Research article Special Issues

Backward bifurcation of a plant virus dynamics model with nonlinear continuous and impulsive control


  • Roguing and elimination of vectors are the most commonly seen biological control strategies regarding the spread of plant viruses. It is practically significant to establish the mathematical models of plant virus transmission and regard the effect of removing infected plants as well as eliminating vector strategies on plant virus eradication. We proposed the mathematical models of plant virus transmission with nonlinear continuous and pulse removal of infected plants and vectors. In terms of the nonlinear continuous control strategy, the threshold values of the existence and stability of multiple equilibria have been provided. Moreover, the conditions for the occurrence of backward bifurcation were also provided. Regarding the nonlinear impulsive control strategy, the stability of the disease-free periodic solution and the threshold of the persistence of the disease were given. With the application of the fixed point theory, the conditions for the existence of forward and backward bifurcations of the model were presented. Our results demonstrated that there was a backward bifurcation phenomenon in continuous systems, and there was also a backward bifurcation phenomenon in impulsive control systems. Moreover, we found that removing healthy plants increased the threshold R1. Finally, numerical simulation was employed to verify our conclusions.

    Citation: Guangming Qiu, Zhizhong Yang, Bo Deng. Backward bifurcation of a plant virus dynamics model with nonlinear continuous and impulsive control[J]. Mathematical Biosciences and Engineering, 2024, 21(3): 4056-4084. doi: 10.3934/mbe.2024179

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  • Roguing and elimination of vectors are the most commonly seen biological control strategies regarding the spread of plant viruses. It is practically significant to establish the mathematical models of plant virus transmission and regard the effect of removing infected plants as well as eliminating vector strategies on plant virus eradication. We proposed the mathematical models of plant virus transmission with nonlinear continuous and pulse removal of infected plants and vectors. In terms of the nonlinear continuous control strategy, the threshold values of the existence and stability of multiple equilibria have been provided. Moreover, the conditions for the occurrence of backward bifurcation were also provided. Regarding the nonlinear impulsive control strategy, the stability of the disease-free periodic solution and the threshold of the persistence of the disease were given. With the application of the fixed point theory, the conditions for the existence of forward and backward bifurcations of the model were presented. Our results demonstrated that there was a backward bifurcation phenomenon in continuous systems, and there was also a backward bifurcation phenomenon in impulsive control systems. Moreover, we found that removing healthy plants increased the threshold R1. Finally, numerical simulation was employed to verify our conclusions.



    Plant virus diseases are transmitted between plant individuals through vectors, including the tomato leaf roll virus and cassava mosaic virus. Plant viruses seriously affect crop yields, which can result in great economic losses and increased poverty, especially in developing countries [1]. According to [2], the annual wheat rust can cause yield loss of up to 80 percent, posing a great threat to global wheat production. Moreover, it is estimated that 12.5 million hectares were disturbed annually from 2003 to 2012 by plant disease, mostly Asia and Europe [3]. Therefore, the control of plant virus transmission remains a major economic and agricultural issue. The effective methods to control plant viruses include gene control, chemical control, and biological control. It is extremely effective to improve the antiviral ability of plants by changing genes, but it cannot be greatly used due to the high cost. Chemical control aims to control the vector by spraying insecticide. However, long-term use will not only pollute the environment, but also lead to drug resistance and increase the incidence of plant viruses. Biological control is widely used to control the spread of diseases by manually removing infected plants or introducing vector natural enemies, having significant effects and causing no pollution to the environment. Thus, biological control is increasingly used as an efficient method to control plant viruses.

    Although biological control of plant viruses provides an effective method for controlling the spread of plant viruses, a successful control plan needs to understand the transmission mechanism of plant viruses in plants and vectors as well as the interaction between biological controllers and plants and vectors. Mathematical modeling has exerted a main and probably greater role in the epidemiology of plant viruses. Mathematical models enable us to comprehend the observed transmission routes and disease control of plant viruses, as well as grasp a deeper understanding of the mechanism of plant disease transmission dynamics. At present, there are numerous research achievements on biological control of plant viruses. As mentioned in [4], the authors consider the evolution of within-plant virus titer as the response to the application of various disease control methods. Moreover, they demonstrate that the process of new and improved disease control methods for viral diseases of vegetatively propagated staple food crops need to consider the evolutionary responses of the virus. According to [5], the authors concentrate on the contribution made by developing mathematical models with a scope of techniques that is extensive and their use for investigating plant virus disease epidemics. Their focus is on the extent to which models can help answer biological questions and raised questions in association with the epidemiology and ecology of plant viruses and the caused diseases. Based on [6,7], the authors applied optimization theory to investigate the plant vector virus model with continuous replanting and roguing, aiming to maximize the harvest of healthy plants and examine the best strategy to fight plant virus disease. In [8], the authors concentrate on a plant disease model with a latent period and nonautonomous phenomenon. Their focus is to explore the long-run behavior of the epidemic model.

    In reality, considering that people control plant viruses at discrete time points, the establishment of impulse differential equations to reveal the propagation dynamics and control strategies of plant viruses has been favored by numerous scholars in recent years. In [9], the authors proposed and analyzed a plant virus disease model under periodic environment and pulse roguing. Based on their research, with the high infection rate, it can probably not be possible to eliminate the disease by easily roguing the infectious plant, and elevating the replanting rate is not conducive to the control of disease. In [10], to detect and design suitable plant disease control strategies, the authors proposed and investigated the dynamics of plant disease models by adopting continuous and impulsive cultural control strategies. Under a certain parameter space, it can be demonstrated that a nontrivial periodic solution occurs through a supercritical bifurcation. In [11], the authors formulated an epidemiological model for mosaic diseases considering plant and vector populations. They discovered that roguing is the most cost effective and beneficial management for mosaic disease eradication of plants if used at an appropriate rate and interval. In [12,13], with the purpose of eradicating plant diseases or keeping the number of infected plants below the economic threshold, the authors proposed and analyzed the plant disease models, including nonlinear impulsive functions and cultural control strategies. In [14], based on the purpose of minimizing losses and maximizing returns, the authors introduce a discontinuous plant disease model, which incorporates a threshold policy control. Their results suggest that we can adopt the proper replanting and roguing rates for designing the threshold policy, and thus the number of infected plants is within an acceptable level. Recently, pulse control methods can be applied in almost every field of applied science. The theoretical research and application analysis of pulse control can be found in many studies [15,16,17,18].

    However, most of the above studies consider only (1) Continuous removal of infected plants or replanting of healthy plants, without considering removal of infected plants at discrete time points. (2) Pulse removes the infected plants without destroying the vector on the plants. Therefore, considering the shortcomings of the above literature, based on the above two points, we aim to develop a plant virus disease transmission model with nonlinear continuous and pulse control, analyze its dynamic behavior, and discuss the impact on plant virus disease control when removing infected plants and vectors on plants.

    According to the model proposed in [19], we develop a vector-borne plant virus disease model with roguing control. The basic idea is that removing the infected plants will also eliminate the vectors on the plants. For plant populations, we classified plants into healthy plants X(t), infected plants Y(t), and the total number of plant populations at time t with N(t)=X(t)+Y(t). The recruitment rate of plant population is constant r. The newly recruited plants enter healthy plant populations X(t). The plant harvesting or death rate is denoted as g. Healthy plants are infected through contact with infection vectors, and the infection rate is k1V(t)N(t), where k1 is the probability of transmission from infected vectors to healthy plant. The expression of infection rate k1V(t)N(t) is acquired as follows. The probability that a vector selects a particular plant is assumed to be 1N(t), which is the probability of each plant selected by total N(t). Therefore, a plant receives in average M(t)N(t) contacts per unit of time. Later, the infection rate per healthy plant is offered by M(t)N(t)k1V(t)M(t). Considering using removal control on infected plants, the removal rate of diseased plants is α.

    Vector populations are divided into non-infective U(t) and infective vectors V(t), and the total number of vector populations at time t denote M(t)=U(t)+V(t). The recruitment rate of vector population is constant μ. The newly recruited vectors enter non-infective vector populations U(t). The vector death rate is c. Uninfected vectors become infected through contact with infected plants, and the infection rate is k2Y(t)N(t), where k2 is the probability of transmission from infected plants to non-infective vectors.

    The elimination of vectors occurs when the infected plants are removed with elimination rate of ραY(t)N(t) which is explained as follows. The average number of vector on each plant is M(t)N(t), and parameter ρ represents the elimination ratio of vectors on infected plants while removing each infected plant. Since the number of infected plants removed is αY(t), with the removal of the infected plants, the total number of eliminated vector is ραY(t)M(t)N(t). Thus, the elimination rate of vectors is ραY(t)M(t)N(t)1M(t). Therefore, our basic plant disease model reads as

    {dX(t)dt=rk1X(t)V(t)N(t)gX(t),dY(t)dt=k1X(t)V(t)N(t)gY(t)αY(t),dU(t)dt=μk2U(t)Y(t)N(t)cU(t)ραU(t)Y(t)N(t),dV(t)dt=k2U(t)Y(t)N(t)cV(t)ραV(t)Y(t)N(t). (2.1)

    We suppose that all parameter values are strictly positive.

