# Ecoepidemiological Model and Analysis of MSV Disease Transmission Dynamics in Maize Plant

Authors: Haileyesus Tessema Alemneh; Oluwole Daniel Makinde; David Mwangi Theuri
Journal: International Journal of Mathematics and Mathematical Sciences
 (2019)

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## Abstract

In this paper, an ecoepidemiological deterministic model for the transmission dynamics of maize streak virus (MSV) disease in maize plant is proposed and analysed qualitatively using the stability theory of differential equations.The basic reproduction number with respect to the MSV free equilibrium is obtained using next generation matrix approach. The conditions for local and global asymptotic stability of MSV free and endemic equilibria are established. The model exhibits forward bifurcation and the sensitivity indices of various embedded parameters with respect to the MSV eradication or spreading are determined. Numerical simulation is performed and dispalyed graphically to justify the analytical results.


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## Body

## 1. Introduction

Maize (Zea mays L.) is grown globally across temperate and tropical zones, spanning all continents [1]. It is the most widely grown and consumed staple crop in Africa with more than 100 million Africans depending on it as their main food source which is annually planted over an area of 15.5 million hectares [2, 3]. In Ethiopia, maize is the most cereal gown which ranks first in yield per hectare and is grown in all 11 administrative regions [4]. It is the primary food, averaging slightly more than 20% of daily caloric intake. About 9 million smallholder farmers grow maize and 75% of the maize produced is consumed as food [5, 6]. The reports of Ethiopian Commodity Exchange show that three-fourth of maize produced is used for household expenditure; only about ten percent is marketed and the remaining is used for seed, in-kind expenses for labor and animal feed [6].Maize production is constrained by many abiotic and biotic factors of which maize streak disease (MSD) is the major biotic threat in Ethiopia. It is the most destructive and devastating disease of maize in Sub-Saharan Africa which is caused by maize streak virus (MSV; genusMastrevirus, familyGeminiviridae) [3, 7] including Ethiopia [4]. MSV is a major constraint to maize and over 80 other crops species [8] including oats, wheat, sorghum, millet, finger millet, and sugarcane [2, 9]. MSD is a major threat to cereal crops amongst smallholder farmers in Sub-Saharan Africa causing up to US $ 480 million losses annually [10]. MSD is a viral disease which has single-component, circular, ssDNA [2, 3].MSV has been reported to be the most economically significant causing 100% yield loss if infection occurs in the first three weeks of planting maize [3, 11]. It is irregular in nature and transmitted in a persistent manner by leafhoppers in the genusCicadulina[2, 5]. Globally, 22 species of Cicadulina leafhoppers have been reported, of which 18 are found in Africa.Cicadulina mbila is the most predominant vector and the most important in the epidemiology of the virus [2] from the 8 known vectors of MSV in the genus. In Ethiopia, five of these 22 known species ofCicadulina have been recorded [4].Mathematical modeling is an important tool used in analyzing the dynamics of infectious diseases. Several models have been formulated and analyzed to explain the dynamics of plant disease transmission. The authors in [12] investigated the impacts of foliar diseases on maize plant population dynamics from the developed epidemiological mathematical model. They also applied optimal control theory with chemical, cultural, and disease resistance as a control intervention. In [13], the authors derived a continuous epidemiological model of African cassava mosaic virus disease in which the dynamics are within a locality, of healthy and infected cassava and of infective and noninfective whitefly vectors. The authors in [14] reviewed a differential equation model on plants by adjusting to a specific plant disease model which was a general model of [15]. Optimal control theory to a continuous deterministic epidemiological Cassava brown streak disease model with chemical and uprooting as control measures was applied by the author [16]. In [17], the authors proposed and analyzed an SI-SEI maize disease model which is a combination of both the host and the vector population models and has been formulated to study and analyse the dynamics of maize lethal necrosis disease.Motivated by references [12, 16, 17], in this paper, we present a deterministic model to study and analyze the dynamics of MSV in the maize plant population. We believe that the results of our research work will be useful indicating suitable means of controlling the disease transmission or rather eradicate it. This may ensure maximum maize harvest for farmers for food security. The organization of this paper is as follows. In Section 2, the mathematical analysis of the model including determining the invariant region where the model is mathematically and epidemiologically well posed is presented; basic reproduction number R0 is obtained by using the next generation matrix method; the local stability of the equilibria is obtained by the Jacobian matrix method; and, by the method of Castillo- Chavez, the global stability of the disease-free equilibrium is investigated; we determine the endemic equilibrium point of the model and the local stability of the equilibria by the Jacobian matrix method; and, by the method of constructing a Lyapunov function, the global stability of the endemic equilibrium is investigated. At the end of Section 3, we determine the bifurcation and sensitivity analysis. In Section 4, we give some numerical simulations to prove our theoretical results with a brief discussion. In the final section, there is a conclusion.


## 2. Model Description and Formulation

The model we introduce consists of two populations: the maize population and the leafhopper vector population. Both populations have two subclasses: susceptible and infected. At time t, let S(t) denote the density of the susceptible maize and I(t) denote the density of the infected maize. The susceptible and infected leafhopper vector densities are denoted by H(t) and Y(t), respectively.If there is no leafhopper predator population and no disease, the host population grows logistically with intrinsic growth rater and environmental carrying capacity K(K>0). In the presence of the disease in maize population, the infected host population contributes to the susceptible host population growth towards the carrying capacity K(K>0). The susceptible leafhopper vectors are recruited at rate q and move to infected leafhopper subgroup by eating infected maize plant at a rate β2 and its natural death rate is μ2. The infected leafhopper has a natural death rate μ3. The disease spreads to the susceptible host when it comes in contact with the MSV pathogen infected leafhopper vector. Susceptible host plants move to the infected class following contact with infected leafhopper at a per capita rate β1. The host, once became infected, never recovers and gives no or very low yield of maize. The infected maize plants have death rate μ1 due to the disease. Furthermore, the disease can not be transmitted horizontally and vertically in both populations and it is not genetically inherited. The predation functional response of the leafhopper towards susceptible maize is assumed to follow Michaelis-Menten kinetics and is modelled using a Holling type II [18] functional form with predation and infection coefficients β1, β2 and half saturation constants A and C. All the parameters and their descriptions are listed in Table 1.Table 1
Description of parameters of the MSV model (1).
parameter Description β1 Predation and infection rate of Infected Leafhopper on Susceptible Maize plant β2 Predation and infection rate of Susceptible Leafhopper on Infected Maize plant b Conversion rate of Infected Leafhopper q Recruitment rate of Susceptible Leafhopper K Carrying capacity C Half saturation rate of Susceptible Leafhopper with Infected Maize plant A Half saturation rate of Susceptible maize with Infected plant μ1 Death rate of infected maize μ2 Death rate of susceptible leafhopper μ3 Death rate of infected leafhopper r Intrinsic growth rate of MaizeWith regard to the above considerations, we have the compartmental flow diagram shown in Figure1. From the flow chart (Figure 1), the model will be governed by the following system of differential equation equations:(1)dSdt=rS1-S+IK-β1SYA+SdIdt=β1SYA+S-μ1IdHdt=q-β2IHC+I-μ2HdYdt=bβ2IHC+I-μ3YWith the initial condition(2)S0=S0≥0,I0=I0≥0,H0=H0≥0,Y0=Y0≥0Figure 1
Compartmental diagram for the transmission dynamics of MSV.


