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Research article

Reproduction number and sensitivity analysis of cassava mosaic disease spread for policy design


  • Received: 06 April 2021 Accepted: 31 May 2021 Published: 07 June 2021
  • We develop a mathematical model for the dynamics of Cassava Mosaic Disease (CMD), which is driven by both planting of infected cuttings and whitefly transmission. We use the model to analyze the dynamics of a CMD outbreak and to identify the most cost-effective policy for controlling it. The model uses the reproduction number R0 as a threshold, calculated using the Next-Generation Method. A locally-asymptotically-stable disease-free equilibrium is established when R0<1, proved by the Routh-Hurwitz criterion. The globally-asymptotically-stable disease-free and endemic-equilibrium points are obtained using Lyapunov's method and LaSalle's invariance principle. Our results indicate that the disease-free equilibrium point is globally-asymptotically-stable when R01, while the endemic-equilibrium point is globally-asymptotically-stable when R0>1. Our sensitivity analysis shows that R0 is most sensitive to the density of whitefly. Numerical simulations confirmed the effectiveness of whitefly control for limiting an outbreak while minimizing costs.

    Citation: Phongchai Jittamai, Natdanai Chanlawong, Wanyok Atisattapong, Wanwarat Anlamlert, Natthiya Buensanteai. Reproduction number and sensitivity analysis of cassava mosaic disease spread for policy design[J]. Mathematical Biosciences and Engineering, 2021, 18(5): 5069-5093. doi: 10.3934/mbe.2021258

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  • We develop a mathematical model for the dynamics of Cassava Mosaic Disease (CMD), which is driven by both planting of infected cuttings and whitefly transmission. We use the model to analyze the dynamics of a CMD outbreak and to identify the most cost-effective policy for controlling it. The model uses the reproduction number R0 as a threshold, calculated using the Next-Generation Method. A locally-asymptotically-stable disease-free equilibrium is established when R0<1, proved by the Routh-Hurwitz criterion. The globally-asymptotically-stable disease-free and endemic-equilibrium points are obtained using Lyapunov's method and LaSalle's invariance principle. Our results indicate that the disease-free equilibrium point is globally-asymptotically-stable when R01, while the endemic-equilibrium point is globally-asymptotically-stable when R0>1. Our sensitivity analysis shows that R0 is most sensitive to the density of whitefly. Numerical simulations confirmed the effectiveness of whitefly control for limiting an outbreak while minimizing costs.



    Cassava Mosaic Disease (CMD) is a plant disease that reduces tuber size and starch percentage upstream in the cassava supply chain, reducing sales of the cassava crop [1]. This leads to downstream economic impacts since cassava is a major industrial raw material. The financial losses due to CMD in the African continent were estimated at $1.2-2.3 billion in 1997 [2], increasing to $1.9-2.7 billion in 2009 [3]. A CMD outbreak can be controlled in four ways: (1) by removing infected cassava from the plantation, (2) by promoting the use of virus-free cuttings, (3) by using resistant varieties, or (4) by killing infected whitefly in the plantation area [4,5,6]. These methods incur huge planning effort and expense. However, there is no clear approach to mapping the major spread factors for appropriate control planning. In this study, we analyzed the CMD spread factors and proposed an epidemic model to isolate those factors that have a major impact in an outbreak. The output may be used to formulate optimal strategies for outbreak control.

    Using such an epidemic model, the authors of [7] showed how tomato yields in India suffered from whitefly-vectored Tomato Leaf Curl Geminivirus (TLCV). The most effective disease-control strategy was to distribute netting treated with persistent insecticide. This significantly controlled spread by decreasing vector immigration to the plantation and increasing vector mortality. The authors of [8] studied the outbreak of Huanglongbing disease (HLB) caused by psyllid bacteria, which seriously impacts citrus greening. They tested different intervention strategies over a short time-frame and showed that the best insecticide intervention was to spray over an optimal number of days.

    Most epidemic models in the literature are based on the pioneering work in [9]. The models identify the major factors that contribute to the outbreak by applying differential equations. This yields the basic reproduction number (R0), used to indicate the severity of spread [10,11,12]. Analysis of the sensitivity of R0 determines which parameters have the greatest impact on disease control and should be targeted by prevention policies [13,14].

