{m}}) }: = e_4 , \; \liminf\limits_{t\rightarrow +\infty}I_m(x, t)\geq \frac{f\underline{ \omega} e_3 }{3 \overline{\mu_{m}} }: = e_5 , \; uniformly \; for \; all \; x\in \bar{\Omega}.

    So, there is a t_{16} > t_{15} such that

    S_m(x, t)\geq \frac{1}{3 } e_4, \; I_m(x, t)\geq \frac{1}{3 } e_5, \; \forall t\geq t_{16}, \; x\in \bar{\Omega}.

    By Lemma 3.3, we know

    \begin{eqnarray*} \begin{aligned} \liminf\limits_{t\rightarrow \infty } S_{h}(x, t, \phi)\geq \zeta_0, \; uniformly \; for \; x\in\bar{\Omega}. \end{aligned} \end{eqnarray*}

    Hence, there is a t_{17} > t_{16} such that

    S_h(x, t)\geq \frac{1}{3 }\zeta_0, \; \forall t\geq t_{17}, \; x\in \bar{\Omega}.

    Therefore, letting \zeta_2 = \frac{1}{3}\max\{\zeta_0, \zeta_1, e_1, e_2, e_3, e_4, e_5\} , we can obtain that (5.8) holds. This completes the proof of Lemma 5.1.

    Theorem 5.3. Let

    \begin{equation*} \begin{aligned} &\mathbb{M}_{0}: = \{(S_{a}, I_{a}, S_{m}, I_{m}, S_{h}, I_{h}, V)^{T}\in \mathbb{X}_{K} : I_{a}(\cdot)\not \equiv 0, \; I_{m}(\cdot)\not \equiv 0, \; I_{h}(\cdot)\not \equiv 0, \; V(\cdot)\not \equiv 0 \}, \\ &\partial\mathbb{M}_{0}: = \mathbb{X}_{K}\backslash \mathbb{M}_{0}. \end{aligned} \end{equation*}

    If R_{0}^{m} > 1 and R_{0} > 1 , then system (2.3) is uniformly persistent, i.e., and there is a constant \varsigma > 0 such that, for any initial value \phi\in\mathbb{M}_{0} , we can obtain

    \begin{eqnarray} \begin{aligned} &\liminf\limits_{t\rightarrow +\infty}S_{a}(x, t, \phi)\geq \varsigma, \liminf\limits_{t\rightarrow +\infty}I_{a}(x, t, \phi)\geq\varsigma, \liminf\limits_{t\rightarrow +\infty}S_{m}(x, t, \phi)\geq\varsigma, \\ &\liminf\limits_{t\rightarrow +\infty}I_{m}(x, t, \phi)\geq\varsigma, \liminf\limits_{t\rightarrow +\infty}S_{h}(x, t, \phi)\geq\varsigma, \liminf\limits_{t\rightarrow +\infty}I_{h}(x, t, \phi)\geq\varsigma, \\ &\liminf\limits_{t\rightarrow +\infty}V(x, t, \phi)\geq\varsigma, \; uniformly\; for\; all \; x\in\bar{\Omega}. \end{aligned} \end{eqnarray} (5.9)

    Proof. The following four steps are taken to prove this result.

    Step I M_0 is invariant under \Psi(t).

    For any initial value \phi\in M_0 , from Lemma 3.3, we can obtain

    I_{a}(x, t, \phi) > 0, I_{m}(x, t, \phi) > 0, I_{h}(x, t, \phi) > 0, V(x, t, \phi) > 0, \; \; \forall \; t > 0, \; x\in\bar{\Omega}.

    Then \Psi(t)\phi\in M_0. So M_0 is invariant under \Psi(t).

    Step II For any \phi \in \partial M_0 , one obtains the \omega- limit set \omega(\phi) = \{E_{01}\}\cup\{E_{02}\}, where \omega(\phi) is the omega limit set of the forward orbit \gamma^+: = \{\Psi(t)\phi:t\geq 0\}.

    Define

    \Gamma_{\partial}: = \{\phi\in\partial M_0\; :\; \Psi(t)\phi\in\partial M_0, \forall \; t\geq0 \}.

