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

Authors: Abayneh Kebede Fantaye
Journal: Journal of Applied Mathematics
 (2022)

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## Abstract

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 differential 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 findings show that lowering the infection rate of cotton vectors has a significant impact on controlling cotton leaf curl virus (CLCuV) in the time frame given.


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## Body

## 1. 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 second-largest 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 loss-making 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.


## 2. 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.At represents susceptible cotton, and Bt represents infected cotton. Similarly, Ct represents the susceptible vector, and Dt represents the infected vector. The model assumed recruitment rate of susceptible vectors by k2 and moves to infected vectors D with ω2 rate after consuming ill plants or cotton. The susceptible cotton A also replanted at rate k1 and the diseases spread to cotton, when infected vectors D react with susceptible cotton A at rate of ω1 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.Table 1
Parameters of the model.
ParameterDescriptionk1Replanting rate of cottonk2Recruitment rate of vectorω1Infection rate of cottonω2Infection rate of vectorαNatural death rate of cottonβNatural death rate of vectorUsing the assumptions and flow chart of the model in Figure1, we can derive the following nonlinear ordinary differential equations:
(1)dAdt=k1−ω1AD−αA,(2)dBdt=ω1AD−αB,(3)dCdt=k2−ω2BC−βC,(4)dDdt=ω2BC−βD,with
(5)A0=A0>0,B0=B0≥0,C0=C0≥0,D0=D0≥0.Figure 1
Flow chart of the model.


## 3. 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), (2), (3), and (4).Theorem 1.
LetΩ=A,B,C,D∈R4:A0>0,B0>0,C0>,D0>0. Then, the solution set At,Bt,Ct,Dt of system of Equations (1), (2), (3), and (4) are positive for all t≥0.Proof.
From the first equation of system Equations (1), (2), (3), and (4), we have that
(6)dAdt=k1−ω1D+αA≥−ω1D+αA.
Using separation variables and by integrating Equation (6)
(7)∫dAA≥−∫ω1D+αdt,lnAt≥−∫ω1D+αdt+c1,At≥e−∫ω1D+αdt+c1,At≥A0e−∫ω1D+αdt>0.
From the second equation of system Equations (1), (2), (3), and (4), we have that:
(8)dBdt=ω1AD−αB≥−αB.
Using separation variables and by integrating Equation (8)
(9)∫dBB≥−∫αdt,lnBt≥−αt+c2,Bt≥e−αt+c2,Bt≥B0e−αt≥0.
Furthermore, using similar procedure on the above, we have(10)Ct≥C0e−∫ω2B+βdt≥0,Dt≥D0e−βt≥0.
Hence, all the solution sets are positive fort≥0, that is the model is meaningful and well posed.


### 3.2. 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, Nc=A+B with respect to time, we have
(11)dNcdt=k1−αNc.By re-arranging and multiplying Equation (11) by integrating factor e∫αdt=eαt, we have
(12)dNcdteαt+αNceαt=k1eαt.That is(13)ddtNceαt=k1eαt.By integrating and solving Equation (13), we obtain
(14)Nct=k1α+e−αtNc0−k1α.Taking the limit ast⟶∞ to Equation (14), we obtain
(15)Ωc=A,B∈R+2:Nc≤k1α.And differentiating the cotton leaf curl virus population,Nv=C+D, we have
(16)dNvdt=k2−βNv.Using similar procedure on the above(17)Nvt=k2β+e−βtNv0−k2β.Taking the limit ast⟶∞ to Equation (17), we obtain
(18)Ωv=C,D∈R+2:Nv≤k2β.As a result, the feasible solution set for the CLCuV model is given by:(19)Ω=Ωc×Ωv=A,B,C,D∈R+4:Nc≤k1α:Nv≤k2βthat is positively invariant inside in which the model is considered to be ecoepidemiologically meaningful and mathematically well posed.


