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Case report

Shark bite survivors advocate for non-lethal shark mitigation measures in Australia

  • Received: 30 August 2021 Accepted: 17 November 2021 Published: 22 November 2021
  • As the annual number of shark-related human casualties in Australia increases, there is a need for policymakers to grasp how policy is created in the discourse of shark bite incidences. This is discussed in relation to individuals who have been most affected, i.e., shark bite survivors. The defined argument, being that, victims should feel the most animosity towards sharks, therefore if they show signs of discontent towards culling programs, the government should be compelled to change their strategy. The paper reinforces and challenges assumptions that contribute to the flow of commonly accepted knowledge of shark-human relations by illustrating how shark bite survivors are unlikely marine conservation advocates who support non-lethal shark mitigation methods. Shark bite victims were contacted via two Australian-based organizations and a total of six qualitative semi-structured interviews were conducted. Government shark mitigation practices are perceived as heavy handed and further perception- and conservation-based research is needed.

    Citation: Michael J. Rosciszewski-Dodgson, Giuseppe T. Cirella. Shark bite survivors advocate for non-lethal shark mitigation measures in Australia[J]. AIMS Environmental Science, 2021, 8(6): 567-579. doi: 10.3934/environsci.2021036

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  • As the annual number of shark-related human casualties in Australia increases, there is a need for policymakers to grasp how policy is created in the discourse of shark bite incidences. This is discussed in relation to individuals who have been most affected, i.e., shark bite survivors. The defined argument, being that, victims should feel the most animosity towards sharks, therefore if they show signs of discontent towards culling programs, the government should be compelled to change their strategy. The paper reinforces and challenges assumptions that contribute to the flow of commonly accepted knowledge of shark-human relations by illustrating how shark bite survivors are unlikely marine conservation advocates who support non-lethal shark mitigation methods. Shark bite victims were contacted via two Australian-based organizations and a total of six qualitative semi-structured interviews were conducted. Government shark mitigation practices are perceived as heavy handed and further perception- and conservation-based research is needed.



    During the process of viral infection, the interactions among uninfected cells, infected cells, virus, and immune responses play a crucial role in controlling the virus propagation and antiviral defence. Establishing a virus model and analyzing it can effectively predict disease development trends[1,2,3]. Wang et al. [4], Zhang et al. [5], Georgescu et al. [6], Yuan et al. [7], and Hattaf [8] proposed different nonlinear incidence rates describing the infection process in detail in order to comprehensively characterize biological systems, further explaining different biological phenomena in depth.

    It is mentioned in [9,10,11] that different modes of infection have varying impacts on the infection process, such as exhaustion of the immune system, organ damage, and increased antibiotic resistance. Komarova et al. [12], Sigal et al. [13], and Iwami et al. [14] studied the spread of HIV models with two transmission modes. As is known to all, latently infected cells are one of the main reasons why AIDS cannot be completely eradicated. Meanwhile, latently infected cells are not only unaffected by drugs, but can also be activated by antigens. Wang et al. [15] proposed an HIV latent infection model with cell-to-cell transmission, but the humoral immune response has been ignored. Shu et al. [16], Lai et al. [17], and Yang et al. [18] incorporated two modes of viral models without the latently infected cells and humoral immune response. Viral models including logistic growth, multi-stages, and cell-to-cell transmission were also analyzed to exhibit complex dynamic behavior [19,20].

    The immune system protects us from various virus infections. Mathematical modeling of virus infection dynamics is critical to the understanding of complex interaction between immune response and viral infection. In [21], Elaiw et al. considered humoral immunity virus models including latently infected cells, without involving cell-to-cell infection and diffusion. Meanwhile, Lin et al. [22] studied the global dynamics of an HIV infection model which incorporated the cell-to-cell transmission and adaptive immunity. The model presented in [22] has neglected the latently infected cells and diffusion.

    Spatial diffusion can be a specific drug for preventing and treating certain diseases, providing precise guidance on drug carriers. Wang et al. [23] proposed a delayed and diffusive model with linear incidence. Thus, cell mobility plays an important role in different virus infections. Many diffusion viral infection models were studied in [24,25,26]. However, to our knowledge, there are few works that simultaneously consider factors such as latently infected cells, time delays, diffusion, and humoral immune response.