## 3. Model Analysis

### 3.1. Positivity of Solutions

For model (1) to be ecoepidemiologically meaningful and well posed, it is necessary to prove that all solutions of system with positive initial data will remain positive for all times t>0. This will be established by the following theorem.Theorem 1.
LetΩ={(S,I,H,Y)∈R4:S(0)>0,I(0)>0,H(0)>0,Y(0)>0}. Then the solution set (S(t),I(t),H(t),Y(t)) of system (1) is positive for all t≥0.Proof.
From the first equation of the system(3)dSdt=rS1-S+IK-β1SYA+S≤rS1-SK Then we have(4)dSS1-S/K≤rdt⟹St≤KS0e-rtK-S0+S0 As t approachers ∞, we obtain 0≤S(t)≤K. By using the same procedure, we obtained(5)It≥I0e-μ1t≥0,Ht≥H0e-μ2t≥0,Yt≥Y0e-μ3t≥0 Thus the model is meaningful and well posed. Therefore, it is sufficient to study the dynamics of the model in Ω.


### 3.2. Invariant Region

Let us determine a region in which the solution of model(1) is bounded. For this model the total maize population is N1(S,I)=S(t)+I(t). Then, after differentiating N1 with respect to time and substituting the expression of dS/dt,dI/dt, we obtain(6)dN1dt=rS1-S+IK-μ1I≤rS-μ1I=Sr+1-S+μ1I≤k^r+1-αN1 where k^=max⁡{S(0),K} and α=min⁡{1,μ1}. Then(7)dN1dt+dN1≤k^r+1 After solving Eq. (7) and evaluating it as t→∞, we got(8)Ωh=S,IεR+2:N1t≤k^αr+1 Similarly, for leafhopper population N2(H,Y)=H(t)+Y(t), we get(9)dN2dt=q-μ2H-μ3Y≤q-mN2 Where m=min⁡(μ1,μ2). Then(10)dN2dt+mN2≤q After solving Eq. (10) and evaluating it as t→∞, we got(11)Ωv=H,YεR+2:N2t≤qm Therefore the feasible solution set for the MSV model given by(12)Ω=Ωh×Ωv=S,I,H,Y∈R+4:N1t≤k^αr+1;N2t≤qm is positively invariant, inside which the model is considered to be epidemiologically meaningful and mathematically well posed.


### 3.3. Disease-Free Equilibrium Point (DFE)

The disease-free equilibrium of model (1) is obtained by equating all equations of the model to zero and then letting I=0 and Y=0. Then we get(13)E0=K,0,qμ2,0


### 3.4. Basic Reproduction Number

We compute the basic reproduction numberR0 for the model, to analyze the stability of the equilibrium points. The basic reproduction number, R0, measures the expected number of secondary infections that result from one newly infected individual introduced into a susceptible population [16]. We calculate the basic reproduction number R0 of the system by applying the next generation operator approach as laid out by [19] and so it is the spectral radius of the next-generation matrix. The first step to get R0 is rewriting the model equations starting with newly infective classes:(14)dIdt=β1SYA+S-μ1IdYdt=bβ2IHC+I-μ3YThen, by the principle of next-generation matrix, we obtained(15)
F
=
β
1
S
Y
A
+
S
b
β
2
I
H
C
+
I
,
V
=
μ
1
I
μ
3
YTherefore, the basic reproduction number is given as(16)R0=bβ1β2Kqμ1μ2μ3CA+KR0 is a threshold parameter that represents the average number of infected vectors and infected hosts caused by a cross-infection of one infectious maize plant and one infectious leafhopper vector when the other population consists of only susceptible population [19]. Two generations are required for transmission of MSV to take place in the maize field; that is why the square root is found in R0, that is, from an infectious maize plant to a susceptible leafhopper vector and then from an infectious leafhopper vector to susceptible maize [16]. It is too clear when R0 is rewritten as follows:(17)R0=β1Kμ1A+Kbβ2qCμ2μ3=R0h×R0v Where(i)
R0h=β1K/μ1(A+K) is the maize plants contribution when they infect the leafhopper, and(ii)
R0v=bβ2q/Cμ2μ3 is the contribution of the leafhopper population when it infects maize plants.


### 3.5. Local Stability of DFE

Theorem 2.
The DFE point is locally asymptotically stable ifR0<1 and unstable if R0>1.Proof.
To proof this theorem let us first find the Jacobian matrix of system (1):(18)
J
=
r
1
-
2
S
+
I
K
-
β
1
A
Y
A
+
S
2
-
r
S
K
0
-
β
1
S
A
+
S
β
1
A
Y
A
+
S
2
-
μ
1
0
-
β
1
S
A
+
S
0
-
C
β
2
H
C
+
I
2
-
β
2
I
C
+
I
-
μ
2
0
0
b
C
β
2
H
C
+
I
2
b
β
2
I
C
+
I
-
μ
3 Evaluating Eq. (18) at the disease-free equilibrium E0=(K,0,q/μ2,0), we get (19)
J
=
-
r
-
r
0
-
β
1
K
A
+
K
0
-
μ
1
0
β
1
K
A
+
K
0
-
β
2
q
C
μ
2
-
μ
2
0
0
b
β
2
q
C
μ
2
0
-
μ
3 From the Jacobian matrix we obtained a characteristic polynomial as(20)-λ-r-λ-μ2λ2+a1λ+a2=0 Where (21)a1=μ1+μ3a2=μ1μ3-bβ1β2Kqμ2CA+K From Eq. (20), we see that(22)-λ-r⇒λ1=-r<0,-λ-μ2⇒λ2=-μ2<0 From the last expression, that is(23)λ2+a1λ+a2=0 we applied Routh-Hurwitz criteria, and by the principle Eq. (23) has strictly negative real root if a1>0 and a2>0. Clearly we see that a1>0 because it is the sum of positive parameters and also (24)a2=μ1μ3-bβ1β2Kqμ2CA+K=1-R02>0 Hence the DFE is locally asymptotically stable if R0<1.