    Next, we indicate the boundedness of system (2.1), and it can follow from system (2.1) that:

    dN(t)dt=rgX(t)(α+g)Y(t)rgN(t),

    which suggests:

    N(t)N(0)exp(gt)+rg(1exp(gt))rg,fort.

    Similarly, we have

    dM(t)dt=μcM(t)ραM(t)Y(t)N(t)μcM(t),

    and thus, we obtain:

    M(t)M(0)exp(ct)+μc(1exp(ct))μc,fort.

    Therefore, the total plant population N(t) and vector population M(t) remain uniformly bounded. Moreover, the region

    Ω={(X(t),Y(t),U(t),V(t))R4+:X(t)+Y(t)rg;U(t)+V(t)μc} (2.2)

    is positively-invariant, indicating that this study concentrates on the dynamics of system (2.1) on the set Ω in the following.

    With no disease in both of the populations(i.e., Y(t)=0, V(t)=0), the model (2.1) has one disease-free equilibrium E0=(N0,0,M0,0)=(rg,0,μc,0). By [20], we define the basic reproductive number

    R0=k1k2μgc2r(g+α). (2.3)

    We note that

    R0=R01R02,

    where

    R01=k1c

    refers to the expected number of plants that one vector infects via its infectious life time. R01 can be provided by the product of the infection rate of infectious vectors k1 and the average duration in the infectious stage 1c.

    Similarly, we have

    R02=k2M0N01(g+α)=k2gμcr(g+α),

    which is the expected number of vectors that one plant infects via its infectious life time. R02 is given by the product of the infection rate of infectious plants k2M0N0 and the average duration in the infectious stage 1g+α.

    The basic reproduction number equals to the geometric mean of R01 and R02 since infection from plant to plant goes via one generation of vectors. In addition, the local asymptotic stability result of equilibrium E0 is provided:

    Theorem 2.1. When R0<1, the disease-free equilibrium E0=(rg,0,μc,0) remains locally asymptotically stable in Ω.

    Proof. The Jacobian matrix of system (2.1) at disease-free equilibrium E0 can be provided by:

    JE0=(g00k10(g+α)0k10(k2+ρα)μgcrc00k2μgcr0c).

    The characteristic polynomial of JE0 can be expressed by:

    P(λ)=(λ+g)(λ+c)φ(λ),

    where φ(λ)=λ2+a1λ+a2, with a1=c(g+α) and a2=c(g+α)(1R20).

    The roots of P(λ) are λ1=g, λ2=c and the others roots are the roots of φ(λ). Because R0<1, all coefficients of φ(λ) are always positive. Now, we just have to verify that the Routh-Hurwitz criterion holds for polynomial φ(λ). Therefore, setting H1=a1, H2=|a110a2|=a1a2. We have H1>0 and H2>0 if R0<1. Thus, by the Routh-Hurwitz criterion the trivial equilibrium E0 is locally asymptotically stable when R0<1.

    In general, the biological implication of Theorem 2.1 is that with the basic reproduction number R0 being less than 1, a small number of infected vectors are introduced into the plant population, which will not cause disease outbreak, and the disease disappears in time. Nevertheless, in the following subsection, we present that the disease can persist even with R0<1.

    With the purpose of determining whether there is an endemic equilibrium, we need to find the solution of the algebraic system of equations acquired through equating the right sides of system (2.1) to zero.

    rk1X(t)V(t)N(t)gX(t)=0,k1X(t)V(t)N(t)gY(t)αY(t)=0,μk2U(t)Y(t)N(t)cU(t)ραU(t)Y(t)N(t)=0,k2U(t)Y(t)N(t)cV(t)ραV(t)Y(t)N(t)=0. (2.4)

    For the sake of easier readability, we express the quantities as follows,

    λp=VN,λv=YN.

    By addressing the equations in the system (2.4) in terms of λp and λv, we can obtain:

    X=rk1λp+g,Y=rk1λp(g+α)(k1λp+g), (2.5)

    and

    U=μc+(k2+ρα)λv,V=k2μλv(c+ραλv)((k2+ρα)λv+c). (2.6)

    By substituting (2.5) and (2.6) into the expression of λp and λv, we have

    A(λp)2+Bλp+C=0, (2.7)

    where

    A=k21r(c+ρα)(c+ρα+k2)>0,B=k1(g+α)2c2rg(R2ρR20),C=c2r(g+α)2(1R20),Rρ=(2(c+ρα)+k2)r2c(g+α).

    We aimed to investigate the existence of endemic equilibria in the following cases:

    (1) There is a unique endemic equilibrium if

    (C<0) or (B<0 and C=0) or (B<0 and C>0 and B24AC=0);

    (2) There are two endemic equilibria if

    B<0 and C>0 and B24AC>0;

    (3) There are no endemic equilibria otherwise.

    Hence, we present the result in the theorem below.

    Theorem 2.2. For system (2.1),

    (1) If R0>1, there is a unique endemic equilibrium E.

    (2) If R0=1 and B<0 there is a unique endemic equilibrium E. Otherwise, there exists no endemic equilibrium.

    (3) If R0<1, and

    (a) B<0 and B24AC>0, there are two equilibria E1 and E2.

    (b) B<0 and B24AC=0, there exists a unique endemic equilibrium.

    (c) There are no endemic equilibria otherwise.

    Obviously, case (3) (item(a)) of Theorem 2.2 indicates the possibility of backward bifurcation in the model (2.1). Next, by applying the method of literature [21] to model (2.1), we present a rigorous proof that model (2.1) experiences a backward bifurcation.

    Theorem 2.3. If cαg>r+k2+2ρα, the direction of the bifurcation of system (2.1) at R0=1 is backward.

    Proof. Let k1 be the bifurcation parameter. To apply the method in reference[21], we introduce the notation x1=X, x2=Y, x3=U, x4=V, the system (2.1) becomes

    {dx1dt=rk1x1x4x1+x2gx1:=f1(x1,x2,x3,x4),dx2dt=k1x1x4x1+x2gx2αx2:=f2(x1,x2,x3,x4),dx3dt=μk2x2x3x1+x2cx3ραx2x3x1+x2:=f3(x1,x2,x3,x4),dx4dt=k2x2x3x1+x2cx4ραx2x4x1+x2:=f4(x1,x2,x3,x4), (2.8)

    with R0=1 corresponding to k1=k1=c2r(g+α)k2μg. The disease-free equilibrium is E0=(rg,0,μc,0). The linearization matrix of system (2.8) around the disease-free equilibrium when k1=k1 is

    DE0f=(g00k10(g+α)0k10(k2+ρα)μgcrc00k2μgcr0c),

    where f=(f1,f2,f3,f4). Clearly, 0 indicates a simple eigenvalue of DE0f. A right eigenvector related to 0 eigenvalue is ω=[cr(g+α)k2μg2,crk2μg,k2+ραck2,1c], and the left eigenvector υ meeting ωυ=1 is υ=[0,k2μgr(c+g+α),0,c(g+α)(c+g+α)]. Algebraic calculations demonstrate that

    2f2x2x4=2f2x4x2=cr(g+α)k2μ,2f4x2x1=2f4x1x2=k2μg2cr2,2f4x22=2k2μg2cr2,2f4x3x2=2f4x2x3=k2gr,2f4x4x2=2f4x2x4=ραgr,2f2x2k1=1.

    The remaining of the second derivatives appear in the formula for a and b are all zero. The a and b presented in Theorem 4.1 of [21] are

    a=nk,i,j=1υkωiωj2fxixj(0,0),b=nk,i=1υkωi2fxiϕ(0,0).

    Hence,

    a=2c(g+α)k2μ(c+g+α)(cαg(r+k2+2ρα)),b=k2μgcr(c+g+α)>0.

    By [21], if cαg>r+k2+2ρα, then a>0, and the direction of the bifurcation of system (2.1) at R0=1 is backward.

    The backward bifurcation phenomenon indicates that with the basic reproductive number R0<1, the disease-free equilibrium is locally stable. Therefore, it is insufficient to only reduce the basic reproductive number R0<1 in order to eliminate the disease. Figure 1 depicts the associated bifurcation diagram. This clearly demonstrates the coexistence of two locally-asymptotically stable equilibria with R0<1, proving that the model (2.1) reveals the phenomenon of backward bifurcation. Therefore, to eradicate plant diseases, it is of necessity to increase α, and thus R0 is less than R0(α0), where R0(α0) is the threshold for the appearance of two endemic equilibria.

