We investigate two models of vector-borne plant virus transmission. Both of these are basic, and the objective is to explore how different techniques of introducing the delay affect the outcomes. There are two plant densities in the first, model A: susceptible [S(t)], healthier and susceptible to infection, and infectious [I(t)], previously infected. Because we assume that plants may not recover, we should not have a recovered class. There are also two vector populations: susceptible [X(t)] and infectious [Y(t)]. This model is a simplified form of the models provided in [49, 50].

Model A assumes that: plants as well as vectors that are new to this field, are susceptible, and the overall plant density remains stable at N because a farmer may replace any dead plants with healthy new ones, that the interaction among both the vector as well as plant is a mass movement, that the viruses decapitate plants but not the vectors who do not contract the disease, and the disease cannot be recovered from either plants or vectors. The model's parameters are the γ infection ratio of a susceptible plant by an infected vector, γ₁ infection ratio of vectors by infected plants, υ plants' natural fatality rate, c plants' increased fatality rate owing to illness, r vectors' natural fatality rate, and vector replenishment rate (according to birth or/and emigration).

Model A is represented by the system of ODEs as follows [25]: (1)S′(t)=υ(N-S(t))+cI(t)-γY(t)S(t), (2)I′(t)=γY(t)S(t)-(c+υ)I(t), (3)X′(t)=Ω-γ1I(t)X(t)-rX(t), (4)Y′(t)=γ1I(t)X(t)-rY(t). There are two delays in virus transmission via a vector. One is being the time required for the virus to propagate throughout the plant after it has been infected. The other is the time required for the virus to propagate within the vector after it has been infected. Because the virus is not reproducing in the vector, the second is significantly smaller than the first. For the sake of simplicity, we'll assume that the second delay is zero.

We will incorporate the delays in two ways: the first is based on the premise that a susceptible needs the time delay to become infectious after coming into contact with an infectious [50, 51]. This is supposed to be model A1 [25]: (5)S′(t)=υ(N-S(t))+cI(t)-γY(t-δ)S(t-δ), (6)I′(t)=γY(t-δ)S(t-δ)-(c+υ)I(t), (7)X′(t)=Ω-γ1I(t)X(t)-rX(t), (8)Y′(t)=γ1I(t)X(t)-rY(t). It would be possible to replace the exposed density [E(t)] with a delay-accounting density. After coming into contact with an infectious, a susceptible become exposed or dormant, unable to infect. The exposed becomes infectious at the rate η = 1/δ. Then the model A2 will be: (9)S′(t)=υ(N-S(t))+cI(t)-γY(t)S(t)+υE(t), (10)E′(t)=γY(t)S(t)-υE(t)-ηE(t), (11)I′(t)=ηE(t)-(c+υ)I(t), (12)X′(t)=Ω-γ1I(t)X(t)-rX(t), (13)Y′(t)=γ1I(t)X(t)-rY(t). Epidemic models involving an exposed class are widely known for plant virus propagation [52, 53]. Models containing exposed densities have the advantage of not requiring the initial/staring conditions to be presented at an interval equal to the delay, as delay differential equations (DDEs) require.

We construct a further plant virus propagation model which is based on the models presented in [54, 55], but revised to include healthy vectors and mass response interactions for the disease. It takes into account four different densities: susceptible plants [S(t)], infectious plants [I(t)], susceptible vectors [X(t)] and infectious vectors [Y(t)]. Because the plants grow in a logistical manner, the overall plant density does not remain constant. All emerging vectors are subject to susceptible, and their growth rate is continuously attributed to births as well as emigration. Plants are unable to recover and insects do not contract the disease, as it does in model A.