    Given the above discussion, the diffusive and delayed latent virus infection model with humoral immunity is described by the following nonlinear system:

    T(x,t)t=d1ΔT(x,t)+n(T(x,t))π1(T(x,t),V(x,t))π2(T(x,t),G(x,t)),L(x,t)t=d2ΔL(x,t)+(1η)ea1τ1[π1(T(x,tτ1),V(x,tτ1))+π2(T(x,tτ1),G(x,tτ1))](μ+α)h1(L(x,t)),G(x,t)t=d3ΔG(x,t)+ηea1τ2[π1(T(x,tτ2),V(x,tτ2))+π2(T(x,tτ2),G(x,tτ2))]+αh1(L(x,t))σh2(G(x,t)),V(x,t)t=d4ΔV(x,t)+γh2(G(x,t))kh3(V(x,t))ρh3(V(x,t))h4(Z(x,t)),Z(x,t)t=d5ΔZ(x,t)+χ+δh3(V(x,t))h4(Z(x,t))βh4(Z(x,t)), (1.1)

    with initial conditions

    T(x,θ)=ϕ1(x,θ)0,L(x,θ)=ϕ2(x,θ)0,G(x,θ)=ϕ3(x,θ)0,V(x,θ)=ϕ4(x,θ)0,Z(x,θ)=ϕ5(x,θ)0,xˉΩ,θ[τ,0],τ=max{τ1,τ2}, (1.2)

    and homogeneous Neumann boundary conditions

    Tn=Ln=Gn=Vn=Zn=0,t>0,xΩ, (1.3)

    where Ω is a bounded domain in Rn with smooth boundary Ω, and n denotes the outward normal derivative on Ω. Δ is the Laplacian operator. di(i=1,2,3,4,5) are the diffusion coefficients. T(x,t), L(x,t), G(x,t), V(x,t), and Z(x,t) denote the concentration of uninfected cells, latently infected cells, infected cells, viruses, and B cells at position x and time t, respectively. η(0<η<1) is the probability that the uninfected cell will turn into an infected cell. α is the conversion rate. n(T) denotes the growth of the uninfected cells. μh1(L), σh2(G), kh3(V), and βh4(Z) are the death rates of the latently infected cells, infected cells, viruses, and B cells, which only depend on its concentration. γ is the production rate. π1(T,V) and π2(T,G) are the virus-to-cell and cell-to-cell incidence rates, respectively. Let ρh3(V)h4(Z) and δh3(V)h4(Z) be the neutralization rates of viruses and activation rate of B cells, respectively. ea1τ1 and ea1τ2 represent the probability of an infected cell surviving to the stage of τ1 and τ2, respectively. χ is the generation rate of B cells.

    Define

    π11(T)=limV0π1(T,V)V=π1(T,0)V,π21(T)=limG0π2(T,G)G=π2(T,0)G.

    In this paper, we first introduce the following assumptions:

    (A1) n(T) is continuously differentiable, and there exists T0>0 such that n(T0)=0 and n(T0)<0.

    (A2) πi(T,θ) is continuously differentiable; πi(T,θ)>0 for T(0,),θ(0,); πi(T,θ)=0 if and only if T=0 or θ=0. πi(T,θ)T>0 and πi(T,θ)θ>0, for all T>0 and θ>0,i=1,2. πi1(T)>0 and πi1(T)>0 for all T>0,i=1,2.

    (A3) hi is strictly increasing on [0,+), hi(0)=0, hi(0)=1, limθ0hi(θ)=, and there exists ϱi>0 such that hi(θ)ϱiθ for any θ0,i=1,2,3,4.

    (A4) π1(T,V)h3(V) is non-increasing with respect to V(0,+) and π2(T,G)h2(G) is non-increasing with respect to G(0,+).

    In this paper, the purpose is to investigate the dynamical properties of model (1.1). The organization of our paper is as follows: In Section 2, the basic properties of solutions and the existence of equilibria are discussed. In Section 3, the global stability is stated. In Section 4, the numerical simulations are presented to further illustrate the dynamical behavior of the model. Finally, we will give a conclusion.

    Let Y=C(¯Ω,R5) be the Banach space with the supremum norm. For τ0, define C=C([τ,0],Y), which is a Banach space of continuous functions from [τ,0] into Y with the norm φ=maxε[τ,0]φ(ε)Y and let C+=C([τ,0],Y+) with Y+=C(¯Ω,R5+). We will say that ΦC if Φ is a function from ¯Ω×[τ,0] to R5 and is defined by Φ(x,s)=Φ(s)(x). Also, for ζ>0, a function ν():[τ,ζ)Y induces functions νtC for t[0,ζ), defined by νt(κ)=ν(t+κ),κ[τ,0].

