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Research article Special Issues

Pseudo-Stieltjes calculus: αpseudo-differentiability, the pseudo-Stieltjes integrability and applications

  • In this paper, the concepts of the αpseudo-differentiability and the pseudo-Stieltjes integrability are proposed, and the corresponding transformation theorems and Newton–Leibniz formula are established. The obtained results provide a framework for analyzing nonlinear differential equations.

    Citation: Caiqin Wang, Hongbin Xie, Zengtai Gong. Pseudo-Stieltjes calculus: αpseudo-differentiability, the pseudo-Stieltjes integrability and applications[J]. Electronic Research Archive, 2024, 32(11): 6467-6480. doi: 10.3934/era.2024302

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  • In this paper, the concepts of the αpseudo-differentiability and the pseudo-Stieltjes integrability are proposed, and the corresponding transformation theorems and Newton–Leibniz formula are established. The obtained results provide a framework for analyzing nonlinear differential equations.



    In this paper, we are concerned with the following susceptible-infected-susceptible(SIS) model

    {St=(dSSxa(x)S)xβ(x)SIS+I+γ(x)I,  0<x<L, t>0,It=(dIIxa(x)I)x+β(x)SIS+Iγ(x)I,  0<x<L, t>0,dSSxa(x)S=dIIxa(x)I=0,  x=0,L, t>0,S(x,0)=S0(x),  I(x,0)=I0(x),0<x<L. (1.1)

    Here S(x,t) and I(x,t) denote the density of susceptible and infected individuals in a given spatial interval (0,L), dS and dI are positive constants which stand for the diffusion coefficients for the susceptible and infected populations, a(x) is a smooth nonnegative function which represents the advection speed rate, while β(x) and γ(x) represent the rates of disease transmission and recovery at location x, which are Hölder continuous functions on (0,L). In addition, S0(x) and I0(x) are continuous and satisfy

    (A1)S0(x)0 and I0(x)0 for x(0,L),L0I0(x)dx>0.

    We would like to give the survey of some results on SIS model. In [1], Allen et al. investigated a discrete SIS model, in [2], they also proposed the SIS model with no advection in a given spatial region Ω, where they dealt with the existence, uniqueness and asymptotic behaviors of the endemic equilibrium as the diffusion rate of the susceptible individuals approaches to zero. Many authors also considered the SIS reaction–diffusion model, including the global stability of the endemic equilibrium, the effects of large and small diffusion rates of the susceptible and infected population on the persistence and extinction of the disease, discuss how the disease vanish or spreading in high-risk or low-risk domain, and so on. For the dynamics and asymptotic profiles of steady states of an epidemic model in advective environments, we can see[3]. For A SIS reaction-diffusion-advection model in a low-risk and high-risk domain, we can see [4]. For Dynamics of an SIS reaction-diffusion epidemic model for disease transmission, we can see[5], For Concentration profile of endemic equilibrium of a reaction- diffusion-advection SIS epidemic model, we can see [6]. For the varying total population enhances disease persistence, we can see [7]; For the asymptotic profiles of the positive steady state for an SIS epidemic reaction-diffusion model, we can see [8]. For the global stability of the steady states of an SIS epidemic reaction-diffusion model, we can see [9]. For the asymptotic profile of the positive steady state for an SIS epidemic reaction- diffusion model: effects of epidemic risk and population movement, we can see [10]; For reaction-diffusion SIS epidemic model in a time-periodic environment, we can see [11]. For the global dynamics and traveling waves for a periodic and diffusive chemostat model with two nutrients and one microorganism, we can see [12]. For more information about dynamical systems in population biology, we also can refer to see [13] and the references therein. Recently, Cui and Lou studied (1.1) when a(x)q for x[0,L] in [14], that is, it is a constant advection. Besides establishing the asymptotic stability of the unique disease-free equilibrium(DFE) when R0<1 and the existence of the endemic equilibrium when R0>1, they found that the DFE changes its stability at most once as dI varies from zero to infinity, which is strong contrast with the case of no advection. Since (1.1) has vary advection, an natural and interesting question is whether we can establish the similar results on (1.1) to those in the case of no advection or not.

    Since the functions a(x), β(x), γ(x), S0(x) and I0(x) are continuous in (0,L), by the standard theory for a system of semilinear parabolic equations, (1.1) is locally wellposedness in (0,Tmax). Noticing (A1), by the maximum principle, S(x,t) and I(x,t) are positive and bounded for x[0,L] and t(0,Tmax). Hence, by the results in [15], Tmax= and (1.1) posses a unique classical solution (S(x,t),I(x,t)) for all time.

    It is easy to verify that

    L0[S(x,t)+I(x,t)]dx=L0[S(x,0)+I(x,0)]dx:=N>0,t>0. (1.2)

    Inspired by [2] and [14], we say that (0,L) is a low-risk domain if L0β(x)dx<L0γ(x)dx and high-risk domain if L0β(x)dx>L0γ(x)dx.

    The corresponding equilibrium system of (1.1) is

    {(dS˜Sxa(x)˜S)xβ(x)˜S˜I˜S+˜I+γ(x)˜I=0,  0<x<L,(dI˜Ixa(x)˜I)x+β(x)˜S˜I˜S+˜Iγ(x)˜I=0,  0<x<L,dS˜Sxa(x)˜S=dI˜Ixa(x)˜I=0,  x=0,L. (1.3)

    The half trivial solution (˜S(x),0) of (1.3) is called a disease-free equilibrium(DFE), while the solution (˜S(x),˜I(x)) of (1.3) is called endemic equilibrium(EE) if ˜I(x)>0 for some x(0,L).

    We also introduce the following basic reproduction number as those in literatures [2] and [14]. We also can refer to [16] and see the definition and the computation of the basic reproduction ratio R0 in models for infectious diseases in heterogeneous populations, refer to [17] and see reproduction numbers and sub-threshold endemic equilibria for compartmental models of disease transmission, see basic reproduction numbers for reaction-diffusion epidemic models [18].

    R0=supφH1((0,L)),φ0{L0β(x)ea(x)dIφ2dxdIL0ea(x)dIφ2xdx+L0γ(x)ea(x)dIφ2dx}. (1.4)

    Our first result is concerned with the qualitative properties for R0.

    Theorem 1.1. Let ˆR0 be the basic reproduction number when a(x)0 which was introduced in [2]. Then the following conclusions hold.

    (1) For any given a(x)>0, R0β(L)γ(L) as dI0 and R0L0β(x)dxL0γ(x)dx as dI+;

    (2) For any given dI>0, R0ˆR0 as maxx[0,L]a(x)0 and R0β(L)γ(L) as minx[0,L]a(x)+;

    (3) If β(x)>(<)γ(x) on [0,L], then R0>(<1) for any given dI>0 and a(x)>0.

    Our second result deals with the stability of DFE, which will extend those of [2] and [14].

    Theorem 1.2. The DFE is unstable if R0>1 while it is globally asymptotically stable if R0<1.

    We will analyze (1.1) under the following assumptions on β(x) and γ(x):

    (C1) β(0)γ(0)<0<β(L)γ(L), i.e., β(x)γ(x) changes sign from negative to positive,

    or

    (C2) β(0)γ(0)>0>β(L)γ(L), i.e., β(x)γ(x) changes sign from positive to negative.

    In the point view of biological,

    (C1) all lower-risk sites are located at the upstream and all high-risk sites are at the downstream,

    or

    (C2) all high-risk sites are distributed at the upstream and lower-risk sites are at the downstream.

    To state other results, in convenience, let q=maxx[0,L]a(x) and denote a(x)=q˜a(x) sometimes in the sequels.