    Most viral crop plant diseases radiate in vectors [15,16]. The models must therefore represent the dynamics of plant-virus epidemics in order to reveal relationships between vectors and infected plant numbers [17,18,19]. In 1994, African cassava mosaic virus (ACMV) caused widespread loss of production. The authors in [20] incorporated vector-population dynamics into their statistical model to empirically derive plant-virus relationships. In [21], an epidemiological model of ACMV was introduced to describe the dynamics of infected cassava and infective whitefly. The authors derived a strategy in which cassava yields are maximized by reducing the whitefly population. However, this approach is not cost-effective, so farmers are advised to select uninfected cuttings for planting to prevent a collapse of the healthy cassava population. This is an economical strategy that is capable of controlling an outbreak. The authors [22] developed a mathematical model for the spread of Cassava Brown Streak Disease (CBSD) caused by whitefly, using an approach similar to that in [21]. They simulated two control policies: uprooting and burning of infected cassava, and killing of whitefly in the plantation. They concluded that the optimal policy was uprooting and burning.

    In previous works [21,22], the same authors formulated a mathematical model for vector-host dynamics using a single spread factor (whitefly). However, CMD infection requires both whitefly (Bemisia tabaci) transmission and the presence of virus-infected cuttings [23,24]. Outbreaks have spread beyond the African continent to countries in Asia including India and Sri Lanka following imports of virus-infected cuttings. The CMD model must incorporate this route when devising optimal strategies for outbreak control. The mathematical model developed in [25] for an outbreak of ACMV took account of diseased cuttings, following [21]. The model suggested that a strategy combining rouging and spraying performed better than those that apply a single control mechanism.

    When infection levels are high, symptomatic cassava can be readily detected and removed from the plantation. However, the symptoms take five weeks to appear [20], and during this period the asymptomatic cassava may spread the disease. Our goal was to develop a better understanding of the relationship between asymptomatic plants and the severity of CMD spread. Our model added a latent state to that in [25], in an effort to analyze the comparative contribution of whitefly transmission and planting of infected cuttings. We applied sensitivity analysis to identify the most important parameters in determining the severity of an outbreak.

    The rest of this paper is organized as follows. Section 2 discusses CMD, the construction of our model, and the most important parameters in CMD spread. In section 3 we introduce the positivity and boundedness of solutions to initial conditions. Section 4 concerns the stability of the disease-free and endemic-equilibrium points, and introduces the threshold parameter (R0), obtained using the Next-Generation method. The sensitivity analysis described in section 5 identified the most sensitive parameter with respect to R0, key to improving the design of CMD-control policies. We conducted numerical simulations to explore the behavior of the system with and without control policies. The simulation is introduced in section 6, and the results reported in section 7. Section 8 presents a summary and our conclusions.

    Few mathematical models of viral cassava disease (East African Cassava Mosaic Virus (EACMV), ACMV, or CBSD) [21,22,26] have analyzed how the disease spreads within a plantation. To understand the dynamics of an outbreak, we used ordinary differential equations (ODEs).

    The dynamics of ACMV spread were analyzed in [21] and [25], and the "SI-type epidemic model" was developed, which categorizes plants as infected or uninfected. In [26], an epidemic model was developed in which two viruses, ACMV and EACMV, are carried by whitefly into the plantation. The model was based on the "m-group SI-type epidemic model" of [27], which incorporates four plant states: uninfected, infected by ACMV, infected by EACMV, or infected by both. The authors of [22] developed an epidemic model based on the SEIR-type model of [28], and used it to explore the dynamics of CBSD. It assumed four states: healthy, latent, exposed, and rouging. The latent stage, which was missing from earlier models, increases the transmission rate from infected plants to whitefly and vice versa. Our proposed model takes account of the spread of CMD by both by planting and whitefly transmission. The design process is comprised of three steps:

    (1) The dynamics are based on infection by transplanting and by whitefly transmission [23,24]. Defining these factors as parameters yields the ODEs, which we use to calculate equilibrium points and to establish the stability of the outbreak. The Next-Generation method is used to construct R0.

    (2) The local stability of the disease-free equilibrium is proved by the Routh-Hurwitz criterion. The system is locally asymptotically stable if R0<1 and unstable if R0>1. Global stability of the disease-free equilibrium and the endemic equilibrium point are proved by Lyapunov's method and LaSalle's invariance principle. When R01, the disease-free equilibrium is globally asymptotically stable. If R0>1 then the endemic equilibrium is globally asymptotically stable.