    Theorem 2.1. For any given initial condition ψC satisfying (1.2), there exists a unique non-negative solution of model (1.1)(1.3) defined on ˉΩ×[0,+) and this solution remains bounded for all t0.

    Proof: For any ψ=(ψ1,ψ2,ψ3,ψ4,ψ5)TC and x¯Ω, we define H=(H1,H2,H3,H4,H5):CY by

    H1(ψ)(x)=n(ψ1(x,0))π1(ψ1(x,0),ψ4(x,0))π2(ψ1(x,0),ψ3(x,0)),H2(ψ)(x)=(1η)ea1τ1[π1(ψ1(x,τ1),ψ4(x,τ1))+π2(ψ1(x,τ1),ψ3(x,τ1))](μ+α)h1(ψ2(x,0)),H3(ψ)(x)=ηea1τ2[π1(ψ1(x,τ2),ψ4(x,τ2))+π2(ψ1(x,τ2),ψ3(x,τ2))]+αh1(ψ2(x,0))σh2(ψ3(x,0)),H4(ψ)(x)=γh2(ψ3(x,0))kh3(ψ4(x,0))ρh3(ψ4(x,0))h4(ψ5(x,0)),H5(ψ)(x)=χ+δh3(ψ4(x,0))h4(ψ5(x,0))βh4(ψ5(x,0)).

    After that, model (1.1)–(1.3) can be written as the following abstract functional differential equation:

    W(t)=BW+H(Wt),t>0,W(0)=ψX, (2.1)

    where W=(T,L,G,V,Z)T, ψ=(ψ1,ψ2,ψ3,ψ4,ψ5)T, and BW=(d1ΔT,d2ΔL,d3ΔG,d4ΔV,d5ΔZ)T. Obviously, H is locally Lipschitz in Y. From [27,28,29,30,31], we deduce that model (2.1) has a unique local solution on [0,Tmax), where Tmax is the maximal existence time for the solution of model (2.1).

    It is obvious that a lower-solution of the model (1.1)–(1.3) is 0=(0,0,0,0,0). So, we have T(x,t)0,L(x,t)0,G(x,t)0,V(x,t)0, and Z(x,t)0.

    From the first equation of model (1.1), we have T(t)n(T(t))mˉmT(t), which gives limt+supT(t)mˉm. Let

    G1(x,t)=(1η)ea1τ1T(x,tτ1)+ηea1τ2T(x,tτ2)+L(x,t)+G(x,t)+σ2γV(x,t)+σρ2γδZ(x,t),

    and then, it can be obtained that

    G1(x,t)t(1η)ea1τ1d1ΔT(x,tτ1)+ηea1τ2d1ΔT(x,tτ2)+d2ΔL(x,t)+d3ΔG(x,t)+σ2γd4ΔV(x,t)+σρ2γδd5ΔZ(x,t)+σρχ2γδ+δ[(1η)ea1τ1+ηea1τ2]m1G1(x,t)=(1η)ea1τ1d1ΔT(x,tτ1)+ηea1τ2d1ΔT(x,tτ2)+d2ΔL(x,t)+d3ΔG(x,t)+σ2γd4ΔV(x,t)+σρ2γδd5ΔZ(x,t)+Am1G1(x,t),

    where

    m1=min{ˉm,μ,σ2,k,β},A=σρχ2γδ+δ[(1η)ea1τ1+ηea1τ2].

    Therefore, G1(x,t)max{Am1,B}, where

    B=maxx¯Ω{(1η)ea1τ1ψ1(x,τ1)+ηea1τ2ψ1(x,τ2)+ψ2(x,0)+ψ3(x,0)+σ2γψ4(x,0)+σρ2γδψ5(x,0)},

    for (x,t)¯Ω×[0,Tmax). Thus, (T(x,t),L(x,t),G(x,t),V(x,t),Z(x,t)) are bounded on ¯Ω×[0,Tmax). Therefore, by the standard theory for semilinear parabolic systems [32], we have Tmax=+.

    Next, we discuss the existence of equilibria of model (1.1). Inspired by the method in [33,34], we consider the infection and viral production, and define matrices F and V as

    F=(0(1η)ea1τ1π2(T0,0)G(1η)ea1τ1π1(T0,0)V0ηea1τ2π2(T0,0)Gηea1τ2π1(T0,0)V000),

    and

    V=(μ+α00ασ00γk).