### 3.3. Disease Free Equilibrium Point of the Model

The model’s disease-free equilibrium points,E0, are stationary solutions in which there is no infection. It is obtained by equating Equations (1), (2), (3), and (4) to zero and using B=0 and D=0. Then, disease free equilibrium point, E0 of our model Equation (1) is given by
(20)E0=A0,B0,C0,D0=k1α,0,k2β,0.The basic reproduction numberR0: The basic reproduction number, R0, 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 R0 can be obtained as
(21)dBdt=ω1AD−αB,dDdt=ω2BC−βD.Then, we consider(22)f=ω1ADω2BC,v=αBβD.Now, the Jacobian matrix off and v with respect to B and D at disease free equilibrium point, E0=k1/α,0,k2/β,0, is
(23)F=0ω1k1αω2k2β0,V=α00β.Then, by the principle of next generation matrix, the basic reproduction numberR0 is the dominant eigen value of the FV−1 or spectral radius of FV−1 where
(24)FV−1=0ω1k1αω2k2β01α001β=0ω1k1αβω2k2βα0.The characteristic equations of Equation (24) becomes
(25)λ2−k1k2ω1ω2α2β2=0.That is(26)λ=±k1k2ω1ω2α2β2.Since, basic reproduction numberR0 is the maximum eigen values of FV−1 or the spectral radius of FV−1. As a result,
(27)R0=k1k2ω1ω2α2β2.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 R0. It implies that
(28)R0=k1k2ω1ω2α2β2=k1ω1α2k2ω2β2=k1ω1α2k2ω2β2=R0c×R0v.where R0c=k1ω1/α2 is the cotton plants contribution when they infect the vector and R0v=k2ω2/α2 is the contribution of the vector population when it infects cotton plants.


### 3.4. Local Stability of the Disease Free Equilibrium Point

The linearization system of Equations (1), (2), (3), and (4) at E0 can be used to find the local stability of the model at disease-free equilibrium point, E0.Theorem 2.
Disease free equilibrium point,E0 of system of Equations (1), (2), (3), and (4) are locally asymptotically stable, if R0<1.Proof.
The Jacobian matrix of system of Equations (1), (2), (3), and (4) is
(29)J=−ω1D+α00−ω1Aω1D−α0ω1A0−ω2C−ω2B+β00ω2Cω2B−β.
Evaluating the Jacobian matrix of system of Equation (29) at disease free equilibrium point E0=k1/α,0,k2/β,0 is
(30)JE0=−α00−ω1k1α0−α0ω1l1α0−ω2k2β−β00ω2k2β0−β.
The characteristic equation of Jacobian matrix of Equation (30) at disease free equilibrium point, E0, is JE0−λI4=0. That is
(31)−α−λ00−ω1k1α0−α−λ0ω1k1α0−ω2k2β−β−λ00ω2k2β0−β−λ=0.
Evaluating Equation (31) simplifying it, we get
(32)−α−λ−β−λλ2+d1λ+d2=0,where
(33)d1=α+β,d2=αβ−k1k2ω1ω2αβ=αβ1−k1k2ω1ω2α2β2=αβ1−R02.
Clearly, from Equation (32), we observe that
(34)λ1=−α<0,λ2=−β<0,and from the last expression of Equation (32), that is
(35)λ2+d1λ+d2=0,
By using the Routh-Hurwitz criteria, Equation (35) has strictly a negative real root if d1>0 and d2>0. Clearly, we observe that d1=α+β>0 and
(36)d2=αβ1−R02>0,if 1−R02>0. That is, R02<1 implies that R0<1. As a result, our model Equations (1), (2), (3), and (4) at E0 offers all eigenvalues with a negative real part, and so it is locally asymptotically stable if R0<1.


