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Research article

A priori estimates of solutions to nonlinear fractional Laplacian equation

  • In this paper, we focus on the research of a priori estimates of several types of semi-linear fractional Laplacian equations with a critical Sobolev exponent. Employing the method of moving planes, we can achieve a priori estimates which are closely connected to the existence of solutions to nonlinear fractional Laplacian equations. Our result can extend a priori estimates of the second order elliptic equation to the fractional Laplacian equation and we believe that the method used here will be applicable to more general nonlocal problems.

    Citation: Tao Zhang, Tingzhi Cheng. A priori estimates of solutions to nonlinear fractional Laplacian equation[J]. Electronic Research Archive, 2023, 31(2): 1119-1133. doi: 10.3934/era.2023056

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  • In this paper, we focus on the research of a priori estimates of several types of semi-linear fractional Laplacian equations with a critical Sobolev exponent. Employing the method of moving planes, we can achieve a priori estimates which are closely connected to the existence of solutions to nonlinear fractional Laplacian equations. Our result can extend a priori estimates of the second order elliptic equation to the fractional Laplacian equation and we believe that the method used here will be applicable to more general nonlocal problems.



    Viral dynamics is a field of applied mathematics which employs mathematical models to describe the changes over time of infected cells and the viral load. Nowak et al. [1] and Nowak and May [2] proposed the following basic viral dynamic model,

    {du(t)dt=sdu(t)βu(t)v(t),dw(t)dt=βu(t)v(t)δw(t),dv(t)dt=Nδw(t)cv(t), (1.1)

    where u(t), w(t), and v(t) are the numbers of uninfected cells, productively infected cells, and virus particles at time t, respectively. See the references for the biological meanings of the parameters. This basic model has been modified to study different viral infections, which include hepatitis C virus (HCV) [3,4], human immunodeficiency virus (HIV) [5,6,7,8], human T-cell leukemia virus (HTLV) [9,10,11], and so on.

    During the process of viral infection, specific immune response plays an important role. Specific immune response includes cell-mediated immunity (which depends on cytotoxic T lymphocytes response (CTLs)) and humoral immunity (which depends on antibody response). Since the work of Nowak and Bangham [12], much has been done on mathematical models on immune response against infected cells [13,14,15,16].

    Nowadays, time delays have been taken into account in order to better understand viral dynamics. Usually, distributed time delays [17,18,19] and discrete time delays [20,21,22] have been incorporated into viral dynamic models. In particular, based on (1.1), Zhu and Zou [20] proposed the following viral dynamic model with time delay and CTL immune response,

    {du(t)dt=sdu(t)βu(t)v(t),dw(t)dt=βemˆτu(tˆτ)v(tˆτ)δw(t)pw(t)z(t),dv(t)dt=Nδw(t)cv(t),dz(t)dt=qw(t)z(t)bz(t), (1.2)

    where z(t) denotes the density of immune effectors at time t. Here the delay ˆτ represents the time from a virus entering a target cell to the production of new free virus particles. We refer to [20] for the meanings of the other parameters.

    Note that both models (1.1) and (1.2) and most existing ones are described by ordinary differential equations. The cells and free virus particles are assumed to be uniform in location. In other words, the effect of spatial heterogeneity is ignored. For example, the lymphoid tissues are among the primary sites of HIV infection and replication. The lymphoid tissues consist of many lymph nodes with different sizes. The different tissue architecture and composition and biophysical parameters can influence the spread and replication of the virus [23]. To understand the viral pathogenesis better, it is necessary to consider the spatial aspects of the tissues. In [24], Wang and Wang proposed the following mathematical model of HBV infection with spatial dependence,

    {u(x,t)t=sdu(x,t)βu(x,t)v(x,t),w(x,t)t=βu(x,t)v(x,t)δw(x,t),v(x,t)t=DΔv(x,t)+Nδw(x,t)cv(x,t), (1.3)

    where u(x,t), w(x,t), and v(x,t) are the densities of uninfected cells, productively infected cells, and free virus particles at spatial position x and time t, respectively. D is the diffusion coefficient and Δ is the Laplace operator. Using the geometric singular perturbation method, they studied the existence of traveling waves. Since then a lot of works have followed in this direction (see, for example, [25,26,27,28,29]).

    In (1.1), (1.2), and (1.3), only the cell-free transmission (newly released free virus particles infect uninfected cells [2]) is considered. Recent experimental studies [30,31] prove that a healthy cell can be infected when it comes with close contact of an infected cell (cell-to-cell transmission [32,33]). Sigal et al. [34] found that the cell-to-cell spread of HIV can still permit ongoing replication even with an antiretroviral therapy. Consequently, viral dynamic models incorporating both transmission modes have been formulated and studied (to name a few, see [35,36,37,38,39]). We should mention that the incidences in these works are bilinear. Incidence is the number of new infections per unit of time. It depends on the infectivity of viruses and behavior of cells. Thus it is reasonable to be nonlinear in general. For example, the saturated incidence rate βuv1+αv is used in [40] and the Beddington-DeAngelis incidence function is used in [41]. In a recent work, Sun and Wang [42] also used a general incidence f(u,v) in a diffusive viral dynamic model.

