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Research article Special Issues

Complex network near-synchronization for non-identical predator-prey systems

  • In this paper, we analyze the properties of a complex network of predator-prey systems, modeling the ecological dynamics of interacting species living in a fragmented environment. We consider non-identical instances of a Lotka-Volterra model with Holling type II functional response, which undergoes a Hopf bifurcation, and focus on the possible synchronization of distinct local behaviours. We prove an original result for the near-synchronization of non-identical systems, which shows how to and to what extent an extinction dynamic can be driven to a persistence equilibrium. Our theoretical statements are illustrated by appropriate numerical simulations.

    Citation: Guillaume Cantin, Cristiana J. Silva. Complex network near-synchronization for non-identical predator-prey systems[J]. AIMS Mathematics, 2022, 7(11): 19975-19997. doi: 10.3934/math.20221093

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  • In this paper, we analyze the properties of a complex network of predator-prey systems, modeling the ecological dynamics of interacting species living in a fragmented environment. We consider non-identical instances of a Lotka-Volterra model with Holling type II functional response, which undergoes a Hopf bifurcation, and focus on the possible synchronization of distinct local behaviours. We prove an original result for the near-synchronization of non-identical systems, which shows how to and to what extent an extinction dynamic can be driven to a persistence equilibrium. Our theoretical statements are illustrated by appropriate numerical simulations.



    Since a basic within-host viral infection model introduced by Nowak et al. [1], the dynamics of viral infection such as hepatitis B virus (HBV), hepatitis C virus (HCV) and human immunodeficiency virus (HIV) infection models have been widely studied by incorporating various biological factors. Consider age as a continuous variable, writing the production rate of viral particles and the death rate of productively infected cells as two continuous functions of age, Nelson et al. [2] studied a HIV infection model with infection-age, the model is described as follows:

    {dT(t)dt=Λμ1T(t)βT(t)V(t),(t+a)i(t,a)=δ(a)i(t,a),dV(t)dt=0p(a)i(t,a)daμ2V(t) (1.1)

    with the boundary and initial condition

    {i(t,0)=βT(t)V(t),T(0)=T0>0,  V(0)=V0>0  and  i(0,a)=i0(a)L1+(0,), (1.2)

    where T(t) and V(t) denote the densities of uninfected target cell and free viruses at time t, respectively; i(t,a) denote the density of infected cells at time t with infection-age a. The parameters of model (1.1) are biologically explained in Table 1.

    Table 1.  Parameters and their biological meaning in model (1.1). All these parameters are assumed to be positive.
    Parameter Interpretation
    Λ Constant recruitment rate;
    β Virus infection rate;
    μ1 Mortality rate of uninfected target cell;
    μ2 Mortality rate of free viruses;
    δ(a) Mortality rate of infected cell with age a;
    p(a) Production rate of viral particles.

     | Show Table
    DownLoad: CSV

    Nelson et al. analyzed the local stability of the model by evaluating eigenvalues and its related characteristic equation. In [3], Rong et al. extended the model with combination antiretroviral therapy, and analyzed the local stability of the model. Huang et al. [4] have been further investigated the global stability of the model (1.1) with (1.2) by using Lyapunov direct method and LaSalle invariance principle. For some recent works on viral models with age structure, we refer readers to the papers[5,6,7,8,9,10,11,12,13].

    Recently, experimental work [14] shows that direct cell-to-cell transmission also contributes to the viral persistence. In a more recent work [15], the authors reveals that environmental restrictions limit infection by cell-free virions but promote cell-associated HIV-1 transmission. In fact, cell-to-cell transmission could be also found in other viral infection for human and animals. For example, hepatitis C virus [16]; bovine viral diarrhea virus [17]; vaccinia virus [18]. Due to this fact, Lai and Zou [19] formulated a HIV-1 viral model with direct cell-to-cell transmission and studied the global threshold dynamics. Yang et al. [20] studied a cell-to-cell virus model with three distributed delays, they also obtained the global stability of each equilibrium for the model. Wang et al. [21] investigated an age-structured HIV model with virus-to-cell infection and cell-to-cell transmission, the model takes the following form:

    {dT(t)dt=Λμ1T(t)βT(t)V(t)0k(a)T(t)i(t,a)da,(t+a)i(t,a)=δ(a)i(t,a),dV(t)dt=0p(a)i(t,a)daμ2V(t), (1.3)

    with the boundary and initial condition

    {i(t,0)=βT(t)V(t)+0k(a)T(t)i(t,a)da,T(0)=T0>0,  V(0)=V0>0  and  i(0,a)=i0(a)L1+(0,). (1.4)

    By constructing suitable Lyapunov functional, Wang et al. were able to complete a global analysis for the model (1.3). In [22], Zhang and Liu studied the Hopf bifurcation of an age-structured HIV model with cell-to-cell transmission and logistic growth.

    In viral infection, the host immune system play a critical part on the progress of the infection. The role of the immune system is to fight off pathogenic organisms within the host, for example, cytotoxic T lymphocyte cells (CTLs) attack infected cells, and antibody cells attack viruses (humoral immunity response). In [23], Murase et al. studied an viral infection model with humoral immunity response:

    {dT(t)dt=Λμ1T(t)βT(t)V(t),dI(t)dt=βT(t)V(t)aI(t),dV(t)dt=arI(t)μ2V(t)kV(t)Z(t),dZ(t)dt=hV(t)Z(t)μ3Z(t), (1.5)

    where T(t), I(t), V(t) and Z(t) denote the densities of uninfected cells, infected cells, free viruses and humoral immunity response released by B cells, respectively; the viruses are removed at rate kZ by the humoral immunity response; the humoral immunity response are activated in proportion to hV(t) and removed at rate μ3. The global dynamics of model (1.5) were obtain in [23]. Consider the delay between viral entry into a cell and the maturation delay of the newly produced viruses, Wang et al. [24] studied a virus model with two delays and humoral immunity response. They established the global dynamics based on two threshold parameters, and they found that the three equilibria are globally asymptotically stable under some conditions. For another delay, which is the time that antigenic stimulation needs for generating immunity response, Wang et al. [25] considered another virus model with delay and humoral immunity response, they found that this delay could lead to a Hopf bifurcation at the infected equilibrium with immunity. In [26], Kajiwara et al. proposed a age-structured viral infection model contains humoral immunity response and the effect of absorption of pathogens into uninfected cells, they also proved the global stability of each equilibria. Duan and Yuan [30] considered an infection-age viral model with saturation humoral immune response, the local and global stability of this model are obtained. Additionally, for the virus model with CTL immune response, we refer readers to the papers [27,28,29,31,32,33,34] and the reference therein.