    We can get further properties of R0 when L0β(x)dx>L0γ(x)dx.

    Theorem 1.3. Assume that L0β(x)dx>L0γ(x)dx. Denote R0=R0(dI,q).

    (i) If (C1) holds, then the DFE is unstable for any q>minx[0,L]a(x)>0 and dI>0;

    (ii) If (C2) holds, then there exists a unique curve in dIq plane

    Γ1={(dI,ρ1(dI)):R0(dI,ρ1(dI))=1,dI(0,+)}

    with the function ρ1=ρ1(dI):(0,+)(0,+) satisfying

    limdI0+ρ1(dI)=0,limdI+ρ1(dI)dI=θ1,

    and such that for every dI>0, the DFE is unstable for 0<minx[0,L]a(x)<q<ρ1(dI) and it is globally and asymptotically stable for q>minx[0,L]a(x)>ρ1(dI).

    Here θ1 is the unique positive solution of

    L0[β(x)γ(x)]eθ1˜a(x)dx=0.

    Similarly, we can get further properties of R0 when L0β(x)dx<L0γ(x)dx.

    Theorem 1.4. Assume that L0β(x)dx<L0γ(x)dx. Let dI is the unique positive root of the equation ˆR0=1, where ˆR0 was introduced in [2].

    (1) If (C1) holds, then the DFE is unstable for any q>minx[0,L]a(x)>0 and dI(0,dI], while for dI(dI,+) there exists a unique curve in dIq plane

    Γ2={(dI,ρ2(dI)):R0(dI,ρ2(dI))=1,dI(dI,+)}

    with the monotone function ρ2=ρ2(dI):(dI,+)(0,+) satisfying

    limdIdI+ρ2(dI)=0,limdI+ρ2(dI)dI=θ2,

    and such that the DFE is unstable for 0<minx[0,L]a(x)<q<ρ2(dI) and it is globally asymptotically stable for q>minx[0,L]a(x)>ρ2(dI).

    Here θ2 is the unique positive solution of

    L0[β(x)γ(x)]eθ2˜a(x)dx=0.

    (2) If (C2) holds, then for dI(0,dI), there exists a unique curve in dIq plane

    Γ3={(dI,ρ3(dI)):R0(dI,ρ3(dI))=1,dI(0,dI)}

    with the function ρ3=ρ3(dI):(0,dI)(0,+) satisfying

    limdI0+ρ3(dI)=0,limdIdIρ3(dI)=0,

    and such that the DFE is unstable for 0<minx[0,L]a(x)<q<ρ3(dI) and it is globally and asymptotically stable for q>minx[0,L]a(x)>ρ3(dI), while for dI(dI,+), the DFE is globally and asymptotically stable for any q>minx[0,L]a(x)>0.

    The following theorem deals with the existence of EE.

    Theorem 1.5. Assume that β(x)γ(x) changes sign once in (0,L). If R0>1, then problem (1.3) possesses at least one EE.

    The last theorem will consider the results on (1.1) when β(x)γ(x) changes sign twice in (0,L).

    Theorem 1.6. Assume that β(x)γ(x) changes sign twice in (0,L).

    (1) If L0β(x)dx>L0γ(x)dx and β(L)<γ(L), then there exists some positive constant Λ which is independent of dI and q such that for every dI>Λ, we can find a positive constant Q which depends on dI such that R0>1 when 0<minx[0,L]a(x)<q<Q and R0<1 when q>Q.

    (2) If L0β(x)dx>L0γ(x)dx and β(L)>γ(L), then there exists some positive constant Λ which is independent of dI and q such that for every dI>Λ one of the following conclusions holds:

    (i) R0>1 for any q>minx[0,L]a(x)>0;

    (ii) There exists a positive constant ˆQ which is independent of dI and satisfies that R0>1 for qˆQ and R0=1 when q=ˆQ;

    (iii) There exist two positive constants Q2>Q1 both depending on dI such that R0>1 when q(0,Q1)(Q2,+) while R0<1 when q(Q1,Q2).

    (3) If L0β(x)dx<L0γ(x)dx and β(L)>γ(L), then there exists some positive constant Λ>dI which is independent of dI and q such that for every dI>Λ, we can find a positive constant Q which depends on dI such that R0<1 when 0<minx[0,L]a(x)<q<Q and R0>1 when q>Q.

    (4) If L0β(x)dx<L0γ(x)dx and β(L)<γ(L), then there exists some positive constant Λ>dI which is independent of dI and q such that for every dI>Λ one of the following conclusions holds:

    (iv) R0<1 for any q>minx[0,L]a(x)>0;

    (v) There exists a positive constant ˆQ which is independent of dI and satisfies that R0<1 for qˆQ and R0=1 when q=ˆQ;

    (vi) There exist two positive constants Q2>Q1 both depending on dI and satisfy that R0<1 when q(0,Q1)(Q2,+) while R0>1 when q(Q1,Q2).

    The rest of this paper is organized as follows. In Section 2, we give the proofs of Theorem 1.1 and Theorem 1.2. In Section 3, we will prove Theorem 1.3. In Section 4, we will prove Theorem 1.4. In Section 5, we will prove Theorem 1.5. In Section 6, we will prove Theorem 1.6.

    In this section, we first give some qualitative properties of R0, then we deal with the stability of DFE, and we can finish the proofs of Theorem 1.1 and Theorem 1.2.

    By the definition of R0, there exits some positive function Φ(x)C2([0,L]) such that

    {[dIΦxa(x)Φ]x+γ(x)Φ=1R0β(x)Φ,0<x<L,dIΦx(0)a(0)Φ(0)=0,dIΦx(L)a(L)Φ(L)=0. (2.1)

    Letting φ(x)=ea(x)dIΦ(x), we have

    {dIφxxa(x)φx+γ(x)φ=1R0β(x)φ,0<x<L,φx(0)=0,φx(L)=0. (2.2)

    Linearizing (1.1) around (ˆS,0) and letting ˉξ(x,t)=S(x,t)ˆS(x,t), ˉη(x,t)=I(x,t), we have

    {ˉξt=(dSˉξxa(x)ˉξ)x[β(x)γ(x)]ˉη,0<x<L, t>0,ˉηt=(dIˉηxa(x)ˉη)x+[β(x)γ(x)]ˉη,0<x<L, t>0.

    For the linear system, seeking for the solution which is separation of variables, i.e., ˉξ(x,t)=eλtξ(x) and ˉη(x,t)=eλtη(x), we have

    {(dSξxa(x)ξ)x[β(x)γ(x)]η+λξ=0,0<x<L,(dIηxa(x)η)x+[β(x)γ(x)]η+λη=0,0<x<L, (2.3)

    subject to boundary conditions

    {dSξx(0)a(0)ξ(0)=0,dSξx(L)a(L)ξ(L)=0,dSηx(0)a(0)η(0)=0,dSηx(L)a(L)η(L)=0. (2.4)

    By the conservation of total population, we need to impose that

    L0[ξ(x)+η(x)]dx=0. (2.5)

    Noticing that the second equation of (2.3) is independent of ξ, letting ζ(x)=ea(x)dIη(x), we only need to consider the following eigenvalue problem

    {dIζxx+a(x)ζx+[β(x)γ(x)]ζ(x)+λζ(x)=0,0<x<L,ζx(0)=ζx(L)=0. (2.6)

    By the results of [19], all the eigenvalues are real, the smallest eigenvalue λ1(dI,q) is simple, and its corresponding eigenfunction ϕ1 can be chosen positive.

    We will show a fact below.