    For any given \phi\in\Gamma_{\partial} , we have \Psi(t)\phi\in\partial M_0, \forall \; t\geq0. That is, for every t\geq0 , we have

    I_{a}(\cdot, t, \phi) \equiv 0\; or \; I_{m}(\cdot, t, \phi) \equiv 0\; or \; I_{h}(\cdot, t, \phi) \equiv 0\; or \; V(\cdot, t, \phi) \equiv 0.

    We first consider the case I_{m}(\cdot, t, \phi) \equiv 0 for all t\geq0 . From the sixth equation of system (2.3), we can get that I_{h}(x, t, \phi) satisfies system (5.3). Thus,

    \lim\limits_{t\rightarrow \infty } I_{h}(x, t, \phi) = 0, \; \rm{uniformly \; for \; all} \; x\in \bar{\Omega} .

    From the seventh equation of system (2.3), and according to Corollary 4.3 in [42], one has

    \lim\limits_{t\rightarrow \infty }V(x, t, \phi) = 0, \; uniformly \; for \; all \; x\in \bar{\Omega} .

    Similarly, we can obtain

    \lim\limits_{t\rightarrow \infty }I_a(x, t, \phi) = 0, \; uniformly \; for \; all \; x\in \bar{\Omega} .

    In addition, S_{h}(\cdot, t, \phi) in system (2.3) is asymptotic to system (5.2). Thus,

    \lim\limits_{t\rightarrow \infty } S_{h}(x, t, \phi) = H(x), \; uniformly \; for \; all\; x\in \bar{\Omega} .

    In the case I_{m}(\cdot, t, \phi)\not \equiv 0 for all t\geq0 , S_{a}(\cdot, t, \phi) and S_{m}(\cdot, t, \phi) are as follows.

    (i) S_{a}(\cdot, t, \phi) \equiv 0 and S_{m}(\cdot, t, \phi) \equiv 0 for all t\geq0 .

    (ii) S_{a}(\cdot, t, \phi) \equiv 0 for all t\geq0 , and S_{m}(\cdot, t_{21}, \phi)\not\equiv 0 for some t_{21} > 0.

    In this case, from the third equation of system (2.3), we can get

    \lim\limits_{t\rightarrow \infty } S_{m}(x, t, \phi) = 0, \; uniformly \; for \; all \; x\in \bar{\Omega} .

    (iii) S_{m}(\cdot, t, \phi) \equiv 0 for all t\geq0 , and S_{a}(\cdot, t_{22}, \phi)\not\equiv 0 for some t_{22} > 0.

    In this case, from the first equation of system (2.3), we can obatin

    \lim\limits_{t\rightarrow \infty } S_{a}(x, t, \phi) = 0, \; uniformly \; for \; all \; x\in \bar{\Omega} .

    (iv) S_{a}(\cdot, t_{23}, \phi)\not\equiv 0 and S_{m}(\cdot, t_{23}, \phi)\not\equiv 0 for some t_{23} > 0.

    In this case, (S_{a}(x, t, \phi), S_{m}(x, t, \phi)) in system (2.3) is asymptotic to system (3.8). From Lemma 4.2, we have

    \lim\limits_{t\rightarrow \infty }( S_{a}(x, t, \phi), \; S_{m}(x, t, \phi)) = (A^{*}(x), \; M^{*}(x)), \; uniformly \; for \; all \; x\in \bar{\Omega} .

    Thus, we obtain \omega(\phi) = \{E_{01}\}\cup\{E_{02}\}.

    Next, we assume I_{m}(\cdot, t_{24}, \phi)\not \equiv 0 for some t_{24} > 0. Form Lemma 4.2, one has I_{m}(\cdot, t, \phi) > 0 for all t > t_{24}. Then, we get

    I_{a}(\cdot, t, \phi) \equiv 0\; or \; I_{h}(\cdot, t, \phi) \equiv 0\; or \; V(\cdot, t, \phi) \equiv 0, \; for\; all \; t > t_{24}.

    Here, we only show the case I_{a}(\cdot, t, \phi) \equiv 0 for all t > t_{24} . It follows from the second equation of system (2.3) that

    S_{a}(\cdot, t, \phi) \equiv 0\; or \; V(\cdot, t, \phi) \equiv 0, \; for\; all \; t > t_{24}.

    If S_{a}(\cdot, t, \phi) \equiv 0, \; for\; all \; t > t_{24}, then, from the first equation of system (2.3), we have I_{m}(\cdot, t, \phi) \equiv 0, \; for\; all \; t > t_{24}, which contradicts our assumption.