    Based on the above facts, we propose an age-structured viral infection model with cell-to-cell transmission and general humoral immune response in this paper. Precisely, we study the following model:

    {dT(t)dt=Λμ1T(t)βT(t)V(t)0k(a)T(t)i(t,a)da,(t+a)i(t,a)=δ(a)i(t,a),dV(t)dt=0p(a)i(t,a)daμ2V(t)qV(t)f(Z(t)),dZ(t)dt=cV(t)f(Z(t))μ3Z(t) (1.6)

    with the boundary and initial condition

    {i(t,0)=βT(t)V(t)+0k(a)T(t)i(t,a)da,T(0)=T0>0,  V(0)=V0>0,  Z(0)=Z0>0  and  i(0,a)=i0(a)L1+(0,), (1.7)

    where L1+ is the set of integrable functions from (0,+) into [0,+). T(t), V(t) and Z(t) denote the densities of uninfected target cell, free viruses and antibody responses released from B cells at time t, respectively; i(t,a) denotes the density of infected cells at time t with infection-age a; k(a) denote the infection rate of productively infected cells with age a; qV(t)f(Z(t)) is the neutralization rate of viruses and cV(t)f(Z(t)) is the activation rate of antibody responses. The antibody responses vanish at rate μ3. Other parameters of model (1.6) have the same biological meaning in the Table 1.

    We made the following assumption on parameters and nonlinear function f:RR.

    (A1) k(a), δ(a), θ(a), p(a), c(a)L+(0,), with respective essential supremums ˉk, ˉδ,ˉθ, ˉp, ˉc and respective essential infimums ˜k, ˜δ, ˜θ, ˜p, ˜c.

    (A2) f(Z)0 for Z0, f(Z)=0 if and only if Z=0; f is Lipschitz continuous on R+.

    (A3) f(Z) is differentiable such that f(Z)>0 and f(Z) is concave down on R+.

    Here are some examples on function f(Z) satisfies (A2) and (A3):

    (ⅰ) f(Z(t))=Z(t) which is the bilinear function (see [23]);

    (ⅱ) Saturation immune response function f(Z(t))=Z(t)h+Z(t) (see [30]).

    The paper is organized as follows. In Section 2, we introduce the existence and uniqueness of the solutions to system (1.6), the steady state and reproduction numbers are also determined in this section; In Section 3, we show that system (1.6) is asymptotically smooth; Section 4 is devoted to proving the local stability of each steady state; uniform persistence and global stability of each steady state is considered in Section 5; We perform a numerical simulation of a special case in Section 6; Section 7 provide some brief discussions.

    In this section, we show the existence and uniqueness of the solutions to system (1.6) by a standard method [36] (see also [37,38]), which is to rewrite system (1.6) as an abstract Cauchy problem.

    For convenience, we first denote the following notations.

    Γ(a)=ea0δ(τ)dτ,  P=0p(a)Γ(a)da,  K=0k(a)Γ(a)da.

    It is easy to see that

    Γ(0)=1   and   Γ(a)=δ(a)Γ(a).

    Set the following spaces:

    X:=R×L1(R+,R)×R×R, X+:=R+×L1+(R+,R)×R+×R+,

    with the following norm

    φ(),ϕ1,ϕ2,ϕ3X=φL1+|ϕ1|+|ϕ2|+|ϕ3|,

    Furthermore, define

    X0:={0}×L1(R+,R)×R×R×R, X0+:={0}×L1+(R+,R)×R×R+×R+,

    Let A:Dom(A)XX be the following linear operator:

    A((0φ)ϕ1ϕ2ϕ3)=((φ(0)φδφ)μ1ϕ1μ2ϕ2μ3ϕ3) (2.1)

    with Dom(A)=R×{0}×W1,1(0,+)×R×R. In the following, we apply the method in [36] since ¯Dom(A)=X0 is not dense in X. Consider the nonlinear map F:Dom(A)X defined by

    F((0φ)ϕ1ϕ2ϕ3)=((βϕ1ϕ2+0k(a)ϕ1φ(a)da0)Λβϕ1ϕ20k(a)ϕ1φ(a)da0p(a)φ(a)daqϕ2f(ϕ3)cϕ2f(ϕ3)).

    One can see that F is Lipschitz continuous on bounded sets. Let

    u(t)=(T(t),(0i(t,)),V(t),Z(t))T,

    where T represents transposition. Then we can rewrite system (1.6) as the following abstract Cauchy problem:

    {du(t)dt=Au(t)+F(u(t)),  t0,u(0)=u0X0+. (2.2)

    In order to use the method in [36], we need to show that A is a Hille-Yosida operator. Denote ρ(A) be the resolvent set of A. The definition of Hille-Yosida operator is:

    Definition 2.1. (See [36,Definition 2.4.1]) A linear operator A:Dom(A)XX on a Banach space (X,) (densely defined or not) is called a Hille-Yosida operator if there exist real constants M1, and ωR, such that (ω,+)ρ(A), and

    (λA)nM(λω)n,    for  nN+   and all  λ>ω.

    Now, we prove the following lemma.

    Lemma 2.1. The operator A defined in (2.1) is a Hille-Yosida operator.

    Proof. Let

    (λIA)1((ˆφ0ˆφ(a))ˆϕ1ˆϕ2ˆϕ3)=((0φ)ϕ1ϕ2ϕ3),

    by some simple calculations, we have

    ϕ1=^ϕ1λ+μ1,  ϕ2=^ϕ2λ+μ2,  ϕ3=^ϕ3λ+μ3

    and

    φ(a)=ˆφ0ea0(λ+δ(s))ds+0ˆφ(τ)eaτ(λ+δ(s))dsdτ.