    Lemma 2.1.1. For any dI and q>minx[0,L]a(x)>0, λ1(dI,q)<0 if R0>1, λ1(dI,q)=0 if R0=1 and λ1(dI,q)>0 if R0<1.

    Proof. Note that (λ1(dI,q),ϕ1) satisfies

    {dI(ϕ1)xxa(x)(ϕ1)x+[γ(x)β(x)]ϕ1(x)=λ1(dI,q)ϕ1(x), 0<x<L,(ϕ1)x(0)=(ϕ1)x(L)=0. (2.7)

    Multiplying (2.1) by ea(x)dIϕ1 and (2.7) by ea(x)dIΦ, integrating by parts in (0,L), and subtracting the resulting equations, we get

    L0(1R01)β(x)Φ(x)ϕ1(x)dx=L0λ1(dI,q)Φ(x)ϕ1(x)dx.

    Using the mean value theorem of integrating, we have

    (1R01)β(x1)Φ(x1)ϕ1(x1)=λ1(dI,q)Φ(x2)ϕ1(x2)

    for some 0x1L and 0x2L. Using β(x1)Φ(x1)ϕ1(x1)>0 and Φ(x2)ϕ1(x2)>0, we know that

    (1R01)has the same sign ofλ1(dI,q),

    which implies the conclusions are true.

    Lemma 2.1.2. If dIq0 and dIq20, ˜a(x)>δ>0 for some constant δ, then R0β(L)γ(L).

    Proof. Let w(x)=eqdIA˜a(x)Φ(x), where Φ(x) is the solution of (2.1), A is a constant which will be chosen later. It is easy to verify that w satisfies

    { [q2A(A1)dI(˜a(x))2+q(A1)˜a(x)+1R0β(x)γ(x)]w=dIwxx+(12A)a(x)wx,0<x<L, t>0,dIwx(0)=a(0)(1A)w(0),dIwx(L)=a(L)(1A)w(L). (2.8)

    First we chose A=1+C1dIq2, where C1 is a positive constant to be chosen later. Then (2.8) becomes

    { [C1(1+C1dIq2)(˜a(x))2+q(1+C1dIq2)˜a(x)+1R0β(x)γ(x)]w=dIwxx(1+2C1dIq2)a(x)wx,0<x<L, t>0,dIwx(0)=C1dIq˜a(0)w(0),dIwx(L)=C1dIq˜a(L)w(L).

    Assume that w(x)=minx[0,L]w(x). We will show that x=L below. wx(0)<0 implies that x0. If x(0,L), then wxx(x)0 and wx(x)=0, (2.9) means that

    [C1(1+C1dIq2)(˜a(x))2+q(1+C1dIq2)˜a(x)+1R0β(x)γ(x)]0

    Taking C1=Kq with K large enough, we can get a contradiction. Therefore, x=L and w(x)w(L) for x[0,L], which implies that

    Φ(x)Φ(L)eqdI(1+C1dIq2)[˜a(L)˜a(x)]. (2.9)

    Next, we chose A=1C2dIq2, where C2 is a positive constant to be chosen later. Then (2.8) becomes

    { [C2(1C2dIq2)(˜a(x))2+q(1C2dIq2)˜a(x)+1R0β(x)γ(x)]w=dIwxx(12C2dIq2)a(x)wx,0<x<L, t>0,dIwx(0)=C2dIq˜a(0)w(0),dIwx(L)=C2dIq˜a(L)w(L).

    Assume that w(x)=maxx[0,L]w(x). We will show that x=L below. wx(0)>0 implies that x0. If x(0,L), then wxx(x)0 and wx(x)=0, (2.10) means that

    [C2(1C2dIq2)(˜a(x))2+q(1C2dIq2)˜a(x)+1R0β(x)γ(x)]0

    Taking C2=Kq with K large enough, we can get a contradiction. Therefore, x=L and w(x)w(L) for x[0,L], which implies that

    Φ(x)Φ(L)eqdI(1C2dIq2)[˜a(L)˜a(x)]. (2.10)

    Dividing (2.1) by Φ(L) and integrating the result in (0,L), we have

    L0γ(x)Φ(x)Φ(L)dx=1R0L0β(x)Φ(x)Φ(L)dx. (2.11)

    Letting y=q[˜a(L)˜a(x)]dI, i.e., x=˜a1[˜a(L)dIyq], we have

    e(1+C1dIq)yΦ(˜a1[˜a(L)dIyq])Φ(L)e(1C2dIq)y (2.12)

    and

    q[˜a(L)˜a(0)]dI0γ(˜a1[˜a(L)dIyq])Φ(˜a1[˜a(L)dIyq])˜a(˜a1[˜a(L)dIyq])Φ(L)dy=1R0q[˜a(L)˜a(0)]dI0β(˜a1[˜a(L)dIyq])Φ(˜a1[˜a(L)dIyq])˜a(˜a1[˜a(L)dIyq])Φ(L)dy. (2.13)

    Using (2.12), by Lebesgue dominant convergence theorem, then passing to the limit in (2.13), we get

    limdI/q0,dI/q20R0=limdI/q0,dI/q20q[˜a(L)˜a(0)]dI0β(˜a1[˜a(L)dIyq])Φ(˜a1[˜a(L)dIyq])˜a(˜a1[˜a(L)dIyq])Φ(L)dyq[˜a(L)˜a(0)]dI0γ(˜a1[˜a(L)dIyq])Φ(˜a1[˜a(L)dIyq])˜a(˜a1[˜a(L)dIyq])Φ(L)dy=0β(L)˜a(L)eydy0γ(L)˜a(L)eydy=β(L)γ(L). (2.14)

    We have the following corollary.

    Corollary 2.1.1. The following statements hold.

    (i) Given dI>0, R0ˆR0 as q0;

    (ii) Given dI>0, R0β(L)γ(L) as q+;

    (iii) Given q>0, R0β(L)γ(L) as dI0;

    (iv) Given q>0, R0L0β(x)dxL0γ(x)dx as dI+.

    Proof. (i) For any fixed φH1((0,L)), φ0, we have

    limq0dIL0ea(x)dIφ2xdx+L0γ(x)ea(x)dIφ2dxL0β(x)ea(x)dIφ2dx=dIL0φ2xdx+L0γ(x)φ2dxL0β(x)φ2dx.

    Taking infφH1((0,L)),φ0 both sides, we have 1R01ˆR0 as q0.

    (ii) and (iii) are the direct conclusions of Lemma 2.2.

    (iv) By the definition of 1R0, for φ1, we have

    1R0L0γ(x)ea(x)dIdxL0β(x)ea(x)dIdxmaxx[0,L]γ(x)minx[0,L]β(x),

    which implies that 1R0 is uniformly bounded for dI>0, passing to a subsequence if necessary, it has a finite limit 1ˉR0 as dI.

    On the other hand, by the standard elliptic regularity and the Sobolev embedding theorem, Φ is uniformly bounded for all dI1. Dividing both sides of (2.1) by dI and letting dI+, we have Φxx0 for x(0,L) and Φx(0)0, Φx(L)0. Consequently, there exists a positive constant ˉΦ such that Φ(x)ˉΦ as dI+. Integrating (2.1) by parts over (0,L), we can get

    qdIL0ea(x)dI[dIΦxa(x)Φ(x)]dx+L0ea(x)dIγ(x)Φ(x)dx=1R0L0ea(x)dIβ(x)Φ(x)dx.

    Letting dI+, we obtain ˉR0=L0β(x)dxL0γ(x)dx.

    Lemma 2.1.3. The following statements hold.