### 3.6. Global Stability of DFE

To investigate the global stability of DFE, we used technique implemented by Castillo-Chavez and Song [20] as done in the paper [16]. Thus we rewrite our model (1) in the form(25)dXdt=FX,Z,dZdt=GX,Z,GX,0=0 where X=(S,H)∈R2 denotes uninfected populations and Z=(I,Y)∈R2 denotes the infected population. E0=(X∗,0) represents the disease-free equilibrium of this system. E0 is a globally asymptotically stable equilibrium for the model if it satisfies conditions (i) and (ii) below:(i)
FordX/dt=F(X,0),X∗ is globally asymptotically stable.(ii)
dZ/dt=DZG(X∗,0)Z-G^(X,Z),G^(X,Z)≥0forall(X,Z)∈ΩwhereDZG(X∗,0) is an M-matrix (the off diagonal elements are nonnegative) and is also the Jacobian of G(X,Z) taken in (I, Y) and evaluated at (X∗,0)=(K,q/μ2,0,0). If system (25) satisfies the above conditions, then the following theorem holds.Theorem 3.
The equilibrium pointE0=(X∗,0) of system (25) is globally asymptotically stable if R0≤1 and conditions (i) and (ii) are satisfied.Proof.
From system (1) we can get F(X,Z) and G(X,Z):(26)
F
X
,
Z
=
r
s
1
-
S
+
I
K
-
β
1
S
Y
A
+
S
q
-
β
2
I
H
C
+
I
-
μ
2
H
,
G
X
,
Z
=
β
1
S
Y
A
+
S
-
μ
1
I
b
β
2
I
H
C
+
I
-
μ
3
YNow we consider the reduced system dX/dt=F(X,0) from condition (i)(27)dSdt=rs1-SK,dHdt=q-μ2HX∗=(K,q/μ2) is a globally asymptotically stable equilibrium point for the reduced system dX/dt=F(X,0). This can be verified from the solution of the first equation in Eq. (27); we obtain S(t) = S(0)K/e-rt(K-S(0))+S(0) which approaches K as t→∞ and from the second equation of Eq. (27) we get H(t)=q/μ2+(H(0)-q/μ2)e-μ2t which approaches q/μ2 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 (27) is global in Ω. Now we compute(28)
D
Z
G
X
∗
,
0
=
-
μ
1
β
1
K
A
+
K
b
β
2
q
C
μ
2
-
μ
3 Then, G(X,Z) can be written as (29)GX,Z=DZGX∗,0Z-G^X,Z and we want to show G^(X;Z)≥0, which is obtained as(30)
G
^
X
,
Z
=
β
1
Y
K
A
+
K
-
S
A
+
S
b
β
2
I
q
C
μ
2
-
H
C
+
I Here K≥S and q/μ2≥H. Hence it is clear that G^(X,Z)≥0 for all (X,Z) ∈Ω. Therefore, this proves that DFE is globally asymptotically stable when R0≤1.


### 3.7. The Endemic Equilibrium Point

In the presence of MSD,S(t)≥0;I(t)≥0;H(t)≥0,Y(t)≥0 the model has an equilibrium point called endemic equilibrium point denoted by E∗=(S∗,I∗,H∗,Y∗). E∗ is the steady state solution where MSD persist in the population of maize plants. It can be obtained by equating each equation of the model equal to zero; that is,(31)dSdt=dIdt=dHdt=dYdt=0Then we obtain (32)I∗=rS∗R02μ1μ2μ3CA-S∗bβ1β2q-R02μ1μ2μ3R02μ12μ2μ3CA+rS∗bβ1β2q-R02μ1μ2μ3H∗=qcμ1+rS∗/K+qrS∗1-S∗/Kβ2rS∗1-s/K+β2μ2μ1+rS∗/KY∗=qcμ3(1-μ2β1μ1+rS∗/K+rS∗1-S∗/Kβ2rS∗1-S∗/K+μ2μ1+rS∗/K) and S∗ is the positive root of the equation(33)Q1S4+Q2S3+Q3S2+Q4S+Q5=0 where (34)Q1=-μ1μ3β2r2Q2=μ1μ3β2r2μ2-A+2K+β1qcr2μ1-β2Q3=μ1μ2μ3β2rAr-Kr+μ1K-β1β2qcrμ1K-Kr-μ2r-β1μ1qcrβ1r-K+Kr+μ1μ3β2Kr22A-1Q4=β1β2μ1qcrKK+2μ2-β1μ12qcrK2β1+K-μ1μ3β2rKArK+μ2r+μ1μ2μ3β2rKμ1A-KQ5=μ12K2β1qcβ2μ2β1μ1-μ2μ3β2rA


### 3.8. Local Stability of Endemic Equilibrium

Theorem 4.
The endemic equilibriumE∗ of system (1) is locally asymptotically stable in Ω if the following conditions hold for R0>1:(35)r1-2S∗+I∗K>β1AY∗A+S∗2+μ1,-μ1+μ2C+I∗>β2I∗,bCβ1β2S∗H∗A+S∗C+I∗2>μ1μ2Proof.
Let us first obtain the Jacobian matrix of system (1):(36)
J
=
r
1
-
2
S
+
I
K
-
β
1
A
Y
A
+
S
2
-
r
S
K
0
-
β
1
S
A
+
S
β
1
A
Y
A
+
S
2
-
μ
1
0
β
1
S
A
+
S
0
-
C
β
2
H
C
+
I
2
-
β
2
I
C
+
I
-
μ
2
0
0
b
C
β
2
H
C
+
I
2
b
β
2
I
C
+
I
-
μ
3 Evaluating this at the endemic equilibrium E∗=(S∗,I∗,H∗,Y∗), we get (37)J=A11A12A13A14A21A22A23A24A31A32A33A34A41A42A43A44 where (38)A11=r1-2S∗+I∗K-β1AY∗A+S∗2,A12=-rS∗K,A13=0,A14=-β1S∗A+S∗A21=β1AY∗A+S∗2,A22=-μ1,A23=0,A24=β1S∗A+S∗A31=0,A32=-Cβ2H∗C+I∗2,A33=-β2I∗C+I∗-μ2,A34=0A41=0,A42=bCβ2H∗C+I∗2,A43=bβ2I∗C+I∗,A44=-μ3The characteristic equation of the Jacobian matrix is given by(39)λ4+Aλ3+Bλ2+Cλ+D=0where (40)A=-A11+A22+A33+A44B=A11A22+A11A33+A22A44+A11+A33A22+A44+A21A12+A42A24C=A33A44A11+A22+A11A22A33+A44+A24A42A11+A33+A12A21A33+A44-A24A32A43-A21A14A42D=A11A33A22A44-A24A42-A21A14A32A43-A42A33+A11A24A43A32-A21A12A33A44The sufficient conditions for A>0,C>0,D>0,AB>C,ABC>C2+A2D are as follows:(41)r1-2S∗+I∗K>β1AY∗A+S∗2+μ1,-μ1+μ2C+I∗>β2I∗,bCβ1β2S∗H∗A+S∗C+I∗2>μ1μ2Thus, according to the Routh Hurwitz criterion, the characteristic equation (39) will have negative roots or imaginary roots with negative real part for R0>1; the endemic equilibrium E∗ is locally asymptotically stable.


### 3.9. Global Stability of Endemic Equilibrium

Theorem 5.
IfR0>1, the endemic equilibrium E∗ of the model (1) is globally stable.Proof.
To establish the global stability of the endemic equilibriumE∗, we consider the following Lyapunov function:(42)V=S-S∗22+σ1I-I∗22+σ2H-H∗22+σ3Y-Y∗22where σ1,σ2,σ3>0 are to be chosen properly such that dV/dt(E∗)=0 and V(S,I,H,Y)>0 for all (S,I,H,Y)∖E∗.
By direct calculation, the derivative ofV along the solution curve of system (1) yields(43)dVdt=S-S∗dSdt+σ1I-I∗dIdt+σ2H-H∗dHdt+σ3Y-Y∗dYdt Now substituting equations of model (1), we get (44)=S-S∗rS1-S+IK-β1SYA+S+σ1I-I∗β1SYA+S-μ1I+σ2H-H∗q-β2IHC+I-μ2H+σ3Y-Y∗bβ2IHC+I-μ3YBy rearranging we obtain(45)dVdt=-S-S∗2r-1+S+IK+β1YA+S-σ1I-I∗2-β1SYIA+S+μ1-σ2H-H∗2-qH+β2IC+I+μ2-σ3Y-Y∗2-bβ2IHYC+I+μ3We now choose (46)σ1=IA+Sμ1IA+S-β1SY,σ2=HC+Iμ2HC+I-qC+I+β2IH,σ3=YC+Iμ3YC+I-bβ2IH Thus, dV/dt(S,I,H,Y)≤0 and an endemic equilibrium point is globally stable. Also dV/dt=0, if and only if S=S∗,I=I∗,H=H∗,Y=Y∗. Therefore, the largest compact invariant set in {(S∗,I∗,H∗,Y∗)∈Ω:dV/dt=0} is the singleton E∗, where E∗ is the endemic equilibrium of the system (1). By LaSalle’s invariant principle [21], it implies that E∗ is globally asymptotically stable in Ω.


