Research article

Insecticidal Efficacy of Syzygium aromaticum, Tephrosia vogelii and Croton dichogamus Extracts against Plutella xylostella and Trichoplusia ni on Brassica oleracea crop in Northern Tanzania

  • The insecticidal efficacy of 10%, 5% and 1% w/v of Tephrosia vogelii, Croton dichogamus and Syzygium aromaticum aqueous plant extracts were assessed against larvae of Plutella xylostella and Trichoplusia ni on Brassica oleracea var. capitata crop field. Synthetic organophosphate pesticide (Chlorpyrifos) was used as a positive control and the negative controls were water and water plus soap. It was revealed that, aqueous plant extracts significantly (P ≤ 0.05) controlled the number of P. xylostella and T. ni larvae compared to negative controls. The 10% concentration of aqueous plant extracts showed significant higher efficacy in terms of reducing the insect population and their damage than 1% and 5 % concentration. The population of P. xylostella larvae per B. oleracea in five weeks of treatment applications at 10% w/v of T. vogelii, C. dichogamus and S. aromaticum aqueous plant extracts were 0.08, 0.15, 0.13, 0.05 and 0.08; 0.08, 0.20, 0.15, 0.13 and 0.18; 0.03, 0.05, 0.15, 0.18 and 0.13, respectively which was significantly (P ≤ 0.05) lower than in water (1.13, 1.68, 2.28, 2.20 and 3.28) and water plus soap (0.75, 1.60, 2.58, 1.83 and 3.30) negative controls respectively. The number of T. ni larvae per B. oleracea in five weeks of treatment applications at 10% w/v of T. vogelii, C. dichogamus and S. aromaticum of aqueous plant extracts were 0.00, 0.03, 0.05, 0.03 and 0.00; 0.03, 0.08, 0.05, 0.08 and 0.08; 0.05, 0.03, 0.00, 0.05 and 0.03, respectively which was significantly (P ≤ 0.05) lower than in water (0.50, 0.63, 0.60, 0.48 and 0.78) and water plus soap (0.30, 0.48, 68, 0.65 and 0.80) negative controls. The percentage damage of B. oleracea in five weeks of treatment applications at 10% w/v of T. vogelii, C. dichogamus and S. aromaticum aqueous plant extracts were 10.0, 6.3, 7.5, 7.5 and 5.6; 11.3, 11.3, 11.3, 11.3 and 12.5; 8.8, 8.1, 6.9, 8.1 and 9.4, respectively compared with water (33.8, 33.1, 38.8, 45.0 and 70.6) and water plus soap (30.0, 31.9, 41.3, 41.3 and 56.3). These pesticidal plants can be recommended for smallholder farmers to significantly control P. xylostella and T. ni larvae in B. oleracea crop.

    Citation: Nelson Mpumi, Revocatus L. Machunda, Kelvin M. Mtei, Patrick A. Ndakidemi. Insecticidal Efficacy of Syzygium aromaticum, Tephrosia vogelii and Croton dichogamus Extracts against Plutella xylostella and Trichoplusia ni on Brassica oleracea crop in Northern Tanzania[J]. AIMS Agriculture and Food, 2021, 6(1): 185-202. doi: 10.3934/agrfood.2021012