    Figure 1.  The backward bifurcation curves for system of (2.1) in the (R0,Y) and (R0,V) planes. The parameter α varied within the range (0.03, 0.08) in order to allow R0 to be different in the rang (0, 1.04). Red lines suggest stable equilibria and blue lines indicate unstable equilibria. There is a positive value R0(α0), and when R0 is less than R0(α0), there is no positive equilibrium. The used parameter values are: k1=k2=0.003, g=0.002, μ=400, c=0.2, ρ=0.86.

    Figure 2 presents the occurrence of the backward bifurcation. In this study, R0<1, while according to the initial condition, the solution of the model (2.1) can approach either the endemic equilibrium point or the disease-free equilibrium point.

    Figure 2.  Solution of model (2.1) of the number of infectious plants Y(t) and vectors V(t), for parameter values provided in the bifurcation diagram in Figure 1 with α=0.0315, and thus R0=0.9975<1, for two different sets of initial conditions. The first initial conditions (corresponding to the red line) is (7,0.5,750,30) and the second initial conditions (conforming to the blue line) is (70,0.5,750,30).

    The current section extends model (2.1) by replacing the continuous infected plants with a periodic pulse roguing control strategy, which is shown to be more realistic. Therefore, our impulsive control model of plant virus diseases reads as

    {dX(t)dt=rk1X(t)V(t)N(t)gX(t),dY(t)dt=k1X(t)V(t)N(t)gY(t),dU(t)dt=μk2U(t)Y(t)N(t)cU(t),dV(t)dt=k2U(t)Y(t)N(t)cV(t),}tnT,X(t+)=(1p)X(nT),Y(t+)=(1α)Y(nT),U(t+)=(1ραY(nT)N(nT)ρpX(nT)N(nT))U(nT),V(t+)=(1ραY(nT)N(nT))V(nT),}t=nT,X(0+)=X0,Y(0+)=Y0,U(0+)=U0,V(0+)=V0, (3.1)

    where T indicates a fixed positive constant and suggests the periodic of the impulsive effect, where nN represents the positive integer set. The parameter α suggests the proportion of the infected plants, which is rogued at each pulse perturbation. Considering the inevitable impact of control measures on healthy plants, we use p representing the proportion of the healthy plants, which is rogued at each pulse perturbation. Here, we assume that p<α. Similar to the continuous model, the ratio of removing uninfected and infected vectors while removing plants is ραY(nT)N(nT). As there are only uninfected vectors on healthy plants, only uninfected vectors can be eliminated while removing healthy plants. The elimination ratio is ρpX(nT)N(nT), where ρ represents the vectors elimination rate of vectors, which probably means that some of the uninfected vectors and infected vectors will be occasionally eliminated due to the rogued plants.

    When Y(t)=0, V(t)=0, then model (3.1) is the subsystem below:

    {dX(t)dt=rgX(t),dU(t)dt=μcU(t),}tnT,X(nT+)=(1p)X(nT),U(nT+)=(1ρp)U(nT),}t=nT,X(0+)=X0,U(0+)=U0, (3.2)

    which presents the dynamics of the system in the absence of the infected plants and vectors. For system (3.2), we can address it in any impulsive interval (nT,(n+1)T] and obtain:

    {X(t)=(X(nT+)rg)exp(g(tnT))+rg,U(t)=(U(nT+)μc)exp(c(tnT))+μc. (3.3)

    Denote Xn=X(nT+) and Un=U(nT+), then:

    {Xn+1=(1p)Xnexp(gT)+r(1p)g(1exp(gT)),Un+1=(1pρ)Xnexp(cT)+μ(1pρ)c(1exp(cT)). (3.4)

    There is a steady state (X,U), which suggests that system (3.2) has a positive periodic solution (X(t),U(t)), where X(t)=(Xrg)exp(g(tnT))+rg, U(t)=(Uμc)exp(c(tnT))+μc with X=r(1p)(1exp(gT))g(1(1p)exp(gT)) and U=μ(1pρ)(1exp(cT))c(1(1pρ)exp(cT)). Hence, the following lemma can be obtained.

    Lemma 3.1. System (3.2) has a positive periodic solution (X(t),U(t)). For any solution of (3.2), this study obtains X(t)X(t), U(t)U(t) as t.

    Proof. From (3.3) and (3.4), we obtain the solution of (3.2) as

    {X(t)=(X0(1p)nexp(ngT)+(1(1p)nexp(ngT))r(1p)(1exp(gT))g(1(1p)exp(gT))rg)exp(g(tnT))+rg,U(t)=(U0(1pρ)nexp(ncT)+(1(1p)nexp(ncT))μ(1pρ)(1exp(ncT))c(1(1pρ)exp(cT))μc)exp(c(tnT))+μc, (3.5)

    which indicates that X(t)X(t) and U(t)U(t) as n and t(nT,(n+1)T].

    Therefore, system (3.1) has a disease-free periodic solution (X(t),0,U(t),0) and its stability has been solved.

    To determine the stability of disease-free periodic solution (X(t),0,U(t),0) of system (3.1), we first calculate the basic reproduction number for the impulsive model (3.1) with the application of the next infection operator for the piecewise continuous periodic system being proposed in [22]. A(t) can be denoted as a n×n matrix, ΦA(.)(t) is suggested as the fundamental solution matrix of the linear ordinary differential system χ=A(t)χ, and r(ΦA(.)(t)) is indicated as the spectral radius of ΦA(.)(t). Define X(t)=x(t)+X(t), Y(t)=y(t), U(t)=u(t)+U(t), V(t)=v(t), χ(t)=(x(t),u(t),y(t),v(t)). The corresponding linear system (3.1) reads as

    {χ(t)=G(t)χ(t),tnT,χ(t+)=Hχ(t),t=nT, (3.6)

    where

    G(t)=(G1(t)G2(t)OF(t)Vr(t)),H=(H1H2OH3) (3.7)

    with

    G1(t)=(g00c),G2(t)=(0k1k2U(t)X(t)0),F(t)=(0k1k2U(t)X(t)0),Vr=(g00c) (3.8)

    and

    H1=(1p001ρp),H2=(00U(nT)ρ(pα)X(nT)0),H3=(1α001). (3.9)

    Let ΦG(t)=Φij, 1i,j2, the fundamental solution matrix of system (3.6). Then, we have ΦG(t)=G(t)ΦG(t) with the initial value ΦG(0)=I4. Based on the further computation, it is suggested that:

    ΦG(t)=(exp(G1t)Φ12OΦFVr(t)), (3.10)

    then we have

    HΦG(T)=(H1exp(G1T)H2Φ12OH3ΦFVr(T)). (3.11)

    It is obvious that r(H1exp(G1T))<1, by [22], the basic reproduction number for system (3.6) is provided as follow

    R1=r(H3ΦFVr(T)). (3.12)

    Using Floquet theory, if R1<1, we obtain the following result.

    Theorem 3.2. The disease-free periodic solution (X(t),0,U(t),0) of model (3.1) is locally asymptotically stable when R1<1.

    According to the persistence of the system, it can be found that plants (susceptible and infected) and vectors (susceptible and infected) can coexist, indicating that some conditions are satisfied, and the disease will not disappear. If we aim to eliminate the disease, the persistence indicates that control strategies are inefficient. Moreover, the permanent condition acquired from the exploration of the system can offer scientific support for determining the key factors causing failure and the effectiveness of control strategies, which later benefit us in establishing a good control program.