### 3.10. Bifurcation Analysis

A bifurcation is a qualitative change in the nature of the solution trajectories due to a parameter change. The point at which this change takes place is called a bifurcation point. At the bifurcation point, a number of equilibrium points, or their stability properties, or both, change. We investigate the nature of the bifurcation by using the method introduced in [22], which is based on the use of the center manifold theory in [22].Theorem 6 (Castillo-Chavez & Song [22]).
Let us consider a general system of ODE’s with a parameterϕ:(47)dxdt=fx,ϕ,f:Rn×R→Rn,f∈C2Rn×R where x=0 is an equilibrium point for the system in Eq. (47). That is f(0,ϕ)≡0 for all ϕ. Assume the following.
M 1: A=Dxf(0,0)=(∂f/∂xj(0,0)) is the linearization matrix of the system given by (47) around the equilibrium 0 with ϕ evaluated at 0. Zero is a simple eigenvalue of A and other eigenvalues of A have negative real parts.
M 2: Matrix A has a nonnegative right eigenvector w and a left eigenvector v corresponding to the zero eigenvalue. Let fk be the kth component of f and(48)a=∑k,i,j=1nvkwiwj∂2fk∂xi∂xj0,0b=∑k,i=1nvkwi∂2fk∂xi∂ϕ0,0 The local dynamics of (52) around 0 are totally determined by a and b. (i)
a>0,b>0. When ϕ<0 with |ϕ|≪1, 0 is locally asymptotically stable and there exists a positive unstable equilibrium; when 0<ϕ≪1,0 is unstable and there exists a negative, locally asymptotically stable equilibrium.(ii)
a<0,b<0. When ϕ<0 with |ϕ|≪1, 0 is unstable; when 0<ϕ≪1,0 is locally asymptotically stable equilibrium, and there exists a positive unstable equilibrium.(iii)
a>0,b<0. When ϕ<0 with |ϕ|≪1,0 is unstable, and there exists a locally asymptotically stable negative equilibrium; when 0<ϕ≪1,0 is stable, and a positive unstable equilibrium appears.(iv)
a<0,b>0. When ϕ changes from negative to positive, 0 changes its stability from stable to unstable. Correspondingly a negative unstable equilibrium becomes positive and locally asymptotically stable.
In particular, ifa<0 and b>0, then the bifurcation is forward; if a>0 and b>0, then the bifurcation is backward. Using this approach, the following result may be obtained.Theorem 7.
The model in system (1) exhibits forward bifurcation at R0=1.Proof.
We prove, using center manifold theorem [22], the possibility of bifurcation at R0=1. Let S=x1, I=x2, H=x3 and Y=x4. In addition, using vector notation x=(x1,x2,x3,x4)T, and dx/dt=F(x), with F=(f1,f2,f3,f4)T, then model in system (1) is rewritten in the form(49)dx1dt=rx11-x1+x2K-β1x1x4A+x1dx2dt=β1x1x4A+x1-μ1x2dx3dt=q-β2x2x3C+x2-μ2x3dx4dt=bβ2x2x3C+x2-μ3x4We consider the predation and transmission rate β1 as bifurcation parameters so that R0=1 if(50)β1=β1∗=μ1μ2μ3CA+Kbβ2qK The disease-free equilibrium is given by (x1=K,x2=0;x3=q/μ2,x4=0). Then the linearizion matrix of Eq. (49) at a disease-free Equilibrium is given by(51)
J
=
-
r
-
r
0
-
β
1
∗
K
A
+
K
0
-
μ
1
0
β
1
∗
K
A
+
K
0
-
β
2
∗
q
C
μ
2
-
μ
2
0
0
b
β
2
∗
q
C
μ
2
0
-
μ
3 The right eigenvector, w=(w1,w2,w3,w4)T, associated with this simple zero eigenvalue can be obtained from Jw=0. The system becomes(52)-rw1-rw2-β1∗KA+Kw4=0-μ1w2+β1∗KA+Kw4=0β2∗qCμ2w2-μ2w3=0bβ2∗qCμ2w2-μ3w4=0From Eq. (52) we obtain(53)w1=-1+μ1rw2,w2=w2>0,w3=β2qCμ12w2,w4=bβ2qCμ2μ3w2 Here we have taken into account the expression for β1∗. Next we compute the left eigenvector, v=(v1,v2,v3,v4); associated with this simple zero eigenvalue can be obtained from vJ=0 and the system becomes(54)-rv1=0-rv1-μ1v2-β2∗qCμ2v3+bβ2∗qCμ2v4=0-μ2v3=0-β1∗KA+Kv1+β1∗KA+Kv2-μ3v4=0From equation of Eq. (54), we obtain (55)v1=v3=0,v4=Cμ1μ2bβ2qv2, Here we have taken into account the expression for β1∗, where v2 is calculated to ensure that the eigenvectors satisfy the condition v.w=1. Since the first and third components of v are zero, we do not need the derivatives of f1 and f3. From the derivatives of f2 and f4, the only ones that are nonzero are the following:(56)∂2f2∂x1∂x4=∂2f2∂x4∂x1=β1∗AA+x1∗2,∂2f4∂x2∂x3=∂2f4∂x3∂x2=bβ2C∂2f4∂x22=-2bβ2Cx3∗C+x2∗3,∂2f2∂x4∂β1=x1∗A+x1∗and all the other partial derivatives of f2 and f4 are zero. The direction of the bifurcation at R0=1 is determined by the signs of the bifurcation coefficients a and b, obtained from the above partial derivatives, given, respectively, by(57)a=v2w4w1β1∗AA+x1∗2+v2w1w4β1∗AA+x1∗2+v4w3w2bβ2C+v4w3w2bβ2C+v4w2w2-2bβ2Cx3∗C+x2∗3=2v2w4w1β1∗AA+x1∗2+2v4w3w2bβ2C-2v4w2w2bβ2Cx3∗C+x2∗3Hence(58)a=-2μ1v2w22A1+μ1/rKA+K-β2Cμ2+μ1C<0,(59)b=v2w4x1∗A+x1∗=v2w4KA+K>0 As the coefficient b is always positive and the sign of the coefficient a is negative, MSD model exhibits a forward bifurcation and there exists at least one stable endemic equilibrium when R0>1. Using expression for I∗ in the endemic equilibrium, we plotted a forward bifurcation diagram in Figure 7. We used a set of estimated and assumed parameters in Table 3.


### 3.11. Sensitivity Analysis of Model Parameters

We carried out sensitivity analysis, on the basic parameters, to check and identify parameters that can impact the basic reproductive number. Sensitivity analysis notifies us on how significant each parameter is to disease transmission. To go through sensitivity analysis, we followed the approach defined by [23] like in [24]. This technique develops a formula to obtain the sensitivity index of all the basic parameters, defined as follows.Definition 8.
The normalized forward sensitivity index of a variable,g, that depends differentiably on a parameter, p, is defined as(60)ΛPg=∂g∂p×pg for p represents all the basic parameters. Here we have R0=bβ1β2Kq/μ1μ2μ3C(A+K). For example the sensitivity index of R0 to β1 is(61)Λβ1R0=∂R0∂β1×β1R0=12bβ1β2Kq/μ1μ2μ3A+Kbβ2Kqμ1μ2μ3CA+Kβ1R0=12≥0 And we do this in a similar fashion for the remaining parameters.