Model B, which propagates plant viruses is as follows [25]: (14)S′(t)=mS(t)(1-S(t)+I(t)N)                -γS(t)Y(t)1+βS(t)+aY(t), (15)I′(t)=γS(t)Y(t)-(r+c)I(t), (16)X′(t)=Ω-γ1I(t)X(t)-rX(t), (17)Y′(t)=γ1I(t)X(t)-rY(t). Here, m represents the plants' proliferation rate, N their maximum capacity of carrying, and γ the rate of infection of a susceptible plant by an infected vector, r represents the plants' natural fatality rate, and c represents the virus's additional fatality rate. Ω represents the rate at which susceptible vectors are recruited, γ₁ represents the rate at which an infected plant infects a susceptible vector, and r represents the vectors' natural fatality rate.

We will incorporate the delay in two different ways, just like we did with model A. The first assumes that a susceptible takes the time delay to get infected after coming into contact with an infected [50, 51]. Then model B^* may be expressed in the form [25]: (18)S′(t)=mS(t)(1-S(t)+I(t)N)             -γS(t-δ)Y(t-δ)1+βS(t-δ)+aY(t-δ), (19)I′(t)=γS(t-δ)Y(t-δ)1+βS(t-δ)+aY(t-δ)-(r+c)I(t), (20)Y′(t)=γI(t)-rY(t). In the alternative version, we uphold [54, 55] in which the plant ceases being susceptible immediately after interaction with an infected insect, but it requires a delay period to become infected. Because the plant could die at any time, the surviving rate is directly proportional e^−rδ, where r is the plant's fatality rate and δ is the delay. Then model B1 will be of the form [25]: (21)S′(t)=mS(t)(1-S(t)+I(t)N)            -γS(t)Y(t)1+βS(t)+aY(t), (22)I′(t)=e-rδγS(t-δ)Y(t-δ)1+βS(t-δ)+aY(t-δ)              -(r+c)I(t), (23)X′(t)=Ω-γ1I(t)X(t)-rX(t), (24)Y′(t)=γ1I(t)X(t)-rY(t). A susceptible plant is infected by an infected vector at time (t − δ) in model B^*, as well as the susceptible plant becomes infective at time t. A susceptible plant is infected by an infected vector that takes t to infect in model B1, with e^−rδ indicating the average rate of infectious susceptible who survived in time t.

Incorporating an exposed density [E(t)], as before, is an alternate with the same boon and bane. The model B2 with exposed class is as follows [25]: (25)S′(t)=mS(t)(1-S(t)+I(t)N)           -γS(t)Y(t)1+βS(t)+aY(t), (26)E′(t)=γY(t)S(t)-υE(t)-ηE(t), (27)I′(t)=ηE(t)-(c+υ)I(t), (28)X′(t)=Ω-γ1I(t)X(t)-rX(t), (29)Y′(t)=γ1I(t)X(t)-rY(t).

The Adams method (ADM) is a two-step procedure for solving an ODE [56–61]. First, to use an explicit approach, the predictive step determines a crude approximation of the target number. The corrector step streamlines the preceding approximation using a different mechanism, usually an implicit one.

The predictor-corrector technique, which is based on set of Equations (1)–(4), is represented as follows: (30)dSdt=f(t,S,I,Y),S(t0)=S(0),dIdt=f(t,Y,S,I),I(t0)=I(0),dXdt=f(t,I,X),X(t0)=X(0),dYdt=f(t,I,X,Y),Y(t0)=Y(0), To obtain a two-step predictor solution for first equation of set (30) for the non-linear plant virus propagation model by vector, use the following expression: Sk+1=Sk+64hf(tk,Sk)-12hf(tk-1,Sk-1), We have the following two-step corrector equation after evaluating the first equation in the nonlinear plant virus propagation model by vector: Sk+1=Sk+12hf(tk+1,Sk+1)+f(tk,Sk),

The backward differentiation formula (BDF) is a collection of implicit approaches for solving ordinary differential equations numerically [62–64]. They are linear multi-step algorithms that use information from previously determined time points to approximating the derivative of a function for a particular function and time, improving the precision of the approximations. These techniques are particularly useful for solving stiff differential equations [65]. In 1952, Charles F. Curtiss and Joseph O. Hirschfelder introduced the methods for the first time.