    If V(\cdot, t, \phi) \equiv 0, \; for\; all \; t > t_{24}, then, from the sixth equation of system (2.3), one has I_{m}(\cdot, t, \phi) \equiv 0, \; for\; all \; t > t_{24}, which contradicts our assumption. Thus, the step II is proved.

    Step III W^s(\{E_{01}(x)\})\cap M_0 = \emptyset and W^s(\{E_{02}(x)\})\cap M_0 = \emptyset . In this step, we will show the following two claims.

    Claim 1 E_{01}(x) is a uniform weak repeller for M_0. That is, there exists \varepsilon_1 > 0 such that

    \begin{eqnarray} \begin{aligned} \limsup\limits_{t\rightarrow +\infty}\|\Psi(t)\phi- E_{01}(x)\|\geq \varepsilon_1, \; for\; all\; \phi\in M_0. \end{aligned} \end{eqnarray} (5.10)

    Claim 2 E_{02}(x) is a uniform weak repeller for M_0. That is, there exists \varepsilon_2 > 0 such that

    \begin{eqnarray} \begin{aligned} \limsup\limits_{t\rightarrow +\infty}\|\Psi(t)\phi- E_{02}(x)\|\geq \varepsilon_2, \; for\; all\; \phi\in M_0. \end{aligned} \end{eqnarray} (5.11)

    Here, we just prove Claim 1 . Claim 2 can be similarly proven.

    If (5.10) does not hold, then

    \begin{eqnarray} \begin{aligned} \limsup\limits_{t\rightarrow +\infty}\|\Psi(t)\tilde{\phi}- E_{01}(x)\| < \varepsilon_1, \; for\; some\; \tilde{\phi}\in M_0. \end{aligned} \end{eqnarray} (5.12)

    That is, there exists t_{25} > 0 such that

    \begin{eqnarray} \begin{aligned} &0 < S_a(x, t, \tilde{\phi}) < \varepsilon_1, \; 0 < I_a(x, t, \tilde{\phi}) < \varepsilon_1, \; 0 < S_m(x, t, \tilde{\phi}) < \varepsilon_1, \; 0 < I_m(x, t, \tilde{\phi}) < \varepsilon_1, \\ &H(x)-\varepsilon_1 < S_h(x, t, \tilde{\phi}) < H(x)+\varepsilon_1, \; 0 < I_h(x, t, \tilde{\phi}) < \varepsilon_1, \; 0 < V(x, t, \tilde{\phi}) < \varepsilon_1, \; for\; \forall\; t\geq t_{25}, \; x\in \bar{\Omega}. \end{aligned} \end{eqnarray} (5.13)

    Thus, S_a(x, t, \tilde{\phi}) and S_m(x, t, \tilde{\phi}) satisfy

    \begin{eqnarray*} \left\{ \begin{aligned} &\frac{\partial S_a}{\partial t}\geq \rho(x)\left( 1-\frac{2\varepsilon_1}{K(x)}\right)S_m -(\beta_v(x)\varepsilon_1+ \omega(x) +\mu_{a}(x))S_a, \; &x\in\Omega, \; t\geq t_{25}, \\ &\frac{\partial S_m}{\partial t}\geq d_{m}\triangle S_m+ \omega(x)S_a -\left(\mu_{m}(x)+ \frac{\alpha(x)\beta_1(x)}{H(x)}\varepsilon_1\right)S_m, \; &x\in\Omega, \; t\geq t_{25}, \\ &\frac{\partial M}{\partial n} = 0, \; &x\in\partial\Omega, \; t\geq t_{25}.\\ \end{aligned} \right. \end{eqnarray*}

    Consider the following auxiliary linear system

    \begin{eqnarray} \left\{ \begin{aligned} &\frac{\partial \nu}{\partial t} = \tilde{B}_m(\varepsilon_1)\nu, \; &x\in\Omega, \; t\geq t_{25}, \\ &\frac{\partial \nu_2}{\partial n} = 0, \; &x\in\partial\Omega, \; t\geq t_{25}, \\ \end{aligned} \right. \end{eqnarray} (5.14)

    where \nu = (\nu_1, \nu_2)^T , and

    \begin{eqnarray*} \tilde{B}_{m}(\varepsilon_1) = \left[ \begin{array}{cc} -(\beta_v(x)\varepsilon_1+ \omega(x) +\mu_{a}(x)) & \rho(x)\left( 1-\frac{2\varepsilon_1}{K(x)}\right) \\ \omega(x) & d_{m}\triangle-\left(\mu_{m}(x)+ \frac{\alpha(x)\beta_1(x)}{H(x)}\varepsilon_1\right) \\ \end{array} \right]. \end{eqnarray*}