    Denote ζ=((ˆφ0ˆφ(a)),ˆϕ1,ˆϕ2,ˆϕ3)T, then

    (λIA)1ζX=|0|+0φ(a)da+|ϕ1|+|ϕ2|+|ϕ3|=0φ(a)da+|ˆϕ1||λ+μ1|+|ˆϕ2||λ+μ2|+|ˆϕ3||λ+μ3||ˆφ0||λ+μ|+ˆφ(a)L1|λ+μ|+|ˆϕ1||λ+μ|+|ˆϕ2||λ+μ|+|ˆϕ3||λ+μ|=1λ+μζX.

    where μ=min{μ1,μ2,μ3,˜δ}. By the Definition 2.1, the operator A is a Hille-Yosida operator. This ends the proof.

    Let X0=(T0,(0i0),V0,Z0)TX0+, by using [36,Theorem 5.2.7] (see also in [37,39]), we have the following theorem.

    Theorem 2.1. There exists a uniquely determined semi-flow {U(t)}t0 on X0+ such that for each X0, there exists a unique continuous map UC([0,+),X0+) which is an integrated solution of Cauchy problem (2.2), that is

    {t0U(s)X0dsDom(A),  t0,U(t)X0=X0+At0U(s)X0ds+0F(U(s)X0)ds,  t0. (2.3)

    Let

    Ω={(T,(0,i()),V,Z)X0+ | T(t)++0i(t,a)daΛμ0, V(t)+qcZ(t)Λˉpμ0ˆμ}, (2.4)

    where μ0=min{μ1,˜δ} and ˆμ=min{μ2,μ3}. We show that Ω is a positively invariant set under semi-flow {U(t)}t0.

    Theorem 2.2. Ω is a positively invariant set under semi-flow {U(t)}t0. Moreover the semi-flow {U(t)}t0 is point dissipative and Ω attracts all positive solutions of (2.2) in X0+.

    Proof. Integrating the second equation of (1.6) along the characteristic line ta=constant, yields

    i(t,a)={i(ta,0)Γ(a),  t>a>0,i0(at)Γ(a)Γ(at),  a>t>0. (2.5)

    Then

    0i(t,a)da=t0i(ta,0)Γ(a)da+ti0(at)Γ(a)Γ(at)da=t0i(σ,0)Γ(tσ)dσ+0i0(a)Γ(t+a)Γ(a)da.

    Note that Γ(0)=1 and Γ(a)=δ(a)Γ(a), thus

    ddt0i(t,a)da=tt0i(σ,0)Γ(tσ)dσ+ddt0i0(a)Γ(t+a)Γ(a)da=i(t,0)0δ(a)i(t,a)da.

    One has that

    ddt(T(t)++0i(t,a)da)=Λμ1T(t)0δ(a)i(t,a)daΛμ0(T(t)0δ(a)i(t,a)da).

    We have

    lim supt{T(t)+0i(t,a)da}Λμ0,  t0.

    From the third and forth equations of (1.6), it is easy to check

    lim supt(V(t)+qcZ(t))Λˉpμ0ˆμ,  t0.

    Hence

    U(t)X0X+Π,

    where Π=Λμ0(1+ˉpˆμ+cˉPqˆμ). Therefore, for any solution of (2.2) satisfying X0Ω and U(t)X0Ω for all t0, Ω is a positively invariant set under semi-flow {U(t)}t0. Moreover the semi-flow {U(t)}t0 is point dissipative and Ω attracts all positive solutions of (2.2) in X0+.

    In this subsection, we concern with the existence of steady states for system (1.6). Obviously, the system (1.6) always has a virus-free steady state E0=(T0,0,0,0)=(Λμ1,0,0,0). E0 is the unique equilibrium if 01, where 0=βT0Pμ2+T0K is the basic reproduction number of system (1.6). If 0>1, there exists an immune-inactivated infection steady state E1=(T1,i1(a),V1,0), which is the same situation in [21], that is,

    T1=T00,   i1(a)=Λ(110)Γ(a),   V1=1μ20p(a)i1(a)da. (2.6)

    There also exists another immune-activated infection steady state E2=(T2,i2(a),V2,Z2), which is satisfies

    {Λμ1T2=i2(0)=βT2V2+0k(a)T2i2(a)da,di2(a)da=δ(a)i2(a),0p(a)i2(a)daϖ(Z2)V2=0,cV2f(Z2)μ3Z2=0, (2.7)

    where

    ϖ(Z2)=μ2+qf(Z2).

    By some calculations, we have

    i2(a)=i2(0)Γ(a). (2.8)

    From the third equation of (2.7), yields

    V2=0p(a)i2(a)daϖ(Z2). (2.9)

    Substituting (2.8) and (2.9) into the first equation of (2.7) one has that

    T2=ϖ(Z2)βP+ϖ(Z2)K (2.10)

    and

    i2(0)=Λμ1T2=Λϖ(Z2)βP+ϖ(Z2)K. (2.11)

    In the following, we show that Z2>0. In fact, combining the last two equations of (2.7) give us

    0p(a)(Λμ1ϖ(Z2)βP+ϖ(Z2)K)Γ(a)daμ3Z2ϖ(Z2)cf(Z2)=0. (2.12)

    Denote

    Φ(Z)=0p(a)(Λμ1ϖ(Z2)βP+ϖ(Z2)K)Γ(a)daμ3Z2ϖ(Z2)cf(Z2)

    and

    1:=ΛcPf(0)μ2μ3(110).

    Then

    limZ20Φ(Z2)>01>1.

    It is easy to check dΦ(Z)dZ<0 and limZ+Φ(Z). Hence, there is only one positive root for (2.12) if 1>1. By the expressions of 0 and 1, we have 1>00>1, then there is the following theorem on the existence of steady states.

    Theorem 2.3. For system (1.6), there are two threshold parameters 0 and 1 such that

    (ⅰ) if 01, there exists only one positive steady state E0;

    (ⅱ) if 1<1<0, there exists two positive steady states E0 and E1;

    (ⅲ) if 1>1, there exists three positive steady states E0, E1 and E2.

    The following lemma on immune-inactivated infection steady state and immune-activated infection steady state will be used in the proof of global stability.