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  • The insecticidal efficacy of 10%, 5% and 1% w/v of Tephrosia vogelii, Croton dichogamus and Syzygium aromaticum aqueous plant extracts were assessed against larvae of Plutella xylostella and Trichoplusia ni on Brassica oleracea var. capitata crop field. Synthetic organophosphate pesticide (Chlorpyrifos) was used as a positive control and the negative controls were water and water plus soap. It was revealed that, aqueous plant extracts significantly (P ≤ 0.05) controlled the number of P. xylostella and T. ni larvae compared to negative controls. The 10% concentration of aqueous plant extracts showed significant higher efficacy in terms of reducing the insect population and their damage than 1% and 5 % concentration. The population of P. xylostella larvae per B. oleracea in five weeks of treatment applications at 10% w/v of T. vogelii, C. dichogamus and S. aromaticum aqueous plant extracts were 0.08, 0.15, 0.13, 0.05 and 0.08; 0.08, 0.20, 0.15, 0.13 and 0.18; 0.03, 0.05, 0.15, 0.18 and 0.13, respectively which was significantly (P ≤ 0.05) lower than in water (1.13, 1.68, 2.28, 2.20 and 3.28) and water plus soap (0.75, 1.60, 2.58, 1.83 and 3.30) negative controls respectively. The number of T. ni larvae per B. oleracea in five weeks of treatment applications at 10% w/v of T. vogelii, C. dichogamus and S. aromaticum of aqueous plant extracts were 0.00, 0.03, 0.05, 0.03 and 0.00; 0.03, 0.08, 0.05, 0.08 and 0.08; 0.05, 0.03, 0.00, 0.05 and 0.03, respectively which was significantly (P ≤ 0.05) lower than in water (0.50, 0.63, 0.60, 0.48 and 0.78) and water plus soap (0.30, 0.48, 68, 0.65 and 0.80) negative controls. The percentage damage of B. oleracea in five weeks of treatment applications at 10% w/v of T. vogelii, C. dichogamus and S. aromaticum aqueous plant extracts were 10.0, 6.3, 7.5, 7.5 and 5.6; 11.3, 11.3, 11.3, 11.3 and 12.5; 8.8, 8.1, 6.9, 8.1 and 9.4, respectively compared with water (33.8, 33.1, 38.8, 45.0 and 70.6) and water plus soap (30.0, 31.9, 41.3, 41.3 and 56.3). These pesticidal plants can be recommended for smallholder farmers to significantly control P. xylostella and T. ni larvae in B. oleracea crop.


    Middle East respiratory syndrome (MERS) is a viral respiratory disease caused by Middle East respiratory syndrome coronavirus (MERS-CoV). The intermediate host of MERS-CoV is probably the dromedary camel, a zoonotic virus [1]. Most MERS cases are acquired by human-to-human transmission. There is no vaccine or specific treatment available, and approximately 35% of patients with MERS-CoV infection have died [2]. There has been extensive works on infectious disease models and viral infection models associated with MERS that can help in disease control and provide strategies for potential drug treatments [3,4,5,6,7,8].

    Dipeptidyl peptidase-4 (DPP4) plays an important role in viral infection [2]. Based on classic viral infection models developed in [9,10,11], a four-dimensional ordinary differential equation model is proposed and studied in [8]. The model in [8] describes the interaction mechanisms among uninfected cells, infected cells, DPP4 and MERS-CoV.

    Recently, taking into account periodic factors such as diurnal temperature differences and periodic drug treatment, the model in [8] has been further extended a periodic case in [12], and then the existence of positive periodic solutions is studied by using the theorem in [13].

    It is well-known that CTL immune responses play a very critical role in controlling viral load and the concentration of infected cells. Thus, many scholars have considered CTL immune responses in various viral infection models and have achieved many excellent research results [14,15,16,17,18]. CTL cells can kill virus-infected cells and are important for the control and clearance of MERS-CoV infections [19]. Inspired by the above research works, we consider the following periodic MERS-CoV infection model with CTL immune response:

    {˙T(t)=λ(t)β(t)D(t)v(t)T(t)d(t)T(t),˙I(t)=β(t)D(t)v(t)T(t)d1(t)I(t)p(t)I(t)Z(t),˙v(t)=d1(t)M(t)I(t)c(t)v(t),˙D(t)=λ1(t)β1(t)β(t)D(t)v(t)T(t)γ(t)D(t),˙Z(t)=q(t)I(t)Z(t)b(t)Z(t). (1.1)

    In model (1.1), T(t), I(t), v(t), D(t) and Z(t) represent the concentrations of uninfected cells, infected cells, free virus, DPP4 on the surface of uninfected cells and CTL cells at time t, respectively. CTL cells increase at a rate bilinear rate q(t)I(t)Z(t) by the viral antigen of the infected cells and decay at rate b(t)Z(t); infected cells are killed by the CTL immune response at rate p(t)I(t)Z(t). Except for p(t), q(t) and b(t), all the remaining parameters of model (1.1) have the same biological meanings as in [12].

    Throughout the paper, it is assumed that the functions λ(t), β(t), d(t), d1(t), p(t), M(t), c(t), λ1(t), γ(t), q(t) and b(t) are positive, continuous and ω periodic (ω>0); the function β1(t) is non-negative, continuous and ω periodic.

    From point of view in both biology and mathematics, it is one of the most significant topics to study the existence of periodic oscillations of a system (see, for example, [12,20,21,22,23,24,25,26] and the references therein).