### 3.12. Interpretation of Sensitivity Indices

The sensitivity indices of the basic reproductive number with respect to main parameters are found in Table2. Those parameters that have positive indices (b1,β1,β2,Kandq) show that they have great impact on expanding the disease in the community if their values are increasing. Due to the reason that the basic reproduction number increases as their values increase, it means that the average number of secondary cases of infection increases in the community. And also those parameters in which their sensitivity indices are negative (C,A,μ1,μ2andμ3) have an effect of minimizing the burden of the disease in the community as their values increase while the others are left constant. And also as their values increase, the basic reproduction number decreases, which leads to minimizing the endemicity of the disease in the community.Table 2
Sensitivity indices table.
parameter symbol Sensitivity indices β1 0.5 β2 0.5 b 0.5 q 0.5 K 0.39998 × 1 0 - 4 μ1 -0.5 μ2 -0.5 μ3 -0.5 C -0.5 A - 0.39998 × 1 0 - 4Table 3
Parameter values for the MSV model.
parameter symbol Value  day-1 Source β1 0.45 [2] β2 0.04 [2] q 0.02 Assumed K 10,000 Assumed μ1 0.008 Assumed μ2 0.0303 [3] μ3 0.0303 [3] b 0.45 Assumed A 0.4 Assumed C 0.6 Assumed r 0.005 Assumed


## 3.1. Positivity of Solutions

For model (1) to be ecoepidemiologically meaningful and well posed, it is necessary to prove that all solutions of system with positive initial data will remain positive for all times t>0. This will be established by the following theorem.Theorem 1.
LetΩ={(S,I,H,Y)∈R4:S(0)>0,I(0)>0,H(0)>0,Y(0)>0}. Then the solution set (S(t),I(t),H(t),Y(t)) of system (1) is positive for all t≥0.Proof.
From the first equation of the system(3)dSdt=rS1-S+IK-β1SYA+S≤rS1-SK Then we have(4)dSS1-S/K≤rdt⟹St≤KS0e-rtK-S0+S0 As t approachers ∞, we obtain 0≤S(t)≤K. By using the same procedure, we obtained(5)It≥I0e-μ1t≥0,Ht≥H0e-μ2t≥0,Yt≥Y0e-μ3t≥0 Thus the model is meaningful and well posed. Therefore, it is sufficient to study the dynamics of the model in Ω.


## 3.2. Invariant Region

Let us determine a region in which the solution of model(1) is bounded. For this model the total maize population is N1(S,I)=S(t)+I(t). Then, after differentiating N1 with respect to time and substituting the expression of dS/dt,dI/dt, we obtain(6)dN1dt=rS1-S+IK-μ1I≤rS-μ1I=Sr+1-S+μ1I≤k^r+1-αN1 where k^=max⁡{S(0),K} and α=min⁡{1,μ1}. Then(7)dN1dt+dN1≤k^r+1 After solving Eq. (7) and evaluating it as t→∞, we got(8)Ωh=S,IεR+2:N1t≤k^αr+1 Similarly, for leafhopper population N2(H,Y)=H(t)+Y(t), we get(9)dN2dt=q-μ2H-μ3Y≤q-mN2 Where m=min⁡(μ1,μ2). Then(10)dN2dt+mN2≤q After solving Eq. (10) and evaluating it as t→∞, we got(11)Ωv=H,YεR+2:N2t≤qm Therefore the feasible solution set for the MSV model given by(12)Ω=Ωh×Ωv=S,I,H,Y∈R+4:N1t≤k^αr+1;N2t≤qm is positively invariant, inside which the model is considered to be epidemiologically meaningful and mathematically well posed.


## 3.3. Disease-Free Equilibrium Point (DFE)

The disease-free equilibrium of model (1) is obtained by equating all equations of the model to zero and then letting I=0 and Y=0. Then we get(13)E0=K,0,qμ2,0


## 3.4. Basic Reproduction Number

We compute the basic reproduction numberR0 for the model, to analyze the stability of the equilibrium points. The basic reproduction number, R0, measures the expected number of secondary infections that result from one newly infected individual introduced into a susceptible population [16]. We calculate the basic reproduction number R0 of the system by applying the next generation operator approach as laid out by [19] and so it is the spectral radius of the next-generation matrix. The first step to get R0 is rewriting the model equations starting with newly infective classes:(14)dIdt=β1SYA+S-μ1IdYdt=bβ2IHC+I-μ3YThen, by the principle of next-generation matrix, we obtained(15)
F
=
β
1
S
Y
A
+
S
b
β
2
I
H
C
+
I
,
V
=
μ
1
I
μ
3
YTherefore, the basic reproduction number is given as(16)R0=bβ1β2Kqμ1μ2μ3CA+KR0 is a threshold parameter that represents the average number of infected vectors and infected hosts caused by a cross-infection of one infectious maize plant and one infectious leafhopper vector when the other population consists of only susceptible population [19]. Two generations are required for transmission of MSV to take place in the maize field; that is why the square root is found in R0, that is, from an infectious maize plant to a susceptible leafhopper vector and then from an infectious leafhopper vector to susceptible maize [16]. It is too clear when R0 is rewritten as follows:(17)R0=β1Kμ1A+Kbβ2qCμ2μ3=R0h×R0v Where(i)
R0h=β1K/μ1(A+K) is the maize plants contribution when they infect the leafhopper, and(ii)
R0v=bβ2q/Cμ2μ3 is the contribution of the leafhopper population when it infects maize plants.


## 3.5. Local Stability of DFE

Theorem 2.
The DFE point is locally asymptotically stable ifR0<1 and unstable if R0>1.Proof.
To proof this theorem let us first find the Jacobian matrix of system (1):(18)
J
=
r
1
-
2
S
+
I
K
-
β
1
A
Y
A
+
S
2
-
r
S
K
0
-
β
1
S
A
+
S
β
1
A
Y
A
+
S
2
-
μ
1
0
-
β
1
S
A
+
S
0
-
C
β
2
H
C
+
I
2
-
β
2
I
C
+
I
-
μ
2
0
0
b
C
β
2
H
C
+
I
2
b
β
2
I
C
+
I
-
μ
3 Evaluating Eq. (18) at the disease-free equilibrium E0=(K,0,q/μ2,0), we get (19)
J
=
-
r
-
r
0
-
β
1
K
A
+
K
0
-
μ
1
0
β
1
K
A
+
K
0
-
β
2
q
C
μ
2
-
μ
2
0
0
b
β
2
q
C
μ
2
0
-
μ
3 From the Jacobian matrix we obtained a characteristic polynomial as(20)-λ-r-λ-μ2λ2+a1λ+a2=0 Where (21)a1=μ1+μ3a2=μ1μ3-bβ1β2Kqμ2CA+K From Eq. (20), we see that(22)-λ-r⇒λ1=-r<0,-λ-μ2⇒λ2=-μ2<0 From the last expression, that is(23)λ2+a1λ+a2=0 we applied Routh-Hurwitz criteria, and by the principle Eq. (23) has strictly negative real root if a1>0 and a2>0. Clearly we see that a1>0 because it is the sum of positive parameters and also (24)a2=μ1μ3-bβ1β2Kqμ2CA+K=1-R02>0 Hence the DFE is locally asymptotically stable if R0<1.