Consider the initial value problem as: dzdt=g(t,z),       z(t0)=z0, BDF can be written in generic form as follows: ∑m=0lcmzn+m=hαg(tn+l,zn+l), where the step size is denoted by h, g is being calculated for an unknown z_n+l. BDF techniques are implicit and may require non-linear equations to be solved at each step. The coefficients c_m as well as α are considered to obtain order l, which is the highest feasible order.

Table 1 [25] lists the default settings for the non-linear PVPMV parameters, while the nomenclature describes the parameters. These default settings utilizing in all of the scenarios of non-linear PVPMV.

The three different models of plant virus propagation by a vector based on the system of ODEs without delay (model A), with delay (model A1), and without delay but including exposed class [E(t)] (model A2) as presented in Equations (1–4), (5–8), and (9–13) are numerically solved employing the ADM and BDF methods invoking he Mathematica routine with inputs [0, 30] and step size 0.1. Numerical outcomes and simulations are determined for five distinct scenarios of each model comprising cases 1–5 for non-linear PVPMV and selected random scenarios from each model for discussion. We first presented the dynamical behavior of S(t), I(t), X(t) and Y(t) classes of scenario 2 for model A of non-linear PVPMV. The numerical outcomes of non-linear PVPMV model A for case-1 of scenario 2 against the classes S(t), I(t), X(t) and Y(t) are listed in Table 3.

Figures 2A,B illustrate the dynamics of susceptible plants utilizing the ADM and BDF methods, respectively, for the variation in infection ratio of the vectors by infected plants, i.e., γ₁ for model A. It has been discovered that increasing the value of γ₁ causes the susceptible density of plants to drop. The impacts of infected plants are shown in Figures 3A,B for varied values of γ₁. As can be seen from the graph, increasing the value of γ₁ increases the density of infected plants. Figures 4A,B demonstrate that how the behavior of susceptible vectors changes as the value of γ₁ changes. For higher values of γ₁ there is an increase in the density of infected vectors. Figures 5A,B show the effects of infected vectors for various values of γ₁. Increasing the value of γ₁ increases the density of infected plants, as shown in the graphic.

The dynamics of plants' natural fatality rate i.e., υ is explored for all four classes S(t), I(t), X(t) and Y(t) using the strength of ADM and BDF methods for scenario 5 of the model A1. As seen in Figures 6A,B, raising the value of υ causes the density of susceptible plants to grow. The density of infected plants decreased as the value of υ increased, as seen in Figures 7A,B. Figures 8A,B show the effects of plants' natural mortality rate i.e., υ for class X(t) of model A1. As can be seen in the graphs, increasing the value of υ will increase the number of susceptible vectors. For class Y(t) of model A1, the influence of plants' natural fatality rate, i.e., υ is also computed. The rate of infected vectors reduces as the value of the infected vectors increases, as seen in Figures 9A,B. Table 4 shows the numerical outcomes of non-linear PVPMV model A1 for case-1 of scenario 5 against the classes S(t), I(t), X(t) and Y(t).

Similarly, the dynamics for all four classes S(t), I(t), X(t) and Y(t)are analyzed by varying the infection ratio of a susceptible plant by an infected vector which is denoted by γ for scenario 1 of model A2 and graphical illustrations are presented in Figures 10–13, respectively. Numerical outcomes classes S(t), I(t), X(t) and Y(t) in model A2 for case-1 of scenario 1 are computed and provided in Table 5. Figures 10A,B depict the influence of the infection ratio of a susceptible plant by an infected vector on susceptible plants using the ADM and BDM methods, respectively. It is permissible to observe that when the value of γ rises, the density of susceptible plants decreases. Figures 11A,B describe the effects of the infection ratio of a susceptible plant by an infected vector on infected plants. One may observe that the density of infected plants increased in correlation with the value of γ. Figures 12A,B illustrate progressive increase in the density of susceptible vectors as the value of γ increases, whereas Figures 13A,B demonstrate the opposing behavior in the case of infected vectors.