    Thus, the basic reproductive number, R0, can be defined as the spectral radius of the next generation operator FV1, where

    FV1=(a11a12a13a21a22a23000),

    where

    a11=(1η)ea1τ1αμ+α(1σπ2(T0,0)G+γkσπ1(T0,0)V),a12=(1η)ea1τ1(1σπ2(T0,0)G+γkσπ1(T0,0)V),a13=(1η)ea1τ1kπ1(T0,0)V,a21=ηea1τ2αα+μ(1σπ2(T0,0)G+γkσπ1(T0,0)V),a22=ηea1τ2(1σπ2(T0,0)G+γkσπ1(T0,0)V),a23=ηea1τ2kπ1(T0,0)V.

    Therefore,

    R0=[ηea1τ2+(1η)ea1τ1αα+μ](γkσπ1(T0,0)V+1σπ2(T0,0)G),

    which biologically describes the average number of secondary infections produced by one infected cell at the beginning of infection. In the above expression of R0, divided into parts as R0=R01+R02, where R01=[ηea1τ2+(1η)ea1τ1αα+μ]γkσπ1(T0,0)V is the basic reproduction number via the virus-to-cell infection and R02=[ηea1τ2+(1η)ea1τ1αα+μ]1σπ2(T0,0)G is the basic reproduction number via the cell-to-cell transmission, respectively.

    To find the equilibria of model (1.1), we need to solve

    n(T)π1(T,V)π2(T,G)=0,(1η)ea1τ1[π1(T,V)+π2(T,G)](μ+α)h1(L)=0,ηea1τ2[π1(T,V)+π2(T,G)]+αh1(L)σh2(G)=0,γh2(G)kh3(V)ρh3(V)h4(Z)=0,χ+δh3(V)h4(Z)βh4(Z)=0. (2.2)

    When V=0, the second and fourth equations of (2.2) lead to G=0 and L=0. From the first equation of (2.2), we obtain n(T)=0T=T0. Solving Z from (2.2) yields χβh4(Z)=0Z0=h14(χβ). It always has an infection-free equilibrium E0=(T0,0,0,0,h14(χβ)).

    Now, we assume that there exists V1(0,h13(βδ)), the fifth equation of (2.2) leads to Z1=h14(χβδh3(V1)), and the fourth equation of (2.2) leads to G1=h12(kγh3(V1)+ργh3(V1)h4(Z1)).

    Define

    F(T)=n(T)π1(T,V1)π2(T,G1),

    and then, F(0)=n(0)>0 and F(T0)=n(T0)π1(T0,V1)π2(T0,G1)=π1(T0,V1)π2(T0,G1)<0. According to (A1) and (A2), F(T) is a strictly decreasing function of T, which implies that there exists a unique T1(0,T0) such that F(T1)=0. From the second equation, we obtain L1=h11((1η)ea1τ1(π1(T1,V1)+π2(T1,G1))μ+α). Hence, model (1.1) has unique endemic equilibrium E1=(T1,L1,G1,V1,Z1), where

    T1(0,T0),L1=h11((1η)ea1τ1(π1(T1,V1)+π2(T1,G1))μ+α),G1=h12(kγh3(V1)+ργh3(V1)h4(Z1)),V1(0,h13(βδ)),Z1=h14(χβδh3(V1)).

    In this section, the global stability of the equilibria E0 and E1 of model (1.1)-(1.3) will be investigated. Let H(ξ)=ξ1lnξ,ξ(0,+), and it is observed that H(ξ)0, ξ>0. H(ξ)=0 ξ=1. For convenience, for any solution (T(x,t),L(x,t),G(x,t),V(x,t),Z(x,t)) of model (1.1), we set

    T(x,t)=T,T(x,tτ1)=Tτ1,T(x,tτ2)=Tτ2,L(x,t)=L,G(x,t)=G,G(x,tτ1)=Gτ1,G(x,tτ2)=Gτ2,V(x,t)=V,V(x,tτ1)=Vτ1,V(x,tτ2)=Vτ2,Z(x,t)=Z.

    To state the global stability on E0, we need an additional assumption:

    (A5)limV0π1(T0,V)π1(T,V)limG0π2(T,G)Gπ21(T)/π11(T)π21(T0)/π11(T0)π21(T0)π21(T0)forTT0.

    Theorem 3.1. Assume that (A1)(A5) hold. If R01, then infection-free equilibrium E0 is globally asymptotically stable.