    We know \tilde{B}_{m}(0) = B_m , where B_m is defined by (4.3). From Lemma 4.1, if R_{0}^{m} > 1, then \lambda_m^* = s(B_m) > 0. \tilde{B}_{m}(\varepsilon_1) is continuous for small \varepsilon_1 . So, when \varepsilon_1 is small enough, we have s(\tilde{B}_{m}(\varepsilon_1)) > 0. Denote \lambda_{m\varepsilon_1}^*: = s(\tilde{B}_{m}(\varepsilon_1)) . Obviously, \lambda_{m\varepsilon_1}^* > 0 .

    Let (\tilde{\varphi}_1, \tilde{\varphi}_2) be the positive eigenfunction corresponding to \lambda_{m\varepsilon_1}^* . Then auxiliary system (5.14) admits a solution (\nu_1(x, t), \nu_2(x, t)) = e^{\lambda_{m\varepsilon_1}^*}(\tilde{\varphi}_1, \tilde{\varphi}_2) . Due to S_a(x, t, \tilde{\phi}) > 0 , S_m(x, t, \tilde{\phi}) > 0 for \forall\; t\geq t_{25}, there exists \varrho_1 > 0 such that

    (S_a(x, t_{25}, \tilde{\phi}), S_m(x, t_{25}, \tilde{\phi}))\geq \varrho_1 (\tilde{\varphi}_1, \tilde{\varphi}_2).

    According to the comparison principle, we can obtain

    (S_a(x, t, \tilde{\phi}), S_m(x, t, \tilde{\phi}))\geq \varrho_1 e^{\lambda_{m\varepsilon_1}^*(t-t_{25})}(\tilde{\varphi}_1, \tilde{\varphi}_2), \; for\; \forall\; t\geq t_{25}, \; x\in \bar{\Omega} .

    Since \lambda_{m\varepsilon_1}^* > 0 , we get

    \lim\limits_{t\rightarrow \infty } S_{a}(\cdot, t, \tilde{\phi}) = +\infty, \; \lim\limits_{t\rightarrow \infty } S_{m}(\cdot, t, \tilde{\phi}) = +\infty,

    which contradicts with (5.13).

    The above discussion implies that \{E_{01}(x)\} and \{E_{02}(x)\} are isolated invariant sets in M_0 , and W^s(\{E_{01}(x)\})\cap M_0 = \emptyset , W^s(\{E_{02}(x)\})\cap M_0 = \emptyset . Clearly, every orbit in \Gamma_{\partial} converges to either E_{01}(x) or E_{02}(x) , and there are no subsets of \{ E_{01}(x), E_{02}(x)\} forms a cycle in \Gamma_{\partial} . From Theorem 4.1 in [43], system (2.3) is uniformly persistent if R^m_{0} > 1 and R_{0} > 1 .

    Step IV Define a continues function \bf{g}:\; \mathbb{X}_{K}\rightarrow [0, +\infty) with

    \bf{g}(\phi): = \min\left\{\min\limits_{x\in\bar{\Omega}}\phi_2(x), \min\limits_{x\in\bar{\Omega}}\phi_4(x), \min\limits_{x\in\bar{\Omega}}\phi_6(x), \min\limits_{x\in\bar{\Omega}}\phi_7(x)\right\}, \; \forall\; \phi\in\mathbb{X}_{K}.

    It follows from Lemma 3.3 that \bf{g}^{-1}(0, +\infty)\subseteq M_0. If \bf{g}(\phi) > 0 , or \phi\in M_0 with \bf{g}(\phi) = 0 , then \bf{g}(\Psi(t)\phi) > 0 , \forall\; t > 0 . That is, \bf{g} is a generalized distance fuction for the semiflow \Psi(t) (see [44]). From Lemma 3.2 and according to Theorem 3 in [44], there exists a \varrho_2 > 0 such that

    \min\limits_{\psi\in \omega(\phi)} > \varrho_2, \; \forall\; \phi\in M_0.