    (3) The use of sensitivity analysis to determine the parameters that most strongly affect the outbreak, and the use of these parameters to simulate outbreak control by addressing risk factors.

    In our model, the parameters in the CMD outbreak represent disease spread factors. The values and ranges of the parameters were set with reference to previous studies or by analysis of CMD outbreaks. The parameters are listed in Table 1.

    Table 1.  Parameters for outbreak analysis.
    Symbol Description Value Range Source
    h harvesting rate of cassava plants 0.003 [0.002, 0.004] [21]
    β CMD latent rate 0.008 [0.008, 0.05] [29]
    γ rouging rate of symptomatic cassava 0.03 [0.03, 0.1] [22]
    K amaximum plantation capacity (m2) 15, 000 [10,000, 20,000] estimated
    r replanting rate of cassava 0.025 [0.025, 0.1] [21]
    a average number of cassava plants
    visited by uninfected whitefly 200 [50, 300] estimated
    Λ constant birth rate of whitefly 0.2 [0.1, 0.3] [30]
    μ natural death rate of whitefly 0.0142 [0.0142, 0.06] [22]
    p1 probability of transmission to healthy 0.0033 [0, 1.0] [22]
    cassava plants from infected whitefly
    p2 probability of transmission to uninfected 0.0033 [0, 1.0] [22]
    whitefly from latent or symptomatic cassava
    p3 probability of planting infected cassava cuttings 0.1 [0, 1.0] [21]
    *Note: aassuming 1 plant per m2. The unit of h,β,γ,r and a is per day, and the unit of Λ is #vector per day and μ is per day.

     | Show Table
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    The model tracks the dynamics of the cassava and whitefly populations. Cassava may be uninfected X, latent Y (infected but asymptomatic), or infected and symptomatic Z. Whitefly may be infected R or uninfected Q. The total cassava population N is therefore N=X+Y+Z at time t. Similarly, the total whitefly population NV is NV=Q+R.

    The transmission dynamics of CMD from an initial state to a second state are governed by the parameters in Table 1. Infection by whitefly is given by ap1XNR. Infection of whitefly from infected plants is given by ap2QY+ZN. CMD infection by planting of infected cuttings is r(1NK)p3Y. This term was not used in previous works.

    dXdt=r(1NK)Xap1XNRhX, (2.1)
    dYdt=r(1NK)p3Y+ap1XNR(β+h)Y, (2.2)
    dZdt=βY(γ+h)Z, (2.3)
    dQdt=Λap2QY+ZNμQ, (2.4)
    dRdt=ap2QY+ZNμR. (2.5)

    The size of the cassava population is increased by replanting into the plantation area. r(1NK)X in Eq (2.1) and r(1NK)p3Y in Eq (2.2) represent the replanting terms of healthy and latent cassavas. Cassavas are removed from the system by constant harvesting (h) and rouging (γ). The change from state X (non-infected) to Y reflects the plants that become infected after planting. The change from Y to Z reflects the number of infected cuttings that begin to show CMD symptoms during the period.

    The whitefly population is driven by two factors: birth rate (Λ) and death rate (μ). Transmission within the population causes a change from state Q to state R, as whitefly visit infected plants and acquire the virus. Equations (2.6) and (2.7) give the total cassava and whitefly populations:

    dNdt=r(1NK)(X+p3Y)γZhN, (2.6)
    dNVdt=ΛμNV. (2.7)

    The model makes the following assumptions:

    (1) All model parameters are positive.

    (2) The growth rate of the cassava population is positive, i.e., rh>0, where r is the replanting rate and h is the harvesting rate.

    (3) The cassava population increases by one of the logistic growth equations, r(1NK)X or r(1NK)p3Y.

    (4) The whitefly birth rate (Λ) is constant.

    To confirm the biological validity of the model, we must prove that solutions to the ODEs are positive and bounded for all time values. Furthermore, the cassava and whitefly populations must remain finite since they are bounded by the plantation area. In the next section we analyze our model and demonstrate the positivity and boundedness of the system of ODEs.

    Theorem 3.1. Let t0. In the model, if the initial conditions satisfy

    X(0),Y(0),Z(0),Q(0),R(0)>0, (3.1)

    then X(t),Y(t),Z(t),Q(t),andR(t) will remain positive in R5+.