    Lemma 2.2. The immune-inactivated infection steady state (T1,i1(a),V1,0) satisfies

    0[βT1μ2p(a)i1(a)(1i1(0)T(t)V(t)i(t,0)T1V1)+T1k(a)i1(a)(1i1(0)T(t)i(t,a)i1(t,0)T1i1(a))]da=0, (2.13)

    and immune-activated infection steady state (T2,i2(a),V2,Z2) satisfies

    0[βT2μ2+qf(Z2)p(a)i2(a)(1i2(0)T(t)V(t)i(t,0)T2V2)+T2k(a)i2(a)(1i2(0)T(t)i(t,a)i2(t,0)T2i2(a))]da=0. (2.14)

    Proof. For the immune-inactivated infection steady state (T1,i1(a),V1,0), it follows from the third equation of (2.6), we have

    0βT1μ2p(a)i1(a)i1(0)T(t)V(t)i(t,0)T1V1da= βi1(0)T(t)V(t)i(t,0)μ2V10p(a)i1(a)da= βT(t)V(t)i(0)i(t,0).

    Recall that i(t,0)=βT(t)V(t)+0k(a)T(t)i(t,a)da in (1.7), hence

     0βT1μp(a)i1(a)i1(0)T(t)V(t)i(t,0)T1V1da+0T1k(a)i1(a)i1(0)T(t)i(t,a)ii(t,0)T1i1(a)da= βT(t)V(t)i1(0)i(t,0)+T(t)0k(a)i(t,a)dai1(0)i(t,0)= i1(0)= βT1V1+0k(a)T1i1(a)da,

    Thus, (2.13) holds true. The proof of Eq (2.14) is similar to (2.13), so we omitted it. This ends the proof.

    In this section, we show that the semi-flow {U(t)}t0 is asymptotically smooth. Since the state space X0+ is the infinite dimensional Banach space, we need the semi-flow {U(t)}t0 is asymptotically smooth to proof the global stability of each steady states. Rewrite U:=Φ+Ψ, where

    Φ(t)X0:=(0,ϖ1(,t),0,0), (3.1)
    Ψ(t)X0:=(T(t),ϖ2(,t),V(t),Z(t)), (3.2)

    with

    ϖ1(,t)={0,         t>a0,i(t,a),   at0,   and   ϖ2(,t)={i(t,a),   t>a0,0,         at0.

    We are now in the position to prove the following theorem.

    Theorem 3.1. For any X0Ω, {U(t)X0:t0} has compact closure in X if the following two conditions hold:

    (ⅰ) There exists a function Δ:R+×R+R+ such that for any r>0, limtΔ(t,r)=0, and if X0Ω with X0Xr, then Φ(t)X0XΔ(t,r) for t0;

    (ⅱ) For t0, Ψ(t)X0 maps any bounded sets of Ω into sets with compact closure in X.

    Proof. Step Ⅰ, to show that (ⅰ) holds.

    Let Δ(t,r):=e˜δtr, it is obvious that limtΔ(t,r)=0. Then for X0Ω satisfying X0Xr, we have

    Φ(t)X0X=|0|+0|ϖ1(a,t)da|+|0|+|0|=t|i0(at)Γ(a)Γ(at)|da=0|i0(s)Γ(s+t)Γ(s)|dse˜δt0|i0(s)|dse˜δtX0XΔ(t,r),  t0.

    This completes the proof of condition (ⅰ).

    Step Ⅱ, to show that (ⅱ) holds.

    We just have to show that ϖ2(t,a) remains in a precompact subset of L1+(0,). In order to prove it, we should show the following conditions hold [40,Theorem B.2]:

    (a) The supremum of 0ϖ2(t,a)da with respect to X0Ω is finite;

    (b) limuuϖ2(t,a)da=0 uniformly with respect to X0Ω;

    (c) limu0+0(ϖ2(t,a+u)ϖ2(t,a)da=0 uniformly with respect to X0Ω;

    (d) limu0+uϖ2(t,a)da=0 uniformly with respect to X0Ω.

    Conditions (a), (b) and (d) hold since ϖ2(t,a)(βˉpˆμ+ˉk)Λ2μ20. Next, we verify condition (c). For sufficiently small u(0,t), set K(t)=0k(a)i(t,a)da, we have

    0|ϖ2(t,a+u)ϖ2(t,a)|da=tu0|(βT(tau)V(tau)+K(tau)T(tau))Γ(a+u)(βT(ta)V(ta)+K(ta)T(ta))Γ(a)|da+ttu|0βT(ta)V(ta)+K(ta)T(ta))Γ(a)|datu0(βT(tau)V(tau)+K(tau)T(tau))|Γ(a+u)Γ(a)|da+tu0|βT(tau)V(tau)βT(ta)V(ta)|Γ(a)da+tu0|K(tau)T(tau)K(ta)T(ta)|Γ(a)da+u(Λμ0)2(βˉpˆμ+˜k).

    Since Γ(a) is non-increasing function with respect to a and 0Γ(a)1, we have

    tu0|Γ(a+u)Γ(a)|da=tu0(Γ(a)Γ(a+u))da=tu0Γ(a)datuΓ(a)datu0Γ(a)da+utuΓ(a)dau.

    Then

    0|ϖ2(t,a+u)ϖ2(t,a)|da2u(Λμ0)2(βˉpˆμ+ˉk)+Ξ,

    where

    Ξ=tu0|βT(tau)V(tau)βT(ta)V(ta)|Γ(a)da+tu0|K(tau)T(tau)K(ta)T(ta)|Γ(a)da.

    Thanks to the argument in [41,Proposition 6], T()V() and T()K() are Lipschitz on R+. Let M1 and M2 be the Lipschitz coefficients of T()V() and T()K() respectively. Then

    Ξ(βM1+M2)utu0Γ(a)da(βM1+M2)utu0Γ(a)dau(βM1+M2)˜δ.

    Hence

    0|ϖ2(t,a+u)ϖ2(t,a)|da2u(Λμ0)2(βˉpμ2+ˉk)+u(βM1+M2)˜δ,

    which converges to 0 as u0+, the condition (c) holds. Let YX be a bounded closed set and B be a bound for Y, where BA. It is easy to check M1 and M2 are only depend on A, that is M1 and M2 are independent on X. Consequently, ϖ2(t,a) remains in a precompact subset Y of L+1(0,+) and thus

    Ψ(t,Y)[0,B]×Y×[0,B]×[0,B],

    which has compact closure in X. The proof is completed.

    In this section, we show the local stability of system (1.6) at each steady states.

    Theorem 4.1. If 0<1, then the virus-free steady state E0 of system (1.6) is locally asymptotically stable.