    Proof: Define a Lyapunov functional

    U1(t)=Ω{α(1η)ea1τ1+η(μ+α)ea1τ2μ+α(TT0T(t)T0limV0π1(T0,V)π1(θ,V)dθ)+αμ+αL(t)+G(t)+σ(1R02)γV(t)+σρ(1R02)γδ(ZZ0Z(t)Z0h4(Z0)h4(θ)dθ)+α(1η)ea1τ1μ+α0τ1[π1(T(t+θ),V(t+θ))+π2(T(t+θ),G(t+θ))]dθ+ηea1τ20τ2[π1(T(t+θ),V(t+θ))+π2(T(t+θ),G(t+θ))]dθ}dx.

    Calculating the derivative of U1(t) along the positive solution of model (1.1), we obtain

    dU1(t)dt=Ω{α(1η)ea1τ1+η(μ+α)ea1τ2μ+α(1π1(T0,V)π1(T,V))Tt+αμ+αLt+Gt+σ(1R02)γVt+σρ(1R02)γδ(1h4(Z0)h4(Z))Zt+α(1η)ea1τ1μ+α[π1(T,V)+π2(T,G)π1(Tτ1,Vτ1)π2(Tτ1,Gτ1)]+ηea1τ2[π1(T,V)+π2(T,G)π1(Tτ2,Vτ2)π2(Tτ2,Gτ2)]}dx,=Ω{α(1η)ea1τ1+η(μ+α)ea1τ2μ+α(1limV0π1(T0,V)π1(T,V))(n(T)n(T0))+α(1η)ea1τ1+η(μ+α)ea1τ2μ+α(π1(T,V)+π2(T,G))limV0π1(T0,V)π1(T,V)R02σh2(G)σ(1R02)γh3(V)(k+ρh4(Z0))+σρ(1R02)γδ(1h4(Z0)h4(Z))(βh4(Z0)βh4(Z))+α(1η)ea1τ1+η(μ+α)ea1τ2μ+α(1limV0π1(T0,V)π1(T,V))d1ΔT+αμ+αd2ΔL+d3ΔG+σ(1R02)γd4ΔV+σρ(1R02)γδ(1h4(Z0)h4(Z))d5ΔZ}dx.

    From assumptions (A3) and (A4), we have

    α(1η)ea1τ1+η(μ+α)ea1τ2μ+απ1(T,V)limV0π1(T0,V)π1(T,V)σ(1R02)γ(k+ρh4(Z0))h3(V)=σ(k+ρh4(Z0))γ(R01)h3(V),

    and

    α(1η)ea1τ1+η(μ+α)ea1τ2μ+απ2(T,G)limV0π1(T0,V)π1(T,V)R02σh2(G)=α(1η)ea1τ1+η(μ+α)ea1τ2μ+α(limV0π1(T0,V)π1(T,V)limG0π2(T,G)Gπ2(T0,0)G)G0.

    From assumptions (A1) and (A2), we have

    (1limV0π1(T0,V)π1(T,V))(n(T)n(T0))<0.

    Moreover, by utilizing assumption (A5), we obtain

    limV0π1(T0,V)π1(T,V)limG0π2(T,G)Gπ21(T)/π11(T)π21(T0)/π11(T0)π21(T0)π21(T0),forTT0.

    Therefore, we obtain

    dU1(t)dtΩ{σ(k+ρh4(Z0))γ(R01)h3(V)σρβ(1R02)γδ(h4(Z)h4(Z0))2h4(Z)+α(1η)ea1τ1+η(μ+α)ea1τ2μ+α(1limV0π1(T0,V)π1(T,V))d1ΔT+αμ+αd2ΔL+d3ΔG+σ(1R02)γd4ΔV+σρ(1R02)γδ(1h4(Z0)h4(Z))d5ΔZ}dx.

    Using the divergence theorem and the homogeneous Neumann boundary conditions, we get

    ΩΔTdx=ΩTndx=0,ΩΔTπ1(T,V1)dx=ΩT2π21(T,V1)dx,ΩΔLdx=ΩLndx=0,ΩΔGdx=ΩGndx=0,ΩΔVdx=ΩVndx=0,ΩΔZdx=ΩZndx=0,ΩΔZh4(Z)dx=ΩZ2h24(Z)dx.