## 3.6. Global Stability of DFE

To investigate the global stability of DFE, we used technique implemented by Castillo-Chavez and Song [20] as done in the paper [16]. Thus we rewrite our model (1) in the form(25)dXdt=FX,Z,dZdt=GX,Z,GX,0=0 where X=(S,H)∈R2 denotes uninfected populations and Z=(I,Y)∈R2 denotes the infected population. E0=(X∗,0) represents the disease-free equilibrium of this system. E0 is a globally asymptotically stable equilibrium for the model if it satisfies conditions (i) and (ii) below:(i)
FordX/dt=F(X,0),X∗ is globally asymptotically stable.(ii)
dZ/dt=DZG(X∗,0)Z-G^(X,Z),G^(X,Z)≥0forall(X,Z)∈ΩwhereDZG(X∗,0) is an M-matrix (the off diagonal elements are nonnegative) and is also the Jacobian of G(X,Z) taken in (I, Y) and evaluated at (X∗,0)=(K,q/μ2,0,0). If system (25) satisfies the above conditions, then the following theorem holds.Theorem 3.
The equilibrium pointE0=(X∗,0) of system (25) is globally asymptotically stable if R0≤1 and conditions (i) and (ii) are satisfied.Proof.
From system (1) we can get F(X,Z) and G(X,Z):(26)
F
X
,
Z
=
r
s
1
-
S
+
I
K
-
β
1
S
Y
A
+
S
q
-
β
2
I
H
C
+
I
-
μ
2
H
,
G
X
,
Z
=
β
1
S
Y
A
+
S
-
μ
1
I
b
β
2
I
H
C
+
I
-
μ
3
YNow we consider the reduced system dX/dt=F(X,0) from condition (i)(27)dSdt=rs1-SK,dHdt=q-μ2HX∗=(K,q/μ2) is a globally asymptotically stable equilibrium point for the reduced system dX/dt=F(X,0). This can be verified from the solution of the first equation in Eq. (27); we obtain S(t) = S(0)K/e-rt(K-S(0))+S(0) which approaches K as t→∞ and from the second equation of Eq. (27) we get H(t)=q/μ2+(H(0)-q/μ2)e-μ2t which approaches q/μ2 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 (27) is global in Ω. Now we compute(28)
D
Z
G
X
∗
,
0
=
-
μ
1
β
1
K
A
+
K
b
β
2
q
C
μ
2
-
μ
3 Then, G(X,Z) can be written as (29)GX,Z=DZGX∗,0Z-G^X,Z and we want to show G^(X;Z)≥0, which is obtained as(30)
G
^
X
,
Z
=
β
1
Y
K
A
+
K
-
S
A
+
S
b
β
2
I
q
C
μ
2
-
H
C
+
I Here K≥S and q/μ2≥H. Hence it is clear that G^(X,Z)≥0 for all (X,Z) ∈Ω. Therefore, this proves that DFE is globally asymptotically stable when R0≤1.


## 3.7. The Endemic Equilibrium Point

In the presence of MSD,S(t)≥0;I(t)≥0;H(t)≥0,Y(t)≥0 the model has an equilibrium point called endemic equilibrium point denoted by E∗=(S∗,I∗,H∗,Y∗). E∗ is the steady state solution where MSD persist in the population of maize plants. It can be obtained by equating each equation of the model equal to zero; that is,(31)dSdt=dIdt=dHdt=dYdt=0Then we obtain (32)I∗=rS∗R02μ1μ2μ3CA-S∗bβ1β2q-R02μ1μ2μ3R02μ12μ2μ3CA+rS∗bβ1β2q-R02μ1μ2μ3H∗=qcμ1+rS∗/K+qrS∗1-S∗/Kβ2rS∗1-s/K+β2μ2μ1+rS∗/KY∗=qcμ3(1-μ2β1μ1+rS∗/K+rS∗1-S∗/Kβ2rS∗1-S∗/K+μ2μ1+rS∗/K) and S∗ is the positive root of the equation(33)Q1S4+Q2S3+Q3S2+Q4S+Q5=0 where (34)Q1=-μ1μ3β2r2Q2=μ1μ3β2r2μ2-A+2K+β1qcr2μ1-β2Q3=μ1μ2μ3β2rAr-Kr+μ1K-β1β2qcrμ1K-Kr-μ2r-β1μ1qcrβ1r-K+Kr+μ1μ3β2Kr22A-1Q4=β1β2μ1qcrKK+2μ2-β1μ12qcrK2β1+K-μ1μ3β2rKArK+μ2r+μ1μ2μ3β2rKμ1A-KQ5=μ12K2β1qcβ2μ2β1μ1-μ2μ3β2rA


## 3.8. Local Stability of Endemic Equilibrium

Theorem 4.
The endemic equilibriumE∗ of system (1) is locally asymptotically stable in Ω if the following conditions hold for R0>1:(35)r1-2S∗+I∗K>β1AY∗A+S∗2+μ1,-μ1+μ2C+I∗>β2I∗,bCβ1β2S∗H∗A+S∗C+I∗2>μ1μ2Proof.
Let us first obtain the Jacobian matrix of system (1):(36)
J
=
r
1
-
2
S
+
I
K
-
β
1
A
Y
A
+
S
2
-
r
S
K
0
-
β
1
S
A
+
S
β
1
A
Y
A
+
S
2
-
μ
1
0
β
1
S
A
+
S
0
-
C
β
2
H
C
+
I
2
-
β
2
I
C
+
I
-
μ
2
0
0
b
C
β
2
H
C
+
I
2
b
β
2
I
C
+
I
-
μ
3 Evaluating this at the endemic equilibrium E∗=(S∗,I∗,H∗,Y∗), we get (37)J=A11A12A13A14A21A22A23A24A31A32A33A34A41A42A43A44 where (38)A11=r1-2S∗+I∗K-β1AY∗A+S∗2,A12=-rS∗K,A13=0,A14=-β1S∗A+S∗A21=β1AY∗A+S∗2,A22=-μ1,A23=0,A24=β1S∗A+S∗A31=0,A32=-Cβ2H∗C+I∗2,A33=-β2I∗C+I∗-μ2,A34=0A41=0,A42=bCβ2H∗C+I∗2,A43=bβ2I∗C+I∗,A44=-μ3The characteristic equation of the Jacobian matrix is given by(39)λ4+Aλ3+Bλ2+Cλ+D=0where (40)A=-A11+A22+A33+A44B=A11A22+A11A33+A22A44+A11+A33A22+A44+A21A12+A42A24C=A33A44A11+A22+A11A22A33+A44+A24A42A11+A33+A12A21A33+A44-A24A32A43-A21A14A42D=A11A33A22A44-A24A42-A21A14A32A43-A42A33+A11A24A43A32-A21A12A33A44The sufficient conditions for A>0,C>0,D>0,AB>C,ABC>C2+A2D are as follows:(41)r1-2S∗+I∗K>β1AY∗A+S∗2+μ1,-μ1+μ2C+I∗>β2I∗,bCβ1β2S∗H∗A+S∗C+I∗2>μ1μ2Thus, according to the Routh Hurwitz criterion, the characteristic equation (39) will have negative roots or imaginary roots with negative real part for R0>1; the endemic equilibrium E∗ is locally asymptotically stable.