### 3.5. Global Stability of the Disease Free Equilibrium Point

To establish the global stability of the disease free equilibrium pointE0, we use the method proposed by Castillo-Chavez et al. [24] and Fantaye and Birhanu [25]. Based on these, we have written the system of Equations (1), (2), (3), and (4) in the following form:
(37)dMdt=JM,L,(38)dLdt=PM,L,(39)PM,0=0,where M=A,C∈R2 represent the number of uninfected classes, while L=B,D∈R2 represent the number of infected classes and E0=M∗,0 represents the disease-free equilibrium of this system. The disease-free equilibrium E0 is globally asymptotically stable equilibrium for the model if the following conditions are fulfilled:
(1)
dM/dt=JM,0,M∗ is globally asymptotically stable:(40)dLdt=DLPM∗,0L−P^M,L,P^M,L≥0∀M,L∈Ω,where DLPM∗,0 is an M-matrix and PM,L taken in B,D and evaluated at M∗,0=k1/α,k2/β,0,0. If system of Equation (37) satisfies the above conditions, then the following theorem holds.Theorem 3.
The disease free equilibrium point,E0=M∗,0 of system of Equation (37) is globally asymptotically stable if R0≤1, and Conditions (1) and (5) are holds.Proof.
From our model of Equations (1), (2), (3), and (4), we can obtain JM,L and PM,L:
(41)JM,L=k1−ω1AD−αAk2−ω2BC−βC,PM,L=ω1AD−αBω2BC−βD.
Now, we consider the reduced systemdM/dt=JM,0 from Condition (1):
(42)dAdt=k1−αA,(43)dCdt=k2−βC.
M∗=k1/α,k2/β is a globally asymptotically stable equilibrium point for the reduced system dM/dt=JM,0. This can be verifed from the solution of Equation (42); we get At=k1/α+A0−k1/αe−αt which approaches k1/α as t⟶∞, and from Equation (43), we obtain Ct=k2/βk2/β+C0−k2/βk2/βe−βt which approaches k2/β as t⟶∞, We note that this asymptomatic dynamics is independent of the initial conditions in Ω; therefore, the convergence of the solutions of the reduced system (42) and (43) is global in Ω. Now, we compute
(44)DLPM∗,0=−αk1ω1αk2ω2β−β.
Then,PM,L can be written as
(45)PM,L=DLPM∗,0L−P^M,L,and we want to show P^M,L, which is obtained as
(46)P^M,L=ω1Dk1α−Aω2Bk2β−C.
Herek1/αk1/α≥A and k2/βk2/β≥B. Hence, it is clear that P^M,L≥0, ∀M,L∈Ω. Thus, this proves that disease free equilibrium point E0 is globally asymptotically stable when R0≤1.


### 3.6. Disease Endemic Equilibrium Point of the Model

The endemic equilibrium point,E1, 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,
(47)k1−ω1A∗D∗−αA∗=0,(48)ω1A∗D∗−αB∗=0,(49)k2−ω2B∗C∗−βC∗=0,(50)ω2B∗C∗−βD∗=0,From first equation of (47), we get
(51)A∗=k1ω1D∗+α.From second equation of (47), we have
(52)B∗=ω1A∗D∗α.Substituting the value ofA∗ from Equation (51) into Equation (52), we obtain
(53)B∗=k1ω1D∗αω1D∗+μ.From third equation of (47), we have
(54)C∗=k2ω2B∗+β.Substituting the value ofB∗ from Equation (53) into Equation (54), we obtain
(55)C∗=k2αω1D∗+αk1ω1ω2D∗+βαω1D∗+α.From the last Equation (47), we have
(56)ω2B∗C∗=βD∗.Substituting the valueB∗ from Equation (53) and the value C∗ from Equation (55) in Equation (56), we have
(57)k1k2ω1ω2k1ω1ω2D∗+βαω1D∗+α=β.By re-arranging and simplifying Equation (57), we get
(58)D∗=α2βR02−1ω1k1ω2+αβ.Thus, by substituting Equation (58) into Equations (51), (53), and (55), we obtain
(59)A∗=k1k1ω2+αβα2βR02−1+αk1ω2+αβ,B∗=k1βR02−1αβR02−1+k1ω2+αβ,C∗=k2αβR02−1+k1ω2+αββk1ω2+αβR02−1+k1ω2+αβ.