    Thus, we obtain

    dU1(t)dtΩ{σ(k+ρh4(Z0))γ(R01)h3(V)σρβ(1R02)γδ(h4(Z)h4(Z0))2h4(Z)}dx(α(1η)ea1τ1+η(μ+α)ea1τ2)d1μ+αlimV0π1(T0,V)π1(T,V)ΩT2π21(T,V1)dxσρ(1R02)d5h4(Z0)γδΩZ2h24(Z)dx.

    Therefore, dU1(t)dt0. dU1(t)dt=0 T=T0, L=0, G=0, V=0, and Z=Z0. By LaSalle's invariance principle [31], E0 is globally asymptotically stable when R01.

    Assume that π1(T,V), π2(T,G), and h3(V) satisfy

    (A6)(π1(T,V)π1(T,V1)h3(V)h3(V1))(1π1(T,V1)π1(T,V))0,(π2(T,G)π1(T1,V1)π2(T1,G1)π1(T,V1)h3(V)h3(V1))(1π1(T,V1)π2(T1,G1)π1(T1,V1)π2(T,G))0.

    Theorem 3.2. If R0>1, and (A1)-(A6) hold, then the endemic equilibrium E1 is globally asymptotically stable.

    Proof: Define a Lyapunov functional

    U2(t)=Ω{α(1η)ea1τ1+η(μ+α)ea1τ2μ+α(TT1T(t)T1π1(T1,V1)π1(θ,V1)dθ)+αμ+α(LL1L(t)L1h1(L1)h1(θ)dθ)+(GG1G(t)G1h2(G1)h2(θ)dθ)+σγ(VV1V(t)V1h3(V1)h3(θ)dθ)+σργδ(ZZ1Z(t)Z1h4(Z1)h4(θ)dθ)+α(1η)ea1τ1μ+απ1(T1,V1)0τ1H(π1(T(t+θ),V(t+θ))π1(T1,V1))dθ+ηea1τ2π1(T1,V1)0τ2H(π1(T(t+θ),V(t+θ))π1(T1,V1))dθ+α(1η)ea1τ1μ+απ2(T1,G1)0τ1H(π2(T(t+θ),G(t+θ))π2(T1,G1))dθ+ηea1τ2π2(T1,G1)0τ2H(π2(T(t+θ),G(t+θ))π2(T1,G1))dθ}dx.

    Calculating the derivative of U2(t) along the positive solution of model (1.1), it follows that

    dU2(t)dt=Ω{α(1η)ea1τ1+η(μ+α)ea1τ2μ+α(1π1(T1,V1)π1(T,V1))d1ΔT+α(1η)ea1τ1+η(μ+α)ea1τ2μ+α(1π1(T1,V1)π1(T,V1))(n(T)n(T1))+α(1η)ea1τ1+η(μ+α)ea1τ2μ+απ1(T1,V1)(1π1(T1,V1)π1(T,V1))+α(1η)ea1τ1+η(μ+α)ea1τ2μ+απ2(T1,G1)(1π1(T1,V1)π1(T,V1))+α(1η)ea1τ1+η(μ+α)ea1τ2μ+απ1(T,V)π1(T1,V1)π1(T,V1)+α(1η)ea1τ1+η(μ+α)ea1τ2μ+απ2(T,G)π1(T1,V1)π1(T,V1)+αμ+α(1h1(L1)h1(L))d2ΔL+αh1(L1)+(1h2(G1)h2(G))d3ΔGα(1η)ea1τ1μ+αh1(L1)h1(L)(π1(Tτ1,Vτ1)+π2(Tτ1,Gτ1))αh2(G1)h2(G)h1(L)+σh2(G1)+σγ(1h3(V1)h3(V))d4ΔVηea1τ2h2(G1)h2(G)(π1(Tτ2,Vτ2)+π2(Tτ2,Gτ2))σkγh3(V)σh2(G)h3(V1)h3(V)+σkγh3(V1)+σργh3(V1)h4(Z)+σργδ(1h4(Z1)h4(Z))d5ΔZ+σργδ(1h4(Z1)h4(Z))(βh4(Z1)βh4(Z))σργh3(V1)h4(Z1)σργh3(V)h4(Z1)+σργh3(V1)h4(Z1)h4(Z1)h4(Z)+α(1η)ea1τ1μ+απ1(T1,V1)ln(π1(Tτ1,Vτ1)π1(T,V))+ηea1τ2π1(T1,V1)ln(π1(Tτ2,Vτ2)π1(T,V))+α(1η)ea1τ1μ+απ2(T1,G1)ln(π2(Tτ1,Gτ1)π2(T,G))+ηea1τ2π2(T1,G1)ln(π2(Tτ2,Gτ2)π2(T,G))}dx.