    Based on the above discussion, in this paper, we propose and study the following delayed diffusive viral dynamic model with cell-mediated immunity, cell-to-cell transmission, and general incidences,

    {u(x,t)t=sdu(x,t)f(u(x,t),v(x,t))g(u(x,t),w(x,t)),w(x,t)t=emτf(u(x,tτ),v(x,tτ))+emτg(u(x,tτ),w(x,tτ))δw(x,t)pw(x,t)z(x,t),v(x,t)t=D1Δv(x,t)+Nδw(x,t)cv(x,t), xΩ, t>0,z(x,t)t=D2Δz(x,t)+qw(x,t)z(x,t)bz(x,t), xΩ, t>0, (1.4)

    where z(x,t) denotes the densities of immune effectors at spatial position x and time t. Ω is a general open bounded domain in Rn with smooth boundary Ω. We consider model (1.4) with the homogeneous Neumann boundary conditions

    vn=0, zn=0, xΩ, t>0, (1.5)

    where n denotes the outward normal derivative on Ω. We also assume the initial conditions

    u(x,θ)=ϕ1(x,θ)0,w(x,θ)=ϕ2(x,θ)0,v(x,θ)=ϕ3(x,θ)0,z(x,θ)=ϕ4(x,θ)0,(x,θ)¯Ω×[τ,0], (1.6)

    where ϕi's (i=1, 2, 3, 4) are bounded and uniformly continuous functions on ¯Ω×[τ,0].

    In (1.4), intracellular delays for both transmission modes are assumed to be the same. In general, the intracellular delay in the cell-to-cell transmission is less than that in the cell-free infection [30,39]. However, the difference is not large enough. As for some existing studies (for example, [35,38]), for simplicity of presentation, we make the above assumption on the delays.

    In (1.4), the incidences due to the cell-free transmission and the cell-to-cell transmission are given by the nonlinear functions f(u,v) and g(u,w), respectively. As in [26], we always make the following assumption on them in the sequel.

    (A1) The nonlinear incidence functions f and g satisfy the following properties.

    (i) f(u,v)0 and g(u,w)0 for u0, v0, and w0, and the equalities hold if and only if uv=0 and uw=0;

    (ii) There exists η1>0 and η2>0 such that f(u,v)η1u and g(u,w)η2u for u0, v0, and w0;

    (iii) f(u,v)u and g(u,w)u are continuous with f(u,v)u>0 and g(u,w)u>0 for u0, v>0, and w>0;

    (iv) f(u,v)v and g(u,w)w are continuous with f(u,v)v0 and g(u,w)w0 for u0, v0, and w0;

    (v) vf(u,v)vf(u,v)0 and wg(u,w)wg(u,w)0 for u0, v0, and w0.

    Note that, by Assumption (A1), for any u>0,

    (f(u,v)v)v=f(u,v)vvf(u,v)v2<0,

    which implies that f(u,v)v is decreasing on (0,). In particular, for any u>0 and v>0,

    f(u,v)vlimv0+f(u,v)v=f(u,0)v.

    An analog also holds for g. Thus we have

    f(u,v)f(u,0)vv and g(u,w)g(u,0)ww for u0, v0, and w0. (1.7)

    The rest of the paper is organized as follows. In section 2, we consider the existence, uniqueness, positivity, and boundedness of solutions to system (1.4)–(1.6). Then we study the existence of homogeneous steady states in section 3, which depend on the basic reproduction number of infection and the basic reproduction number of immunity. The main part is section 4, where we discuss the local and global dynamics of system (1.4)–(1.6) by analyzing the characteristic equations and constructing suitable Lyapunov functionals. These results are supported with numerical simulations in section 5. The paper ends with a brief conclusion.

    Let X:=C(¯Ω,R4) be the Banach space equipped with the supremum norm X. For τ0, define C=C([τ,0],X), which is a Banach space equipped with the norm ϕ=maxθ[τ,0]ϕ(θ)X. If σ>0 and U:[τ,σ)X, then for t[0,σ), UtC is defined by Ut(θ)=U(t+θ) for θ[τ,0]. Denote X+=C(¯Ω,R4+) and C+=C([τ,0],X+). Then both (X,X+) and (C,C+) are strongly ordered spaces. According to Corollary 4 in [43], we have the following result on the well-posedness. The arguments are standard and hence are omitted here. Interested readers can refer to, for example, a recent paper by Gao and Wang [44].

    Theorem 2.1. For each ϕ=(ϕ1,ϕ2,ϕ3,ϕ4)C+, system (1.4)–(1.6) has a unique solution U(,t,ϕ)=(u(,t,ϕ),w(,t,ϕ),v(,t,ϕ),z(,t,ϕ)) on [0,) with U0(,ϕ)=ϕ. Moreover, Ut(,ϕ)C+ for t0 and U(,t,ϕ) is a classical solution.

    Let Φ(t):C+C+ be the solution semiflow associated with (1.4)–(1.6), that is, Φ(t,ϕ)=Ut(,ϕ), where U(,t,ϕ) is the solution of (1.4)–(1.6) with the initial condition ϕC+.

    The following result gives some properties of solutions.

    Lemma 2.2. For ϕC+, the following statements hold for the solution U(,t,ϕ) of (1.4)–(1.6).

    (i) lim suptu(x,t,ϕ)sd, lim suptw(x,t,ϕ)emτ(η1+η2)sdδ, lim suptv(x,t,ϕ)Nemτ(η1+η2)sdc, and lim suptz(x,t,ϕ)emτ(η1+η2)sdmin{δ,b} uniformly for all xΩ.

    (ii) u(,t,ϕ)>0 for t>0 and lim inftu(x,t,ϕ)sd+η1+η2 uniformly for all xΩ.

    (iii) If w(,t0,ϕ)0 or v(,t0,ϕ)0 for some t00, then w(x,t,ϕ)>0 and v(x,t,ϕ)>0 for all xΩ and t>t0+τ.

    (iv) If z(,t0,ϕ)0 for some t00, then z(x,t,ϕ)>0 for all xΩ and t>t0.