### 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.Theorem 4.
WhenR0>1, the model’s endemic equilibrium point, E1, is locally asymptotically stable.Proof.
The local stability of the endemic equilibrium,E1, is determined based on the signs of the eigenvalues of the Jacobian matrix which is computed at the disease endemic equilibrium, E1. Now, the Jacobian matrix of the model at E1 is given by
(60)J=−ω1D∗+α00−ω1A∗ω1D∗−α0ω1A∗0−ω2C∗−ω2B∗+β00ω2C∗ω2B∗−β.
The characteristic equation of Jacobian matrix of Equation (60) at disease endemic equilibrium point, E1, is JE1−λI4=0. That is:
(61)−ω1D∗+α−λ00−ω1A∗ω1D∗−α−λ0ω1A∗0−ω2C∗−ω2B∗+β−λ00ω2C∗ω2B∗−β−λ=0.
Equation (61) can be simplified as
(62)Pλ=f4λ4+f3λ3+f2λ2+f1λ+f0,where
(63)f4=1,f3=2α+2β+ω1D∗+ω2B∗,f2=αβ+α+βα+βαβ+2β+αω1D∗+2α+βω2B∗+ω1ω2B∗−A∗C∗,f1=μβα+β+α+βαβ+αβ+α+βαω2B∗+α+ββω1D∗+α+βω1ω2B∗D∗−ω1ω2α+βA∗C∗,f0=βω1ω2αB∗D∗+ω1β2αD∗+βαω2αB∗+α2β2−αβω1ω2A∗C∗.
Now, the characteristic polynomial of Equation (62) can be analyzed by Routh-Hurwitz criteria. The coefficients f4,f3,f2,f1,f0 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:
(64)s4s3s2s1s0f4f2f0f3f10g1g2g3h1h2h3k1k2k3,where f4,f3,f2,f1,f0 are the coefficients of the characteristic polynomial and the remaining elements in the array are obtained as follows:
(65)g1=−1f3f4f2f3f1=f3f2−f1f3>0,g2=−1f3f4f0f30=f0,g3=−1f3f40f30=0,h1=−1g1f3f1g1g2=f1g1−f3f0g1>0,h2=−1g1f30g10=0,h3=−1g1f30g10=0,k1=−1h1g1g2h10=f0,k2=−1h1g10h10=0,k3=−1h1g10h10=0.
Since the coefficients of the characteristic polynomial,f4,f3,f2,f1,f0 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 E1 is locally asymptotically stable if R0>1.


### 3.8. Global Stability of Disease Endemic Equilibrium Point

Theorem 5.
ForR0>1, then the system of Equations (1), (2), (3), and (4) at E1 is globally asymptotical stable.Proof.
To investigate the global stability of the endemic equilibrium pointE1, we consider the following Lyapunov function for model of Equations (1), (2), (3), and (4):
(66)Vt=K1A−A∗22+K2B−B∗22+K3C−C∗22+K4D−D∗22,where K1,K2,K3,K4 are chosen. By differentiating (66) with respect to time, we have
(67)dVdt=K1A−A∗dAdt+K2B−B∗dBdt+K3C−C∗dCdt+K4D−D∗dDdt.
Using system Equations (1), (2), (3), and (4), Equation (67) can becomes
(68)dVdt=K1A−A∗k1−ω1D+αA+K2B−B∗ω1AD−αB+K3C−C∗k2−ω2B+βC+K4D−D∗ω2BC−βD.
By re arranging Equation (68), we have
(69)dVdt=−K1A−A∗A−k1A+ω1D+α−K2B−B∗B−ω1ADB+α−K3C−C∗C−k2C+ω2B+β−K4D−D∗D−ω2BCD+β.
Here, we can choose(70)K1=Aω1D+μA−k1,K2=BαB−ω1AD,K3=Cω2B+βC−k2K4=DβD−ω2BC.
Hence, we observe thatdV/dt<0 and an endemic equilibrium point, E1, of the model is globally stable. Moreover, dV/dt=0 if and only if either A=A∗,B=B∗,C−C∗,D−D∗ or A=B=C=D=0. Thus, using LaSalle [26], E1 is global asymptotical stable whenever R0>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,R0. 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 aR0 that depends differentiably on a parameter, xi, is given as
(71)ΠxiR0=∂R0∂xi.xiR0,where xi represent all the basic parameters. For example,
(72)Mk1R0=∂R0∂k1.k1R0,=12k1b2ω1ω2/α2β2k2ω1ω2α2β2.k1k1b2ω1ω2/α2β2,=12.
Using the same method, we calculate the sensitivity indexes for the other parameters. Table2 presents the criteria in decreasing order of sensitivity.Table 2
The parameter values and sensitivity index of model.
ParameterValueSensitivity indexk10.97+vek20.27+veω10.000007+veω20.000005+veα0.04-veβ0.03-ve


### 3.10. Interpretation of Sensitivity Indices

Table2 displays the sensitivity indices of the basic reproductive number in relation to the important parameters. Positive indices k1,k1,ω1,andω2 indicate that parameters with increasing values have a significant impact on the spread of the disease. Since the basic reproduction number rises as their values rise, so does the average number of secondary cases of infection. Additionally, those parameters with negative sensitivity indices α,β have the effect of reducing disease burden when their values rise, while the others remain constant. Additionally, as their values rise, the basic reproduction number falls, resulting in a reduction in the disease’s endemic areas.