    (i) If β(x)>γ(x) on [0,L], then R0>1 for any dI>0 and q>minx[0,L]a(x)>0;

    (i) If β(x)<γ(x) on [0,L], then R0<1 for any dI>0 and q>minx[0,L]a(x)>0.

    Proof. (i) If β(x)>γ(x) on [0,L], by the definition of 1R0, for φ1, we have

    1R0L0γ(x)ea(x)dIdxL0β(x)ea(x)dIdx<1,

    i.e., R0>1.

    (ii) Subtracting both sides of (2.2) by β(x)φ, multiplying by ea(x)dIφ, we have

    dIφxxea(x)dIφa(x)φxea(x)dIφ+[γ(x)β(x)]ea(x)dIφ2=(1R01)β(x)ea(x)dIφ2.

    Integrating it by parts over (0,L), using φx(0)=φx(L)=0, we obtain

    dIL0ea(x)dI(φx)2dx+L0[γ(x)β(x)]ea(x)dIφ2dx=(1R01)L0β(x)ea(x)dIφ2dx.

    Since β(x)<γ(x) on [0,L], the left side of the above equality is positive, and

    (1R01)L0β(x)ea(x)dIφ2dx>0,

    which implies that R0<1.

    Proof. Theorem 1.1 is the direct results of Lemma 2.1.2, Corollary 2.1.1 and Lemma 2.1.3.

    Next we will consider the stability of DFE.

    Lemma 2.1.4. The DFE is stable if R0<1, while it is unstable if R0>1.

    Proof. 1. Assume contradictorily the DFE is unstable if R0<1. Then we can find (λ,ξ,η) which is a solution of (2.3)–(2.4) subject to (2.5), with at least one of ξ and η is not identical zero, and (λ)0. Suppose that η0, then ξ0 on [0,L]. Using (2.3)–(2.4), we have

    {(dSξxa(x)ξ)x=λξ,0<x<L,dSξx(0)a(0)ξ(0)=0,dSξx(L)a(L)ξ(L)=0. (2.15)

    It is easy to see that λ is real and nonnegative, and therefore λ=0. We find that ξ=ξ0eqdI˜a(x), where ξ0 is some constant to be determined later. By (1.2), we impose that L0[ξ(x)+η(x)]dx=0, ξ0=0, i.e., ξ0 on [0,L]. This is a contradiction. Then we conclude that η0 on [0,L]. From (2.6), λ must be real and λ0. Since λ1(dI,q) is the principal eigenvalue, then λ1(dI,q)λ0. Lemma 2.1 implies that R01, which is a contradiction. Then we conclude that if (λ,ξ,η) is a solution of (2.3)–(2.4), with at least one of ξ and η not identical zero on [0,L], then (λ)>0. This proves the linear stability of the DFE.

    2. Suppose that R0>1. Since (λ1(dI,q),ϕ1) is the principal eigen-pair of (2.6), (λ1(dI,q),ea(x)dIϕ1) satisfies

    {[dI(ϕ1)xa(x)ϕ1]x+[β(x)γ(x)]ϕ1+λ1(dI,q)ϕ1=0,0<x<L,dI(ϕ1)xa(x)ϕ1=0,x=0, L.

    By the result of Lemma 2.1.1, λ1(dI,q)<0. On the other hand,

    {(dSξxa(x)ξ)x+λξ=[β(x)γ(x)]ea(x)dIϕ1,0<x<L,dSξx(0)a(0)ξ(0)=0,dSξx(L)a(L)ξ(L)=0. (2.16)

    There exists a unique solution ξ1 of (2.16). And (2.5) becomes

    L0[ξ1(x)+ea(x)dIϕ1(x)]dx=0,

    which implies that (2.3)–(2.4) has a solution (λ1(dI,q),ξ1,ea(x)dIϕ1(x)) satisfying λ1(dI,q)<0 and ea(x)dIϕ1(x)>0 in (0,L). Therefore, the DFE is linearly unstable.

    Lemma 2.1.5. If R0<1, then (S,I)(ˆS,0) in C([0,L]) as t+.

    Proof. If R0<1, letting u(x,t)=Meλ1(dI,q)tea(x)dIϕ1(x), then we have

    {ut=[dIuxa(x)u]x+[β(x)γ(x)]u,0<x<L,t>0,dIux(0,t)a(0)u(0,t)=0,dIux(L,t)a(L)u(L,t)=0, t>0.

    Here (λ1(dI,q),ϕ1) is the principal eigen-pair, λ1(dI,q)>0 and ϕ1(x)>0 on [0,L]. M is large enough such that I(x,0)u(x,0) for every x(0,L). Noticing that

    {It=[dIIxa(x)I]x+[β(x)γ(x)]I,0<x<L,t>0,dIux(0,t)a(0)u(0,t)=0,dIux(L,t)a(L)u(L,t)=0, t>0.

    By the comparison principle, we have I(x,t)u(x,t) for every x(0,L) and t0. Obviously, u(x,t)0 for every x(0,L) as t, which implies that I(x,t)0 for every x(0,L) as t.

    Now we will show that SˆS as t+. Since

    St=(dSSxa(x)S)xβ(x)SIS+I+γ(x)I, 0<x<L, t>0,

    we have

    |St(dSSxa(x)S)x|(β+γ)ICeλ1(dI,q)t,

    for 0<x<L, t>0. Noticing that

    limt+eλ1(dI,q)t0

    as t+, we know that there exists a positive function ˜S(x) such that

    limt+S(x,t)=˜S(x),L0˜S(x)dx=N.

    Therefore, limt+S(x,t)=˜S(x)=ˆS(x).

    Proof. Theorem 1.2 is the direct results of Lemma 2.1.4 and Lemma 2.1.5.

    In this section, we will study further properties of R0 in the case of β(x)γ(x) changing sign once.

    Lemma 2.2.1. Assume that ϕ1 is a positive eigenfunction corresponding to R0=1, β(x)γ(x) changes sign once in (0,L). If assumption (C1)(or (C2)) holds, then (ϕ1)x>0(or (ϕ1)x<0) in (0,L).

    Proof. If β(x)γ(x) changes sign once in (0,L) and assumption (C1) holds, then there exists some x0(0,L) such that β(x)γ(x)<0 in (0,x0), β(x0)=γ(x0) and β(x)γ(x)>0 in (x0,L).

    By the definition of ϕ1, we have

    {dI(ϕ1)xxa(x)(ϕ1)x=[β(x)γ(x)]ϕ1,0<x<L,(ϕ1)x(0)=(ϕ1)x(L)=0. (2.17)

    Multiplying (2.17) by ea(x)dI, we obtain

    dI(ea(x)dI(ϕ1)x)x=[β(x)γ(x)]ea(x)dIϕ1.

    Under the assumptions on β(x) and γ(x), we can obtain (ea(x)dI(ϕ1)x)x>0 in (0,x0), (ea(x)dI(ϕ1)x)x=0 at x0 and (ea(x)dI(ϕ1)x)x<0 in (x0,L). That is, ea(x)dI(ϕ1)x is strictly increasing in (0,x0) and strictly decreasing in (x0,L). Noticing that (ϕ1)x(0)=(ϕ1)x(L)=0, we can get ea(x)dI(ϕ1)x>0 in (0,L). So (ϕ1)x>0 in (0,L).

    Similarly, if β(x)γ(x) changes sign once in (0,L) and assumption (C2) holds, (ϕ1)x<0 in (0,L). We omit the details here.

    Now we prove two general lemmas below.

    For any continuous function m(x) on [0,L], define

    F(η)=L0˜a(x)eη˜a(x)m(x)dx,0η<.

    Lemma 2.2.2. Assume that m(x)C1([0,L]) and m(L)>0(or m(L)<0). Then there exists some positive constant M such that F(η)>0(or F(η)<0) for any η>M.