The three segregated models of plant virus transmission by a vector, as described in Equations (14–17), (21–24), and (25–29), are numerically solved employing the ADM and BDF methods invoking the Mathematica routine. We construct five distinct scenarios incorporating cases 1–5 for non-linear PVPMV and chosen random scenarios from each model are used to determine numerical outcomes and simulations. For model B of non-linear PVPMV. We first described the dynamical behavior of the S(t), I(t), X(t), and Y(t) classes in scenario 3 for the variation in disease's additional fatality rate i.e., c. for model B. For all four classes S(t), I(t), X(t), and Y(t) numerical outcomes are determined and provided in Table 6 for case-1 of scenario 3 of model B. Figures 14A,B illustrate the dynamics of susceptible plants using the ADM and BDF methods for the variability in the disease's additional fatality rate i.e., c. It has been discovered that as the value of c is elevated, the susceptible density of plants increases. The impact of disease's additional fatality rate i.e., c on infected plants can be seen in Figures 15A,B. It is clear from Figures that increasing the value of c will result in reduction the density of infected plants. Figures 16A,B demonstrate the behavior of susceptible vectors for the variation in disease's additional fatality rate of model B. One may see that the density of susceptible vectors will increase as the value of c is increased. The influence of disease's additional fatality rate on infected vectors is presented in Figures 17A,B. It is observed from Figures that increasing the value of c causes the density of infected vectors to decrease.

Secondly, the dynamics of susceptible vectors' recruited rate i.e., Ω is investigated for all four classes S(t), I(t), X(t), and Y(t) utilizing the strength of ADM and BDF methods for scenario 5 of the model B1 and numerical outcomes of all four classes S(t), I(t), X(t), and Y(t) for the case-1 of scenario 5 is listed in Table 7. Figures 18A,B portrayed the behavior of susceptible plants density for the different values of Ω, and it is noticed that the number of susceptible plants decreases for the higher values of Ω. Figures 19A,B illustrated that as the value of Ω increases, the number of infected plants goes up. The dynmics of susceptible vectors for the variation in vectors' recruited rate i.e., c are presented in Figures 20A,B. One may witness that in Figures 20A,B the density of susceptible vectors goes in continous behavior for the first two cases and next three cases vectors density increased in the range of 0 to 10 days then steadily decreased and shows their steady behavior for next 20–30 days. As a result, for higher values of Ω, the density of susceptible vectors increases. Figures 21A,B portrayed the impact of susceptible vectors' recruited rate on infected vectors for model B1. The infected vectors show a steady behavior for the first two cases, also a steady behavior for the next three cases in 0–5 days, and then a gradual increase in 5–30 days, as shown in Figures 21A,B.

Finally, the dynamics of vectors' natural fatality rate i.e., r is investigated for all four classes S(t), I(t), X(t), and Y(t) utilizing the strength of ADM and BDF methods for scenario 4 of the model B2. The respective numerical outcomes for case-1 of scenario 4 is provided in Table 8. The impact of vectors' natural fatality rate on susceptible plants is presented in Figures 22A,B As observed in graphical representation, the density of susceptible plants increased up to 90, then decreased between 3 and 10 days before returning to their steady state behavior. Also, the density of susceptible plants increased for the higher value of r as shown in Figures 22A,B, while the infected plants depicted reverse behavior as shown in Figures 23A,B. The influence of vectors' natural fatality rate r on susceptible vectors can be observed in Figures 24A,B for model B2. The number of susceptible vectors appears to decrease as the value of r increases. Similarly, the dynamics of infected vectors is portrayed in Figures 25A,B utilizing the ADM and BDF fro model B2, respectively. The number of infected vectors dropped as the natural fatality rate r of the vectors increased in model B2.