## 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), (2), (3), and (4).Theorem 1.
LetΩ=A,B,C,D∈R4:A0>0,B0>0,C0>,D0>0. Then, the solution set At,Bt,Ct,Dt of system of Equations (1), (2), (3), and (4) are positive for all t≥0.Proof.
From the first equation of system Equations (1), (2), (3), and (4), we have that
(6)dAdt=k1−ω1D+αA≥−ω1D+αA.
Using separation variables and by integrating Equation (6)
(7)∫dAA≥−∫ω1D+αdt,lnAt≥−∫ω1D+αdt+c1,At≥e−∫ω1D+αdt+c1,At≥A0e−∫ω1D+αdt>0.
From the second equation of system Equations (1), (2), (3), and (4), we have that:
(8)dBdt=ω1AD−αB≥−αB.
Using separation variables and by integrating Equation (8)
(9)∫dBB≥−∫αdt,lnBt≥−αt+c2,Bt≥e−αt+c2,Bt≥B0e−αt≥0.
Furthermore, using similar procedure on the above, we have(10)Ct≥C0e−∫ω2B+βdt≥0,Dt≥D0e−βt≥0.
Hence, all the solution sets are positive fort≥0, that is the model is meaningful and well posed.


## 3.2. 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, Nc=A+B with respect to time, we have
(11)dNcdt=k1−αNc.By re-arranging and multiplying Equation (11) by integrating factor e∫αdt=eαt, we have
(12)dNcdteαt+αNceαt=k1eαt.That is(13)ddtNceαt=k1eαt.By integrating and solving Equation (13), we obtain
(14)Nct=k1α+e−αtNc0−k1α.Taking the limit ast⟶∞ to Equation (14), we obtain
(15)Ωc=A,B∈R+2:Nc≤k1α.And differentiating the cotton leaf curl virus population,Nv=C+D, we have
(16)dNvdt=k2−βNv.Using similar procedure on the above(17)Nvt=k2β+e−βtNv0−k2β.Taking the limit ast⟶∞ to Equation (17), we obtain
(18)Ωv=C,D∈R+2:Nv≤k2β.As a result, the feasible solution set for the CLCuV model is given by:(19)Ω=Ωc×Ωv=A,B,C,D∈R+4:Nc≤k1α:Nv≤k2βthat is positively invariant inside in which the model is considered to be ecoepidemiologically meaningful and mathematically well posed.


## 3.3. Disease Free Equilibrium Point of the Model

The model’s disease-free equilibrium points,E0, are stationary solutions in which there is no infection. It is obtained by equating Equations (1), (2), (3), and (4) to zero and using B=0 and D=0. Then, disease free equilibrium point, E0 of our model Equation (1) is given by
(20)E0=A0,B0,C0,D0=k1α,0,k2β,0.The basic reproduction numberR0: The basic reproduction number, R0, 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 R0 can be obtained as
(21)dBdt=ω1AD−αB,dDdt=ω2BC−βD.Then, we consider(22)f=ω1ADω2BC,v=αBβD.Now, the Jacobian matrix off and v with respect to B and D at disease free equilibrium point, E0=k1/α,0,k2/β,0, is
(23)F=0ω1k1αω2k2β0,V=α00β.Then, by the principle of next generation matrix, the basic reproduction numberR0 is the dominant eigen value of the FV−1 or spectral radius of FV−1 where
(24)FV−1=0ω1k1αω2k2β01α001β=0ω1k1αβω2k2βα0.The characteristic equations of Equation (24) becomes
(25)λ2−k1k2ω1ω2α2β2=0.That is(26)λ=±k1k2ω1ω2α2β2.Since, basic reproduction numberR0 is the maximum eigen values of FV−1 or the spectral radius of FV−1. As a result,
(27)R0=k1k2ω1ω2α2β2.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 R0. It implies that
(28)R0=k1k2ω1ω2α2β2=k1ω1α2k2ω2β2=k1ω1α2k2ω2β2=R0c×R0v.where R0c=k1ω1/α2 is the cotton plants contribution when they infect the vector and R0v=k2ω2/α2 is the contribution of the vector population when it infects cotton plants.