    In the next section, some sufficient criteria are given for the existence of positive periodic oscillations of model (1.1). It should be mentioned here that, in the proofs of the main results in the following section, a new technique is developed to obtain a lower bound of the state variable Z(t) characterizing CTL immune response in model (1.1).

    For some function f(t) which is continuous and ω-periodic on R, let us define the following notations:

    fU=maxt[0,ω]f(t),fl=mint[0,ω]f(t),ˆf=1ωω0f(t)dt.

    Moreover, for convenience, let us give the following parameters:

    R=ˆλβlexp{L3+L4}^d1exp{M2}(βlexp{L3+L4}+dU)>1,ω=ˆb2ˆλˆq,δ=^d12ˆp(R1),M1=ln(λUdl),M2=ln(ˆbˆq+2ˆλω),M3=ln(^(d1M)ˆc)+M2+2ˆcω,M4=ln(λU1γl),M5=ln(ˆβexp{M1+M3+M4}ˆpexp{L2})+2ˆbω,L1=ln(λlβUexp{M3+M4}+dU),L2=ln(ˆbˆq2ˆλω),L3=ln((^d1M)ˆc)+L22ˆcω,L4=ln(λl1(β1β)Uexp{M1+M3}+γU),L5=ln(δ)2ˆbω.

    The following theorem is the main result of this paper.

    Theorem 2.1. If R>1 and ω<ω, then model (1.1) has at least one positive ω-periodic solution.

    Proof. Making the change of variables T(t)=exp{u1(t)}, I(t)=exp{u2(t)}, v(t)=exp{u3(t)}, D(t)=exp{u4(t)}, Z(t)=exp{u5(t)}, then model (1.1) can be rewritten as

    {˙u1(t)=λ(t)exp{u1(t)}β(t)exp{u3(t)+u4(t)}d(t),˙u2(t)=β(t)exp{u1(t)+u3(t)+u4(t)}exp{u2(t)}d1(t)p(t)exp{u5(t)},˙u3(t)=d1(t)M(t)exp{u2(t)}exp{u3(t)}c(t),˙u4(t)=λ1(t)exp{u4(t)}β1(t)β(t)exp{u1(t)+u3(t)}γ(t),˙u5(t)=q(t)exp{u2(t)}b(t). (2.1)

    Thus, we only need to consider model (2.1).

    Let us set

    X=Y={u=(u1(t),u2(t),u3(t),u4(t),u5(t))TC(R,R5)|u(t)=u(t+ω)}

    with the norm

    ||u||=maxt[0,ω]|u1(t)|+maxt[0,ω]|u2(t)|+maxt[0,ω]|u3(t)|+maxt[0,ω]|u4(t)|+maxt[0,ω]|u5(t)|.

    It can be shown that X and Y are Banach spaces. Define

    Nu=[λ(t)exp{u1(t)}β(t)exp{u3(t)+u4(t)}d(t)β(t)exp{u1(t)+u3(t)+u4(t)}exp{u2(t)}d1(t)p(t)exp{u5(t)}d1(t)M(t)exp{u2(t)}exp{u3(t)}c(t)λ1(t)exp{u4(t)}β1(t)β(t)exp{u1(t)+u3(t)}γ(t)q(t)exp{u2(t)}b(t)]:=[N1(t)N2(t)N3(t)N4(t)N5(t)](uX),
    Lu=˙u(uDomL),Pu=1ωω0u(t)dt(uX),Qu=1ωω0u(t)dt(uY),

    here DomL={uX,˙uX}. It easily has that KerL={uX|uR5} and ImL={uY|ω0u(t)dt=0}. Further, it is clear that ImL is closed in Y and dimKerL=codimImL=5. Hence, L is a Fredholm mapping with index zero.

    For μ(0,1), let us consider the equation Lu=μNu, i.e.,

    {˙u1(t)=μ[λ(t)exp{u1(t)}β(t)exp{u3(t)+u4(t)}d(t)],˙u2(t)=μ[β(t)exp{u1(t)+u3(t)+u4(t)}exp{u2(t)}d1(t)p(t)exp{u5(t)}],˙u3(t)=μ[d1(t)M(t)exp{u2(t)}exp{u3(t)}c(t)],˙u4(t)=μ[λ1(t)exp{u4(t)}β1(t)β(t)exp{u1(t)+u3(t)}γ(t)],˙u5(t)=μ[q(t)exp{u2(t)}b(t)]. (2.2)