    Thus, \liminf\limits_{t\rightarrow +\infty}I_{h}(\cdot, t, \phi)\geq \varrho_2, \forall\; \phi\in M_0. From Lemma 5.1, there exists some \varsigma > 0 such that (5.9) holds. This completes the proof.

    Remark 3. Biologically, Theorem 5.3 shows that mosquito population and the disease will persist when the basic offspring number R_{0}^m > 1 and basic reproduction number R_{0} > 1 .

    In this section, we implement numerical simulations to confirm the analytic results. For the sake of convenience, we concentrate on one dimensional domain \Omega , which can be taken, without loss of generality, to be (0, \pi) . For the sake of simplicity, we focus on model (2.3) and fix some coefficients and functions as follows: H(x) = 100 , K(x) = 500 , \vartheta(x) = 5 , l = 1/2 , \omega(x) = 0.05 , \mu_{a}(x) = 0.15 , \mu_{m}(x) = 0.05 , \mu_{h}(x) = \frac{1}{75\times365} , \Lambda(x) = H(x)\times\mu_{h}(x) , r(x) = \frac{1}{7} , a(x) = 0.3 , \theta(x) = 0.1 , \delta(x) = 0.3 , d_m = 0.01, d_h = 0.1,

    \beta_{1}(x) = 0.12(1+\cos(2x)), \; \; \; \beta_{2}(x) = 0.15(1+\cos(2x)).

    We set initial data

    \begin{eqnarray} \begin{aligned} &S_a(x, 0) = 200+\sin(x)-200\cos(4x), \; I_a(x, 0) = 5+0.1\sin(x)-5\cos(4x), \\ &S_h(x, 0) = 21+0.5\sin(x)-20\cos(4x), \; I_h(x, 0) = 0.5+0.01\sin(x)-0.5\cos(4x), \\ &S_m(x, 0) = \omega(x)S_a(x, 0), \; \; I_m(x, 0) = \omega(x)I_a(x, 0), \; \; V(x, 0) = \theta(x) I_{h}(x, 0), \; x\in(0, \pi). \end{aligned} \end{eqnarray} (6.1)

    Next, we will change the parameters s and \beta_v(x) and then observe the longtime behavior of the solution to model (2.3).

    Example 6.1. Choose s = 0.04 and \beta_{v}(x) = 0.00035(1+\cos(2x)) . It follows from (4.5) that R_{0}^{m} = 0.52 < 1. Theorem 5.1 shows that the disease free equilibrium E_{01}(x) is globally attractive when R_{0}^{m} < 1 . We can confirm this in Figure 2. Figure 2 shows that the evolutions of S_{a}(x, t), S_{m}(x, t), I_{a}(x, t), I_{m}(x, t), I_{h}(x, t), V(x, t) decay to zero. Biologically, mosquito population will vanish, and Zika virus will eradicate in human population and contaminated aquatic environment.

    Figure 2.  The evolution diagram of numerical solutions of model (2.3) for R_{0}^{m} < 1 . Diagrams (a)-(f) show that the evolutions of S_{a}(x, t), S_{m}(x, t), I_{a}(x, t), I_{m}(x, t), I_{h}(x, t), V(x, t) decay to zero. It mean that mosquito population will vanish, and Zika virus will eradicate in human population and contaminated aquatic environment.

    Example 6.2. Choose s = 0.4 and \beta_{v}(x) = 0.00035(1+\cos(2x)) . It follows from (4.5) that R_{0}^{m} = 5 > 1 . According to the method of calculating the regeneration numbers in [38], we can get R_{0} = 0.98 < 1 . Theorem 5.2 shows that the disease free equilibrium E_{02}(x) is globally attractive when R_{0}^{m} > 1 and R_{0} < 1 . We can confirm this in Figure 3. Figure 3 shows that the evolutions of S_{a}(x, t), S_{m}(x, t) tend to steady state A^*(x), M^*(x) , and the evolutions of diseased compartments I_{a}(x, t), I_{m}(x, t), I_{h}(x, t), V(x, t) decay to zero. Biologically, mosquito population is present, and Zika virus will eradicate in mosquito population, human population and contaminated aquatic environment.