    Proof. Let us consider Y(t) for all t0.

    From Eq (2.2), we have

    dYdt=r(1NK)p3Y+ap1XNR(β+h)Yr(1NK)p3Y(β+h)Y.

    The integration of the inequality is

    Y(t)Y(0)e(β+hr(1NK)p3)t>0,for allt0. (3.2)

    From the initial conditions (3.1), we have Y(t)>0 for all t0.

    Next, we consider Z(t) for all t0. From Eq (2.3),

    dZdt=βY(γ+h)Z(γ+h)Z.

    A comparison argument shows that

    Z(t)Z(0)e(γ+h)t>0,for allt0. (3.3)

    Since Z(0)>0, we conclude that Z(t) is always positive for all t0.

    We prove that Q(t)>0 for all t0 by contradiction. Let t1 be the first point that satisfies Q(t1)=0. Equation (2.4) yields

    dQdt|t=t1=Λ>0. (3.4)

    This means that Q(t)<0 for t(t1θ,t1), where θ is an arbitrarily small positive constant. This leads to a contradiction. Hence, Q(t)>0 for all t0.

    Next, we prove that R(t) is positive for all t0. From Eq (2.5),

    dRdt=ap2QY+ZNμRμR,

    it then follows that

    R(t)R(0)eμt>0,for allt0. (3.5)

    Since R(0)>0, we know that R(t) remains positive for all t0.

    Finally, we prove that X(t)>0 for all t0. From Eq (2.1), we have

    dXdt=r(1NK)Xap1XNRhXap1XNRhX.

    The integration of the inequality is

    X(t)X(0)ehtt0ap1R(s)Nds>0,for all0tt1. (3.6)

    From the initial conditions (3.1), we have X(t)>0 for all t0.

    We conclude that the solutions of the model are positive in R5+.

    We show the boundedness of the system with the initial conditions (3.1).

    Theorem 3.2. All solutions of the system (2.1) to (2.5) with positive initial conditions (3.1) are ultimately bounded.

    Proof. From Theorem 3.1, the solutions to the system are positive for all t0. Since N=X+Y+Z. From Eq (2.6), we have

    dNdt=r(X+p3Y)(1NK)γZhN.

    As can be observed, the solution is bounded by logistic growth

    dNdtr(1NK)N.

    The integration of the inequality is

    N(t)N(0)KN(0)+(KN(0))ert

    (assuming N(0)>0), and hence limtN(t)K. This gives the feasible solution set of the cassava entering the region:

    ΩH={X,Y,ZR3+NK}for allt0.

    Next, note that NV=Q+R. From Eq (2.7),

    dNVdt=ΛμNV.

    The integration of the equation is

    NV(t)=Λ+(NV(0)μΛ)eμtμ

    (assuming NV(0)>0), which implies that limtNV(t)=Λμ. Thus, the feasible solution set for the CMD system is given by

    ΩV={Q,RR2+NVΛμ}for allt0.

    The solutions of the system in R5+ are confined to the region Ω. Here

    Ω={(X,Y,Z,Q,R)R5+NK,NVΛμ}for allt0.

    Therefore, all solutions of the system (2.1) to (2.5) with initial conditions (3.1) remain positive invariant in the region Ω for all t0.

    To ensure that disease eradication is independent of the initial size of the susceptible population, locally- and globally-stable disease-free equilibria and endemic equilibria are established when the region Ω is positivity invariant.

    Our model admits two equilibrium points: a disease-free equilibrium point (DFE) and an endemic equilibrium point (EE). The notation of DFE is

    E0=(X,Y,Z,Q,R)=((rh)Kr,0,0,Λμ,0).

    The EE is denoted E1=(ˉX,ˉY,ˉZ,ˉQ,ˉR), where

    ˉX=rN(K(γ+h)(β+γ+h)ˉY)K(γ+h)(ap1ˉR+hN)r(γ+h)N,ˉY=ap1KˉR(γ+h)((rh)Nap1ˉR)rN((γ+h)(β+hhp3)N+ap1ˉR(β+γ+hp3(γ+h))),ˉZ=βγ+hˉY,ˉQ=Λ(γ+h)Nap2(β+γ+h)ˉY+μ(γ+h)N,ˉR=aΛp2(β+γ+h)ˉYμ(ap2(β+γ+h)ˉY+μ(γ+h)N).