    For any solution u=(u1(t),u2(t),u3(t),u4(t),u5(t))TX of (2.2), it has

    {ω0[λ(t)exp{u1(t)}β(t)exp{u3(t)+u4(t)}d(t)]dt=0,ω0[β(t)exp{u1(t)+u3(t)+u4(t)}exp{u2(t)}d1(t)p(t)exp{u5(t)}]dt=0,ω0[d1(t)M(t)exp{u2(t)}exp{u3(t)}c(t)]dt=0,ω0[λ1(t)exp{u4(t)}β1(t)β(t)exp{u1(t)+u3(t)}γ(t)]dt=0,ω0[q(t)exp{u2(t)}b(t)]dt=0. (2.3)

    From the first two equations in (2.2), it has

    ˙u1(t)exp{u1(t)}=μ[λ(t)β(t)exp{u1(t)+u3(t)+u4(t)}d(t)exp{u1(t)}],

    and

    ˙u2(t)exp{u2(t)}=μ[β(t)exp{u1(t)+u3(t)+u4(t)}d1(t)exp{u2(t)}p(t)exp{u2(t)+u5(t)}].

    Hence, by integrating the above two equations on [0,ω], it has

    ω0[λ(t)β(t)exp{u1(t)+u3(t)+u4(t)}d(t)exp{u1(t)}]dt=0 (2.4)

    and

    ω0[β(t)exp{u1(t)+u3(t)+u4(t)}d1(t)exp{u2(t)}p(t)exp{u2(t)+u5(t)}]dt=0. (2.5)

    Note that I(t):=exp{u2(t)} satisfies

    ˙I(t)=˙u2(t)exp{u2(t)}=μ[β(t)exp{u1(t)+u3(t)+u4(t)}d1(t)p(t)exp{u2(t)+u5(t)}].

    Then, from (2.4) and (2.5), it has

    ω0|˙I(t)|dtμω0[β(t)exp{u1(t)+u3(t)+u4(t)}+d1(t)+p(t)exp{u2(t)+u5(t)}]dt2ω0β(t)exp{u1(t)+u3(t)+u4(t)}dt2ˆλω. (2.6)

    From the third and the fifth equations of (2.2), it has

    ω0|˙u3(t)|dtμ[ω0d1(t)M(t)exp{u2(t)}exp{u3(t)}dt+ω0c(t)dt]<2ˆcω,ω0|˙u5(t)|dtμ[ω0q(t)exp{u2(t)}dt+ω0b(t)dt]<2ˆbω. (2.7)

    Note that uX, there exist ξi,ηi[0,ω] (i=1,2,3,4,5), such that

    ui(ξi)=mint[0,ω]ui(t),ui(ηi)=maxt[0,ω]ui(t)(i=1,2,3,4,5).

    From (2.2), ˙u1(η1)=0 and ˙u4(η4)=0, it has

    λ(η1)exp{u1(η1)}β(η1)exp{u3(η1)+u4(η1)}d(η1)=0,λ1(η4)exp{u4(η4)}β1(η4)β(η4)exp{u1(η4)+u3(η4)}γ(η4)=0,

    which imply that

    u1(t)u1(η1)ln(λ(η1)d(η1))ln(λUdl)=M1,u4(t)u4(η4)ln(λ1(η4)γ(η4))ln(λU1γl)=M4. (2.8)

    From the last equation of (2.3), it has

    ω0q(t)exp{u2(ξ2)}dtˆbωω0q(t)exp{u2(η2)}dt,

    which implies that

    I(ξ2)=exp{u2(ξ2)}ˆbˆqexp{u2(η2)}=I(η2).

    Then, from (2.6) and ω<ω, it has

    I(t)I(ξ2)+ω0|˙I(t)|dtˆbˆq+2ˆλω,I(t)I(η2)ω0|˙I(t)|dtˆbˆq2ˆλω=2ˆλ(ωω)>0.

    Thus, it has

    u2(t)ln(ˆbˆq+2ˆλω)=M2,u2(t)ln(ˆbˆq2ˆλω)=L2. (2.9)

    From the third equation of (2.3), it has

    ω0d1(t)M(t)exp{M2}exp{u3(ξ3)}dtˆcωω0d1(t)M(t)exp{L2}exp{u3(η3)}dt,

    which implies that

    u3(ξ3)ln(^(d1M)ˆc)+M2,u3(η3)ln(^(d1M)ˆc)+L2.