    Proof. Since m(x) and ˜a(x) is uniformly bounded independent of η, we have

    limη+ηeη˜a(L)F(η)=limη+L0η˜a(x)eη[˜a(x)˜a(L)]m(x)dx=m(L)limη+(m(0)eη[˜a(0)˜a(L)]+L0m(x)eη[˜a(x)˜a(L)]dx)=m(L)limη+(m(0)eη[˜a(0)˜a(L)]+L0m(x)e˜a(ξ)[xL]dx)=m(L)>0(<0).

    Therefore, there exists some positive constant M such that F(η)>0(<0) for η>M.

    Lemma 2.2.3. Assume that m(x) changes sign once in (0,L). Then

    (i) If m(L)>0 and L0˜a(x)m(x)dx>0, then F(η)>0 for any η>0;

    (ii) If m(L)<0 and L0˜a(x)m(x)dx<0, then F(η)<0 for any η>0;

    (iii) If m(L)>0 and L0˜a(x)m(x)dx<0, then there exists a unique η1(0,+) such that F(η1)=0 and F(η1)>0;

    (iv) If m(L)<0 and L0˜a(x)m(x)dx>0, then there exists a unique η1(0,+) such that F(η1)=0 and F(η1)<0.

    Proof. We only prove part (i) and part (iii). The proofs of part (ii) and part (iv) are similar.

    (i) If m(L)>0 and m(x) changes sign once in (0,L), then there exists x1(0,L) such that m(x)<0 for x(0,x1) and m(x)>0 for x(x1,L). Since ˜a(x) is increasing, we have m(x)[˜a(x)˜a(x1)]>0 for x(0,L) and xx1. And

    [e˜a(x1)ηF(η)]=e˜a(x1)η[F(η)˜a(x1)F(η)]=e˜a(x1)ηL0[˜a(x)˜a(x1)]m(x)˜a(x)eη˜a(x)dx>0, (2.18)

    which implies that e˜a(x1)ηF(η) is strictly increasing in η(0,), e˜a(x1)ηF(η)>F(0)=L0˜a(x)m(x)dx>0. Consequently, F(η)>0 for any η>0. Here the prime notation denotes differentiation by η. Part (i) is proved.

    (iii) L0˜a(x)m(x)dx<0 means that F(0)<0, while, by the result of Lemma 2.2.2, m(L)>0 means that F(η)>0 for η>M with M large enough. By continuity, there at least exists a positive root for F(η)=0. But e˜a(x1)ηF(η) is increasing in η(0,), so F(η)=0 only has a unique positive root η1. By (2.18), we have F(η1)>a(x1)F(η1)=0. Part (iii) is proved.

    In this section, we consider the stability of DFE. First we have

    Lemma 2.3.1. Assume that β(x)γ(x) changes sign once in (0,L) and L0β(x)dx>L0γ(x)dx.

    (i) If β(x) and γ(x) satisfy (C1), then R0>1 for dI>0 and q>minx[0,L]a(x)>0;

    (ii) If β(x) and γ(x) satisfy (C2), then for every dI>0, there exists a unique ˉq=ˉq(dI) such that R0>1 for 0<minx[0,L]a(x)<q<ˉq, R0=1 for q=ˉq and R0<1 for q>ˉq.

    Proof. (i) Subtracting both sides of (2.2) by β(x)φ, multiplying by ea(x)dIφ, we have

    [dIφxxa(x)φx]ea(x)dIφ+[γ(x)β(x)]ea(x)dI=(1R01)β(x)ea(x)dI.

    Integrating it by parts over (0,L), using φx(0)=φx(L)=0, we obtain

    dIL0ea(x)dI(φx)2φ2dx+L0[β(x)γ(x)]ea(x)dIdx=(11R0)L0β(x)ea(x)dIdx.

    Using Lemma 2.2.3(i) with m(x)=[β(x)γ(x)]˜a(x), L0[β(x)γ(x)]ea(x)dIdx>0, and

    (11R0)L0β(x)ea(x)dIφ2dx>0,

    which implies that R0>1.

    (ii) Differentiating both sides of (2.2) with respect to q, denoting the differentiation with respect to q by the dot notation, we obtain

    {dI˙φxx˜a(x)φx˜a(x)˙φx+γ(x)˙φ=˙R0R20β(x)φ+1R0β(x)˙φ,0<x<L,˙φx(0)=˙φx(L)=0. (2.19)

    Multiplying (2.19) by ea(x)dIφ and integrating the resulting equation in (0,L), we have

    dIL0ea(x)dI˙φxφxdxL0ea(x)dIφxφ˜a(x)dx+L0γ(x)ea(x)dI˙φφdx=˙R0R20L0β(x)ea(x)dIφ2dx+1R0L0β(x)ea(x)dI˙φφdx. (2.20)

    Multiplying (2.2) by ea(x)dI˙φ and integrating the resulting equation in (0,L), we get

    dIL0ea(x)dI˙φxφxdx+L0γ(x)ea(x)dI˙φφdx=1R0L0β(x)ea(x)dI˙φφdx. (2.21)

    Subtracting (2.20) and (2.21), we obtain

    R0q=R20L0ea(x)dIφxφ˜a(x)dxL0β(x)ea(x)dIφ2dx. (2.22)

    By the result of Corollary 2.1.1, we know that

    limqR0=β(L)γ(L)<1.

    Meanwhile, we have

    limq0R0=ˆR0>1

    for any dI. Then there must exist at least some ˉq such that R0(ˉq)=1. By Lemma 2.1.1, for any ˉq>0 satisfying R0(ˉq)=1, (ϕ1)x<0 in (0,L). Recalling (2.22), we have

    R0ˉq=L0eˉqdI˜a(x)(ϕ1)xϕ1dxL0β(x)eˉqdI˜a(x)(ϕ1)2dx<0,

    which implies that ˉq is the unique point satisfying R0(ˉq)=1.

    The following lemma will tell us that there exists a function q=ρ1(dI) such that R0(dI,ρ1(dI))=1 and give the asymptotic profile of ρ1(dI) if L0β(x)dx>L0γ(x)dx.

    Lemma 2.3.2. Assume that β(x)γ(x) changes sign once in (0,L), L0β(x)dx>L0γ(x)dx, and θ1 is the unique solution of

    L0[β(x)γ(x)]eθ1˜a(x)dx=0.

    Suppose that β(x) and γ(x) satisfy (C2). Then there exists a function ρ1:(0,)(0,) such that R0(dI,ρ1(dI))=1. And ρ1 satisfies

    limdI0ρ1(dI)=0,limdIρ1(dI)dI=θ1.

    Proof. 1. Let's first consider the limit of ρ1(dI)dI as dI. Assume that ρ1(dI)dI as dI. Under the assumption (C2), by Lemma 2.1.4, we have

    limρ1(dI),ρ1(dI)dIR0(dI,ρ1(dI))=β(L)γ(L)<1,

    which is a contradiction to R0(dI,ρ1(dI))=1.

    Next, we will prove that ρ1(dI)dIθ1 as dI. Here θ1 is the unique positive root of L0[β(x)γ(x)]eθ1˜a(x)dx=0. By the discussions above, we know that ρ1(dI)dI is bounded for large dI. Passing to a subsequence if necessary, we suppose that ρ1(dI)dIθ for some nonnegative number θ as dI. Let ˜φ be the unique normalized eigenfunction of the eigenvalue R0(dI,ρ1(dI))=1. Then

    {dI(eρ1(dI)dI˜a(x)˜φx)x+[γ(x)β(x)]eρ1(dI)dI˜a(x)˜φ=0,0<x<L,˜φx(0)=˜φx(L)=0. (2.23)

    Integrating (2.23) in (0,L), we get

    L0[β(x)γ(x)]eρ1(dI)dI˜a(x)˜φdx=0. (2.24)

    Recalling that, up to a subsequence if necessary, ˜φ1 in C([0,1]) as dI. Letting dI in (2.24), we have

    L0[β(x)γ(x)]eθ˜a(x)dx=0.