    The basic reproduction number (R0) is the one of the most useful threshold parameters in epidemiology. It is defined as the expected number of secondary cases produced by a single infection in a completely susceptible population. It is used as an indicator of the stability of E0 and E1. The DFE E0 is asymptotically stable if R0<1, as the disease cannot invade the population, but unstable if R0>1. E1 is asymptotically stable if R0>1. We use the Next-Generation method, following [31,32], to determine R0.

    The appearance of new infections is represented by the vector F and the transfer of existing infections by the vector V. Let xj be an infection state for j=1,2,3, i.e., x1=Y,x2=Z,x3=R. F describes new infections arising in state xj and V represents the transfer of existing infections to state xj.

    The Jacobian matrices generated by differentiating F and V with respect to the relevant subset of variables are calculated at E0. This yields the matrices F and V. The (j,k) entry of the matrix F is the rate at which infected individuals in compartment k produce new infections in compartment j. The (j,k) entry of the matrix V represents the transfer of existing infection.

    R0 is computed from the spectral radius of FV1 at DFE. FV1 is called the Next Generation Matrix [10] and is set as follows:

    R0=ρ(FV1),

    where ρ(M) denotes the spectral radius of a matrix M. The spectral radius of FV1 is equal to the dominant (or maximum) eigenvalue of FV1.

    In our model, the vectors F and V can be derived from Eqs (2.1)-(2.5):

    F=[r(1NK)p3Y+ap1XNR0ap2QY+ZN]andV=[(β+h)Y(γ+h)ZβYμR].

    The Jacobians of F and V with respect to the infectious classes are calculated as F = [Fj(E0)xk] and V = [Vj(E0)xk], for j=1,2,3 and k=1,2,3. We then have

    F=[r(1XK)p30ap1000ap2QXap2QX0]andV=[β+h00βγ+h000μ].

    Therefore, the Next Generation Matrix FV1 is

    FV1(E0)=[hp3(β+h)0ap1μ000raΛp2(β+γ+h)μK(rh)(β+h)(γ+h)raΛp2μK(rh)(γ+h)0]. (4.1)

    Hence, R0 is the dominant eigenvalue of Matrix (4.1).

    R0=max(hp32(β+h)±(hp32(β+h))2+ra2Λp1p2(β+γ+h)μ2K(rh)(γ+h)(β+h))=hp32(β+h)+(hp32(β+h))2+ra2Λp1p2(β+γ+h)μ2K(rh)(γ+h)(β+h). (4.2)

    Theorem 4.1. Under the hypotheses rh>0 and 0p31, if R0<1 then E0 is locally asymptotically stable; otherwise it will be unstable.

    Proof. The local asymptotic stability is determined based on the eigenvalue (λ) of the Jacobian at E0. The E0 is locally asymptotically stable if the real parts of λ are all negative. The Jacobian matrix at E0 is given by

    J(E0)=[(rh)(rh)(rh)0ap10hp3(β+h)00ap10β(γ+h)000raΛp2μK(rh)raΛp2μK(rh)μ00raΛp2μK(rh)raΛp2μK(rh)0μ]. (4.3)

    The characteristic equation is given by

    (λ+μ)(λ+(rh))(λ3+a1λ2+a2λ+a3)=0, (4.4)

    with coefficients

    a1=β+γ+μ+h(2p3),a2=(β+hhp3)(γ+h+μ)+(γ+h)μra2Λp1p2μK(rh),

    and

    a3=μ(γ+h)(β+hhp3)ra2Λp1p2(β+γ+h)μK(rh).

    Under the model assumption r>h, it is clear that the first two eigenvalues of this system are negative, μ and (rh), respectively.

    We now consider the cubic equation from (4.4)

    λ3+a1λ2+a2λ+a3=0.

    As 0p31, we obtain

    a1=β+γ+μ+h(2p3)>0.

    We can observe that if R0=hp32(β+h)+(hp32(β+h))2+ra2Λp1p2(β+γ+h)μ2K(rh)(γ+h)(β+h)<1, then

    ra2Λp1p2(β+γ+h)μ2K(rh)(γ+h)<β+hhp3.

    Therefore, we obtain

    a3=μ(γ+h)(β+hhp3)ra2Λp1p2(β+γ+h)μK(rh)>0.