## 3.4. Local Stability of the Disease Free Equilibrium Point

The linearization system of Equations (1), (2), (3), and (4) at E0 can be used to find the local stability of the model at disease-free equilibrium point, E0.Theorem 2.
Disease free equilibrium point,E0 of system of Equations (1), (2), (3), and (4) are locally asymptotically stable, if R0<1.Proof.
The Jacobian matrix of system of Equations (1), (2), (3), and (4) is
(29)J=−ω1D+α00−ω1Aω1D−α0ω1A0−ω2C−ω2B+β00ω2Cω2B−β.
Evaluating the Jacobian matrix of system of Equation (29) at disease free equilibrium point E0=k1/α,0,k2/β,0 is
(30)JE0=−α00−ω1k1α0−α0ω1l1α0−ω2k2β−β00ω2k2β0−β.
The characteristic equation of Jacobian matrix of Equation (30) at disease free equilibrium point, E0, is JE0−λI4=0. That is
(31)−α−λ00−ω1k1α0−α−λ0ω1k1α0−ω2k2β−β−λ00ω2k2β0−β−λ=0.
Evaluating Equation (31) simplifying it, we get
(32)−α−λ−β−λλ2+d1λ+d2=0,where
(33)d1=α+β,d2=αβ−k1k2ω1ω2αβ=αβ1−k1k2ω1ω2α2β2=αβ1−R02.
Clearly, from Equation (32), we observe that
(34)λ1=−α<0,λ2=−β<0,and from the last expression of Equation (32), that is
(35)λ2+d1λ+d2=0,
By using the Routh-Hurwitz criteria, Equation (35) has strictly a negative real root if d1>0 and d2>0. Clearly, we observe that d1=α+β>0 and
(36)d2=αβ1−R02>0,if 1−R02>0. That is, R02<1 implies that R0<1. As a result, our model Equations (1), (2), (3), and (4) at E0 offers all eigenvalues with a negative real part, and so it is locally asymptotically stable if R0<1.


## 3.5. Global Stability of the Disease Free Equilibrium Point

To establish the global stability of the disease free equilibrium pointE0, we use the method proposed by Castillo-Chavez et al. [24] and Fantaye and Birhanu [25]. Based on these, we have written the system of Equations (1), (2), (3), and (4) in the following form:
(37)dMdt=JM,L,(38)dLdt=PM,L,(39)PM,0=0,where M=A,C∈R2 represent the number of uninfected classes, while L=B,D∈R2 represent the number of infected classes and E0=M∗,0 represents the disease-free equilibrium of this system. The disease-free equilibrium E0 is globally asymptotically stable equilibrium for the model if the following conditions are fulfilled:
(1)
dM/dt=JM,0,M∗ is globally asymptotically stable:(40)dLdt=DLPM∗,0L−P^M,L,P^M,L≥0∀M,L∈Ω,where DLPM∗,0 is an M-matrix and PM,L taken in B,D and evaluated at M∗,0=k1/α,k2/β,0,0. If system of Equation (37) satisfies the above conditions, then the following theorem holds.Theorem 3.
The disease free equilibrium point,E0=M∗,0 of system of Equation (37) is globally asymptotically stable if R0≤1, and Conditions (1) and (5) are holds.Proof.
From our model of Equations (1), (2), (3), and (4), we can obtain JM,L and PM,L:
(41)JM,L=k1−ω1AD−αAk2−ω2BC−βC,PM,L=ω1AD−αBω2BC−βD.
Now, we consider the reduced systemdM/dt=JM,0 from Condition (1):
(42)dAdt=k1−αA,(43)dCdt=k2−βC.
M∗=k1/α,k2/β is a globally asymptotically stable equilibrium point for the reduced system dM/dt=JM,0. This can be verifed from the solution of Equation (42); we get At=k1/α+A0−k1/αe−αt which approaches k1/α as t⟶∞, and from Equation (43), we obtain Ct=k2/βk2/β+C0−k2/βk2/βe−βt which approaches k2/β as t⟶∞, We note that this asymptomatic dynamics is independent of the initial conditions in Ω; therefore, the convergence of the solutions of the reduced system (42) and (43) is global in Ω. Now, we compute
(44)DLPM∗,0=−αk1ω1αk2ω2β−β.
Then,PM,L can be written as
(45)PM,L=DLPM∗,0L−P^M,L,and we want to show P^M,L, which is obtained as
(46)P^M,L=ω1Dk1α−Aω2Bk2β−C.
Herek1/αk1/α≥A and k2/βk2/β≥B. Hence, it is clear that P^M,L≥0, ∀M,L∈Ω. Thus, this proves that disease free equilibrium point E0 is globally asymptotically stable when R0≤1.