    By Lemma 2.2.3 with m(x)=[β(x)γ(x)]˜a(x), F(η) has a unique positive root, i.e., θ=θ1.

    2. Contradictorily, assume that q=ρ1(dI)q>0 or q=ρ1(dI) as dI0. By Lemma 2.1.4, we know that

    limρ1(dI)q,ρ1(dI)dIR0(dI,ρ1(dI))=β(L)γ(L)<1

    or

    limρ1(dI),ρ1(dI)dIR0(dI,ρ1(dI))=β(L)γ(L)<1,

    which is a contradiction to R0(dI,ρ1(dI))=1. Therefore, we have limdI0ρ1(dI)=0.

    To study the properties of R0 when L0β(x)dx<L0γ(x)dx, we need the following results which were stated in [2]:

    Proposition 2.3.1. Assume that β(x)γ(x) changes sign in (0,L).

    (i) ˆR0 is a monotone decreasing function of dI with ˆR0max{β(x)/γ(x):x[0,L]} as dI0 and ˆR0L0β(x)dx/L0γ(x)dx as dI+;

    (ii) ˆR0>1 for all dI>0 if L0β(x)dxL0γ(x)dx;

    (iii) There exists a threshold value dI(0,+) such that ˆR0>1 for dI<dI and ˆR0<1 for dI>dI if L0β(x)dx<L0γ(x)dx.

    Lemma 2.3.3. Assume that β(x)γ(x) changes sign once in (0,L) and L0β(x)dx<L0γ(x)dx. Then there exists some constant dI>0 such that dI is the unique positive root of the equation ^R0(dI)=1 and the following statements hold.

    1. If β(x) and γ(x) satisfy (C1), then

    (i) for dI(0,dI], R0>1 for any q>minx[0,L]a(x)>0;

    (ii) for dI(dI,), there exists a unique ˉq=ˉq(dI) such that R0<1 for any 0<minx[0,L]a(x)<q<ˉq and R0>1 for any q>ˉq.

    2. If β(x) and γ(x) satisfy (C2), then

    (iii) for dI(0,dI], there exists a unique ˉq=ˉq(dI) such that R0>1 for any 0<minx[0,L]a(x)<q<ˉq and R0<1 for any q>ˉq;

    (iv) for dI(dI,), R0<1 for any q>minx[0,L]a(x)>0.

    Proof. (i) Noticing that β(x) and γ(x) satisfy (C1), similar to the proof of (ii) in Lemma 2.1.4, we can prove that there exists a unique ˉq>0 satisfying R0(ˉq)=1 and R0(ˉq)>0. Hence, the conclusion is true for dI(dI,+).

    For dI(0,dI], by the results of Proposition 2.3.1, we have limq0R0=ˆR01. By the results of Corollary 2.1.1, limq+R0=β(L)/γ(L)>1 under the condition (C1). Hence R0>1 for any q>0.

    (ii) The proof of Lemma 2.3.3 under the condition (C2) is similar to that of Lemma 2.1.4, we omit the details here.

    Lemma 2.3.4. Assume that β(x)γ(x) changes sign once in (0,L) and L0β(x)dx<L0γ(x)dx. Then there exists a constant dI>0 such that dI is the unique positive root of the equation ^R0(dI)=1 and the following statements hold.

    1. If β(x) and γ(x) satisfy (C1), then there exists a function ρ2:(dI,)(0,) such that ρ2 is a monotone increasing function of dI and R0(dI,ρ2(dI))=1. Let θ2 be the unique solution of

    L0[β(x)γ(x)]eθ2˜a(x)dx=0.

    Then

    limdIdI+ρ2(dI)=0,limdIρ2(dI)dI=θ2.

    2. If β(x) and γ(x) satisfy (C2), then there exists a function ρ3:(0,dI)(0,) such that R0(dI,ρ3(dI))=1 and

    limdI0+ρ3(dI)=0,limdIdIρ3(dI)dI=0.

    Proof. 1. If we can prove that ρ2(dI)>0 for dI(dI,), then ρ2(dI) is a monotone increasing function of dI. Here the prime notation denotes differentiation by dI. Since R0(dI,ρ2(dI))=1, we can get

    R0qρ2(dI)+R0dI=0. (2.25)

    By Lemma 2.3.1, R0q>0 for R0(dI,ρ2(dI))=1. So we need to prove that R0dI<0.

    Differentiating both sides of (2.2) with respect to dI, denoting the differentiation with respect to dI by the dot notation, we obtain

    {φxxdI˙φxxa(x)˙φx+γ(x)˙φ=˙R0R20β(x)φ+1R0β(x)˙φ,0<x<L,˙φx(0)=˙φx(L)=0. (2.26)

    Multiplying (2.26) by ea(x)dIφ and integrating the resulting equation in (0,L), we obtain

    L0ea(x)dIφxxφdx+dIL0ea(x)dI˙φxφxdx+L0γ(x)ea(x)dI˙φφdx=˙R0R20L0β(x)ea(x)dIφ2dx+1R0L0β(x)ea(x)dI˙φφdx. (2.27)

    Multiplying (2.2) by ea(x)dI˙φ and integrating the resulting equation in (0,L), we get

    dIL0ea(x)dI˙φxφxdx+L0γ(x)ea(x)dI˙φφdx=1R0L0β(x)ea(x)dI˙φφdx. (2.28)

    Subtracting (2.27) and (2.28), we have

    R0dI=R20L0ea(x)dIφxxφdxL0β(x)ea(x)dIφ2dx=R20L0ea(x)dI(φx)2dxL0β(x)ea(x)dIφ2dxR20L0ea(x)dIφxφa(x)dxdIL0β(x)ea(x)dIφ2dx. (2.29)

    By Lemma 2.2.1, for any dI satisfying R0(dI,q)=1, (ϕ1)x>0, we can get

    R0dI=R20L0ea(x)dI[(ϕ1)x]2dxL0β(x)ea(x)dIϕ21dxR20L0ea(x)dI(ϕ1)xϕ1a(x)dxdIL0β(x)ea(x)dIϕ21dx<0. (2.30)

    (2.25) and (2.30) imply that ρ2(dI)>0 for dI(dI,).

    The proof of limdIρ2(dI)dI=θ2(θ2 is the unique solution of L0[β(x)γ(x)]eθ2a(x)dx=0) is similar to the proof of Lemma 2.3.2, we omit the details here.

    Now we will prove that limdIdI+ρ2(dI)=0. Assume that there exists q such that q=ρ2(dI)q as dIdI+. Then there exists a positive function ϕ(x)C2([0,L]) such that

    {dIϕxxq˜a(x)ϕx+γ(x)ϕ=β(x)ϕ,0<x<L,ϕx(0)=ϕx(L)=0. (2.31)

    Noticing that dI is the unique positive root of ˆR0=1 and the definition of ˆR0 implies q=0, there exists a positive function ˆϕ(x)C2([0,L]) such that

    {dIˆϕxx+γ(x)ˆϕ=β(x)ˆϕ,0<x<L,ˆϕx(0)=ˆϕx(L)=0. (2.32)

    Multiplying (2.31) by ˆϕ, (2.32) by ϕ, subtracting the two resulting equations, then integrating by parts over (0,L), we get

    qL0˜a(x)ϕxˆϕdx=0.