    Then, from (2.7), it has

    u3(t)u3(ξ3)+ω0|˙u3(t)|dtln(^(d1M)ˆc)+M2+2ˆcω=M3,u3(t)u3(η3)ω0|˙u3(t)|dtln(^(d1M)ˆc)+L22ˆcω=L3. (2.10)

    From the second equation of (2.3), it has

    ˆpexp{u5(ξ5)}ωω0[β(t)exp{M1+M3+M4}exp{L2}d1(t)]dtexp{M1+M3+M4}exp{L2}ˆβω,

    which implies that

    u5(ξ5)ln(ˆβexp{M1+M3+M4}ˆpexp{L2}):=l5.

    Then, from (2.7), it has

    u5(t)u5(ξ5)+ω0|˙u5(t)|dtl5+2ˆbω=M5.

    From ˙u1(ξ1)=0, ˙u4(ξ4)=0, (2.8) and (2.10), it has

    exp{u1(ξ1)}=λ(ξ1)β(ξ1)exp{u3(ξ1)+u4(ξ1)}+d(ξ1)λlβUexp{M3+M4}+dU,exp{u4(ξ4)}=λ1(ξ4)β1(ξ4)β(ξ4)exp{u1(ξ4)+u3(ξ4)}+γ(ξ4)λl1(β1β)Uexp{M1+M3}+γU.

    Thus, it has

    u1(t)u1(ξ1)ln(λlβUexp{M3+M4}+dU)=L1,u4(t)u4(ξ4)=ln(λl1(β1β)Uexp{M1+M3}+γU)=L4. (2.11)

    Let us give an estimate of the lower bound of the state variable u5(t) related to CTL immune response. It should be mentioned here that a completely different method from that in [12] has been used.

    Claim A If R>1 and ω<ω, then

    exp{u5(η5)}δ.

    If Claim A is not true, then it has that, for any t, exp{u5(t)}exp{u5(η5)}<δ. Hence, it has from (2.3), (2.9)–(2.11) that

    0=ω0[β(t)exp{u1(t)+u3(t)+u4(t)}exp{u2(t)}d1(t)p(t)exp{u5(t)}]dtω0[β(t)exp{u1(t)+L3+L4}exp{M2}d1(t)p(t)exp{u5(η5)}]dtβlexp{L3+L4}exp{M2}ω0exp{u1(t)}dt(ˆd1+ˆpδ)ω,

    which implies that

    ω0d(t)exp{u1(t)}dtdUω0exp{u1(t)}dtdU(ˆd1+ˆpδ)exp{M2}βlexp{L3+L4}ω:=Ψω. (2.12)

    Adding (2.4) and (2.5) together, it has

    ω0[λ(t)d(t)exp{u1(t)}]dt=ω0[d1(t)exp{u2(t)}+p(t)exp{u2(t)+u5(t)}]dtω0exp{M2}[d1(t)+p(t)exp{u5(η5)}]dtexp{M2}(^d1+ˆpδ)ω,

    which implies that

    ω0d(t)exp{u1(t)}dt[ˆλexp{M2}(^d1+ˆpδ)]ω=Ψω+[ˆλΨexp{M2}(^d1+ˆpδ)]ω=Ψω+[ˆλexp{M2}(1+dUβlexp{L3+L4})(^d1+ˆpδ)]ω=Ψω+^d1exp{M2}(1+dUβlexp{L3+L4})(R1ˆp^d1δ)ω=Ψω+^d12exp{M2}(1+dUβlexp{L3+L4})(R1)ω>Ψω,

    which is a contradiction to (2.12). Thus, the claim holds.

    From Claim A and (2.7), it has

    u5(t)u5(η5)ω0|˙u5(t)|dtln(δ)2ˆbω=L5. (2.13)

    Now, for convenience, let us define

    ¯R=(ˆλ^d1ˆbˆq)ˆβ^(d1M)ˆd^d1ˆc^λ1^(β1β)ˆλ^(d1M)ˆbˆdˆcˆq+ˆγ,Zmax=ˆqˆpˆb(ˆλ^d1ˆbˆq).