    By using

    (1η)ea1τ1(π1(T1,V1)+π2(T1,G1))=(μ+α)h1(L1),ηea1τ2(π1(T1,V1)+π2(T1,G1))+αh1(L1)=σh2(G1),γh2(G1)=kh3(V1)+ρh3(V1)h4(Z1),

    and by the divergence theorem,

    ΩΔTdx=ΩTndx=0,ΩΔTπ1(T,V1)dx=ΩT2π21(T,V1)dx,ΩΔLdx=ΩLndx=0,ΩΔLh1(L)dx=ΩL2h21(L)dx,ΩΔGdx=ΩGndx=0,ΩΔGh2(G)dx=ΩG2h22(G)dx,ΩΔVdx=ΩVndx=0,ΩΔCh3(V)dx=ΩV2h23(V)dx,ΩΔZdx=ΩZndx=0,ΩΔZh4(Z)dx=ΩZ2h24(Z)dx.

    Thus, we have

    dU2(t)dt=Ω{α(1η)ea1τ1+η(μ+α)ea1τ2μ+α(1π1(T1,V1)π1(T,V1))(n(T)n(T1))+α(1η)ea1τ1+η(μ+α)ea1τ2μ+απ1(T1,V1)(π1(T,V)π1(T,V1)h3(V)h3(V1))(1π1(T,V1)π1(T,V))+α(1η)ea1τ1+η(μ+α)ea1τ2μ+απ2(T1,G1)×(π2(T,G)π1(T1,V1)π2(T1,G1)π1(T,V1)h3(V)h3(V1))(1π1(T,V1)π2(T1,G1)π1(T1,V1)π2(T,G))σρχγδ(h4(Z)h4(Z1))2h4(Z)h4(Z1)σργh3(V1)h4(Z1)[2h4(Z)h4(Z1)h4(Z1)h4(Z)]α(1η)ea1τ1μ+απ1(T1,V1)[H(π1(T1,V1)π1(T,V1))+H(h1(L)h2(G1)h1(L1)h2(G))+H(h2(G)h3(V1)h2(G1)h3(V))+H(π1(T,V1)h3(V)π1(T,V)h3(V1))+H(h1(L1)π1(Tτ1,Vτ1)h1(L)π1(T1,V1))]ηea1τ2π1(T1,V1)[H(π1(T,V)π1(T,V1))+H(h2(G)h3(V1)h2(G1)h3(V))+H(h2(G1)π1(Tτ2,Vτ2)h2(G)π1(T1,V1))+H(π1(T,V1)h3(V)π1(T,V)h3(V1))]α(1η)ea1τ1μ+απ2(T1,G1)[H(π1(T1,V1)π1(T,V1))+H(h1(L)h2(G1)h1(L1)h2(G))+H(h1(L1)π2(Tτ1,Gτ1)h1(L)π2(T1,G1))+H(h2(G)h3(V1)h2(G1)h3(V))+H(h3(V)π1(T,V1)π2(T1,G1)h3(V1)π1(T1,V1)π2(T,G))]ηea1τ2π2(T1,G1)[H(π1(T1,V1)π1(T,V1))+H(h3(V)π1(T,V1)π2(T1,G1)h3(V1)π1(T1,V1)π2(T,G))+H(h2(G1)π2(Tτ2,Gτ2)h2(G)π2(T1,G1))+H(h2(G)h3(V1)h2(G1)h3(V))]}dx(α(1η)ea1τ1+η(μ+α)ea1τ2)d1π1(T1,V1)μ+αΩT2π21(T,V1)dxαd2h1(L1)μ+αΩL2h21(L)dxd3h2(G1)ΩG2h22(G)dxσd4h3(V1)γΩV2h23(V)dxσρd5h4(Z1)γδΩZ2h24(Z)dx.

    Hence, dU2(t)dt0. dU2(t)dt=0 T(t)=T1,L(t)=L1,G(t)=G1, V(t)=V1, and Z(t)=Z1. From LaSalle's invariance principle [31], we have that E1 is globally asymptotically stable when R0>1.