## 3.9. Global Stability of Endemic Equilibrium

Theorem 5.
IfR0>1, the endemic equilibrium E∗ of the model (1) is globally stable.Proof.
To establish the global stability of the endemic equilibriumE∗, we consider the following Lyapunov function:(42)V=S-S∗22+σ1I-I∗22+σ2H-H∗22+σ3Y-Y∗22where σ1,σ2,σ3>0 are to be chosen properly such that dV/dt(E∗)=0 and V(S,I,H,Y)>0 for all (S,I,H,Y)∖E∗.
By direct calculation, the derivative ofV along the solution curve of system (1) yields(43)dVdt=S-S∗dSdt+σ1I-I∗dIdt+σ2H-H∗dHdt+σ3Y-Y∗dYdt Now substituting equations of model (1), we get (44)=S-S∗rS1-S+IK-β1SYA+S+σ1I-I∗β1SYA+S-μ1I+σ2H-H∗q-β2IHC+I-μ2H+σ3Y-Y∗bβ2IHC+I-μ3YBy rearranging we obtain(45)dVdt=-S-S∗2r-1+S+IK+β1YA+S-σ1I-I∗2-β1SYIA+S+μ1-σ2H-H∗2-qH+β2IC+I+μ2-σ3Y-Y∗2-bβ2IHYC+I+μ3We now choose (46)σ1=IA+Sμ1IA+S-β1SY,σ2=HC+Iμ2HC+I-qC+I+β2IH,σ3=YC+Iμ3YC+I-bβ2IH Thus, dV/dt(S,I,H,Y)≤0 and an endemic equilibrium point is globally stable. Also dV/dt=0, if and only if S=S∗,I=I∗,H=H∗,Y=Y∗. Therefore, the largest compact invariant set in {(S∗,I∗,H∗,Y∗)∈Ω:dV/dt=0} is the singleton E∗, where E∗ is the endemic equilibrium of the system (1). By LaSalle’s invariant principle [21], it implies that E∗ is globally asymptotically stable in Ω.


## 3.10. Bifurcation Analysis

A bifurcation is a qualitative change in the nature of the solution trajectories due to a parameter change. The point at which this change takes place is called a bifurcation point. At the bifurcation point, a number of equilibrium points, or their stability properties, or both, change. We investigate the nature of the bifurcation by using the method introduced in [22], which is based on the use of the center manifold theory in [22].Theorem 6 (Castillo-Chavez & Song [22]).
Let us consider a general system of ODE’s with a parameterϕ:(47)dxdt=fx,ϕ,f:Rn×R→Rn,f∈C2Rn×R where x=0 is an equilibrium point for the system in Eq. (47). That is f(0,ϕ)≡0 for all ϕ. Assume the following.
M 1: A=Dxf(0,0)=(∂f/∂xj(0,0)) is the linearization matrix of the system given by (47) around the equilibrium 0 with ϕ evaluated at 0. Zero is a simple eigenvalue of A and other eigenvalues of A have negative real parts.
M 2: Matrix A has a nonnegative right eigenvector w and a left eigenvector v corresponding to the zero eigenvalue. Let fk be the kth component of f and(48)a=∑k,i,j=1nvkwiwj∂2fk∂xi∂xj0,0b=∑k,i=1nvkwi∂2fk∂xi∂ϕ0,0 The local dynamics of (52) around 0 are totally determined by a and b. (i)
a>0,b>0. When ϕ<0 with |ϕ|≪1, 0 is locally asymptotically stable and there exists a positive unstable equilibrium; when 0<ϕ≪1,0 is unstable and there exists a negative, locally asymptotically stable equilibrium.(ii)
a<0,b<0. When ϕ<0 with |ϕ|≪1, 0 is unstable; when 0<ϕ≪1,0 is locally asymptotically stable equilibrium, and there exists a positive unstable equilibrium.(iii)
a>0,b<0. When ϕ<0 with |ϕ|≪1,0 is unstable, and there exists a locally asymptotically stable negative equilibrium; when 0<ϕ≪1,0 is stable, and a positive unstable equilibrium appears.(iv)
a<0,b>0. When ϕ changes from negative to positive, 0 changes its stability from stable to unstable. Correspondingly a negative unstable equilibrium becomes positive and locally asymptotically stable.
In particular, ifa<0 and b>0, then the bifurcation is forward; if a>0 and b>0, then the bifurcation is backward. Using this approach, the following result may be obtained.Theorem 7.
The model in system (1) exhibits forward bifurcation at R0=1.Proof.
We prove, using center manifold theorem [22], the possibility of bifurcation at R0=1. Let S=x1, I=x2, H=x3 and Y=x4. In addition, using vector notation x=(x1,x2,x3,x4)T, and dx/dt=F(x), with F=(f1,f2,f3,f4)T, then model in system (1) is rewritten in the form(49)dx1dt=rx11-x1+x2K-β1x1x4A+x1dx2dt=β1x1x4A+x1-μ1x2dx3dt=q-β2x2x3C+x2-μ2x3dx4dt=bβ2x2x3C+x2-μ3x4We consider the predation and transmission rate β1 as bifurcation parameters so that R0=1 if(50)β1=β1∗=μ1μ2μ3CA+Kbβ2qK The disease-free equilibrium is given by (x1=K,x2=0;x3=q/μ2,x4=0). Then the linearizion matrix of Eq. (49) at a disease-free Equilibrium is given by(51)
J
=
-
r
-
r
0
-
β
1
∗
K
A
+
K
0
-
μ
1
0
β
1
∗
K
A
+
K
0
-
β
2
∗
q
C
μ
2
-
μ
2
0
0
b
β
2
∗
q
C
μ
2
0
-
μ
3 The right eigenvector, w=(w1,w2,w3,w4)T, associated with this simple zero eigenvalue can be obtained from Jw=0. The system becomes(52)-rw1-rw2-β1∗KA+Kw4=0-μ1w2+β1∗KA+Kw4=0β2∗qCμ2w2-μ2w3=0bβ2∗qCμ2w2-μ3w4=0From Eq. (52) we obtain(53)w1=-1+μ1rw2,w2=w2>0,w3=β2qCμ12w2,w4=bβ2qCμ2μ3w2 Here we have taken into account the expression for β1∗. Next we compute the left eigenvector, v=(v1,v2,v3,v4); associated with this simple zero eigenvalue can be obtained from vJ=0 and the system becomes(54)-rv1=0-rv1-μ1v2-β2∗qCμ2v3+bβ2∗qCμ2v4=0-μ2v3=0-β1∗KA+Kv1+β1∗KA+Kv2-μ3v4=0From equation of Eq. (54), we obtain (55)v1=v3=0,v4=Cμ1μ2bβ2qv2, Here we have taken into account the expression for β1∗, where v2 is calculated to ensure that the eigenvectors satisfy the condition v.w=1. Since the first and third components of v are zero, we do not need the derivatives of f1 and f3. From the derivatives of f2 and f4, the only ones that are nonzero are the following:(56)∂2f2∂x1∂x4=∂2f2∂x4∂x1=β1∗AA+x1∗2,∂2f4∂x2∂x3=∂2f4∂x3∂x2=bβ2C∂2f4∂x22=-2bβ2Cx3∗C+x2∗3,∂2f2∂x4∂β1=x1∗A+x1∗and all the other partial derivatives of f2 and f4 are zero. The direction of the bifurcation at R0=1 is determined by the signs of the bifurcation coefficients a and b, obtained from the above partial derivatives, given, respectively, by(57)a=v2w4w1β1∗AA+x1∗2+v2w1w4β1∗AA+x1∗2+v4w3w2bβ2C+v4w3w2bβ2C+v4w2w2-2bβ2Cx3∗C+x2∗3=2v2w4w1β1∗AA+x1∗2+2v4w3w2bβ2C-2v4w2w2bβ2Cx3∗C+x2∗3Hence(58)a=-2μ1v2w22A1+μ1/rKA+K-β2Cμ2+μ1C<0,(59)b=v2w4x1∗A+x1∗=v2w4KA+K>0 As the coefficient b is always positive and the sign of the coefficient a is negative, MSD model exhibits a forward bifurcation and there exists at least one stable endemic equilibrium when R0>1. Using expression for I∗ in the endemic equilibrium, we plotted a forward bifurcation diagram in Figure 7. We used a set of estimated and assumed parameters in Table 3.