    Figure 3.  The evolution diagram of numerical solutions of model (2.3) for R_{0}^{m} > 1 and R_{0} < 1 . Diagrams (a) and (b) show that the evolutions of S_{a}(x, t), S_{m}(x, t) tend to steady state A^*(x), M^*(x) , respectively. Diagrams (c)–(f) show that the evolutions of diseased compartments I_{a}(x, t), I_{m}(x, t), I_{h}(x, t), V(x, t) decay to zero. It mean that mosquito population is present, and Zika virus will eradicate in mosquito population, human population and contaminated aquatic environment.

    Example 6.3. Choose s = 0.4 and \beta_{v}(x) = 0.035(1+\cos(2x)) . It follows from (4.5) that R_{0}^{m} = 5 > 1 . According to the method of calculating the regeneration numbers in [38], we can get R_{0} = 2.8 > 1 . Theorem 5.3 shows that system (2.3) is uniformly persistent when R_{0}^{m} > 1 and R_{0} > 1 . We can confirm this in Figure 4. Figure 4 shows that the evolutions of S_{a}(x, t), S_{m}(x, t) , I_{a}(x, t), I_{m}(x, t), I_{h}(x, t), V(x, t) keep positive. Biologically, mosquito population is present, and Zika virus will persist in mosquito population, human population and contaminated aquatic environment.

    Figure 4.  The evolution diagram of numerical solutions of model (2.3) for R_{0}^{m} > 1 and R_{0} > 1 . Diagrams (a)–(f) show that the evolutions of S_{a}(x, t), S_{m}(x, t) , I_{a}(x, t), I_{m}(x, t), I_{h}(x, t), V(x, t) keep positive. It mean that mosquito population is present, and Zika virus will persist in mosquito population, human population and contaminated aquatic environment.

    The main contribution of this study, based on experimental proving evidence [25], is that we propose a new Zika model by introducing the environment transmission route in a spatial heterogeneous environment. In contrast to [12,14,24], we consider environment transmission (human-environment-mosquito-human) route in Zika model. From Figures 3 and 4, we can get that increasing environment transmission rate \beta_{v}(x) can induce the outbreak of Zika. Therefore, the environment transmission route is indispensable. Our work is an extension of previous mathematical Zika models that the transmission of Zika virus involves both mosquito-borne transmission (human-mosquito-human) routes [12,14,24]. In fact, environmental transmission route in our model is similar to other waterborne disease models, such as cholera [45]. So, our model analysis can also be applied to other waterborne diseases.

    We derive a biologically meaningful threshold indexes, the basic offspring number R_{0}^{m} and basic reproduction number R_{0} by the theory developed by Wang and Zhao [38], which is characterized as the spectral radius of the next generation operator. Then, we prove that if R_{0}^{m} < 1 , then both mosquitoes and Zika virus will become vanish. If R_{0}^{m} > 1 and R_{0} < 1 , then mosquitoes will persist and Zika virus will die out. If R_{0}^{m} > 1 and R_{0} > 1 , then mosquitoes and Zika virus are persistent presences. Finally, numerical simulations conform these results.

    Our current study has some limitations. In our model, we do not consider sexual transmission route, but it is indeed an important route of spreading of Zika virus [46]. Future, we will study a Zika model which incorporates mosquito-borne transmission, sexual transmission and environment transmission threes routes. In addition, in our model, we assume same diffusive coefficients d_m for both S_m and I_m , same d_h for all of S_h , I_h and R_h . However, due to mobility for S_m and I_m , S_h , I_h and R_h is different, studies of different diffusion coefficients have more realistic implications. Yin [47] studied a mathematical model for an infectious disease such as Cholera with different diffusion coefficients. By the delicate theory of elliptic and parabolic equations, global asymptotic behavior of the solution was obtained. This work makes progress toward the case of different diffusive coefficients. Therefore, it motivates us to consider different diffusive coefficients into our model. We leave it for future investigation.

    The work is partially supported by the Startup Foundation for Introducing Talent of Anhui Polytechnic University (No. 2020YQQ069), Pre-Research National Natural Science Foundation of China of Anhui Polytechnic University (No. Xjky2020134), and Joint Special Funds for Basic Research in Local Undergraduate Universities (Part) of Yunnan Province of China (Nos 2018FH001-113).

    The authors declare that they have no competing interests.



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