    In this section, we present several numerical examples to illustrate the results obtained in Section 3. We will use the finite difference scheme which is proposed in [35,36] for the delayed reaction-diffusion epidemic models. For convenience, we consider model (1.1) under the one-dimensional spatial domain Ω=[0,1]. The homogeneous Neumann boundary conditions and the initial conditions are given by

    Tn=Ln=Gn=Vn=Zn=0,t>0,x=0,1, (4.1)

    and

    T(x,θ)=80,L(x,θ)=0.1,G(x,θ)=0.1,V(x,θ)=0.01,Z(x,θ)=0.001,

    for 0x1, τθ0, τ=max{τ1,τ2}.

    In model (1.1), we choose n(T(t))=sdT(t)+rT(t)(1T(t)K), π1(T(t),V(t))=β1T(t)V(t)(1+η1T(t))(1+η2V(t)), π2(T(t),G(t))=β2T(t)G(t)1+α1G(t), and hi(ξ)=ξ. We can easily verify that (A1)-(A4) hold. For simulations, we take η1=0.01,η2=0.01,α1=0.01,d1=0.1,d2=0.1,d3=0.1,d4=0.1,d5=0.1,τ1=10,τ2=5, and choose β1 and β2 as free parameters. The values of the other parameters are summarized in Table 1.

    Table 1.  List of parameters.
    Parameter Definition Value Source
    s production rate of uninfected cells 10 μl1day1 [37]
    d death rate of uninfected cells 0.01 day1 [37]
    η probability that the uninfected cell
    will become an infected cell 0.49 Assumed
    r logistic growth rate 0.01 day1 [38]
    K carrying capacity 1000 μlday1 [38]
    μ death rate for latently infected cells 0.1 day1 [38]
    α conversion rate 0.05 Assumed
    σ death rate for infected cells 0.1 day1 [6]
    γ production rate 1 cell1day1 [6]
    k clearance rate for free virus 3 day1 [37]
    ρ neutralizing rate of viruses 0.5 μlday1 [7]
    χ generation rate of B cells 1.4 μlday1 [37]
    δ proliferation rate of B cells 1.2 μlday1 [6,7]
    β death rate for B cells 1.2 day1 [6]
    a1 death rate for latently infected cells during [tτ1,t] 0.01 Assumed
    a2 death rate for infected cells during [tτ2,t] 0.01 Assumed

     | Show Table
    DownLoad: CSV

    In the following Figures 1 and 2, (a), (b), (c), (d), and (e) are denoted time-series figures of T(t), L(t), G(t), V(t) and Z(t).

    Figure 1.  Taking β1=0.0001 and β2=0.0001, we have R0=0.8352<1, and the infection-free equilibrium E0=(1000,0,0,0,1.1667) is globally asymptotically stable.
    Figure 2.  Taking β1=0.01 and β2=0.01, we have R0=83.52>1, and the endemic equilibrium E1=(18.16,29.62,113.9,0.9948,223) is globally asymptotically stable.

    In this paper, a diffusive and delayed viral dynamics model with two modes of transmission has been analyzed. Some assumptions about nonlinear functions for n(T), π1(T(t),V(t)), π2(T(t),G(t)), h1(L(t)), h2(G(t)), h3(V(t)), and h4(Z(t)) are made and the global stabilities of model (1.1) are proved. The contribution is to construct suitable Lyapunov functionals for the diffusive virus model considering the humoral cells, cell-to-cell transmission, two delays, and latently infected cells, and we can extend this method to more complicated models. Furthermore, the formula of the basic reproduction number R0 is independent of the diffusion coefficient. Without considering either the virus-to-cell infection or cell-to-cell transmission, R0 could be under-evaluated and the transmission and spread trends of diseases need to be studied.

    Based on the obtained results of this paper, we can directly propose the following questions that need further research. On the one hand, in addition to spatial diffusion, humoral response and delays should be considered, determining whether the results obtained in this paper can be extended to a spatially heterogeneous model with immune response delay, random perturbation effect, and memory effect. On the other hand, the globally asymptotic stability of some classes of multiple infection dynamics models will be a very valuable and significative subject. We leave these problems as possible future works.

    The author declares she has not used Artificial Intelligence (AI) tools in the creation of this article.

    The author would like to express the deepest gratitude to the anonymous referees for their careful reading of the manuscript, several valuable comments, and suggestions for its improvement. This work was supported by the NSFC (Nos. 12371504, 12471471), and the General Research Fund for Shanxi Basic Research Project (Nos. 202403021221218, 202403021221214, 202103021224291).

    The author declares that there are no conflicts of interest.



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