    Note that if R>1, then it has

    ~R:=(ˆλ^d1exp{M2})βlexp{L3+L4}dU^d1exp{M2}>1,

    which implies that

    Zmax>ˆqˆpˆb[ˆλ^d1(ˆbˆq+2ˆλω)]=ˆqˆpˆb(ˆλ^d1exp{M2})>0,¯R(ˆλ^d1ˆbˆq)ˆβ^(d1M)ˆd^d1ˆcλl1(β1β)Uexp{M3+M1}+γU~R>1.

    Let (u1,u2,u3,u4,u5)TR5 be the solution of the following equations:

    {ˆλexp{u1}ˆβexp{u3+u4}ˆd=0,ˆβexp{u1+u3+u4}exp{u2}ˆd1ˆpexp{u5}=0,^(d1M)exp{u2}exp{u3}ˆc=0,ˆλ1exp{u4}^(β1β)exp{u1+u3}ˆγ=0,ˆqexp{u2}ˆb=0.. (2.14)

    Define Γ:[0,Zmax]R, via

    Γ(x)=ˆβ^(d1M)ˆc^λ1Γ1(x)^(β1β)Γ1(x)^(d1M)ˆbˆqˆc+ˆγˆd1ˆpx,

    where

    Γ1(x)=ˆλˆd^d1ˆbˆdˆqˆpˆbˆdˆqx=ˆpˆbˆdˆq(Zmaxx).

    Equation (2.14) can be rewritten as

    exp{u2}=ˆbˆq,exp{u3}=^(d1M)ˆcexp{u2}=^(d1M)ˆbˆqˆc,exp{u1}=ˆλˆd^d1exp{u2}ˆdˆpexp{u2}ˆdexp{u5}=Γ1(exp{u5}),exp{u4}=^λ1^(β1β)exp{u1+u3}+ˆγ=^λ1^(β1β)Γ1(exp{u5})^(d1M)ˆbˆqˆc+ˆγ,ˆβ^(d1M)ˆcexp{u1+u4}ˆd1ˆpexp{u5}=Γ(exp{u5})=0.

    It is obvious that if there is a solution (u1,u2,u3,u4,u5)TR5 for (2.14), it must have 0<exp{u5}<Zmax. In addition, note that Γ(x) is monotonically decreasing with respect to x on [0,Zmax]. It has from Γ(Zmax)=^d1ˆpZmax<0 and

    Γ(0)=ˆβ^(d1M)ˆc^λ1Γ1(0)^(β1β)Γ1(0)^(d1M)ˆbˆqˆc+ˆγ^d1>ˆβ^(d1M)ˆc^λ1(ˆλˆd^d1ˆbˆdˆq)^(β1β)ˆλˆd^(d1M)ˆbˆqˆc+ˆγ^d1=^d1(¯R1)>0

    that there exists a unique positive constant x=Z(0,Zmax) such that Γ(Z)=0.

    The above discussions show that, if R>1, (2.14) has a unique solution (u1,u2,u3,u4,u5)T, here ui=ln(ei) (i=1,2,3,4,5),

    e1=Γ1(Z)>0,e2=ˆbˆq>0,e3=^(d1M)ˆbˆqˆc>0,e4=^λ1^(β1β)Γ1(Z)^(d1M)ˆbˆqˆc+ˆγ>0,e5=Z>0.

    Let us define the following set

    Ω={uX|||u||<U1=1+5i=1(max{|Mi|,|Li|}+|ui|)}X.

    Moreover, by similar arguments as in [12], it has that N is L-compact on ¯Ω.

    Now, let us compute the Leray-Schauder degree deg{QN,ΩKerL,(0,0,0,0,0)T}:=Δ as follows,

    Δ=sign|ˆλe10ˆβe3e4ˆβe3e40ˆβe1e3e4e2ˆβe1e3e4e2ˆβe1e3e4e2ˆβe1e3e4e2ˆpe50^(d1M)e2e3^(d1M)e2e300^(β1β)e1e30^(β1β)e1e3^λ1e400ˆqe2000|=sign{^(d1M)ˆpˆqe22e5e3(ˆλ^λ1e1e4ˆβ^(β1β)e1e23e4)}=sign{^(d1M)ˆpˆqe22e5e3e1e4[ˆde1(^(β1β)e1e3e4+ˆγe4)+ˆβˆγe1e3e24]}=10,

    where ˆλ=ˆβe1e3e4+ˆde1 and ^λ1=^(β1β)e1e3e4+ˆγe4 are used.