    Proof. For simplicity of notation, in the proof here we omit ϕ from the expressions of the solution.

    (ⅰ) First, we have

    u(x,t)tsdu(x,t),

    which implies that lim suptu(x,t)sd uniformly for all xΩ. Next, by Assumption (A1) and the second equation in (1.4), we have

    w(x,t)temτ(η1+η2)u(x,t)δw(x,t).

    Then lim suptw(x,t)emτ(η1+η2)sdδ uniformly for xΩ follows easily from this and lim suptu(x,t)sd uniformly for xΩ. Similarly, adding the second and fourth equations of (1.4) yields

    (w(x,t)+z(x,t))temτ(η1+η2)u(x,t)min{δ,b}(w(x,t)+z(x,t)).

    It follows that lim supt(w(x,t)+z(x,t))emτ(η1+η2)sdmin{δ,b} uniformly for xΩ and hence lim suptz(x,t)emτ(η1+η2)sdmin{δ,b} uniformly for xΩ. Now, lim suptw(x,t)emτ(η1+η2)sdδ uniformly for xΩ together with the third equation of (1.4) (Lemma 1 in [45]), and comparison theorem, gives lim suptv(x,t)Nemτ(η1+η2)sdc uniformly for xΩ.

    (ⅱ) Noting that u(x,t)ts(d+η1+η2)u(x,t), one can easily get

    u(x,t)e(d+η1+η2)tu(x,0)+sd+η1+η2e(d+η1+η2)tsd+η1+η2

    for xΩ and t0. Then (ⅱ) follows immediately.

    (ⅲ) Note that the operator D1ΔcId generates a positive semigroup on C(¯Ω,R), where Id is the identity operator. Thus if w(,t0)0, then from the third equation of (1.4), we see that v(,t)0 for t>t0. Without loss of generality, we assume that v(,t0)0. We first show that v(,t)>0 for t>t0. By Theorem 2.1, v(x,t) satisfies

    {v(x,t)tD1Δv(x,t)cv(x,t),t>t0, xΩ,v(x,t)n=0,t>t0, xΩ.

    Let ˉv(x,t) be the solution of

    {ˉv(x,t)t=D1Δˉv(x,t)cˉv(x,t),t>t0, xΩ,ˉv(x,t)n=0,t>t0, xΩ,ˉv(x,t0)=v(x,t0), x¯Ω.

    Then ˉv(x,t)>0 for xΩ and t>t0. In fact, suppose, by contradiction, there exist x0Ω and ˆt>t0 such that ˉv(x0,ˆt)=0. Then, according to the strong maximum principle [46], ˉv(x,t)0 for each tt0, contradicting with ˉv(,t0)0. Applying the comparison theorem, we know that v(x,t)ˉv(x,t)>0 for t>t0 and xΩ. We now prove that w(x,t)>0 for xΩ and t>t0+τ. Otherwise, there exist ˉxΩ and ˉt>t0+τ such that w(ˉx,ˉt)=0. As w(x,t)0, we have w(ˉx,ˉt)t=0. This is impossible since

    w(ˉx,ˉt)t=emτf(u(ˉx,ˉtτ),v(ˉx,ˉtτ))+emτg(u(ˉx,ˉtτ),w(ˉx,ˉtτ))>0

    by Assumption (A1) (ⅱ) due to u(ˉx,ˉtτ)>0 and v(ˉx,ˉtτ)>0. This proves statement (ⅲ).

    (ⅳ) The proof is similar to that of (ⅲ) on v(x,t)>0 for xΩ and t>t0 and hence we omit it here. This completes the proof.

    Lemma 2.2 tells us that Φ is point dissipative. Then it follows from Theorem 2.1.8 in [47] that Φ(t) is compact for all t>τ. This, together with Theorem 3.4.8 in [48], gives the following result.

    Theorem 2.3. The semiflow Φ has a global compact attractor A in C+. Moreover, u(x,t,ϕ)sd for all x¯Ω, t0, and ϕA.

    System (1.4) with (1.5) always has a unique infection-free steady state P0=(u0,0,0,0), where u0=s/d. Applying the result of Wang and Zhao (Theorem 3.4 in [49]), we can obtain the expression of the basic reproduction number of infection, R0, which is given by

    R0=Nemτcf(sd,0)v+emτδg(sd,0)w.

    Denote

    R01=Nemτcf(sd,0)vandR02=emτδg(sd,0)w.

    Then R01 is the number of secondly infected cells through the cell-free transmission and it is referred to as the basic reproduction number from the cell-free transmission; while R02 is the number of secondly infected cells through the cell-to-cell transmission and it is referred to as the basic reproduction number from the cell-to-cell transmission [38].

    In the following, we discuss the existence of homogeneous steady states for (1.4) (the stability results in section 4 indicate that they are the only possible steady states). Clearly, a homogeneous steady state P=(u,w,v,z) satisfies

    sduf(u,v)g(u,w)=0, (3.1a)
    f(u,v)emτ+g(u,w)emτδwpwz=0, (3.1b)
    Nδwcv=0, (3.1c)
    qwzbz=0. (3.1d)

    It follows from (3.1d) that z=0 (which corresponds to the immunity-free infected steady states) or w=bq (which, when z0, corresponds to the infected-immune steady states).

    We firstly consider the case where z=0. It follows from (3.1c) that v=Nδwc. Multiplying both sides of (3.1b) by emτ and then adding up the resultant and (3.1a) to get u=sδwemτd. It is necessary that w(0,sδemτ). Substituting u=sδwemτd and v=Nδwc into (3.1a), we see that w is a positive zero of H1, where

    H1(w)=f(sδwemτd,Nδwc)+g(sδwemτd,w)δwemτ. (3.2)

    According to Assumption (A1), we have H1(0)=0, H1(sδemτ)s<0, and

    H1(0)=δemτ(Nemτcf(sd,0)v+emτδg(sd,0)w)δemτ=δemτ(R01).