    Proof. Denote ˉT1(t)=T(t)T0, ˉi1(t,a)=i(t,a), ˉV1(t)=V(t) and ˉZ1(t)=Z(t), the linearized equation of (1.6) at E0 as follows:

    {dˉT1(t)dt=μ1ˉT1(t)βT0ˉV1(t)0T0k(a)ˉi1(t,a)da,(t+a)ˉi1(t,a)=δ(a)ˉi1(t,a),dˉV1(t)dt=0p(a)ˉi1(t,a)daμ2ˉV1(t),dˉZ1(t)dt=μ3ˉZ1(t),ˉi1(t,0)=βT0ˉV1(t)+0T0k(a)ˉi1(t,a)da. (4.1)

    Let the solution of (4.1) has the following exponential form:

    ˉT1(t)=ˉT1eλt,  ˉV1(t)=ˉV1eλt,  ˉZ1(t)=ˉZ1eλt  and  ˉi1(t,a)=ˉi1(a)eλt,

    then

    {λˉT1=μ1ˉT1βT0ˉV10T0k(a)ˉi1(a)da,λˉi1(a)+dˉi1(a)da=δ(a)ˉi1(a),λˉV1=0p(a)ˉi1(a)daμ2ˉV1,λˉZ1=μ3ˉZ1,ˉi1(0)=βT0ˉV1+0T0k(a)ˉi1(a)da. (4.2)

    Solve the second equation of (4.2) yields

    ˉi1(a)=ˉi1(0)eλaΓ(a).

    We can write the characteristic equation as following

    |λ+μ1T00k(a)eλaΓ(a)daβT001T00k(a)eλaΓ(a)daβT000p(a)eλaΓ(a)daλ+μ2|=Δ(λ)(λ+μ1)=0,

    where

    Δ(λ):=λ+μ2λT00k(a)eλaΓ(a)daμ2T00k(a)eλaΓ(a)daβT00p(a)eλaΓ(a)da.

    Since λ=μ1<0, then we only need to consider the root of Δ(λ)=0. By way of contradiction, we assume that it has an eigenvalue λ0 with Re(λ0)0. We have

    |λ+μ2|= |(λ+μ2)T00k(a)eλaΓ(a)da+βT00p(a)eλaΓ(a)da| |λ+μ2||T00k(a)eλaΓ(a)da+βT00p(a)eλaΓ(a)daλ+μ2| |λ+μ2|(T00k(a)Γ(a)da+βT00p(a)Γ(a)daμ2).

    Hence,

    T00k(a)Γ(a)da+βT00p(a)Γ(a)daμ21,

    which is impossible because 0=βT0Pμ2+T0K<1. This completes the proof.

    Theorem 4.2. If 1<1<0, then the immune-inactivated steady state E1 of system (1.6) is locally asymptotically stable.

    Proof. Denote ˉT2(t)=T(t)T1, ˉi2(t,a)=i(t,a)i1(a), ˉV1(t)=V(t)V1 and ˉZ2(t)=Z(t), the linearized equation of (1.6) at E1 as follows:

    {dˉT2(t)dt=βT1ˉV2(t)0T1k(a)ˉi2(t,a)da(βV1+μ1+0i1(a)k(a)da)ˉT2(t),(t+a)ˉi2(t,a)=δ(a)ˉi1(t,a),dˉV2(t)dt=0p(a)ˉi2(t,a)daμ2ˉV2(t)qf(0)V1ˉZ2(t),dˉZ2(t)dt=cf(0)V1ˉZ2(t)μ3ˉZ2(t),ˉi2(t,0)=βT1ˉV2(t)+βV1ˉT2(t)+0T1k(a)ˉi2(t,a)da+0i1(a)k(a)ˉT2(t)da. (4.3)

    Let ˉT2(t)=ˉT2eλt, ˉV2(t)=ˉV2eλt, ˉZ2(t)=ˉZ2eλt and ˉi2(t,a)=ˉi2(a)eλt, thus we have the following characteristic equation:

    0= (λcf(0)V1+μ3)(λ+μ1)(λ+μ2)(1T10k(a)eλaΓ(a)da) (λcf(0)V1+μ3)(λ+μ1)βT10p(a)eλaΓ(a)da +(λcf(0)V1+μ3)(λ+μ2)(βV1+0k(a)i1(a)da).

    Note that 1<1 and using (2.6), we can obtain that μ3cf(0)V1>0, then the characteristic equation is equivalent to

    0=(λ+μ1)[(λ+μ2)(1T10k(a)eλaΓ(a)da)βT10p(a)eλaΓ(a)da]+(λ+μ2)(βV1+0k(a)i1(a)da). (4.4)

    By way of contradiction, we assume that it has an eigenvalue λ0 with Re(λ0)0. Obviously, λ=μ1 and λ=μ2 are not the roots of (4.4) and note that

    i1(0)= T10k(a)i1(a)da+βT1μ20p(a)i1(a)da= T10k(a)i1(0)Γ(a)da+βT1μ20p(a)i1(0)Γ(a)da.

    Then we have

    |1+βV1+0k(a)i1(a)daλ0+μ1|=|T10k(a)eλ0aΓ(a)da+1λ0+μ2βT10p(a)eλ0aΓ(a)da||T10k(a)Γ(a)da+βT1μ20p(a)Γ(a)da|=1,

    which is impossible since V1>0 and i1(a)>0. Accordingly, the immune-inactivated steady state E1 of system (1.6) is local asymptotically stable if 1<1<0.

    Theorem 4.3. If 1>1, then the immune-activated steady state E2 of system (1.6) is locally asymptotically stable.

    Proof. Applying similar method in the proof of the Theorem (4.2), we have the characteristic equation as following:

    (λ+μ1)[(λ+μ2+qf(Z2))(λcV2f(Z2)+μ3)+qcV2f(Z2)f(Z2)](10T2k(a)eλΓ(a)da)+(λ+μ1)(λcV2f(Z2)+μ3)βT20p(a)eλaΓ(a)da=[(λ+μ2+qf(Z2))(λcV2f(Z2)+μ3)+qcV2f(Z2)f(Z2)](βV2+0k(a)i2(a)da). (4.5)

    By way of contradiction, we assume that it has an eigenvalue λ0 with Re(λ0)0. From the concavity of function f in (A2), we have μ3cV2f(Z2)0 since μ3Z2cV2f(Z2)=0. Note that λ=μ1, λ=(μ2+qf(Z2)) and λ=cV2f(Z2)μ3 are not the roots of (4.5), then we can rewrite (4.5) as

    1+Ξ=0T2k(a)eλaΓ(a)da. (4.6)

    where

    Ξ=(10T2k(a)eλaΓ(a)da)qcV2f(Z2)f(Z2)(λ+μ2+qf(Z2))(λcV2f(Z2)+μ3)+βT20p(a)eλaΓ(a)daλ+μ2+qf(Z2)+βV2+0k(a)i2(a)daλ+μ1+qcV2f(Z2)f(Z2)(βV2+0k(a)i2(a)da)(λ+μ2+qf(Z2))(λcV2f(Z2)+μ3).