## 3.6. Disease Endemic Equilibrium Point of the Model

The endemic equilibrium point,E1, 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,
(47)k1−ω1A∗D∗−αA∗=0,(48)ω1A∗D∗−αB∗=0,(49)k2−ω2B∗C∗−βC∗=0,(50)ω2B∗C∗−βD∗=0,From first equation of (47), we get
(51)A∗=k1ω1D∗+α.From second equation of (47), we have
(52)B∗=ω1A∗D∗α.Substituting the value ofA∗ from Equation (51) into Equation (52), we obtain
(53)B∗=k1ω1D∗αω1D∗+μ.From third equation of (47), we have
(54)C∗=k2ω2B∗+β.Substituting the value ofB∗ from Equation (53) into Equation (54), we obtain
(55)C∗=k2αω1D∗+αk1ω1ω2D∗+βαω1D∗+α.From the last Equation (47), we have
(56)ω2B∗C∗=βD∗.Substituting the valueB∗ from Equation (53) and the value C∗ from Equation (55) in Equation (56), we have
(57)k1k2ω1ω2k1ω1ω2D∗+βαω1D∗+α=β.By re-arranging and simplifying Equation (57), we get
(58)D∗=α2βR02−1ω1k1ω2+αβ.Thus, by substituting Equation (58) into Equations (51), (53), and (55), we obtain
(59)A∗=k1k1ω2+αβα2βR02−1+αk1ω2+αβ,B∗=k1βR02−1αβR02−1+k1ω2+αβ,C∗=k2αβR02−1+k1ω2+αββk1ω2+αβR02−1+k1ω2+αβ.


## 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.Theorem 4.
WhenR0>1, the model’s endemic equilibrium point, E1, is locally asymptotically stable.Proof.
The local stability of the endemic equilibrium,E1, is determined based on the signs of the eigenvalues of the Jacobian matrix which is computed at the disease endemic equilibrium, E1. Now, the Jacobian matrix of the model at E1 is given by
(60)J=−ω1D∗+α00−ω1A∗ω1D∗−α0ω1A∗0−ω2C∗−ω2B∗+β00ω2C∗ω2B∗−β.
The characteristic equation of Jacobian matrix of Equation (60) at disease endemic equilibrium point, E1, is JE1−λI4=0. That is:
(61)−ω1D∗+α−λ00−ω1A∗ω1D∗−α−λ0ω1A∗0−ω2C∗−ω2B∗+β−λ00ω2C∗ω2B∗−β−λ=0.
Equation (61) can be simplified as
(62)Pλ=f4λ4+f3λ3+f2λ2+f1λ+f0,where
(63)f4=1,f3=2α+2β+ω1D∗+ω2B∗,f2=αβ+α+βα+βαβ+2β+αω1D∗+2α+βω2B∗+ω1ω2B∗−A∗C∗,f1=μβα+β+α+βαβ+αβ+α+βαω2B∗+α+ββω1D∗+α+βω1ω2B∗D∗−ω1ω2α+βA∗C∗,f0=βω1ω2αB∗D∗+ω1β2αD∗+βαω2αB∗+α2β2−αβω1ω2A∗C∗.
Now, the characteristic polynomial of Equation (62) can be analyzed by Routh-Hurwitz criteria. The coefficients f4,f3,f2,f1,f0 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:
(64)s4s3s2s1s0f4f2f0f3f10g1g2g3h1h2h3k1k2k3,where f4,f3,f2,f1,f0 are the coefficients of the characteristic polynomial and the remaining elements in the array are obtained as follows:
(65)g1=−1f3f4f2f3f1=f3f2−f1f3>0,g2=−1f3f4f0f30=f0,g3=−1f3f40f30=0,h1=−1g1f3f1g1g2=f1g1−f3f0g1>0,h2=−1g1f30g10=0,h3=−1g1f30g10=0,k1=−1h1g1g2h10=f0,k2=−1h1g10h10=0,k3=−1h1g10h10=0.
Since the coefficients of the characteristic polynomial,f4,f3,f2,f1,f0 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 E1 is locally asymptotically stable if R0>1.