    If R0>1, then H1(0)>0. This, together with H1(0)=0, implies that H1(w) is positive for all sufficiently small w>0. By the Intermediate Value Theorem, H1 has at least one zero in (0,sδemτ) and hence (1.4) has at least one immunity-free infected steady state. In fact, there is only one such steady state by the claim that H1(w1)<0 for any immunity-free infected steady state, which is proved as follows. Note that δemτ=f(u1,v1)w1+g(u1,w1)w1 and w1=cv1Nδ. By Assumption (A1),

    H1(w1)=δemτdf(u1,v1)u+Nδcf(u1,v1)vδemτdg(u1,w1)u+ g(u1,w1)wNδcv1f(u1,v1)1w1g(u1,w1)=δemτdf(u1,v1)uδemτdg(u1,w1)u+ Nδcv1(v1f(u1,v1)vf(u1,v1))+ 1w1(w1g(u1,w1)wg(u1,w1))<0.

    This proves the claim. Next, we assume that R0<1. Then H1(0)=δemτ(R01)<0, which combined with H1(0)=0 implies that H1(w)<0 for w>0 sufficiently small. Using the above claim, we can easily see that there is no immunity-free infected steady state when R0<1. Moreover, H1(w)<0 for w(0,sδemτ]. Finally, we assume that R0=1. We use contradictive arguments to show that there is no immunity-free infected steady state in this case. Otherwise, assume that H1(w) has a positive zero say w. Then from the above claim H1(w)>0 for w<w and closely enough to w. Note that H1(w) depends continuously on the parameters and H1(w)<0 for w(0,sδemτ] when R0<1. Fix w(0,w). Choose a sequence of parameters such that the basic reproduction number R0<1 and tends to 1. Then H1(w) tends to H1(w)>0, a contradiction to the fact that the limit is less than or equal to 0. This proves that there is no immunity-free infected steady state when R0=1.

    Now we study the case where w=bq. This, together with (3.1c), yields v=Nδbcq. As before, add up (3.1a) and (3.1b) multiplied by emτ to get z=sduδemτwpemτw, which necessarily requires u(0,sdδbdqemτ). Substituting w=bq and z=sduδemτwpemτw into (3.1a), we see that u is a positive zero of H2, where

    H2(u)=f(u,Nδbcq)+g(u,bq)s+du. (3.3)

    With Assumption (A1), we have H2(0)=s<0 and

    H2(u)=f(u,Nδbcq)u+g(u,bq)u+d>0.

    Therefore, in order for model (1.4) to have an infected-immune steady state (if exists there is a unique one), it is necessary and sufficient that H2(sdδbdqemτ)=H1(bq)>0. Recall that when R01, H1(w)<0 for w(0,sδemτ]; while when R0>1, H1(w)>0 for w(0,w1) and H1(w)<0 for w(w1,sδemτ). It follows that H1(bq)>0 if and only if R0>1 and bq<w1. Denote,

    R1=qw1b.

    As q is the average number of immune effectors produced from contacting with a productively infected cell and 1b is the average life of an immune effector, it follows that R1 is the total number of immune effectors produced at the immunity-free infected steady state. Thus R1 is called the basic reproduction number of immunity.

    Summarizing the above discussion, we have obtained the following result on the existence of homogeneous steady states.

    Theorem 3.1. For model (1.4) with (1.5), the following statements on the existence of homogeneous steady states are true.

    (i) If R01, then there is only the infection-free steady state P0.

    (ii) If R11<R0, then besides P0, there is also a unique immunity-free infected steady state P1=(u1,w1,v1,0), where w1 is the only positive zero of H1 defined by (3.2), u1=sδw1emτd and v1=Nδw1c.

    (iii) If R1>1 (it is necessary that R0>1), then in addition to P0 and P1, there is also a unique infected-immune steady state P2=(u2,w2,v2,z2), where u2 is the only positive zero of H2 defined by (3.3), w2=bq, v2=Nδbcq, and z2=sdu2δw2emτpw2emτ.

    In the main part of this paper, we establish the stability of each steady state obtained in Theorem 3.1.

    Let P=(u,w,v,z) be an arbitrary homogeneous steady state. The linearization of (1.4) at P is

    Qt=LΔQ+AQ+BQτ, (4.1)

    where

    L=diag(0,0,D1,D2),Q=(u(x,t),w(x,t),v(x,t),z(x,t)),Qτ=(uτ,wτ,vτ,zτ)=(u(x,tτ),w(x,tτ),v(x,tτ),z(x,tτ)),A=(df(u,v)ug(u,w)ug(u,w)wf(u,v)v00δpz0pw0Nδc00qz0qwb),B=(0000(f(u,v)u+g(u,w)u)emτg(u,w)wemτf(u,v)vemτ000000000).

    Denote 0=μ0<μ1<μ2<<μn< to be all the eigenvalues of the operator Δ on Ω with the homogeneous Neumann boundary condition. Then P is locally asymptotically stable if, for any iN={0,1,2,}, every solution of the characteristic equation

    |λE+μiLABeλτ|=0 (4.2)

    has a negative real part and P is unstable if there exists i0N such that (4.2) has a solution with a positive real part.

    We first study the local stability of P0.

    Proposition 4.1. The infection-free steady state P0 of (1.4) is locally asymptotically stable if R0<1 and unstable if R0>1.