    Then

    |1+Ξ|=|0T2k(a)eλaΓ(a)da|<|1i2(0)(0i2(0)T2k(a)eλaΓ(a)da+βT2μ2+qf(Z2)0i2(0)p(a)Γ(a)da)|1,

    which is contradictory. Here we use the fact:

    10T2k(a)eλaΓ(a)da=1i2(0)(βT2V2+0T2k(a)[1eλa]i2(a))>0.

    This completes the proof.

    In this section, we discuss the global stability of system (1.6) by using Lyapunov direct method and LaSalle invariance principle. We first give the result on uniform persistence.

    Theorem 5.1. Assume that 0>1. Then there exists a constant ζ>0 such that

    lim inft+T(t)ζ,  lim inft+i(,t)L1ζ,  lim inft+V(t)ζ

    for each X0X.

    The proof of Theorem 5.1 is similar with that in [21,Section 4] or [30], so we omit the details. In the following, we proof the global stability of each steady states.

    Theorem 5.2. The virus-free steady state E0 of system (1.6) is globally asymptotically stable if 0<1.

    Proof. Let

    α1(a):=a(βT0μ2p(ϵ)+T0k(ϵ))eϵaδ(s)dsdϵ; (5.1)
    g(x):=x1lnx. (5.2)

    By direct calculations, we have α1(0)=0 and α1(a)=δ(a)α(a)(βT0μ2p(a)+T0k(a)). Define the following Lyapunov functional:

    H(t)=H1(t)+H2(t),

    where

    H1(t)=T0g(T(t)T0)+βT0μ2V(t)+qβT0cμ2Z(t), (5.3)
    H2(t)=0α1(a)i(t,a)da. (5.4)

    Calculating the derivatives of H1(t) and H2(t) along system (1.6), we have

    dH1(t)dt=(1T0T(t))dT(t)dt+βT0μ2dV(t)dt+qβT0cμ2dZ(t)dt=μ1T0(2T0T(t)T(t)T0)i(t,0)qβT0μ3cμ2Z(t)+0k(a)T0i(t,a)da+βT0μ20p(a)i(t,a)da,

    and

    dH2(t)dt=0α1(a)i(t,a)tda=0α1(a)i(t,a)ada0α1(a)δ(a)i(t,a)da=0i(t,0)0(βT0μ2p(a)+T0k(a))i(t,a)da,

    thus

    dH(t)dt=μ1T0(2T0T(t)T(t)T0)+(01)i(t,0)qβT0μ3cμ2Z(t)0 (5.5)

    if 0<1. Note that dH(t)dt|(1.6)=0 implies that T(t)=T0, i(t,0)=0 and Z(t)=0, then the largest invariant subset of {dH(t)dt|(1.6)=0} is {E0}. Therefore, the virus-free steady state E0 of system (1.6) is global asymptotically stable if 0<1 by Lyapunov-LaSalle theorem. This ends the proof.

    Theorem 5.3. The immune-inactivated steady state E1=(T1,i1(a),V1,0) of system (1.6) is globally asymptotically stable if 1<1<0.

    Proof. Let

    α2(a):=a(βT1μ2p(ϵ)+T1k(ϵ))eϵaδ(s)dsdϵ. (5.6)

    Define the following Lyapunov functional:

    W(t):=W1(t)+W2(t)+W3(t),

    where

    W1(t):=T1g(T(t)T1); (5.7)
    W2(t):=0α2(a)i1(a)g(i(t,a)i1(a))da; (5.8)
    W3(t):=βT1μ2V1g(V(t)V1)+qβT1cμ2Z(t). (5.9)

    The derivative of W1(t) is calculated as follows:

    dW1(t)dt=(1T1T(t))dT1(t)dt=(1T1T(t))(Λμ1T(t)βT(t)V(t)0k(a)T(t)i(t,a)da)=μ1T1(2T1T(t)T(t)T1)+(i1(0)i(t,0))(1T1T(t)).

    Note that

    i1(a)dda(i(t,a)i1(a)1lni(t,a)i1(a))=(1i1(a)i(t,a))i(t,a)a+δ(a)i(t,a)(1i1(a)i(t,a)), (5.10)

    which leads to

    0α2(a)(1i1(a)i(t,a))i(t,a)ada=α2(a)i1(a)(i(t,a)i1(a)1lni(t,a)i1(a))|a=a=0+0α(a)δ(a)[i1(a)i(t,a))]da0(i(t,a)i1(a)1lni(t,a)i1(a))(dα2(a)dai1(a)+α2(a)i1(a)a)da=limaα2(a)i(a)g(i(t,a)i(a))α2(0)i1(0)g(i(t,0)i(0))+0α2(a)δ(a)[i1(a)i(t,a))]da0g(i(t,a)i(a))(dα2(a)dai1(a)+α2(a)di1(a)da)da.

    By some calculations, we have

    α2(0)=1,   α2(a)=δ(a)α2(a)(βT1μ2p(a)+T1k(a)),

    and using the fact that

    di1(a)da=δ(a)i1(a).

    Hence

    0α2(a)(1i1(a)i(t,a))i(t,a)ada=limaα2(a)i(a)g(i(t,a)i(a))i1(0)g(i(t,0)i(0))+0α2(a)δ(a)[i1(a)i(t,a))]da+0(T1k(a)i(a)+βT1μ2p(a)i(a))g(i(t,a)i(a))da.

    Then we have the derivative of W2(t) as follows:

    dW2(t)dt=0α(a)(1i(a)i(t,a))i(t,a)tda=0α(a)(1i(a)i(t,a))(i(t,a)a+δ(a)i(t,a))da=limaα(a)i(a)g(i(t,a)i(a))+i(0)g(i(t,0)i(0))0T1k(a)i(a)g(i(t,a)i(a))da0βT1μ2p(a)i(a)g(i(t,a)i(a))da.

