Modelling and Stability Analysis of Cotton Leaf Curl Virus (CLCuV) Transmission Dynamics in Cotton Plant

In this paper, the transmission dynamics of cotton leaf curl virus (CLCuV) disease in cotton plants was proposed and investigated qualitatively using the stability theory of a nonlinear ordinary di ﬀ erential equations. Cotton and vector populations were both taken into account in the models. Cotton population was categorized as susceptible (A) and infected (B). The vector population was also categorized as susceptible (C) and infected (D). We established that all model solutions are positive and bounded by relevant initial conditions. The existence of unique CLCuV free and endemic equilibrium points, as well as the basic reproduction number, which is computed using the next generation matrix approach, are investigated. The conditions for the local and global asymptotic stability of these equilibrium points are then established. When the basic reproduction number is less than one, the system has locally and globally asymptotically stable CLCuV free equilibrium point, but when the basic reproduction number is more than one, the system has locally and globally asymptotically stable endemic equilibrium point. The numerical simulation ﬁ ndings show that lowering the infection rate of cotton vectors has a signi ﬁ cant impact on controlling cotton leaf curl virus (CLCuV) in the time frame given.


Introduction
Cotton (Gossypium spp.) is the most significant fiber, oil, and protein crop on the planet (Monga and Sain [1]). Cotton is well-known for its versatility, performance, attractiveness, natural comfort, and, most importantly, its many applications, which include astronaut in-flight space suits, towels, tarpaulins, tents, sheets, and all types of clothing (Vanitha et al. [2]). It is referred to as "white gold, " and it is one of the most important crops in developing countries, as well as a raw material for the local textile and oil industries (Sain et al. [3]). Cotton is significant since it is not only the world's most important fiber crop, but also the world's secondlargest oilseed crop (Zhang et al. [4]).
The majority of Ethiopian cotton is exported to Africa, Asia, and Europe, with Asia accounting for 67% of total exports. Ethiopian cotton has now been priced by the Textile Industry Development Institute (Zeleke et al. [5]). The leaf is the most susceptible to diseases, which cause plant damage and death (Kumar et al. [6]). Cotton leaf curl virus (CLCuV) disease is caused by a group of viruses in the genus Begomovirus, which is spread by whiteflies and poses a serious threat to the cotton crop (Farooq et al. [7]). It can be found in Africa, Pakistan, and Northwestern India (Sattar et al. [8]).
Cotton symptoms often appear 2-3 weeks after B. tabaci inoculation (and are characterized by a deep downward cupping of the youngest leaves). Cotton plant infected with CLCuV exhibits thickening and yellowing veins, upward and downward curling, enations on the underside of the leaves, and stunting. CLCuV control relies on insecticide treatments against the insect vector Bemisia tabaci (Briddon) [9].
The mathematical modeling of crop disease in plant pathology is presently a rapidly emerging topic. Fouda et al. [10] constructed a mathematical model of bleached cotton plain single jersey knitted fabrics that may be used to anticipate fabric attributes and define fabric geometrical relationships before manufacturing. Furthermore, the fabric measuring method measures the real yarn diameter and estimates the fabric thickness as a result. According to the findings, the thickness of a simple single jersey is related to three times the yarn diameter.
Levins et al. [11] devised a model that describes the differential equations of interactions between prey and predator, crop and pest, and migration effect. The models were used to find answers to environmental questions.
Hernández-Bautista et al. [12] proposed a mathematical model that portrays cotton dyeing in cones on three scales: micro, meso, and macro. To simulate cotton dyeing, the mass and momentum conservation equations were applied; Banks et al. [13] provided a mathematical model and statistical models, including ANOVA-based model comparison tests and residual plot analysis, that they used to make the best selections. They also investigate the statistical assumptions that are frequently made inadvertently during the parameter estimation process, as well as the effects of making erroneous assumptions.
Mamatov et al. [14] introduced a single parabolic-type boundary value problem for calculating the temperature field of raw cotton and air components in drum dryers. The proposed model and numerical technique are shown to accurately explain the raw cotton drying process. Dome et al. [15] developed a mathematical model for calculating input demand in relation to cotton production costs. They also assert that cotton farmers are very vulnerable to swings in input prices, which determine whether they generate profitable or lossmaking output when compared to total costs per hectare.
Monomolecular, Logistic, Gompertz, Richards, Quadratic, and Reciprocal growth models were provided by Sundar Rajan and Palanivel [16] for India Cotton area, output, and productivity statistics from 1980 to 2013. The chosen model had the highest R2, lower residual sum of squares, and mean square error of the six models considered.
Aboukarima et al. [17] created a multiple regression model to forecast the leaf area of a cotton crop for agricultural research purposes. The developed model could be a viable and quick alternative, especially in areas where modern technology or other tools for measuring leaf area are not available.
Su et al. [18] studied leaf area index (LAI) models and the relationships between LAI, dry matter, and yield for cotton cultivated under three soil conditioners in Korla, Xinjiang, China, with the goal of improving water use efficiency and finding optimal soil conditioner application rates. Khan et al. [19] investigated the association between the occurrence of cotton leaf curl virus (CLCuV), environmental circumstances, and the population of silver leaf whiteflies in Pakistan's agricultural sector. The proposed mathematical relationship can anticipate disease incidence in future months, which can aid agriculturists in disease control in Pakistan's agricultural areas.
Ahmad et al. [20] developed a mathematical model of cotton leaf curl viral infection in Pakistan and its link to meteorological variables. They employ mathematics to connect the intensity of the cotton leaf curl virus (CLCuV) to environmental factors including temperature, rainfall, and humidity, as well as the population of whiteflies in Pakistan's agricultural sector. Humidity and rainfall have been connected to the sickness.
Saeed et al. [21] examined cotton leaf curl virus, fiber quality, and yield components in germplasm imported from the United States. 79 cotton genotypes were evaluated using statistical techniques such as correlation analysis, clustering, and principal components. Cotton leaf curl virus demonstrated a substantial negative association with plant height, monopodial and sympodial branches, and a significant positive relationship with fiber fineness, but not with other characteristics.
Inspired by the literature, we develop a new ecoepidemiological model that uses differential equations to investigate and analyze the dynamics of the cotton leaf curl virus (CLCuV) of cotton plants. Moreover, our present model is unique in that it divides the cotton leaf curl virus (CLCuV) model into cotton and vector populations. Cotton and vector populations both have susceptible and infected subclasses. The findings of our study are useful in developing effective methods for preventing or eradicating the spread of a cotton leaf curl virus disease. This research is structured as follows. Part two involves the development of a novel mathematical model for the transmission dynamics of cotton leaf curl virus (CLCuV). The existence and stability of cotton leaf curl virus (CLCuV) equilibria, as well as the positivity and boundedness of solutions, are discussed in part three. Part four addresses numerical simulation. Part five concludes with conclusions.