    Since ϕx is positive(by Lemma 2.2.1), we have q=0. Therefore, limdIdI+ρ2(dI)=0.

    2. Using the arguments above, similar to the proof of Lemma 2.3.2, we can obtain the conclusions.

    In this section, we will show that: If the disease-free equilibrium is unstable, then we can use the bifurcation analysis and degree theory to study the existence of endemic equilibrium.

    Letting ˜S=ea(x)dSˉS, ˜I=ea(x)dIˉI, we have

    {dSˉSxx+a(x)ˉSxβ(x)ea(x)dIˉSˉIea(x)dSˉS+ea(x)dIˉI+γ(x)e(1dI1dS)a(x)ˉI=0,  0<x<L,dIˉIxx+a(x)ˉIx+β(x)ea(x)dSˉSˉIea(x)dSˉS+ea(x)dIˉIγ(x)ˉI=0,  0<x<L,ˉSx(0)=ˉSx(L)=0,ˉIx(0)=ˉIx(L)=0,  L0[ea(x)dSˉS+ea(x)dIˉI]dx=N. (2.33)

    Since the structure of the solution set of (2.33) is the same as that of (1.3), we study (2.33) instead of (1.3). Denote the unique disease-free equilibrium of (2.33) by (ˆˉS,0)=(NL0ea(x)dS,0). We will consider a branch of positive solutions of (2.33) bifurcating from the branch of semi-trivial solutions given by

    ΓS:={(q,(ˆˉS,0)):0<minx[0,L]a(x)<q<}

    through using the local and global bifurcation theorems. For fixed dS, dI>0, we take q as the bifurcation parameter. Let

    X={uW2,p((0,L)):ux(0)=ux(L)=0},Y=Lp((0,L))

    for p>1 and the set of positive solution of (2.33) to be

    O={(q,(S,I))R+×X×X:q>minx[0,L]a(x)>0,S>0,I>0,(q,(S,I)) satisfies (2.33)}.

    Lemma 2.4.1 Assume that dS, dI>0 and β(x)γ(x) changes sign once in (0,L). Then

    1. q>0 is a bifurcation point for the positive solutions of (2.33) from the semi-trivial branch ΓS if and only if q satisfies R0(dI,q)=1. That is,

    (I) If L0β(x)dx>L0γ(x)dx, then such q exists uniquely for any dI>0 if and only if assumption (C2) holds;

    (II) If L0β(x)dx<L0γ(x)dx, let dI be the unique positive root of ˆR0=1, then such q exists uniquely for any dI>0 if and only if either β(x) and γ(x) satisfy condition (C1) and d>dI or they satisfy condition (C2) and 0<d<dI.

    2. There exits some δ>0 such that all positive solutions of (2.33) near (q,(ˆˉS,0)))R×X×X can be parameterized as

    Γ={(q(τ),(ˆˉS+ˉS1(τ),ˉI1(τ))):τ[0,δ)}, (2.34)

    where (q(τ),(ˆˉS+ˉS1(τ),I1(τ))) is a smooth curve with respect to τ and satisfies q(0)=q, ˆS1(0)=I1(0)=0.

    3. There exists a connected component Σ of ˉO satisfying ΓΣ, and Σ possesses some properties as follows.

    Case (I) Assume that L0β(x)dx>L0γ(x)dx and (C2) holds. Then there exists some endemic equilibrium (ˆS,ˆI) of (2.33) when q=0 such that for Σ, the projection of Σ to the q-axis satisfies ProjqΣ=[0,q] and the connected component Σ connects to (0,(ˆS,ˆI)).

    Case (II) Assume that L0β(x)dx<L0γ(x)dx. Then

    (i) If (C1) holds and dI>dI, then (2.33) has no positive solution for 0<minx[0,L]a(x)<q<q and for Σ, the projection of Σ to the q-axis satisfies ProjqΣ=[q,).

    (ii) If (C2) holds and 0<dI<dI, then there exists some endemic equilibrium (ˆS,ˆI) of (2.33) when q=0 such that for Σ, the projection of Σ to the q-axis satisfies ProjqΣ=[0,q] and the connected component Σ connects to (0,(ˆS,ˆI)).

    Proof. 1. Let F:R+×X×XY×Y×R be the mapping as follows.

    F(q,(ˉS,ˉI))=(dSˉSxx+a(x)ˉSxβ(x)ea(x)dIˉSˉIea(x)dSˉS+ea(x)dIˉI+γ(x)e(1dI1dS)a(x)ˉIdIˉIxx+a(x)ˉIx+β(x)ea(x)dSˉSˉIea(x)dSˉS+ea(x)dIˉIγ(x)ˉIL0[ea(x)dSˉS+ea(x)dIˉI]dxN).

    It is to verify that the pair (ˉS,ˉI) is a solution of (2.33) if only if F(q,(ˉS,ˉI))=0. Obviously, F(q,(ˆˉS,0))=0 for any q>minx[0,L]a(x)>0. The Freˊchet derivatives of at are given by

    If is a nontrivial solution of the following problem

    (2.35)

    then is degenerate solution of (2.33). The second equation of (2.33) has a positive solution only if satisfies . And satisfies

    (2.36)

    Obviously, is uniquely determined by in (2.36). Therefore, is the only possible bifurcation point along where positive solutions of (2.33) bifurcates and such exists if and only if . We can obtain the necessary and sufficient conditions for the occurrence of bifurcation by Lemma 2.3.1 and Lemma 2.3.3.

    2. At , the kernel

    where is the solution of (2.35) with . Up to a multiple of constant, is unique. And the range of is given by

    and it is co-dimension one. By the result of Lemma 2.1.1, keeps one sign in and , which implies that

    Therefore, using the local bifurcation theorem in [20] to at , we know that the set of positive solutions of (2.33) is a smooth curve

    satisfying , and . Similar to the procedure in [21] and [22], (also see [23]), we can compute

    Here is the linear functional on defined by .

    3. By the global bifurcation theorem in [23] and [24], we can get the existence of the connected component . Moreover, is either unbounded, or connects to another , or connects to another point on the boundary of .

    Case (I) Assume that and (C2) holds. By Lemma 2.2.1 and the proof of part 2, we see that there exits a unique such that the local bifurcation occurs at and , which means that the bifurcation direction is subcritical. Therefore, there exists some small such that (2.33) has a positive solution if . By Lemma 2.1.4, if for small enough. By Lemma 2.1.5, (2.33) has no positive solution if , which implies that (2.33) has no positive solution if . Consequently, the projection of to the -axis . And must be bounded in because the positive solutions are uniformly bounded in for . So the third option must happen here. Hence must connect to , so . Here is the unique endemic equilibrium of (2.33) when .

    Case (II) Assume that .

    (i) If (C1) holds and , by Lemma 2.2.1 and the bifurcation analysis above, there exists unique bifurcation point satisfying , which means the bifurcation direction is supercritical. Then there exists some small such that (2.33) has a positive solution if . By Lemma 2.3.3, if for some small enough. By Lemma 2.1.5, (2.33) has no positive solution if , which implies that (2.33) has no positive solution if . So the first option must happen here. If there exists some finite such that , then it contradicts to the fact that all positive solutions are uniformly bounded in for . Consequently, the projection of to the -axis .

    (ii) If (C2) holds and , the proof is similar to that of Case (I), we omit the details here.

    We will give the Leray-Schauder degree argument.

    Lemma 2.4.2. For any , there exist two constants and which depend on , , , and such that if , then for any positive solution of (2.33),

    (2.37)

    for any and .