    Theorem 3.3. When R2=r(H3ΦFVr(T))>1, the disease is uniform persistence, i.e., there exists η>0 such that limtinfY(t)η>0,limtinfV(t)η>0. Here,

    H3=(1α001ρα). (3.13)

    Proof. At first, we claim that there exists η>0, and thus:

    limtsupY(t)η>0,limtsupV(t)η>0. (3.14)

    Next, there is a t1>0 such that Y(t)<η,V(t)<η for all tt1. According to the first and third equations in system (3.1), the following can be obtained:

    {dX(t)dtrk1X(t)N(t)ηgX(t),tnT,dU(t)dtμk2U(t)N(t)ηcU(t),tnT,X(t+)=(1p)X(t),t=nT,U(t+)(1ρ(α+p))U(t),t=nT, (3.15)

    and focus on the auxiliary system

    {dz1(t)dt=rk1ηgz1(t),tnT,dz2(t)dt=μk2z2(t)N(t)ηcz2(t),tnT,z1(t+)=(1p)z1(t),t=nT,z2(t+)=(1ρ(α+p))z2(t),t=nT. (3.16)

    Since system (3.16) refers to a quasimonotone increasing system, based on the comparison theorem [23], the following can be obtained:

    X(t)z1(t),U(t)z2(t). (3.17)

    Using the solution of system (3.2), we acquire that model (3.16) admits a globally asymptotically stable positive periodic solution z(t)=(z1(t),z2(t)), and limη0z(t)=(X(t),U(t)). Then, there exists η1 small enough and ε1>0 such that z1(t)X(t)ε1 and z2(t)U(t)ε1for η<η1. Therefore, by the comparison theorem, there is t2t1 and ε2>0, and thus

    X(t)z1(t)z1(t)ε2X(t)ε1ε2,U(t)z2(t)z2(t)ε2U(t)ε1ε2. (3.18)

    By substituting the above inequalities into the second and fourth equations of system (3.1), we acquire:

    {dY(t)dtk1V(t)N(t)(X(t)ε1ε2)gY(t),tnT,dV(t)dtk2Y(t)N(t)(U(t)ε1ε2)cV(t),tnT,Y(t+)=(1α)Y(t),t=nT,V(t+)=(1ραY(t)N(t))V(t),t=nT. (3.19)

    Supplementing the equations of plant population and vector population respectively provides:

    {dN(t)dt=rgN(t),tnT,dM(t)dt=μcM(t),tnT,N(t+)=(1p)X(t)+(1α)Y(t)(1p)N(t),t=nT,M(t+)=(1ρpX(t)N(t)ραY(t)N(t))U(t)+(1ραY(t)N(t))V(t)M(t),t=nT. (3.20)

    Denoting (N1(t),M1(t)) is the solution of system (3.20) and (X1(t),U1(t)) as that of system (3.2). Comparing the equations in system (3.20) and (3.2) yields N1(t)X1(t),M1(t)U1(t). Since X1(t)X(t), U1(t)U(t), then we have N1(t)X(t), M1(t)U(t), and system (3.19) can be modified as

    {dY(t)dtk1V(t)gY(t),tnT,dV(t)dtk2U(t)X(t)Y(t)cV(t),tnT,Y(t+)=(1α)Y(t),t=nT,V(t+)=(1ραY(t)N(t))V(t)(1ρα)V(t),t=nT. (3.21)

    Further, we concentrate on an auxiliary system

    {dz3(t)dt=k1z4(t)gz3(t),tnT,dz4(t)dt=k2U(t)X(t)z3(t)cz4(t),tnT,z3(t+)=(1α)z3(t),t=nT,z4(t+)=(1ρα)z4(t),t=nT. (3.22)

    It can be re-expressed as

    {(dz3(t)dtdz4(t)dt)=(F(t)Vr(t))(z3(t)z4(t)),tnT,(z3(t+)z4(t+))=H3(z3(t)z4(t)),t=nT. (3.23)

    The solution of system (3.23) can be shown as (z3(t),z4(t))T=ΦFVr(tnT)(z3(nT+),z4(nT+))T. Then, (z3((n+1)T+),z4((n+1)T+))T=H3ΦFVr(T)(z3(nT+),z4(nT+))T. While R2>1, z3(t) and z4(t) as t. Consequently, limtY(t)= and limtV(t)=. Moreover, this is a contradiction with the Y(t) and V(t). Thus, the claim is demonstrated, i.e.,

    limtsupY(t)η>0,limtsupV(t)η>0. (3.24)

    We obtain the two possibilities.

    (ⅰ) Y(t)η and V(t)η for all large t;

    (ⅱ) Y(t) and V(t) oscillations concerning η for all large t.

    When case (ⅰ) is true, our proof is completed. Next, we will consider case (ⅱ). Because limtsupY(t)η, limtsupV(t)η, we can select a t1(n1T,(n1+1)T] such that Y(t1)η and V(t1)η. If the oscillation exists, there must be another t2(n2T,(n2+1)T] such that Y(t2)η and V(t2)η, in which n2n10 is finite. Next, this study will focus on the solution of system (3.1) in the interval [t1,t2]:

    {dY(t)dt=k1V(t)N(t)X(t)gY(t)gY(t),tnT,Y(t+)=(1α)Y(t),t=nT. (3.25)

    Which suggests that

    Y(t)η(1α)n2n1exp(g(t2t1))η(1α)n2n1exp(g(n2n1)T). (3.26)

    Moreover, it follows from

    {dV(t)dt=k2Y(t)N(t)U(t)cV(t)cV(t),tnT,V(t+)=(1ραY(t)N(t))V(t)(1ρα)V(t),t=nT. (3.27)

    Then we obtain:

    V(t)η(1ρα)n2n1exp(c(t2t1))η(1ρα)n2n1exp(c(n2n1)T). (3.28)

    Let δ1=min{η(1α)n2n1exp(g(n2n1)T),η(1ρα)n2n1exp(c(n2n1)T)}, then δ1>0 cannot be infinitely small due to the n2n10 is finite, leading to Y(t)δ1>0 and V(t)δ1>0.

    When t>t2, we take the same steps and get another non-infinitesimal positive δ2. Therefore, the sequence {δi}(i=1,2,3,...), in which δi=min{η(1α)ni+1niexp(g(ni+1ni)T),η(1ρα)ni+1niexp(c(ni+1ni)T)} is non-infinitesimal since ni+1ni0 is finite. The solution of system (3.1) Y(t)δk>0 and V(t)δk>0 is true in the time interval [tk,tk+1],tk(nkT,(nk+1)T],tk+1(nk+1T,(nk+1+1)T]. Let η=min{δ1,δ2,...}, then η{δ1,δ2,...} and it is shown that Y(t)η and V(t)η for all tt1. The proof is complete.

    Remark: It becomes easy to observe that R2 is less than R1. Therefore, when R1>R2>1, the disease persists. Nevertheless, due to the complexity of the model, we obtain only the stronger condition for the persistence in the (3.3), that is R2>1, than R1>1.

    In the current subsection, to study the influence of the removal rate of infected plants on the model, we choose α as the bifurcation parameter to investigate the possible behaviors of nontrivial periodic solution. Now, we proceed to explore bifurcation based on the bifurcation theorem of [24] and [25]. Let x1(t)=X(t),x2(t)=U(t),x3(t)=Y(t),x4(t)=V(t). Then, we employ the following notations in model (3.1)

    {dx1(t)dt=rk1x1(t)x4(t)x1(t)+x3(t)gx1(t):=F1(x1(t),x2(t),x3(t),x4(t)),dx2(t)dt=μk2x2(t)x3(t)x1(t)+x3(t)cx2(t):=F2(x1(t),x2(t),x3(t),x4(t)),dx3(t)dt=k1x1(t)x4(t)x1(t)+x3(t)gx3(t):=F3(x1(t),x2(t),x3(t),x4(t)),dx4(t)dt=k2x2(t)x3(t)x1(t)+x3(t)cx4(t):=F4(x1(t),x2(t),x3(t),x4(t)),}tnT,x1(t+)=(1p)x1(t):=Θ1(α,x1(t),x2(t),x3(t),x4(t)),x2(t+)=(1ρpx1(t)x1(t)+x3(t)ραx3(t)x1(t)+x3(t))x2(t):=Θ2(α,x1(t),x2(t),x3(t),x4(t)),x3(t+)=(1α)x3(t):=Θ3(α,x1(t),x2(t),x3(t),x4(t)),x4(t+)=(1ραx3(t)x1(t)+x3(t))x4(t):=Θ4(α,x1(t),x2(t),x3(t),x4(t)),}t=nT,x1(0+)=x1(0),x2(0+)=x2(0),x3(0+)=x3(0),x4(0+)=x4(0). (3.29)

    First, the following notations are introduced: The solution vector ξ(t):=(x1(t),x2(t),x3(t),x4(t)), the mapping F(ξ(t))=(F1(ξ(t)),F2(ξ(t)),F3(ξ(t)),F4(ξ(t))): R4R4 by the right hand side of the first four equations of system (3.29), and the mapping with impulse to be

    Θ(α,ξ(t))=(Θ1(α,ξ(t)),Θ2(α,ξ(t)),Θ3(α,ξ(t)),Θ4(α,ξ(t)))=((1p)x1(t),(1ρpx1(t)x1(t)+x3(t)ραx3(t)x1(t)+x3(t))x2(t),(1α)x3(t),(1ραx3(t)x1(t)+x3(t))x4(t)).