## 3.11. Sensitivity Analysis of Model Parameters

We carried out sensitivity analysis, on the basic parameters, to check and identify parameters that can impact the basic reproductive number. Sensitivity analysis notifies us on how significant each parameter is to disease transmission. To go through sensitivity analysis, we followed the approach defined by [23] like in [24]. This technique develops a formula to obtain the sensitivity index of all the basic parameters, defined as follows.Definition 8.
The normalized forward sensitivity index of a variable,g, that depends differentiably on a parameter, p, is defined as(60)ΛPg=∂g∂p×pg for p represents all the basic parameters. Here we have R0=bβ1β2Kq/μ1μ2μ3C(A+K). For example the sensitivity index of R0 to β1 is(61)Λβ1R0=∂R0∂β1×β1R0=12bβ1β2Kq/μ1μ2μ3A+Kbβ2Kqμ1μ2μ3CA+Kβ1R0=12≥0 And we do this in a similar fashion for the remaining parameters.


## 3.12. Interpretation of Sensitivity Indices

The sensitivity indices of the basic reproductive number with respect to main parameters are found in Table2. Those parameters that have positive indices (b1,β1,β2,Kandq) show that they have great impact on expanding the disease in the community if their values are increasing. Due to the reason that the basic reproduction number increases as their values increase, it means that the average number of secondary cases of infection increases in the community. And also those parameters in which their sensitivity indices are negative (C,A,μ1,μ2andμ3) have an effect of minimizing the burden of the disease in the community as their values increase while the others are left constant. And also as their values increase, the basic reproduction number decreases, which leads to minimizing the endemicity of the disease in the community.Table 2
Sensitivity indices table.
parameter symbol Sensitivity indices β1 0.5 β2 0.5 b 0.5 q 0.5 K 0.39998 × 1 0 - 4 μ1 -0.5 μ2 -0.5 μ3 -0.5 C -0.5 A - 0.39998 × 1 0 - 4Table 3
Parameter values for the MSV model.
parameter symbol Value  day-1 Source β1 0.45 [2] β2 0.04 [2] q 0.02 Assumed K 10,000 Assumed μ1 0.008 Assumed μ2 0.0303 [3] μ3 0.0303 [3] b 0.45 Assumed A 0.4 Assumed C 0.6 Assumed r 0.005 Assumed


## 4. Numerical Simulation

Numerical simulations of the model (1) are carried out, in order to illustrate some of the analytical results of the study. A set of reasonable parameter values is given in Table 3. These parameter values were obtained from literature and some of them were assumed. We used S(0) = 1000, I(0) = 20, H(0) = 100, Y(0) = 0, as initial values and parameter values in Table 3 for simulation of MSV model in addition to parameter values in Table 3.From the left-hand side of Figure2, the susceptible maize population decelerates exponentially to acquire endemic equilibrium level as they die due to infected leafhopper vector population. The infected maize population assumes parabolic curve as it increases exponentially to a certain maximum point before exponential deceleration to the certain endemic level.Figure 2
Simulation results of susceptible and infected population of maize and leafhopper whenR0>1.From the right-hand side of Figure2, the susceptible vectors decrease exponentially due to natural death and acquisition of infestation from severely infected maize and MSV from the environment and finally acquire the endemic equilibrium level and the infected vectors form a parabolic curve as they do raise and drop exponentially to the endemicity level.Figure3 shows the simulation of infected maize and susceptible maize for different value of β1. We can see from the figure that, increasing the infection and predation rate of infected leafhopper on susceptible maize β1, the basic reproduction number increases, which leads to an increase in the number of infective maize and on the other hand the number of susceptible maize population decrease.Figure 3
Simulation of infected maize and susceptible maize with different value ofβ1.Figure4 shows the simulation of infected maize and infected leafhopper for different value of β2. We can see from the figure that, increasing the infection and predation rate of susceptible leafhopper on infected maize β2, the basic reproduction number increases, which will lead to an increase in the number of the infected maizes as well as the number of infected leafhoppers.Figure 4
Simulation of infected maize and infected leafhopper with different value ofβ2.Figure5 shows simulation of infected leafhopper and susceptible leafhopper for a different value of μ2 and μ3. We can see that increasing the death rate of the leafhopper population reduces the basic reproduction number. Due to the indirect relation of R0 and μ2 and μ3 it leads to a decrease in the population of the leafhopper.Figure 5
Simulation of infected leafhopper and susceptible leafhopper with different value ofμ2 and μ3.Figure6 shows simulation of infected maize for different value of μ1. We can see that increasing the death rate of infected and infectious maize μ1 reduces the reproduction number. This leads to a decrease in the infection rate of maize.Figure 6
Simulation of the MSV model with different value ofμ1.Figure 7
Forward bifurcation of MSV model (1).


## 5. Discussions and Conclusions

In this paper, we have proposed and analysed an ecoepidemiological mathematical model of MSV. We considered a Holling type II functional response which is biologically realistic. We showed that the system was uniformly bounded and positive. We found the disease-free and endemic equilibrium points and their local and global stability analysis has been investigated. The bifurcation analysis of the model is shown. The model analysis also shows the sensitivity of parameters to the disease persistence and dying out.Finally, analytical results were confirmed by numerical simulation with realistic parameter values. We showed that increasing the infection and predation rates,β1 and β2, makes an increase of basic reproduction number which leads to the increase of the number of infected maize population. However, increasing death rate of infected maize and leafhopper population decreases the reproduction number which in turn means that the disease dies out from the maize population. Thus, from the results of this paper, control intervention strategies reduces the disease infection of maize population. The model shows that the spread of the disease largely depends on the infection and predation rates β1 and β2; therefore efforts should be made to minimize the contact of infected maize and susceptible leafhopper and MSV infected maize should be treated either using insecticide chemical to reduce the infection rate of leafhoppers and it should be done before the arrival of leafhopper or uprooting and burning the infected maize from the field. This implies that, to get the best and cost-effective control strategy, we should apply optimal control theory. Thus, we come next with a paper applying the optimal and cost-effective strategies to identify the best and cost-effective strategy for this model.



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*Source: 7965232-2019-01-20.xml*