    Proof. means that and are bounded in space. Using the standard theory of elliptic equation, it is easy to see that and have the upper bound depending on , , , and .

    Therefore, we just need to prove that and have lower bounds.

    Suppose contradictorily that there exist a sequence of satisfies and and , and are the corresponding positive solutions of (2.33) satisfying

    and satisfies

    (2.38)

    Up to a subsequence, we assume that and . Note that are uniformly bounded. Letting , we have

    By standard regularity and Sobolev embedding theorem in [25], up to a subsequence, in and there exists such that in and . Since in and implies that is bounded in , using the equation of , we get in . Letting in the equation of , we have

    (2.39)

    Since , (2.39) means that is the principle eigenvalue, which is a contradiction of the assumption of for any and . Therefore, there must exist some positive constant such that . Similar to the argument in [26], by Harnack inequality, we have

    for some constant depending on , , , and , which implies that has uniformly positive lower bound.

    Now we prove that has a uniform positive lower bound. Let . Using the minimum principle in [27], we have

    Consequently,

    and

    which completes the proof.

    Lemma 2.4.3. Assume that changes sign once in and one of the following conditions holds:

    (i) , , and (C2) holds;

    (ii) , , and (C1) holds.

    Then (2.33) has at least an endemic equilibrium.

    Proof. Note that we can extend the ranges of and properly for any nonnegative pair such that the function is Lipschitz continuous for and . Therefore we define the following compact operator family from to :

    (2.40)

    Since the operator is invertible, then for any and , by the second equation of (2.40), is uniquely determined. Substituting this into the first and last equations of (2.40), is also uniquely determined. Therefore, we can define .

    Under conditions (i) and (ii), for any . Here

    By the result of Lemma 2.4.2, for any , there exist two positive constant and depending on , , , , and such that for any solution of (2.40).

    Let

    Then for any and , which implies that Leray-Schauder degree is well defined, and it is independent of . Here is the identity map. Moreover, is a solution of (2.33) if and only if satisfies . If and , then is a positive solution of

    (2.41)

    By the result of [2], (2.41) has a unique positive solution satisfying if the basic reproduction number . Linearizing (2.41) around , we get

    (2.42)

    Adding the first two equations of (2.42) and using the boundary condition , , we get

    Solving it, we have . Substituting this relation into the first equation of (2.42), we obtain

    Since is a positive solution of (2.40), we know that is a positive operator, so . Hence the unique positive solution is linearly stable. Using Leray-Schauder degree index (see Theorem 1.2.8.1 in [28]), we obtain

    Consequently, using the homotopy invariance of Leray-Schauder degree, we have

    for . By the properties of degree, has a fixed point in if , which implies that (2.33) has at least one positive solution.

    In this section, we consider the properties of when changes sign twice. We also need the results on the positive roots of which is defined as

    for any given continuous function on .

    Lemma 2.5.1. Assume that there exists such that , i.e., change sign twice for . Then

    (i) If and , then has a unique positive root for satisfying ;

    (ii) If and , then has a unique positive root for satisfying ;

    (iii) If and , then has at most two positive roots for ;

    (iv) If and , then has at most two positive roots for .

    Proof. We only prove part (i) and part (iii). The proofs of part (ii) and part (iv) are similar.

    (i). Let and the prime notation denote differentiation with respect to . Since and changes sign twice, it is easy to see that for and for . Note that is increasing. We know that

    for and . As a result, for any , we have

    which implies that is a strictly increasing function for . By Lemma 2.2.2 and , for if is large enough. But , so there exits at least a positive root of . Let be the smallest positive one, then .

    If , since

    then

    That is, is a strict local maximum value point of , which is a contradiction. So . Now we will prove that is the unique positive root of . Assume contradictorily that is the first number such that . Since and , then in , which implies that . By the definition of , and noticing that , we have and , which is a contradiction to the fact that is strictly increasing.

    (iii) By Lemma 2.2.2 and , we see that for if is large enough. Then either for any or has positive roots in . Let and be the first positive root of . Similar to the proof of part (i), it is easy to prove that is strictly monotone increasing in and . We discuss in two cases.

    Case 1: . We will show that is the unique positive root of . Since

    then . That is, attains a strict local minimum at . Now we will prove that is the unique positive root of . Assume contradictorily that is the first number such that . Since is a strict local minimum value point, we have in , which implies that . By the definition of , and noticing that , we have and , which is a contradiction to the fact that is strictly increasing. So only has a unique positive root in this case.

    Case 2. . Since , so if and close to enough. By Lemma 3.2 and , for if is large enough. Therefore, there exists at least a root of in . Assume that is the first root of in . Then in and . If , then

    And attains a strict local minimum at , which is a contradiction. Hence .

    We need to show that there is no positive root of for . Assume contradictorily that there exists such that and in . Then . And and , which contradicts the fact that is strictly increasing. Therefore we have proved that there exists a unique such that and .

    Now we give the proof of Theorem 1.6 below.

    Proof. We only prove part(i) and (iii). The proofs of (ii) and (iv) are similar.

    Part (i): Similar to the proofs of Lemma 2.3.2 and 2.3.3, it is easy to prove that there exists some positive constant which is independent of and and for each , there exists some which satisfies and as . Here is the unique positive root of .

    Next, we will prove that

    for any satisfying if is large enough.

    Let be the unique normalized eigenfunction of the eigenvalue , i.e., and

    (2.43)

    By (2.22), we have

    (2.44)

    Multiplying (2.43) by and integrating it over , we get

    Substitute it into (2.44), we obtain

    As , and , we have

    By Lemma 2.5.1(i),

    Hence, there exists some constant (dependent on ) such that for and for .

    Part (iii). According to the results of Lemma 2.5.1(iii), we divide into three cases to prove it.

    Case 1. for any . It is easy to show that there exists some positive constant independent of and such that for every and any .

    Case 2. has a unique positive root for and . Similar to the proof of part (i), we can prove that there exists some positive constant independent of and such that for every , there exists some such that and as , where is the unique positive root of . Moreover, . Therefore there exists some positive constant which is independent of and such that for every , there exists a constant dependent on satisfying for and for .

    Case 3. has two positive roots and () for and , . Similar to the discussion of part (i), for each , there exist and such that and , as . And

    Consequently, there exist two constants which depend on and satisfy that for , for .

    In this section, we will summarize the main results of this paper.

    Theorem 1.1 gives some properties for the basic reproduction number and Theorem 1.2 says that is the watershed for judging whether the DFE is stable or not. Theorem 1.3 and Theorem 1.4 deal with the stable and unstable regions of the DFE. Theorem 1.5 establishes the existence of EE. Theorem 1.6 considers the results on (1.1) when changes sign twice in .

    We only establish the results on (1.1) under the assumption of in this paper. However, it is much more difficult to obtain the results on (1.1) if there exists some satisfying .

    Biologically, the influence of advection is from the upstream to the downstream, small diffusion or large advection tends to force the individuals to concentrate at the downstream end. Therefore, the disease persists for arbitrary advection rate if the habitat is a high-risk domain and the downstream end is a high-risk site. While the advection transports the individuals to a favorable location and thus it can help eliminate the disease if the downstream end is a low-risk site. In conclusion, when advection is strong or the diffusion is small, the disease will be eliminated if the downstream end is a low–risk site, while the disease will persist if the downstream end is a high–risk site.

    The authors thank the anonymous referees for their helpful suggestions.

    Xiaowei An was supported by Natural Science Foundation of China People's Police University(No.ZKJJPY201723).

    All authors declare no conflicts of interest in this paper.



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