    Finally, it has those all the conditions of the continuation theorem in [13] (also see, for example, Lemma 2.1 in [12]) are satisfied. This proves that, if ω<ω and R>1, model (2.1) has at least one ω-periodic solution.

    Let us consider the following classical viral infection dynamic model [9] with CTL immune response:

    {˙T(t)=λ(t)β(t)v(t)T(t)d(t)T(t),˙I(t)=β(t)v(t)T(t)d1(t)I(t)p(t)I(t)Z(t),˙v(t)=d1(t)M(t)I(t)c(t)v(t),˙Z(t)=q(t)I(t)Z(t)b(t)Z(t),(A)

    where, all the coefficients are the same with that in model (1.1).

    Define R1:[0,ω]R, via

    R1(x)=ˆλβl[^(d1M)ˆcexp{2ˆcx}(ˆbˆq2ˆλx)]^d1(ˆbˆq+2ˆλx){dU+βl[^(d1M)ˆcexp{2ˆcx}(ˆbˆq2ˆλx)]}.

    Obviously, R1(x) is monotonically decreasing on [0,ω] and

    R1(0)=ˆλβl^(d1M)ˆq^d1(dUˆcˆq+βl^(d1M)ˆb),R1(ω)=0.

    Therefore, if R1(0)>1, then there exists a unique constant ω(0,ω) such that R1(ω)=1, R1(x)>1 for 0x<ω and R1(x)<1 for ω<xω.

    For model (A), it is not difficult to derive the following result.

    Theorem 2.2. If R1(ω)>1 and ω<ω (i.e. R1(0)>1 and ω<ω<ω), then model (A) has at least one positive ω-periodic solution.

    Remark 2.1. If all the coefficients in model (A) take constants values, i.e., λ(t)λ>0, β(t)β>0, d(t)d>0, d1(t)d1>0, p(t)p>0, M(t)M>0, c(t)c>0, q(t)q>0 and b(t)b>0, then model (A) becomes the classical model which is first proposed by Nowak and Bangham in [9]. the condition ω<ω in Theorem 2.2 is naturally satisfied. Furthermore, it has R1(0)=(λβMq)/(dcq+βd1Mb):=R1. From \emph{[9]}, it has that the condition R1>1 implies the existence of a unique positive equilibrium. This shows that the conditions and conclusion in Theorem 2.2 are reasonable.

    In summary, Theorem 2.1 in the paper successfully extends the main result in [12]) to a MERS-CoV viral infection model with CTL immune response. In the proof of Theorem 2.1, we use a very different method from that in [9] to obtain the lower bound (ln(δ)2ˆbω) of the state variable u5(t). Furthermore, as a special case, Theorem 2.2 gives sufficient conditions for the existence of positive periodic solution of model (A). Model (A) is a natural extension of the classical model in [9]. As the end of the paper, let us give a example to summarize the applications of Theorem 2.1. Let us choose the coefficients in model (1.1) as follows (for the values of some parameters, please refer to [7,27] for the case of some autonomous models), λ(t)=45(1+0.1sin(4πt)), β(t)=1.4×108(1+0.1cos(4πt)), d(t)=0.001(1+0.5cos(4πt)), d1(t)=0.056(1+0.5cos(4πt)), p(t)=0.00092(1+0.5cos(4πt)), M(t)=100000, c(t)=2.1(1+0.3cos(4πt)), λ1(t)=10(1+0.1sin(4πt)), β1(t)=0.001, γ(t)=0.01(1+0.1cos(4πt)), q(t)=0.005(1+0.5sin(4πt)), b(t)=0.5(1+0.4cos(4πt)). Then, with the help of Maple mathematical software, it has ω=0.5<ω1.111111, M111.502875, M24.976734, M314.965318, M47.108426, M518.976215, L10.383569, L2=4.007333, L39.795917, L40.623402, L51.387881, R1.2170332>1. From Theorem 2.1, it follows that model (1.1) has at least one positive ω (ω=0.5)-periodic solution. Figure 1 gives the corresponding numerical simulation, and the initial value is chosen as (T(0),I(0),v(0),D(0),Z(0))T=(12.5,100,265000,995.4,423)T.

    Figure 1.  With the increasing of the time t, the evolution form of the solution of model (1.1).

    This paper is supported by National Natural Science Foundation of China (No.11971055) and Beijing Natural Science Foundation (No.1202019).

    The authors declare there is no conflict of interest.



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