    For W3(t), we have

    dW3(t)dt=βT1μ2(1V1V(t))dV(t)dt+qβT1cμ2dZ(t)dt=βT1μ20p(a)i(t,a)daβT1μ2μ2V(t)βT1μ2V1V(t)0p(a)i(t,a)da+βT1μ2μ2V1+qβT1μ2V1f(Z(t))qβT1cμ2μ3Z(t).

    Hence,

    dW(t)dt=μ1T1(2T1T(t)T(t)T1)βT1V1T1T(t)0k(a)T1i1(a)T1T(t)da+0k(a)T1i(t,a)dalimaα(a)i1(a)g(i(t,a)i1(a))i1(0)lni(t,0)i1(0)0p(a)i1(a)g(i(t,a)i1(a))da+βT1μ20p(a)i(t,a)daβT1μ2V1V(t)0p(a)i(t,a)da+βT1μ2μ2V1+qβT1μ2V1f(Z(t))qβT1cμ2μ3Z(t). (5.11)

    Recalling that V1=0p(a)μ2i1(a)da and i1(0)=βT1V1+0k(a)T1i1(a)da. Substituting (2.13) into (5.11), after some calculations and rearranging the equation yield

    dW(t)dt=μ1T1(2T1T(t)T(t)T1)limaα(a)i1(a)g(i(t,a)i1(a))+0βT1μ2p(a)i1(a)(2T1T(t)V1i(t,a)V(t)i1(a)lni(t,0)i1(0)+lni(t,a)i1(a))da+0T1k(a)i1(a)(1T1T(t)lni(t,0)i1(0)+lni(t,a)i1(a))da+0βT1μ2p(a)i1(a)(1i1(0)T(t)V(t)i(t,0)T1V1)da+0T1k(a)i1(a)(1i1(0)T(t)i(t,a)i(t,0)T1i1(a))da=μ1T1(2T1T(t)T(t)T1)limaα(a)i1(a)g(i(t,a)i1(a))0βT1μ2p(a)i(a){g(T1T(t))+g(V1i(t,a)V(t)i1(a))+g(i1(0)T(t)V(t)i(t,0)T1V1)}da0T1k(a)i1(a){g(T1T(t))+g(i1(0)T(t)i(t,a)i(t,0)T1i1(a))}da+qβT1μ2V1f(Z(t))qβT1cμ2μ3Z(t).

    Note that

    qβT1μ2V1f(Z(t))qβT1cμ2μ3Z(t)qβT1cμ2[ΛcPf(0)μ2μ3(110)1]=qβT1cμ2(11).

    Thus dW(t)dt|(1.6)0 when 1<1<0, dW(t)dt=0 if and only if (T(t),i(t,a),V(t),Z(t))=(T1,i1(a),V1,0). Applying Lyapunov-LaSalle theorem, the immune-inactivated steady state E1=(T1,i1(a),V1,0) of system (1.6) is globally asymptotically stable if 1<1<0.

    Theorem 5.4. The immune-activated steady state E2=(T2,i2(a),V2,Z2) of system (1.6) is globally asymptotically stable if 1>1.

    Proof. Let

    α3(a):=a(βT2ϖ(Z2)p(ϵ)+T2k(ϵ))eϵaδ(s)dsdϵ. (5.12)

    Define the Lyapunov functional as follows

    L(t):=L1(t)+L2(t)+L3(t)+L4(t),

    where

    L1(t):=T2g(T(t)T2);L2(t):=0α3(a)i2(a)g(i(t,a)i2(a))da;L3(t):=βT2μ2V2g(V(t)V2)+qβT2cμ2(Z(t)Z2Z(t)Z2f(Z2)f(τ)dτ).

    Using the results in the proof of Theorem (5.3), after some calculations, we have the derivative of L(t) as follows:

    dL(t)dt=μ1T2(2T2T(t)T(t)T2)limaα3(a)i2(a)g(i(t,a)i2(a))0βT2ϖ(Z2)p(a)i2(a){g(T2T(t))+g(V2i(t,a)V(t)i2(a))+g(i2(0)T(t)V(t)i(t,0)T2V2)}da0T2k(a)i2(a){g(T2T(t))+g(i2(0)T(t)i(t,a)i(t,0)T2i2(a))}da+qβT2V2f(Z2)ϖ(Z2)(Z(t)Z2f(Z(t))f(Z2))(f(Z2)f(Z(t))1).

    It follows from follows (A2) and (A3) that (Z(t)Z2f(Z(t))f(Z2))(f(Z2)f(Z(t))1)0, thus dL(t)dt|(1.6)0 and dL(t)dt=0 if and only if (T(t),i(t,a),V(t),Z(t))=(T2,i2(a),V2,Z2). Therefore, the immune-activated steady state E2 of system (1.6) is global asymptotically stable if 1>1 by Lyapunov-LaSalle theorem.

    In this subsection, as special case for the age-infection model (1.6) and (1.7) with general nonlinear immune response f(Z), we introduce the following age-infection model with saturation immune response function, which have been used for modeling HIV infection in [30,42].

    {dT(t)dt=Λμ1T(t)βT(t)V(t)0k(a)T(t)i(t,a)da,(t+a)i(t,a)=δ(a)i(t,a),dV(t)dt=0p(a)i(t,a)daμ2V(t)qV(t)Z(t)h+Z(t),dZ(t)dt=cV(t)Z(t)h+Z(t)μ3Z(t) (6.1)

    with the boundary and initial condition

    {i(t,0)=βT(t)V(t)+0k(a)T(t)i(t,a)da,T(0)=T0>0,  V(0)=V0>0,  Z(0)=Z0>0  and  i(0,a)=i0(a)L1+(0,). (6.2)

    The model without cell-to-cell transmission of system (6.1) with (6.2) has been studied in [30]. It is easy to see that f(Z)=Z(t)h+Z(t) satisfy (A2) and (A3). System (6.1) with (6.2) is a special case of the original system (1.6) and (1.7).