    Proof. By (4.2), the characteristic equation at P0 is

    (λ+b+μiD2)(λ+d)[(λ+c+μiD1)(λ+δg(sd,0)we(λ+m)τ)Nδf(sd,0)ve(λ+m)τ]=0.

    Obviously, the stability of P0 is determined by

    (λ+c+μiD1)(λ+δg(sd,0)wemτeλτ)Nδf(sd,0)vemτeλτ=0. (4.3)

    Firstly, suppose that R0<1. We claim that all roots of (4.3) have negative real parts. Otherwise, there exists i0N such that (4.3) has a root λ0 with Re(λ0)0. Then

    1=1λ0+δg(sd,0)wemτeλ0τ+Nδ(λ0+δ)(λ0+c+μi0D1)f(sd,0)vemτeλ0τ.

    It follows that

    1=|1λ0+δg(sd,0)wemτeλ0τ+Nδ(λ0+δ)(λ0+c+μi0D1)f(sd,0)vemτeλ0τ||1λ0+δg(sd,0)wemτeλ0τ|+|Nδ(λ0+δ)(λ0+c+μi0D1)f(sd,0)vemτeλ0τ|1δg(sd,0)wemτ+Nc+μi0D1f(sd,0)vemτ1δg(sd,0)wemτ+Ncf(sd,0)vemτ=R0,

    a contradiction to R0<1. This proves the claim and hence P0 is locally asymptotically stable when R0<1.

    Secondly, assume R0>1. For iN, denote

    F(λ,i)=(λ+c+μiD1)(λ+δg(sd,0)wemτeλτ)Nδf(sd,0)vemτeλτ.

    Recall that μ0=0. We have

    F(0,0)=cδ(1R0)<0

    and

    F(λ,0)=λ2+(c+δ)λ+cδ(cδR0+g(sd,0)wemτ)eλτ as λ.

    By the Intermediate Value Theorem, F(λ,0) has a positive zero and hence (4.3) has at least one positive zero for i=0. This means that P0 is unstable when R0>1.

    In fact, P0 is globally stable if it is locally stable.

    Theorem 4.2. If R01, then the infection-free steady state P0 of (1.4) is globally attractive. In particular, P0 is globally asymptotically stable when R0<1.

    Proof. It suffices to show that P0 is globally attractive in A. For this purpose, we consider the Lyapunov functional

    W(t)=Ω(emτw(x,t)+1Nemτv(x,t)+pqemτz(x,t))dx+Ω(ttτf(u(x,θ),v(x,θ))dθ+tτg(u(x,θ),w(x,θ))dθ)dx.

    Calculating the time derivative of W(t) along solutions of model (1.4), we have

    dW(t)dt=Ω(f(u(x,t),v(x,t))cNemτv(x,t)pbqemτz(x,t))dx+1NemτΩD1Δv(x,t)dx.

    It follows from the homogeneous Neumann boundary condition (1.5) and the Divergence Theorem that

    ΩΔv(x,t)dx=Ωv(x,t)ndx=0.

    Moreover, by Theorem 2.3, u(x,t)sd for xΩ and t0. With the help of (1.7), for v(x,t)>0, we have

    f(u(x,t),v(x,t))cNemτv(x,t)=cNemτv(x,t)(Ncemτf(u(x,t),v(x,t))v(x,t)1)cNemτv(x,t)(Ncemτf(sd,v(x,t))v(x,t)1)cNemτv(x,t)(Ncemτf(sd,0)v1)cNemτv(x,t)(R01).

    The above inequality holds automatically for v(x,t)=0 and also observe that the inequality is strict for u(x,t)<sd and v(x,t)>0. Therefore,

    dW(t)dtΩ(cNemτv(x,t)(R01)pbqemτz(x,t))dx0.

    Moreover, dW(t)dt=0 if and only if v(x,t)=0 and z(x,t)=0. In fact, if v(x0,t0)0, then there exists a neighborhood N(x0,t0) of (x0,t0) such that v(x,t)0 for (x,t)N(x0,t0). Then by the observation, u(x,t)=sd for (x,t)N(x0,t0). This, together with the first equation of (1.4) and Assumption (A1), implies that v(x,t)=0 for (x,t)N(x0,t0), a contradiction. Then it is easy to see that the largest invariant subset of dW(t)dt=0 is {P0}. By LaSalle's Invariance Principle (see Theorem 5.3.1 in [50] or Theorem 3.4.7 in [51]), the infection-free steady state P0 is globally attractive. In particular, this together with Proposition 4.1, tells us that P0 is globally asymptotically stable when R0<1.

    Next we consider the stability of the immunity-free infected steady state P1. For convenience of notations, denote

    f1=f(u1,v1),g1=g(u1,w1),f1u=f(u1,v1)u,f1v=f(u1,v1)v,g1u=g(u1,w1)u,g1w=g(u1,w1)w.

    Theorem 4.3. Suppose R0>1. Then the immunity-free infected steady state P1 of (1.4) is locally asymptotically stable if R1<1 and unstable if R1>1.

    Proof. From (4.2), we know that the characteristic equation at P1 is given by

    (λqw1+b+μiD2)ρi(λ)=0, iN,

    where

    ρi(λ)=(λ+d+f1u+g1u)(λ+δ)(λ+c+μiD1)[(λ+d)(λ+c+μiD1)g1w+(λ+d)Nδf1v]e(λ+m)τ.