    Let Φ(t,ξ0) the solution of the system which is consisted of the first four equations of system (3.29), where ξ0=ξ(0). Next, ξ(T)=Φ(T,ξ0)=:Φ(ξ0) and ξ(T+)=Θ(α,Φ(ξ0)). The operator Ψ is defined by:

    Ψ(α,ξ):=(Ψ1(α,ξ),Ψ2(α,ξ),Ψ3(α,ξ),Ψ4(α,ξ))=Θ(α,Φ(ξ)), (3.30)

    and denote by DξΨ the derivative of Ψ with respect to ξ. Later, ξ indicates a periodic solution of period T for system (3.29) if and only if Ψ(α,ξ0)=ξ0. Therefore, to obtain the nontrivial periodic solution of system (3.29), we are required to demonstrate the presence of nontrivial fixed point of Ψ.

    All parameters are fixed except the infective plants removal rate α. Denote that α0 is the critical remove rate, corresponding to R1=r(H3ΦFVR(T))=1. Let ¯ξ=(x1,x2,0,0) be the disease-free periodic solution of system (3.29).

    Denote α=α0+¯α, ξ=ξ0+¯ξ, with ξ0 is the starting point for the disease-free periodic solution with the removal rate α0, and let

    N(¯α,¯ξ)=(N1(¯α,¯ξ),N2(¯α,¯ξ),N3(¯α,¯ξ),N4(¯α,¯ξ))=ξ0+¯ξΨ(α0+α,ξ0+¯ξ).

    Next, the fixed point problem can be expressed:

    N(¯α,¯ξ)=0. (3.31)

    The derivative of N(¯α,¯ξ) with respect to ξ provides:

    DξN(¯α,¯ξ)=E4DξΘ(Φ(t,ξ0))DξΦ(ξ), (3.32)

    where E4 refers to the identity matrix. Based on system (3.29), the equality can be obtained:

    ddt(DξΦ(t,ξ0))=DξF(Φ(t,ξ0))DξΦ(t,ξ0), (3.33)

    with DξΦ(0,ξ0)=E4 and Φ(t,ξ0)=(Φ1(t,ξ0),Φ2(t,ξ0),0,0), then Eq (3.33) takes the form

    ddt(DξΦ(t,ξ0))=G(t)DξΦ(t,ξ0). (3.34)

    The following can be obtained:

    DξN(0,O)=(E2H1eG1TH2Φ120E2H3ΦFVr(T)), (3.35)

    where O=(0,0,0,0). The essential condition for the bifurcation of nontrivial zeros of function N refers to that the determinant of the Jacobian matrix DξN(0,O) is shown to be equal to zeros, for example,

    det(DξN(0,O))=0. (3.36)

    It is not difficult to observe from (3.35) that det(E2H1eG1T)0. Then det(DξN(0,O))=0 can reduce to det(E2H3ΦFVr(T))=0. If R1=r(H3ΦFVr(T))=1, then det(E2H3ΦFVr(T))=0. Now we investigate the sufficient conditions for the existence of bifurcation nontrivial period solutions. With the consideration of convenience, we denote the elements in matrix DξN(0,O) as

    DξN(0,O)=(e00a1b10f0c1d100a0b000c0d0), (3.37)

    from the calculation in Appendix A, we can obtain the expression of each element in the above matrix as follows:

    e0=1(1p)egT,f0=1(1ρp)ecT,a0=1(1α0)Φ3(T,ξ0)ξ3,b0=(1α0)Φ3(T,ξ0)ξ4,c0=Φ4(T,ξ0)ξ3,d0=1Φ4(T,ξ0)ξ4,a1=(1p)Φ1(T,ξ0)ξ3,b1=(1p)Φ1(T,ξ0)ξ4,c1=(1ρp)Φ2(T,ξ0)ξ3Φ2(T,ξ0)ρ(pα0)Φ1(T)Φ3(T,ξ0)ξ3,d1=(1ρp)Φ2(T,ξ0)ξ4Φ2(T,ξ0)ρ(pα0)Φ1(T,ξ0)Φ3(T,ξ0)ξ4.

    Then det(E2H3ΦFVr(T))=0 suggests that there is a constant k0 such that c0=ka0 and d0=kb0. Moreover, since det(DξN(0,O))=0, we cannot employ the Implicit Function Theorem for giving variable ξ as a function of α. By [24] and [25], we perform a Lyapunov-Schmide reduction to obtain a system of equations, where the Implicit Function Theorem can be applied. It is easy to observe that dimKer(DξN(0,O))=1 and a basis in Ker(DξN(0,O)) is

    Y1=(Y11,Y12,Y13,Y14)=(a1b0a0e0b1e0,c1b0a0f0d1f0,b0a0,1).

    Moreover, Y2=(1,0,0,0), Y3=(0,1,0,0) and Y4=(0,0,1,0) compose a basis in Im(DξN(0,O)). Later, for ¯ξR4, based on the decomposition R4=Ker(DξN(0,O))Im(DξN(0,O)), the following can be obtained:

    ¯ξ=α1Y1+α2Y2+α3Y3+α4Y4, (3.38)

    where αiR(i=1,2,3,4) are special. Thus, (3.31) is equivalent to

    Ni(¯α,α1Y1+α2Y2+α3Y3+α4Y4)=0,i=1,2,3,4. (3.39)

    From the first three equations of (3.39), the following can be obtained:

    D(N1,N2,N3)(0,O)D(α2,α3,α4)=a0e0f00. (3.40)

    Thus, from the Implicit Function Theorem, one can deal with equations (3.39) as i=2,3,4 regarding (0,O) with respect to αi, i=2,3,4 as function of ¯α and α1, and find ~αi=~αi(¯α,α1) and thus ~αi(0,0)=0, i=2,3,4, and

    Ni(¯α,αY1+~α2Y2+~α3Y3+~α4Y4)=0, (3.41)

    i=1,2,3. Next, only N(ˉα,ˉξ)=0 if and only if

    f(ˉα,α1)=N4(ˉα,α1Y1+~α2Y2+~α3Y3+~α4Y4)=0. (3.42)

    It can be known that f(ˉα,α1) vanishes at (0,0). Therefore, it becomes essential to compute higher order derivatives of f(ˉα,α1) up to the order i for which Dif(0,0)0. Similar to the calculation in [24] and [25], the first partial derivatives of f about ˉα and α1 satisfy

    f(0,0)ˉα=f(0,0)α1=0. (3.43)

    (See Appendix B). Therefore, we obtain Df(0,0)=(0,0). Then, it becomes essential to calculate the second partial derivative of f.

    Let A=2f(ˉα,α1)ˉα2, B=2f(ˉα,α1)ˉαα1 and C=2f(ˉα,α1)α21. From the calculation in Appendix C, it can be found that A=0,

    B=k(Φ3(ξ0)ξ3Y13+Φ3(ξ0)ξ4Y14) (3.44)

    and

    C=4i=14j=1Y1iY1j(2Φ4(ξ0)ξiξjk(1α0)2Φ3(ξ0)ξiξj). (3.45)

    Therefore, we have

    f(ˉα,α1)=2Bˉαα1+Cα21+o(ˉα,α1)(ˉα2+α21)=α1˜f(ˉα,α1), (3.46)

    where

    ˜f(ˉα,α1)=2Bˉα+Cα1+1α1o(ˉα,α1)(ˉα2+α21). (3.47)

    Furthermore,

    ˜f(0,0)˜α=2B,˜f(0,0)α1=C. (3.48)

    Thus, for B0, we can employ the Implicit Function Theorem, implying that ˉα=φ1(α1), such that for all α1 near 0, ˜f(φ1(α1),α1)=0. Furthermore, for C0, we can also be α1=φ1(ˉα), such that for all ˉα near 0, ˜f(φ2(ˉα),ˉα)=0. Then, if BC0, we have α1ˉα2BC. There is a supercritical bifurcation to a nontrivial periodic solution near the fixed point ξ0, if BC<0, and it is subcritical, if BC>0. We know that the threshold R1 lowers with the increasing α, then a supercritical bifurcation in the ˉαα1 plane indicates a backward bifurcation in the model. Additionally, the subcritical bifurcation equated to a forward bifurcation. Finally, the following theorem below can be concluded.

    Theorem 3.4. If BC<0, system (3.29) admits a backward bifurcation as α passes via the critical value α0. Then, a forward bifurcation will occur BC>0.

    First, the analysis result of model (2.1) suggests that the system has backward bifurcation with the basic reproduction number R0=1. As shown in Figure 1, we fixed other parameters to let α change, and gave the backward bifurcation diagram of the system (2.1). It can be observed from the Figure 1 that when R0(α0)R0<1, the system has a stable positive equilibrium point and an unstable positive equilibrium point. The disease can be eradicated only when R0<R0(α0). Thus, when the infected plants are removed, the disease can be eradicated only if the removal rate α>α0. As presented in Figure 2, we choose the parameter such that R0=0.9975<1 changes the initial value of the system. With the initial value being (7,0.5,750,30), the system has a disease free equilibrium point that remains globally asymptotically stable. When the initial value is (70,0.5,750,30), the system possesses a stable positive solution. Therefore, it is necessary to consider the initial infection when preventing and controlling plant viruses.