Model Formulation
In this section, we divided the cotton leaf curl virus (CLCuV) model into cotton and vector populations. We considered the susceptible and infected subgroups of these populations. AðtÞ represents susceptible cotton, and BðtÞ represents infected cotton. Similarly, CðtÞ represents the susceptible vector, and DðtÞ represents the infected vector. The model assumed recruitment rate of susceptible vectors by k 2 and moves to infected vectors ðDÞ with ω 2 rate after consuming ill plants or cotton. The susceptible cotton ðAÞ also replanted at rate k 1 and the diseases spread to cotton, when infected vectors ðDÞ react with susceptible cotton ðAÞ at rate of ω 1 through eating. Cotton once become infected not ever mends and gives yield or produce very low yield of cotton. The model also assumes that α is the natural death rate for cotton population and β is natural death rate for vector population. All the descriptions of the parameters are listed in Table 1.
Using the assumptions and flow chart of the model in Figure 1, we can derive the following nonlinear ordinary differential equations: Journal of Applied Mathematics with

Model Analysis
3.1. Positivity of Solution. In this subsection, our model (Equations (1), (2), (3), and (4)) to be ecoepidemiologically meaningful and well posed, it is necessary to prove that all state variables of system with positive initial data will remain positive for all times t ≥ 0. The following theorem is used to demonstrate the positivity of the system of Equations (1), , and (4), we have that Using separation variables and by integrating Equation (6) ð dA From the second equation of system Equations (1), (2), (3), and (4), we have that: Using separation variables and by integrating Equation (8) ð dB Furthermore, using similar procedure on the above, we have Hence, all the solution sets are positive for t ≥ 0, that is the model is meaningful and well posed.

Invariant Region.
We determine a region in which the solution of system of Equations (1), (2), (3), and (4) are bounded. Now, differentiating the total cotton population, By re-arranging and multiplying Equation (11) by inte- That is By integrating and solving Equation (13), we obtain Infection rate of cotton ω 2 Infection rate of vector α Natural death rate of cotton β Natural death rate of vector Figure 1: Flow chart of the model.