    Clearly, the eigenvalue λ=b(R11)μiD2<0 for iN when R1<1 but when R1>1, with i=0, we have a positive eigenvalue λ=b(R11). Thus P1 is unstable if R1>1. Now, we assume that R1<1. Then the stability of P1 is determined by the roots of ρi(λ)=0, which is equivalent to

    1=λ+dλ+d+f1u+g1u(g1wλ+δe(λ+m)τ+Nδf1v(λ+δ)(λ+c+μiD1)e(λ+m)τ). (4.4)

    We claim that all solutions of (4.4) have negative real parts. Otherwise, suppose that there exists i1N such that (4.4) has a solution λ1 with Re(λ1)0. Then

    1=|λ1+dλ1+d+f1u+g1u(g1wλ1+δemτeλ1τ+Nδf1v(λ1+δ)(λ1+c+μi1D1)emτeλ1τ)|<|g1wλ1+δemτeλ1τ|+|Nδf1v(λ1+δ)(λ1+c+μi1D1)emτeλ1τ|<g1wδemτ+Nf1vcemτ. (4.5)

    However, from the steady state Eqs (3.1b) and (3.1c), we have

    g(u1,w1)δw1emτ+Nf(u1,v1)cv1emτ=1.

    This and Assumption (A1) (v) together give us

    g1wδemτ+Nf1vcemτg(u1,w1)δw1emτ+Nf(u1,v1)cv1emτ=1,

    which is a contradiction with (4.5). This proves the claim and hence P1 is locally asymptotically stable when R1<1<R0.

    Before studying the global stability of P1, we establish the persistence of infection.

    From the linearized system at P0 (see (4.1)), we have the following cooperative system for (w,v),

    {w(x,t)t=emτf(u0,0)vv(x,tτ)+emτg(u0,0)ww(x,tτ)δw(x,t),v(x,t)t=D1Δv(x,t)+Nδw(x,t)cv(x,t). (4.6)

    With similar arguments as those for Lemma 3 and Lemma 4 in Lou and Zhao [45], we can obtain the following results.

    Lemma 4.4. There exists a principal eigenvalue ˉλ(u0,τ)ˉλ(P0,τ) of (4.6) associated with a strongly positive eigenvector. Moreover, ˉλ(u0,τ) has the same sign as λ(u0)ˉλ(u0,0).

    Lemma 4.5. R01 and λ(u0) have the same sign.

    Theorem 4.6. Suppose R11<R0. Then the infection is persistent, that is, there exists ε>0 such that

    lim inftu(x,t,ϕ)ε,lim inftw(x,t,ϕ)ε,lim inftv(x,t,ϕ)ε

    uniformly for all x¯Ω, where ϕW1:={ϕC+:w(,0)0 and v(,0)0}.

    Proof. Define

    W1:=C+W1={ϕC+:w(,0)0 or v(,0)0}.

    By Lemma 2.2 and the second equation of (1.4), we know that Φ(t)W1W1 for all t0. Denote

    M:={ϕW1:Φ(t)ϕW1 for t0}.

    Claim 1. ω(ϕ)={(u0,0,0,0)} for ϕM, where ω(ϕ) is the omega limit set of the orbit O+(ϕ):={Φ(t)ϕ:t0}.

    Since ϕM, for all t0, either w(x,t,ϕ)0 or v(x,t,ϕ)0. If w(x,t,ϕ)0 for all t0, then limtv(x,t,ϕ)=0 uniformly for x¯Ω from the third equation of (1.4). Now, suppose that w(x,t1,ϕ)0 for some t10. Then by Lemma 2.2, w(x,t,ϕ)>0 for all tt1+τ and xΩ. Thus v(x,t,ϕ)0 for all tt1+τ. This, combined with the third equation of (1.4), implies that w(x,t,ϕ)0 for x¯Ω and tt1+τ. Then, in either case, limtv(x,t,ϕ)=limtw(x,t,ϕ)=0 uniformly for x¯Ω. Thus u is asymptotic to

    u(x,t)t=sdu(x,t).

    By Corollary 4.3 in [52], we get limtu(x,t,ϕ)=u0 uniformly for x¯Ω. The above discussion tells us that w(x,t,ϕ)0 for all t large enough. Then we can easily see from the fourth equation of (1.4) that limtz(x,t,ϕ)=0 uniformly for x¯Ω. This proves ω(ϕ)={(u0,0,0,0)}.

    Since R11<R0, by Lemma 4.4 and Lemma 4.5, there exists a sufficiently small ε0>0 such that the following linear system

    {w(x,t)t=emτ(f(u0,0)vε0)v(x,tτ)+emτ(g(u0,0)wε0)w(x,tτ)(δ+pε0)w(x,t),v(x,t)t=D1Δv(x,t)+Nδw(x,t)cv(x,t)

    has a positive principal eigenvalue ˉλ(u0ε0) with positive eigenfunction (wε0,vε0). By the continuity in Assumption (A1), there exists δ0(0,ε0] such that

    f(u,v)vf(u0,0)vε0andg(u,w)wg(u0,0)wε0

    for all u0δ0uu0+δ0, 0vδ0, and 0wδ0.

    Claim 2. {(u0,0,0,0)} is a uniform weak repeller for W1 in the sense that

    lim suptΦ(t)ϕ(u0,0,0,0)δ0 for ϕW1.