    Second, the disease control threshold condition R1 is obtained by analyzing the model (3.1). However, owing to the complexity of the model, we cannot obtain the explicit expression of the threshold R1. To describe the effect of parameters on the threshold conditions, the method in [26] is used to give a sensitivity analysis of the threshold R1. Owing to the lack of information, we choose uniform distribution for all parameters and possible ranges of parameters in Table 1. It can be observed from Figure 3 that among the four control parameters, the increasing α and ρ can decrease R1, while increasing p and T leads to an increase in R1. As presented in Figure 4, we fixed other parameters except the removal rate p of healthy plants and applied numerical simulation to explore the impact of the removal rate p of healthy plants on the threshold R1. The results demonstrated that the threshold R1 increased with the increasing removal rate p of healthy plants. Therefore, when controlling the spread of plant diseases, we should attempt to avoid removing healthy plants. Moreover, because R1 depends on α and ρ, we use a three-dimensional surface (Figure 5) to illustrate the dependence of R1 on these two parameters. The numerical simulation indicates that high values of ρ and α will guarantee R1<1. As a result, when the control strategy is implemented, not only should the infected plants be removed, but also the vectors should be eliminated together, aiming to ensure that the plant virus disease can be controlled.

    Table 1.  The definitions of all parameters and their baseline values.
    Parameter Interpretation Standard value Range Source
    r Recruitment rate of plant 0.015(day1) 0–0.025 Assumed
    α Plant roguing rate 0.003(day1) 0–0.033 [19]
    g Plant loss/harvesting rate 0.003(day1) 0.002–0.004 [19]
    μ Recruitment rate of vector 400(day1) 0–2500 Assumed
    c Vector mortality 0.12(day1) 0.06–0.18 [19]
    k1 Infection rate 0.008(vector1day1) 0.002–0.032 [19]
    k2 Acquisition rate 0.008(plant1day1) 0.002–0.032 [19]

     | Show Table
    DownLoad: CSV
    Figure 3.  Sensitivity analysis of R1 to all parameters.
    Figure 4.  The impacts of the parameter p on the threshold level R1. The set of parameter values is k1=k2=0.003, μ=400, c=0.12, g=0.002, α=0.9, ρ=0.86 and T=15.
    Figure 5.  The impacts of the parameter α and ρ on the threshold level R1. The set of parameter values is k1=k2=0.003, μ=400, c=0.12, g=0.002 and T=15.

    Finally, it can be found from Theorem 3.4 that system (3.1) may have backward bifurcation. In Figure 6, we select the parameter to make R1=0.9775<1. For different initial values, it is found that (1) the system with initial value (2,0.1,1000,15) has a stable disease eradication periodic solution; and (2) with the initial value being (2,0.1,1000,150), the system has a stable positive periodic solution.

    Figure 6.  Solution of system (3.1) of the number of infectious plants, Y(t), The set of parameter values is k1=k2=0.003, μ=400, c=0.12, g=0.002, α=0.8, T=15, and ρ=0.86, so R1=0.9721<1, for two different sets of initial conditions. The first initial conditions (conforming to the red line) is (7,0.1,1000,150). The second initial conditions (conforming to the blue line) is (7,0.1,1000,15).

    Roguing control is a very vital and effective control strategy for plant virus transmission. In [27], the author considered the removal of infected plants and did not consider the elimination of vectors on plants when removing infected plants. We consider two kinds of plant virus transmissions control strategies to remove infected plants and vectors in a continuous and pulse manner.

    For control strategies with continuous removal of infected plants and vectors, through the analysis of system (2.1), we obtain that when the threshold R0<1, the disease-free equilibrium is locally asymptotically stable, and the system may have two positive equilibrium points. Using the conclusion in [21], we demonstrate the sufficient condition for the existence of backward bifurcation in system (2.1). Based on the results, the disease can be eliminated only when R0<R0(α0), not R0<1.

    For control strategies with an impulse to remove infected plants and vectors, through the analysis of system (3.1), we find that when the threshold R1<1, the disease-free periodic solution of system (3.1) is locally asymptotically stable. Due to the complexity of the model, we use a strong threshold condition threshold R2>1 (R2<R1) to prove the persistence of the disease. Using the nonlinear fixed point theory, we give the conditions for forward and backward bifurcation of impulsive control systems. The results show that the disease can be eradicated only when R1<R1(α0), not R1<1.

    Backward bifurcation of the system indicates that based on different initial conditions, the system may have stable disease free periodic solutions or stable positive periodic solutions, which undoubtedly brings great difficulties to plant disease control. Our results suggest that there is a backward bifurcation phenomenon in continuous systems, and there is also a backward bifurcation phenomenon in impulsive control systems. Therefore, when controlling plant virus diseases, if the initial infection scale is large, we should kill a large number of plants and vectors to control the disease.

    Finally, we note that the introduction of natural enemies will control the number of vectors. This is undoubtedly beneficial to the spread of plant viruses. Therefore, how to establish a plant virus transmission model with the mixed control of natural enemies and the removal of infected plants will be our focus in the later stage.

    The authors declare that they have not used Artificial Intelligence (AI) tools in the creation of this article.

    The current work was supported by the National Natural Science Foundation of Qinghai Province (No.2022-ZJ-T02).

    We declare that we have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.

    From

    DξΨ(α,ξ)=DξΘ(α,Φ(ξ))DξΦ(ξ)=(Θ1(α,ξ)ξ1Θ1(α,ξ)ξ2Θ1(α,ξ)ξ3Θ1(α,ξ)ξ4Θ2(α,ξ)ξ1Θ2(α,ξ)ξ2Θ2(α,ξ)ξ3Θ2(α,ξ)ξ4Θ3(α,ξ)ξ1Θ3(α,ξ)ξ2Θ3(α,ξ)ξ3Θ3(α,ξ)ξ4Θ4(α,ξ)ξ1Θ4(α,ξ)ξ2Θ4(α,ξ)ξ3Θ4(α,ξ)ξ4)(Φ1(ξ)ξ1Φ1(ξ)ξ2Φ1(ξ)ξ3Φ1(ξ)ξ4Φ2(ξ)ξ1Φ2(ξ)ξ2Φ2(ξ)ξ3Φ2(ξ)ξ4Φ3(ξ)ξ1Φ3(ξ)ξ2Φ3(ξ)ξ3Φ3(ξ)ξ4Φ4(ξ)ξ1Φ4(ξ)ξ2Φ4(ξ)ξ3Φ4(ξ)ξ4)

    at ξ0=(ξ10,ξ20,0,0) (ξ10=x1(0),ξ10=x1(0)), we have

    DξΨ(α0,ξ0)=DξΘ(α0,Φ(ξ0))DξΦ(ξ0)=(1p00001ρpρ(pα0)Φ2(ξ0)Φ1(ξ0)0001α000001)(Φ1(ξ0)ξ1Φ1(ξ0)ξ2Φ1(ξ0)ξ3Φ1(ξ0)ξ4Φ2(ξ0)ξ1Φ2(ξ0)ξ2Φ2(ξ0)ξ3Φ2(ξ0)ξ4Φ3(ξ0)ξ1Φ3(ξ0)ξ2Φ3(ξ0)ξ3Φ3(ξ0)ξ4Φ4(ξ0)ξ1Φ4(ξ0)ξ2Φ4(ξ0)ξ3Φ4(ξ0)ξ4)

    From the variational equation related to the first four equation of system (3.29),

    ddt(DξΦ(t,ξ0))=DξF(Φ(t,ξ0))DξΦ(t,ξ0) (A.1)

    with the initial condition DξΦ(0,ξ0)=E4 and Φ(t,ξ0)=(Φ1(t,ξ0),Φ2(t,ξ0),0,0), we obtain

    Φ1(ξ0)ξ1=egT,Φ2(ξ0)ξ2=ecT, (A.2)
    Φ1(ξ0)ξ2=Φ2(ξ0)ξ1=Φ3(ξ0)ξ1=Φ3(ξ0)ξ2=Φ4(ξ0)ξ2Φ2(ξ0)ξ2=0. (A.3)

    Then, from (3.30) and (A.3), we get

    DˉξN(0,O)=(e00a1b10f0c1d100a0b000c0d0), (A.4)

    where

    e0=1(1p)egT,f0=1(1ρp)ecT,a0=1(1α0)Φ3(ξ0)ξ3,d0=1Φ4(ξ0)ξ4,a1=(1p)Φ1(ξ0)ξ3,b1=(1p)Φ1(ξ0)ξ4,c1=(1ρp)Φ2(ξ0)ξ3ρ(pα0)Φ2(ξ0)Φ1(ξ0)Φ3(ξ0)ξ3,d1=(1ρp)Φ2(ξ0)ξ4ρ(pα0)Φ2(ξ0)Φ1(ξ0)Φ3(ξ0)ξ4,b0=(1α0)Φ3(ξ0)ξ4,c0=Φ4(ξ0)ξ3.