Journal of Applied Mathematics
Taking the limit as t ⟶ ∞ to Equation (14), we obtain And differentiating the cotton leaf curl virus population, Using similar procedure on the above Taking the limit as t ⟶ ∞ to Equation (17), we obtain As a result, the feasible solution set for the CLCuV model is given by: that is positively invariant inside in which the model is considered to be ecoepidemiologically meaningful and mathematically well posed.  (1) is given by The basic reproduction number R 0 : The basic reproduction number, R 0 , quantifies the predicted number of secondary infections caused by a single newly infected individual delivered directly into a susceptible group (Kinene et al. [22]). By rewriting the system of Equations (1), (2), (3), and (4) starting with newly infective classes and using the next generation matrix method, the basic reproduction number R 0 can be obtained as Then, we consider Now, the Jacobian matrix of f and v with respect to B and D at disease free equilibrium point, E 0 = ððk 1 /αÞ, 0, ðk 2 / βÞ, 0Þ, is Then, by the principle of next generation matrix, the basic reproduction number R 0 is the dominant eigen value of the FV −1 or spectral radius of FV −1 where The characteristic equations of Equation (24) becomes That is Since, basic reproduction number R 0 is the maximum eigen values of FV −1 or the spectral radius of FV −1 . As a result, The two generation are required to transmission of CLCuV to take place in the cotton field that is from an infectious cotton plant to susceptible vector and then from an infectious vector to susceptible cotton (Van den Driessche and Watmough [23]). That is why the square root found in R 0 . It implies that Journal of Applied Mathematics is the cotton plants contribution when they infect the vector and R 0v = ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi k 2 ω 2 /α 2 p is the contribution of the vector population when it infects cotton plants.  (4) is

Local
Evaluating the Jacobian matrix of system of Equation (29) at disease free equilibrium point E 0 = ðk 1 /α, 0, k 2 /β, 0Þ is The characteristic equation of Jacobian matrix of Equation (30) at disease free equilibrium point, Evaluating Equation (31) simplifying it, we get where Clearly, from Equation (32), we observe that and from the last expression of Equation (32), that is By using the Routh-Hurwitz criteria, Equation (35) has strictly a negative real root if d 1 > 0 and d 2 > 0. Clearly, we observe that d 1 = α + β > 0 and As a result, our model Equations (1), (2), (3), and (4) at E 0 offers all eigenvalues with a negative real part, and so it is locally asymptotically stable if R 0 < 1.

Disease Endemic Equilibrium Point of the Model.
The endemic equilibrium point, E 1 , of the model is the steady state solution where leaf curl virus persist in the population of cotton plants. We can obtain by equating each system of the equation equal to zero; that is, From first equation of (47), we get From second equation of (47), we have Substituting the value of A * from Equation (51) into Equation (52), we obtain From third equation of (47), we have Substituting the value of B * from Equation (53) into Equation (54), we obtain From the last Equation (47), we have Substituting the value B * from Equation (53) and the value C * from Equation (55) in Equation (56), we have By re-arranging and simplifying Equation (57), we get 3.7. Local Stability of the Endemic Equilibrium Point. In this part, we use the Jacobian stability approach to demonstrate the local stability of the disease endemic equilibrium condition.
Proof. The local stability of the endemic equilibrium, E 1 , is determined based on the signs of the eigenvalues of the Jacobian matrix which is computed at the disease endemic equilibrium, E 1 . Now, the Jacobian matrix of the model at E 1 is given by The characteristic equation of Jacobian matrix of Equation (60) at disease endemic equilibrium point, E 1 , is jJðE 1 Þ − λI 4 j = 0. That is: Equation (61) can be simplified as where f 4 = 1, f 3 = 2α + 2β + ω 1 D * + ω 2 B * , Now, the characteristic polynomial of Equation (62) can be analyzed by Routh-Hurwitz criteria. The coefficients f 4 , f 3 , f 2 , f 1 , f 0 of the characteristic polynomial are real positive. As a result, the necessary criterion for the disease endemic equilibrium point's stability is met. The adequate condition for system stability using the Hurwitz array for the characteristic polynomial is then provided as follows: where f 4 , f 3 , f 2 , f 1 , f 0 are the coefficients of the characteristic polynomial and the remaining elements in the array are obtained as follows:

Journal of Applied Mathematics
Since the coefficients of the characteristic polynomial, f 4 , f 3 , f 2 , f 1 , f 0 are real positive, and the first column of the Routh-Hurwitz array has the same positive sign. Therefore, by using the Routh-Hurwitz criteria, all eigenvalues of the characteristics polynomial are negative. Thus, the disease endemic equilibrium point E 1 is locally asymptotically stable if R 0 > 1.
3.9. Sensitivity Analysis of Model Parameters. In order to establish the robustness of model predictions, sensitivity analysis is widely used to test and discover parameters that can alter the basic reproduction number, R 0 . It demonstrates the significance of each parameter in disease transmission. Sensitivity indices measure how much a variable changes when a parameter is altered Rodrigues et al. [27]. When the variable is a differentiable function of the parameter, then the sensitivity index can also be calculated using partial derivatives. Definition 6. The normalized forward sensitivity index of a R 0 that depends differentiably on a parameter, x i , is given as where x i represent all the basic parameters. For example, Using the same method, we calculate the sensitivity indexes for the other parameters. Table 2 presents the criteria in decreasing order of sensitivity.

Numerical Simulation
In this section, we use MATLAB ode45 solvers to numerically validate our work. The simulations investigate the effect of different model parameter combinations on the transmission dynamics of cotton leaf curl virus (CLCuV). The simulation was performed using a variety of parameter values. The source of the collection of parameter values is mostly based on assumptions. The relevant initial circumstances are used in the simulations and analyses: Að0Þ = 600, Bð0Þ = 150, Cð0Þ = 100, Dð0Þ = 180, and the parameters values are displayed in Table 2.
The time series plot of state variables for R 0 < 1 and R 0 > 1 is shown in Figures 2 and 3. Figure 2 shows that susceptible cotton begins to rise asymptotically to the disease free equilibrium point, whereas infected cotton populations drop asymptotically to the disease free equilibrium point. Furthermore, the susceptible vector population grows asymptotically to the disease free equilibrium point, but the infected vector population falls asymptotically to the disease free equilibrium point. In this situation, the disease may eventually eliminate in the long run. The existence of such condition is due to the fact that R 0 = 0:3223 which is less than one. This supports the theorem that the stability of disease free equilibrium point exists when R 0 < 1, that is, if R 0 < 1, then on average, one infected cotton plant produces less than one newly infectious plant over the course of its disease period. Figure 3 shows that susceptible cotton and vector individuals are reduced due to the influence of infected cotton and vector individuals, after which they join the infected class, resulting in an increase in infected cotton and vector people. As a result, there is an increase in infected cotton and vectors, and the disease endemic equilibrium point exists and is stable. The existence of this condition is due to the fact that R 0 = 24:5286 which is greater than one. This supports the theorem that the stability of disease endemic equilibrium point exists when R 0 > 1, that is, if R 0 > 1, each infected cotton and vectors produces, on average more than one new infected cotton and vectors, then disease will be able to spread in the given area.
Susceptible cotton population ðAÞ and infected cotton population ðBÞ with respect to time t for different values of ω 1 are shown in Figure 4. From Figure 4, we observe that as infection rate of cotton, ω 1 , increases, susceptible cotton population ðAÞ decreases, while infected cotton population ðBÞ increased. Also, susceptible vector population ðCÞ and infected vector population ðDÞ with respect to time t for different values of ω 2 are shown in Figure 5. From Figure 5, we observe that as the value of infection rate of vector, ω 2 , increases, susceptible vector population ðCÞ decreases, while infected vector population ðDÞ increased.

Conclusion
In this study, an ecoepidemiological model for the transmission dynamics of cotton leaf curl virus (CLCuV) disease in cotton plant was developed. The model's well-posedness, positivity, and boundedness are all explored. The basic reproduction number and the stability analysis of the model's cotton equilibria were investigated. According to the study, if the basic reproduction number is less than one, the cotton-free equilibrium is locally and globally asymptotically stable; however, if the basic reproduction number is more than one, the endemic equilibrium is locally asymptotically stable. The numerical simulation shows that as infection rate of cotton, ω 1 , increases, susceptible cotton population ðAÞ decreases, while infected cotton population ðBÞ increased. Furthermore, as the value of infection rate of vector, ω 2 , increases, susceptible vector population ðCÞ decreases, while infected vector population ðDÞ increased. Optimal control and the cost-effectiveness analysis of the integrated strategy would be considered in a future study.

Data Availability
The data supporting this nonlinear ecoepidemiological model are taken using assumption which are relevant to this paper.

Conflicts of Interest
The authors declare that they have no conflicts of interest.