## 3.8. Global Stability of Disease Endemic Equilibrium Point

Theorem 5.
ForR0>1, then the system of Equations (1), (2), (3), and (4) at E1 is globally asymptotical stable.Proof.
To investigate the global stability of the endemic equilibrium pointE1, we consider the following Lyapunov function for model of Equations (1), (2), (3), and (4):
(66)Vt=K1A−A∗22+K2B−B∗22+K3C−C∗22+K4D−D∗22,where K1,K2,K3,K4 are chosen. By differentiating (66) with respect to time, we have
(67)dVdt=K1A−A∗dAdt+K2B−B∗dBdt+K3C−C∗dCdt+K4D−D∗dDdt.
Using system Equations (1), (2), (3), and (4), Equation (67) can becomes
(68)dVdt=K1A−A∗k1−ω1D+αA+K2B−B∗ω1AD−αB+K3C−C∗k2−ω2B+βC+K4D−D∗ω2BC−βD.
By re arranging Equation (68), we have
(69)dVdt=−K1A−A∗A−k1A+ω1D+α−K2B−B∗B−ω1ADB+α−K3C−C∗C−k2C+ω2B+β−K4D−D∗D−ω2BCD+β.
Here, we can choose(70)K1=Aω1D+μA−k1,K2=BαB−ω1AD,K3=Cω2B+βC−k2K4=DβD−ω2BC.
Hence, we observe thatdV/dt<0 and an endemic equilibrium point, E1, of the model is globally stable. Moreover, dV/dt=0 if and only if either A=A∗,B=B∗,C−C∗,D−D∗ or A=B=C=D=0. Thus, using LaSalle [26], E1 is global asymptotical stable whenever R0>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,R0. 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 aR0 that depends differentiably on a parameter, xi, is given as
(71)ΠxiR0=∂R0∂xi.xiR0,where xi represent all the basic parameters. For example,
(72)Mk1R0=∂R0∂k1.k1R0,=12k1b2ω1ω2/α2β2k2ω1ω2α2β2.k1k1b2ω1ω2/α2β2,=12.
Using the same method, we calculate the sensitivity indexes for the other parameters. Table2 presents the criteria in decreasing order of sensitivity.Table 2
The parameter values and sensitivity index of model.
ParameterValueSensitivity indexk10.97+vek20.27+veω10.000007+veω20.000005+veα0.04-veβ0.03-ve


## 3.10. Interpretation of Sensitivity Indices

Table2 displays the sensitivity indices of the basic reproductive number in relation to the important parameters. Positive indices k1,k1,ω1,andω2 indicate that parameters with increasing values have a significant impact on the spread of the disease. Since the basic reproduction number rises as their values rise, so does the average number of secondary cases of infection. Additionally, those parameters with negative sensitivity indices α,β have the effect of reducing disease burden when their values rise, while the others remain constant. Additionally, as their values rise, the basic reproduction number falls, resulting in a reduction in the disease’s endemic areas.


## 4. 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:A0=600,B0=150,C0=100,D0=180, and the parameters values are displayed in Table 2.The time series plot of state variables forR0<1 and R0>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 R0=0.3223 which is less than one. This supports the theorem that the stability of disease free equilibrium point exists when R0<1, that is, if R0<1, then on average, one infected cotton plant produces less than one newly infectious plant over the course of its disease period.Figure 2
Time series plot of state variables forR0=0.3223<1.Figure 3
Time series plot of state variables forR0=24.5286>1Figure3 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 R0=24.5286 which is greater than one. This supports the theorem that the stability of disease endemic equilibrium point exists when R0>1, that is, if R0>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 populationA and infected cotton population B with respect to time t for different values of ω1 are shown in Figure 4. From Figure 4, we observe that as infection rate of cotton, ω1, increases, susceptible cotton population A decreases, while infected cotton population B increased. Also, susceptible vector population C and infected vector population D with respect to time tfor different values of ω2 are shown in Figure 5. From Figure 5, we observe that as the value of infection rate of vector, ω2, increases, susceptible vector population C decreases, while infected vector population D increased.Figure 4
Susceptible cotton population (A) and infected cotton population (B) w.r.t. timet for different values of ω1.Figure 5
Susceptible vector population (C) and infected vector population (D) w.r.t. timet for different values of ω2.


## 5. 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,ω1, increases, susceptible cotton population A decreases, while infected cotton population B increased. Furthermore, as the value of infection rate of vector, ω2, increases, susceptible vector population C decreases, while infected vector population D increased. Optimal control and the cost-effectiveness analysis of the integrated strategy would be considered in a future study.



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