    We obtain:

    f(0,O)ˉα=Φ4(ξ0)((ρΦ3(ξ0)+ρ(α0+ˉα)3i=1Φ3(ξ0)ξi˜αi+1ˉα)(Φ1(ξ0)+Φ3(ξ0))ρα0Φ3(ξ0)(3i=1Φ1(ξ0)ξi˜αi+1ˉα+3i=1Φ3(ξ0)ξi˜αi+1ˉα)(Φ1(ξ0)+Φ3(ξ0))2)(1ρα0Φ3(ξ0)Φ1(ξ0)+Φ3(ξ0))3i=1Φ4(ξ0)ξi˜αi+1ˉα. (B.1)

    Since

    Φ3(ξ0)=Φ4(ξ0)=0,Φ3(ξ)ξ1=Φ3(ξ)ξ2=Φ4(ξ)ξ1=Φ4(ξ)ξ2=0, (B.2)

    then

    f(0,O)ˉα=c0~α4ˉα. (B.3)

    Consider Eq (3.41) as i=1, we have

    0=N1(0,O)ˉα=~α0(0,O)ˉα(1p)(Φ1(ξ0)ξ1~α2(0,O)ˉα+Φ1(ξ0)ξ2~α3(0,O)ˉα+Φ1(ξ0)ξ3~α4(0,O)ˉα)=e0~α2(0,O)ˉα+0~α3(0,O)ˉα+a1~α4(0,O)ˉα. (B.4)

    One can similarly acquire from Eq (3.41) as i=2,3 that

    f0~α3(0,O)ˉα+c1~α4(0,O)ˉα=0,a0~α4(0,O)ˉα=0. (B.5)

    It can be deduced from (B.4) and (B.5) that

    ~α2(0,O)ˉα=~α3(0,O)ˉα=~α4(0,O)ˉα=0. (B.6)

    Thus f(0,O)ˉα=0. Additionally,

    0=N1(0,O)ξ1(Y11+~α2(0,O)α1)+N1(0,O)ξ2(Y12+~α3(0,O)α1)+N1(0,O)ξ3(Y13+~α4(0,O)α1)+N1(0,O)ξ4Y11=N1(0,O)ξ1Y11+N4(0,O)ξ2Y12+N1(0,O)ξ3Y13+N1(0,O)ξ4Y14+N1(0,O)ξ1~α2(0,O)α1+N1(0,O)ξ2~α3(0,O)α1+N1(0,O)ξ3~α4(0,O)α1, (B.7)

    because Y1 is a basis in Ker(Dξ(0,O), that is:

    Ni(0,O)ξ1Y11+Ni(0,O)ξ2Y12+Ni(0,O)ξ3Y13+Ni(0,O)ξ4Y14=0,i=1,2,3,4. (B.8)

    Therefore, we can deduce from (12.3) and (B.7) that

    e0~α2(0,O)α1+0~α3(0,O)α1+a1~α4(0,O)α1=0. (B.9)

    Similarly, as i=2,3, we can obtain that

    0~α2(0,O)α1+f0~α3(0,O)α1+c1~α4(0,O)α1=0,0~α2(0,O)α1+0~α3(0,O)α1+a0~α4(0,O)α1=0. (B.10)

    From (B.9) and (B.10) we obtain that

    ~α2(0,O)α1=~α3(0,O)α1=~α4(0,O)α1=0. (B.11)

    Since

    f(0,O)α1=N4(0,O)ξ1(Y11+~α2(0,O)α1)+N4(0,O)ξ2(Y12+~α3(0,O)α1)+N4(0,O)ξ3(Y13+~α4(0,O)α1)+N4(0,O)ξ4Y11=N4(0,O)ξ1Y11+N4(0,O)ξ2Y12+N4(0,O)ξ3Y13+N4(0,O)ξ4Y14+N4(0,O)ξ1~α2(0,O)α1+N4(0,O)ξ2~α3(0,O)α1+N4(0,O)ξ3~α4(0,O)α1=0, (B.12)

    submitting (B.11) into (B.12), we get f(0,O)α1.

    Let A=2f(0,O)ˉα2, B=2f(0,O)ˉαα1, C=2f(0,O)α21.

    Calculation of A.

    2f(0,O)ˉα2=ˉα(f(0,O)ˉα)=3i=1Φ4(ξ0)ξi˜αi+1ˉα((ρΦ3(ξ0)+ρ(α0+ˉα)3i=1Φ3(ξ0)ξi˜αi+1ˉα)(Φ1(ξ0)+Φ3(ξ0))ρα0Φ3(ξ0)(3i=1Φ1(ξ0)ξi˜αi+1ˉα+3i=1Φ3(ξ0)ξi˜αi+1ˉα)(Φ1(ξ0)+Φ3(ξ0))2)+Φ4(ξ0)ˉα((ρΦ3(ξ0)+ρ(α0+ˉα)3i=1Φ3(ξ0)ξi˜αi+1ˉα)(Φ1(ξ0)+Φ3(ξ0))ρα0Φ3(ξ0)(3i=1Φ1(ξ0)ξi˜αi+1ˉα+3i=1Φ3(ξ0)ξi˜αi+1ˉα)(Φ1(ξ0)+Φ3(ξ0))2)3i=1Φ4(ξ0)ξi˜αi+1ˉαˉα(1ρα0Φ3(ξ0)Φ1(ξ0)+Φ3(ξ0))(1ρα0Φ3(ξ0)Φ1(ξ0)+Φ3(ξ0))ˉα(3i=1Φ4(ξ0)ξi˜αi+1ˉα), (C.1)

    submitting (B.2) and (B.6) into (C.1), it can be deduced that

    2f(0,O)ˉα2=Φ4(ξ0)ξ32˜α4ˉα2=c02~α4ˉα2. (C.2)

    Consider Eq (3.41) as i=3, we obtains that

    0=2N3(0,O)ˉα2=ˉα(N3(0,O)ˉα)=ˉα(˜α4(0,O)ˉα+Φ3(ξ0)(1α0ˉα)3i=1Φ3(ξ0)ξi˜αi+1(0,O)ˉα)=2˜α4(0,O)ˉα2+23i=1Φ3(ξ0)ξi˜αi+1(0,O)ˉα(1α0)ˉα(3i=1Φ3(ξ0)ξi˜αi+1(0,O)ˉα)=2˜α4(0,O)ˉα2+23i=1Φ3(ξ0)ξi˜αi+1(0,O)ˉα(1α0)(˜α2(0,O)ˉαˉα(3i=1Φ3(ξ0)ξi˜αi+1(0,O)ˉα)+Φ3(ξ0)ξ12˜α2(0,O)ˉα2+˜α2(0,O)ˉαˉα(3i=1Φ3(ξ0)ξi˜αi+1(0,O)ˉα)+Φ3(ξ0)ξ22˜α3(0,O)ˉα2+˜α4(0,O)ˉαˉα(3i=1Φ3(ξ0)ξi˜αi+1(0,O)ˉα)+Φ3(ξ0)ξ32˜α4(0,O)ˉα2). (C.3)

    Similarly, by submitting (B.2) and (B.6) into (C.3), we can obtain that

    a02˜α4(0,O)ˉα2=0, (C.4)

    then 2˜α4(0,O)ˉα2=0. By substitute the result of the above formula into (C.2), we have that A=2f(0,O)ˉα2=0.

    Calculation of B.