    The virus-free equilibrium of system (6.1) with (6.2) is similar to the previous one, E01=(T0,0,0,0), where T0=Λμ1. By some calculation, we obtain the basic reproduction number and immune-activated reproduction number of system (6.1) with (6.2) as 01=βT0Pμ2+T0K and 11=ΛcPhμ2μ3(110), respectively. We have the following corollary:

    Corollary 6.1. For system (6.1) with (6.2), there are two threshold parameters 01 and 11 with 01>11 such that

    (ⅰ) If 01<1, there exists a virus-free steady state E01, and E01 is globally asymptotically stable;

    (ⅱ) If 11<1<01, there exists a immune-inactivated steady state E11 which is globally asymptotically stable;

    (ⅲ) If 11>1, there exists a immune-activated steady state E21 which is globally asymptotically stable.

    In this subsection, we perform some numerical simulations to the validity of the theoretical result of this paper. Specifically, we focus on the age-infection model with saturation immune response function (see model (6.1)).

    The parameter values will be used in numerical simulation are listed in Table 2. Furthermore, we set the maximum age for the viral production as a=10 and we set

    δ(a)=0.03(1+sin(a5)π10),   p(a)=2.9(1+sin(a5)π10)
    Table 2.  Parameter values for numerical simulations.
    Parameter Value Unite Case 1 Case 2 Case 3 Ref.
    Λ 0100 cells ml-day1 1.2 8 100 [44]
    β 5×1070.5 ml virion-day1 0.001 0.001 0.001 [42]
    μ1 0.0070.1 day 1 0.01 0.01 0.01 [44]
    μ2 2.43 day 1 6 6 6 [44]
    μ3 0.3 day 1 0.3 0.3 0.3 [45]
    q 0.006 ml cell1 day 1 0.006 0.006 0.006 [45]
    c 0.1 ml virion1day 1 0.1 0.1 0.1 [45]
    h 1100 Saturation constant 10 10 100 Assumed

     | Show Table
    DownLoad: CSV

    and

    k(a)=0.0003(1+sin(a5)π10).

    Thus, the averages of δ(a), p(a) and k(a) are 0.03, 2.9 and 0.0003, which are the same in [43,20].

    Numerical simulation shows the following three cases:

    Case 1: Choose parameter values as in Case 1 of Table 2, then we can calculate the basic reproduction number as 00.8093<1. Corollary 1 asserts that the virus-free steady state of system (6.1) with (6.2) is globally asymptotically stable. From Figure 1, one can observe that the levels of all compartmental individuals tend to stable values, where T(t), V(t), i(t,a) and Z(t) converge to a virus-free steady states (100, 0, 0, 0).

    Figure 1.  The long time dynamical behaviors of system (6.1) with (6.2) for 0=0.8093<1, that is, the virus-free steady state of system (6.1) with (6.2) is globally asymptotically stable.

    Case 2: Choose parameter values as in Case 2 of Table 2. By some computing, we can obtain that 05.3952>1>10.9040. From Corollary 1, we derive that immune-inactivated infection steady state is globally asymptotically stable. Numerical simulation illustrates this fact (see Figure 2).

    Figure 2.  The long time dynamical behaviors of system (6.1) with (6.2) for 05.3952 and 10.9040, that is, the immune-activated steady state of system (6.1) with (6.2) is globally asymptotically stable.

    Case 3: Choose parameter values as in Case 3 of Table 2. Similarly, we can obtain that 029.9893>1 and 11.3408>1. Numerical simulation shows that the levels of all compartmental individuals tend to stable values (see Figure 3), that is, immune-inactivated infection steady state is globally asymptotically stable.

    Figure 3.  The long time dynamical behaviors of system (6.1) with (6.2) for 029.9893>1 and 11.3408>1, that is, the immune-activated steady state of system (6.1) with (6.2) is globally asymptotically stable.

    In this paper, we proposed and investigated an age-structured within-host viral infection model with cell-to-cell transmission and general humoral immunity response. We have shown that the global stability of equilibria of model (1.6) are determined by the corresponding basic reproduction numbers 0 and the basic immune reproductive number 1. That is, when 0<1, the virus-free steady state is globally asymptotically stable, which means that the viruses are cleared and immune response is not active; when 1<1<0, the immune-inactivated infection steady state exists and is globally asymptotically stable, which means that viral infection becomes chronic and humoral immune response would not be activated; and when 1>1, the immune-activated infection steady state exists and is globally asymptotically stable, in this case the infection causes a persistent humoral immune response and is chronic.

    Now, we show the relevance of model formulations between our age-structured model (1.6) and the standard ODE models. We consider δ(a)δ, k(a)k and p(a)p in model (1.6). Letting

    I(t)=0i(t,a)da.

    Recall that

    i(t,0)=βT(t)V(t)+kT(t)I(t),

    then we have

    dI(t)dt=0i(t,a)tda= 0(i(t,a)a+δi(t,a))da= i(t,0)0δi(t,a)da= βT(t)V(t)+kT(t)I(t)δI(t),

    here we assume that limai(t,a)=0, which means that there is no biological individual can live forever. Thus, system (1.6) is equivalent to the following ODE model as

    {dT(t)dt=Λμ1T(t)βT(t)V(t)kT(t)I(t),dI(t)dt=βT(t)V(t)+kT(t)I(t)δI(t),dV(t)dt=pI(t)μ2V(t)qV(t)f(Z(t)),dZ(t)dt=cV(t)f(Z(t))μ3Z(t), (7.1)

    which is the model studied by [23] when f(Z)=Z and k=0. In fact, we have not found the above model in any existing literatures, but we think it has the same dynamic behavior with (1.6).

    It is necessary to mention it here, in the proof of Lemma 4.1, there may exists zero eigenvalue if R0=1, and it may lead to more complex dynamic behavior. For example, Qesmi et al. [46] propose a mathematical model describing the dynamics of hepatitis B or C virus infection with age-structure, and they found that when 0=1, the system may undergo a backward bifurcation. In a recent work [22], Zhang and Liu studied an age-structured HIV model with cell-to-cell transmission and logistic growth in uninfected cells. They have shown that there exists Hopf bifurcation of the model by using the Hopf bifurcation theory for semilinear equations with non-dense domain. Introducing logistic growth in uninfected cells to model (1.6), it will be interesting to investigate the existence of a Hopf bifurcation. We leave the above two studies for future consideration.

    The authors are very grateful to the editors and two reviewers for their valuable comments and suggestions that have helped us improving the presentation of this paper. The authors were supported by Natural Science Foundation of China (11871179; 11771374).

    All authors declare no conflicts of interest in this paper.



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