    Suppose, by contradiction, there exists ϕ1W1 such that lim suptΦ(t)ϕ1(u0,0,0,0)<δ0. Then there exists t2>0 such that u(x,t,ϕ1)>u0δ0u0ε0, w(x,t,ϕ1)δ0, and v(x,t,ϕ1)δ0 for tt2 and x¯Ω. It follows from Assumption (A1) and the choice of δ0 that w and v satisfy

    {w(x,t)temτ(f(u0,0)vε0)v(x,tτ)+emτ(g(u0,0)wε0)w(x,tτ)(δ+pε0)w(x,t),tt2, xΩ,v(x,t)t=D1Δv(x,t)+Nδw(x,t)cv(x,t),tt2, xΩ,v(x,t)n=0,tt2,xΩ. (4.7)

    Due to w(x,t,ϕ1)>0 and v(x,t,ϕ1)>0 for t>0 and xΩ, there exists κ1>0 such that (w(x,t2+θ,ϕ1),v(x,t2+θ,ϕ1))κ1eˉλ(u0ε0)(t2+θ)(wε0(x),vε0(x)) for all xΩ and θ[τ,0]. Then it follows from the comparison principle that w(x,t,ϕ1)κ1eˉλ(u0ε0)twε0(x) and v(x,t,ϕ1)κ1eˉλ(u0ε0)tvε0(x) for all xΩ and tt2, a contradiction to the fact that both w(x,t,ϕ1) and v(x,t,ϕ1) are bounded. This proves Claim 2.

    Define a continuous function P1:C+[0,) by

    P1(ϕ):=min{minx¯Ωϕ2(x,0), minx¯Ωϕ3(x,0)} for all ϕC+.

    Clearly, P11(0,)W1, and P1 has the property that if P1(ϕ)=0 and ϕW1 or P1(ϕ)>0, then P1(Φ(t)ϕ)>0 for all t>0. Hence, P1 is a generalized distance function for the semiflow Φ(t) [53]. According to the above discussions, we obtain that any forward orbit of Φ(t) in M converges to (u0,0,0,0), which is isolated in C+ and Ws(u0,0,0,0)W1=, where Ws(u0,0,0,0) is the stable manifold of (u0,0,0,0). Moreover, there is no cycle in W1 from (u0,0,0,0) to (u0,0,0,0). Applying Theorem 3 in [53], we know that there exists an ˉε>0 such that min{P1(ϕ)}>ˉε for any ϕW1. It follows that

    lim inftw(x,t)ˉε and lim inftv(x,t)ˉε uniformly for all x¯Ω.

    This combined with Lemma 2.2 finishes the proof with ε=min{ˉε,sd+η1+η2}.

    In order to study the global stability of P1, define G:(0,)xx1lnx. Obviously, G(x)>0 for x(0,) and G attains its global minimum only at x=1. We also need the following assumption.

    (A2) The nonlinear incidence functions f(u,v) and g(u,w) satisfy the following conditions.

    (i) For any u>0,

    {vv1u1f(u,v)uf(u1,v1)<1if  0<v<v1,1u1f(u,v)uf(u1,v1)<vv1if  v1<v.

    (ii) For any u>0,

    {ww1u1g(u,w)ug(u1,w1)<1if  0<w<w1,1u1g(u,w)ug(u1,w1)<ww1if  w1<w.

    Theorem 4.7. Suppose that R11<R0 and Assumption (A2) are satisfied. Then the immunity-free infected steady state P1 is globally attractive in

    C+1={ϕC+|there exists t3R+ such that w(,t3,ϕ)0 or v(,t3,ϕ)0}.

    In particular, it is globally asymptotically stable in C+1 if further R1<1.

    Proof. According to Lemma 2.2 and Theorem 4.6, we know that there exists ε>0 such that lim inftu(x,t,ϕ)ε, lim inftw(x,t,ϕ)ε, and lim inftv(x,t,ϕ)ε for ϕC+1. Without loss of generality, we define a Lyapunov functional

    L(t)=ΩL(x,t)dx,

    where

    L(x,t)=u1G(u(x,t)u1)+emτw1G(w(x,t)w1)+f(u1,v1)cv1v1G(v(x,t)v1)+pqemτz(x,t)+f(u1,v1)ttτG(f(u(x,θ),v(x,θ))f(u1,v1))dθ+g(u1,w1)ttτG(g(u(x,θ),w(x,θ))g(u1,w1))dθ.

    Calculating the time derivative of L(x,t) along solutions of (1.4) yields

    L(x,t)t=(1u1u(x,t))[sdu(x,t)f(u,v)g(u,w)]+pqemτ[qw(x,t)z(x,t)bz(x,t)]+emτ(1w1w(x,t))[emτf(uτ,vτ)+emτg(uτ,wτ)δw(x,t)pw(x,t)z(x,t)]+f(u1,v1)cv1(1v1v(x,t))[D1Δv(x,t)+Nδw(x,t)cv(x,t)]+pqemτD2Δz(x,t)+f(u,v)f(uτ,vτ)+f(u1,v1)lnf(uτ,vτ)f(u,v)+g(u,w)g(uτ,wτ)+g(u1,w1)lng(uτ,wτ)g(u,w).

    With

    we have

    Clearly,

    and

    Using the Divergence Theorem and the homogeneous Neumann boundary conditions of (1.5), we have

    and

    The latter gives

    and hence

    To summarize, we have obtained

    Note that the monotonicity of on each side of and Assumption (A2) give us

    Thus

    Moreover, if and only if , , , and . Then the largest invariant subset of is . By LaSalle's Invariance Principle (see Theorem 5.3.1 in [50] or Theorem 3.4.7 in [51]), the immunity-free infected steady state is globally attractive in when . This, together with Theorem 4.3, implies that is globally asymptotically stable in if further .

    For convenience of notations, denote

    Theorem 4.8. If , then the infected-immune steady state is locally asymptotically stable.