    First, we calculate the value of 2~α4(0,O)ˉαα1,

    0=2N3(0,O)α1ˉα=α1(˜α4(0,O)ˉα+Φ3(ξ0)(1α0ˉα)3i=1Φ3(ξ0)ξi˜αi+1(0,O)ˉα)=2~α4(0,O)ˉαα1+Φ3(ξ0)α1(1α0)α1(Φ3(ξ0)ξ1˜α2(0,O)ˉα+Φ3(ξ0)ξ2˜α3(0,O)ˉα+Φ3(ξ0)ξ3˜α4(0,O)ˉα)=2˜α4(0,O)ˉαα1+Φ3(ξ0)ξ1(Y11+˜α2(0,O)ˉα)+Φ3(ξ0)ξ2(Y12+˜α3(0,O)ˉα)+Φ3(ξ0)ξ3(Y13+˜α4(0,O)ˉα)+Φ3(ξ0)ξ4Y14(1α0)(˜α2(0,O)ˉαα1(Φ3(ξ0)ξ1)+Φ3(ξ0)ξ12˜α4(0,O)ˉαα1+˜α3(0,O)ˉαα1(Φ3(ξ0)ξ2)+Φ3(ξ0)ξ22˜α4(0,O)ˉαα1+˜α4(0,O)ˉαα1(Φ3(ξ0)ξ3)+Φ3(ξ0)ξ32˜α4(0,O)ˉαα1), (C.5)

    submitting (B.2) and (B.6) into (C.5), we can therefore deduce that

    2˜α4(0,O)ˉαα1=1a0(Φ3(ξ0)ξ3Y13+Φ3(ξ0)ξ4Y14). (C.6)

    It can be calculated that

    2f(0,O)ˉαα1=α1(Φ4(ξ0)((ρΦ3(ξ0)+ρ(α0+ˉα)3i=1Φ3(ξ0)ξi˜αi+1ˉα)(Φ1(ξ0)+Φ3(ξ0))ρα0Φ3(ξ0)(3i=1Φ1(ξ0)ξi˜αi+1ˉα+3i=1Φ3(ξ0)ξi˜αi+1ˉα)(Φ1(ξ0)+Φ3(ξ0))2)(1ρ(α0+ˉα)Φ3(ξ0)Φ1(ξ0)+Φ3(ξ0))3i=1Φ4(ξ0)ξi˜αi+1ˉα)=Φ4(ξ0)α1(ρΦ3(ξ0)+ρ(α0+ˉα)3i=1Φ3(ξ0)ξi˜αi+1ˉαΦ1(ξ0)+Φ3(ξ0)ρ(α0+ˉα)(3i=1Φ1(ξ0)ξi˜αi+1ˉα+3i=1Φ3(ξ0)ξi˜αi+1ˉα)(Φ1(ξ0)+Φ3(ξ0))2)+Φ4(ξ0)(α1(ρΦ3(ξ0)+ρ(α0+ˉα)3i=1Φ3(ξ0)ξi˜αi+1ˉαΦ1(ξ0)+Φ3(ξ0))α1(ρ(α0+ˉα)(3i=1Φ1(ξ0)ξi˜αi+1ˉα+3i=1Φ3(ξ0)ξi˜αi+1ˉα)(Φ1(ξ0)+Φ3(ξ0))2))α1(1ρ(α0+ˉα)Φ3(ξ0)Φ1(ξ0)+Φ3(ξ0))3i=1Φ4(ξ0)ξi˜αi+1ˉα(1ρ(α0+ˉα)Φ3(ξ0)Φ1(ξ0)+Φ3(ξ0))α1(3i=1Φ4(ξ0)ξi˜αi+1ˉα), (C.7)

    submitting (B.2), (B.6) and (C.6) into (C.7), we have

    B=2f(0,O)ˉαα1=1a0(Φ3(ξ0)ξ3Y13+Φ3(ξ0)ξ4Y14)Φ4(ξ0)ξ3=k(Φ3(ξ0)ξ3Y13+Φ3(ξ0)ξ4Y14). (C.8)

    Calculation of C.

    2f(0,O)α21=α1(N4(0,O)ξ1(Y11+˜α2(0,O)α1)+N4(0,O)ξ2(Y12+˜α3(0,O)α1)+N4(0,O)ξ3(Y13+˜α4(0,O)α1)+N4(0,O)ξ4Y14)=(Y11+˜α2(0,O)α1)α1(N4(0,O)ξ1)+N4(0,O)ξ12˜α2(0,O)α21+(Y12+˜α3(0,O)α1)α1(N4(0,O)ξ2)+N4(0,O)ξ22˜α3(0,O)α21+(Y13+˜α4(0,O)α1)α1(N4(0,O)ξ3)+N4(0,O)ξ32˜α4(0,O)α21+Y14α1(N4(0,O)ξ4)=α1(Y11N4(0,O)ξ1+Y12N4(0,O)ξ2+Y13N4(0,O)ξ3+Y14N4(0,O)ξ4)+N4(0,O)ξ32˜α4(0,O)α21=Y11(2N4(0,O)ξ21(Y11+˜α2(0,O)α1)+2N4(0,O)ξ1ξ2(Y12+˜α3(0,O)α1)+2N4(0,O)ξ1ξ3(Y13+˜α4(0,O)α1)+2N4(0,O)ξ1ξ4Y14)+Y12(2N4(0,O)ξ1ξ2(Y11+˜α2(0,O)α1)+2N4(0,O)ξ22(Y12+˜α3(0,O)α1)+2N4(0,O)ξ2ξ3(Y13+˜α4(0,O)α1)+2N4(0,O)ξ2ξ4Y14)+Y13(2N4(0,O)ξ1ξ3(Y11+˜α2(0,O)α1)+2N4(0,O)ξ2ξ3(Y12+˜α3(0,O)α1)+2N4(0,O)ξ23(Y13+˜α4(0,O)α1)+2N4(0,O)ξ3ξ4Y14)+Y14(2N4(0,O)ξ1ξ4(Y11+˜α2(0,O)α1)+2N4(0,O)ξ2ξ4(Y12+˜α3(0,O)α1)+2N4(0,O)ξ3ξ4(Y13+˜α4(0,O)α1)+2N4(0,O)ξ24Y14)+N4(0,O)ξ32˜α4α21=4i=14j=1Y1iY1j2N4(0,O)ξiξj+c02˜α4(0,O)α21. (C.9)

    Consider Eq (3.41) as i=1, we obtains that

    0=2N1(0,O)α21=α1(N1(0,O)ξ1(Y11+˜α2(0,O)α1)+N1(0,O)ξ2(Y12+˜α3(0,O)α1)+N1(0,O)ξ3(Y13+˜α4(0,O)α1)+N1(0,O)ξ4Y14)=(Y11+˜α2(0,O)α1)α1(N1(0,O)ξ1)+N1(0,O)ξ12˜α2(0,O)α21+(Y12+˜α3(0,O)α1)α1(N1(0,O)ξ2)+N1(0,O)ξ22˜α3(0,O)α21+(Y13+˜α4(0,O)α1)α1(N1(0,O)ξ3)+N1(0,O)ξ32˜α4(0,O)α21+Y14α1(N1(0,O)ξ4)=α1(Y11N1(0,O)ξ1+Y12N1(0,O)ξ2+Y13N1(0,O)ξ3+Y14N1(0,O)ξ4)+N1(0,O)ξ12˜α2(0,O)α21+N1(0,O)ξ32˜α4(0,O)α21=Y11(2N1(0,O)ξ21(Y11+˜α2(0,O)α1)+2N1(0,O)ξ1ξ2(Y12+˜α3(0,O)α1)+2N4(0,O)ξ1ξ3(Y13+˜α4(0,O)α1)+2N4(0,O)ξ1ξ4Y14)+Y12(2N1(0,O)ξ1ξ2(Y11+˜α2(0,O)α1)+2N1(0,O)ξ22(Y12+˜α3(0,O)α1)+2N4(0,O)ξ2ξ3(Y13+˜α4(0,O)α1)+2N4(0,O)ξ2ξ4Y14)+Y13(2N1(0,O)ξ1ξ3(Y11+˜α2(0,O)α1)+2N1(0,O)ξ2ξ3(Y12+˜α3(0,O)α1)+2N4(0,O)ξ23(Y13+˜α4(0,O)α1)+2N4(0,O)ξ3ξ4Y14)+Y14(2N1(0,O)ξ1ξ4(Y11+˜α2(0,O)α1)+2N1(0,O)ξ2ξ4(Y12+˜α3(0,O)α1)+2N4(0,O)ξ3ξ4(Y13+˜α4(0,O)α1)+2N4(0,O)ξ24Y14)+N1(0,O)ξ32˜α4α21=4i=14j=1Y1iY1j2N1(0,O)ξiξj+a02˜α2(0,O)α21+a12˜α4(0,O)α21, (C.10)

    then we have that

    e02˜α2(0,O)α21+a12˜α4(0,O)α21=4i=14j=1Y1iY1j2N1(0,O)ξiξj. (C.11)

    Similarly, we can obtain from Eq (3.41) as i=2,3 that

    f02˜α3(0,O)α21+c12˜α4(0,O)α21=4i=14j=1Y1iY1j2N2(0,O)ξiξj,a02˜α4(0,O)α21=4i=14j=1Y1iY1j2N3(0,O)ξiξj. (C.12)

    By solving (C.11) and (C.12), we can get the values of 2˜αi(0,O)α21, i=2,3,4, and submit it as i=4 into (C.9), one can obtain:

    C=2f(0,O)α21=4i=14j=1Y1iY1j(2N4(0,O)ξiξjk2N3(0,O)ξiξj)=4i=14j=1Y1iY1j(2Φ4(ξ0)ξiξjk(1α0)2Φ3(ξ0)ξiξj). (C.13)


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