    Proof. According to (4.2), the characteristic equation at is

    (4.8)

    We claim that all roots of (4.8) have negative real parts. Otherwise, suppose for some , it has a root with . Since , we have

    which implies

    With similar arguments as those in the proof of Theorem 4.3, we can obtain

    Thus we have arrived at

    which is impossible as one can check that .

    This completes the proof.

    To establish the global stability of , we need the persistence of immunity.

    From the linearized system at (see (4.1)), we have the following cooperative system for ,

    (4.9)

    With similar arguments as those for Lemma 3 and Lemma 4 in Lou and Zhao [45], we can obtain the following results.

    Lemma 4.9. There exists a principal eigenvalue of (4.9) associated with a strongly positive eigenvector. Moreover, has the same sign as .

    Lemma 4.10. and have the same sign.

    Theorem 4.11. Suppose that (it is necessary that ) and (A2) holds. Then the immunity is persistent, that is, there exists such that

    uniformly for all , where .

    Proof. The proof is quite similar to that of Theorem 4.6. Denote

    Set and .

    According to Lemma 2.2, we know that , , and for all and , , which implies that for all . Define

    Claim 3. Let . Then or .

    Sine , for any , we have either , or , or . If for some , then by Lemma 2.2, for and . Then either or for each . By the proof of Claim 1, we know that . Now, suppose that for all . If for each , either or , then by Claim 1, . If there exists such that and . Then by Theorem 4.6, there exists such that

    Now consider the reduced system of (1.4) with . Modifying the Lyapunov functional in the proof of Theorem 4.7 by ignoring the term in , we can show that the solution of the reduced system converges to and hence . This proves Claim 3.

    Claim 4. Both and are uniform weak repellers for . Since , by Claim 2, is a uniform repeller for . The proof of being a uniform repeller of is similar as that of Claim 2 by using Lemma 4.9 and Lemma 4.10. Therefore, we omit the detail here.

    Define a continuous function by

    It is easy to see that , and has the property that if and or , then for all . Thus is a generalized distance function for the semiflow . As and are repellers, we know that both and are isolated, and for and . Moreover, no subset of forms a cycle in . By Smith and Zhao [53,Theorem 3], there exists a such that for any . Then as for Theorem 4.6, with finishes the proof.

    As for the global stability of , we make the following assumption to establish the global stability of .

    The nonlinear incidence functions and satisfy the following conditions.

    For any ,

    For any ,

    Theorem 4.12. Suppose that and Assumptions (A2) and (A3) are satisfied. Then the infected-immune steady state is globally asymptotically stable in

    Proof. It follows from Lemma 2.2 and Theorem 4.11 that there exists an such that

    unfiormly in and . Without loss of generality, we define a Lyapunov functional

    where

    Calculate the time derivative of along the solutions of (1.4) to get

    With the following relations,

    we get

    Then

    Similarly as in the proof of Theorem 4.7, we can show

    Therefore, we have . Moreover, if and only if , , , . Then the largest invariant subset of is . By LaSalle's Invariance Principle (see Theorem 5.3.1 in [50] or Theorem 3.4.7 in [51]), the infected-immune steady state is globally attractive in . This, together with Theorem 4.8, implies the global asymptotic stability of in .

    In this section, we perform some numerical simulations to illustrate the results obtained in section 4. Let and . One can easily verify that and satisfy (A1)–(A3). Then the model (1.4) becomes

    (5.1)

    subject to the homogeneous Neumann boundary conditions

    For (5.1), the basic reproduction number of infection is given by

    and the basic reproduction number of immunity is given by

    where

    For simulations, we take , , , and the values of the other parameters are summarized in Table 1. Moreover, and the initial condition used is

    Table 1.  Parameter values for simulation.
    Parameters Ranges value Units References
    10 [54]
    0.01 [22]
    variable [55]
    [39]
    0.05 [55]
    0.5 [55]
    1 [22]
    0.024 [22,55]
    2000 [22]
    23 [39,22]
    0.15 [22]
    0.5 [39,22]

     | Show Table
    DownLoad: CSV

    for and .

    Firstly, we take . Then . By Theorem 4.2, the infection-free steady state is globally asymptotically stable (see Figure 1).

    Figure 1.  When , the infection-free steady state is globally asymptotically stable. Parameter values are given in the text.

    Next, we choose . Then and . From Theorem 4.7, the immunity-free infected steady state is globally asymptotically stable (see Figure 2).

    Figure 2.  When and , the immunity-free infected steady state is globally asymptotically stable. See the text for the parameter values.

    Finally, with , we get and . By Theorem 4.12, the infected-immune steady state is globally asymptotically stable (see Figure 3).

    Figure 3.  When , the infected-immune steady state is globally asymptotically stable. See the text for the parameter values.

    In this paper, we have proposed and studied a reaction-diffusion virus infection model by incorporating time delays, general incidence functions, and cell-to-cell transmission.

    We have proved that the global dynamics of system (1.4)–(1.6) is determined by the basic reproduction number of infection and the basic reproduction number of immunity . By analyzing the characteristic equations and constructing Lyapunov functionals, we have obtained the following conclusions: if , then the infected-free steady state is globally asymptotically stable; if , then the immunity-free infected steady state is globally asymptotically stable under additional Assumption (A2); if , then the infected-immune steady state is globally asymptotically stable under additional Assumptions (A2) and (A3). We mention that most commonly used incidences satisfy (A1)–(A3). Some examples are the Holling type Ⅱ incidence [40], Beddington-DeAnglis incidence [41], and [56].

    Chen is supported by NSERC of Canada. Wang is supported by the NSFC (No. 11771374), the Nanhu Scholars Program for Young Scholars of Xinyang Normal University.

    The authors declare that there is no conflict of interest.



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