    Finally, we consider

    a1a2a3=(β+γ+μ+h(2p3))((β+hhp3)(γ+h+μ)+(γ+h)μra2Λp1p2μK(rh))μ(γ+h)(β+hhp3)+ra2Λp1p2(β+γ+h)μK(rh). (4.5)

    Under hypotheses R0<1, r>h, and 0p31, we obtain

    a1a2a3=(γ+μ+h)(γ+h)μ+ra2Λp1p2(β+γ+h)μK(rh)+(β+γ+μ+h(2p3))((β+hhp3)(γ+h+μ)ra2Λp1p2μK(rh))=G1+G2+a3,

    with coefficients

    G1=(β+γ+h)(β+γ+μ+h(2p3))((γ+μ+h)(γ+h)μ+ra2Λp1p2(β+hhp3)μK(rh))>0,G2=((γ+h)(β+γ+h)+μβ)(β+hhp3)>0.

    This means that a1a2a3>0 whenever R0<1, r>h, and 0p31.

    The Routh-Hurwitz criterion for the third order polynomial indicated that the remaining eigenvalues of the disease-free equilibrium system have negative real parts when a1>0,a3>0, and a1a2>a3.

    Therefore, E0 will be locally asymptotically stable if R0<1. When R0>1, E0 is unstable and the disease will persist.

    The global stability of the DFE and the EE are established using Lyapunov's method and LaSalle's invariance principle to obtain the control condition under which disease can be eradicated.

    Theorem 4.2. (1) If R01, then the disease-free equilibrium point E0 is globally asymptotically stable in Ω. (2) If R0>1, then the endemic equilibrium point E1 is globally asymptotically stable in Ω.

    Proof. Theorems 4.2 (1) and 4.2 (2) can be proved by using a Lyapunov function. We adopt the Lyapunov function used in [33,34].

    First, we prove Theorem 4.2 (1) by assuming that:

    μ=ap1NX. (4.6)

    We define the Lyapunov function

    L1:{(X,Y,Z,Q,R)ΩX,Q>0}R+5

    by

    L1(X,Y,Z,Q,R)=(XXXlnXX)+Y+Z+(QQQlnQQ)+R. (4.7)

    When L1 is C1, a proper positive definite function, and E0 is the global minimum of L1 on Ω, we obtain

    L1(X,Y,Z,Q,R)=0.

    The time derivative of L1 computed along solutions of Eqs (2.1)-(2.5) is

    dL1dt=dXdt(1XX)+dYdt+dZdt+dQdt(1QQ)+dRdt. (4.8)

    Substituting the ODEs (2.1) to (2.5) into Eq (4.8) yields

    dL1dt=(r(1NK)Xap1XNRhX)(1XX)+(r(1NK)p3Y+ap1XNR(β+h)Y)+(βY(γ+h)Z)+(Λap2QY+ZNμQ)(1QQ)+(ap2QY+ZNμR),

    this simplifies to

    dL1dt=(r(1NK)XhX)(1XX)+r(1NK)p3YhY(γ+h)Z+Λ(1QQ)+μQ(1QQ)+ap2NQ(Y+Z)+(ap1NXμ)R. (4.9)

    Since Y+Z=NX, N=X=(rh)Kr, NV=Q=Λμ, and observing the condition in Eq (4.6), Equation (4.9) can be rewritten as

    dL1dt=hp3YhY(γ+h)Z+Λ(1QQ)+Λ(1QQ)+G3(XX), (4.10)

    where G3=ap2NQ. Therefore, Eq (4.10) becomes

    dL1dt=Λ(2QQQQ)(γ+h)Z+G3(XX)+h(p31)Y=Λ(QQ)2QQ(γ+h)Z+G3(XX)+h(p31)Y.0. (4.11)

    All terms in Eq (4.11) are always non-positive since limtX(t)X (Eq (3.6)) and 0p31 (Table 1). Note that dL1dt=0 if and only if X=X, Q=Q, Y=0, and Z=0.

    Therefore, the largest invariant compact invariant set in {(X,Y,Z,Q,R)ΩdL1dt=0} is the singleton, i.e., E0=(X,Y,Z,Q,R). Thus, dL1(t)dt0. This implies that E0 is globally asymptotically stable in Ω.

    We next prove Theorem 4.2 (2). We define the Lyapunov function

    L2:{(X,Y,Z,Q,R)ΩX,Y,Z,Q,R>0}R+5

    by

    L2(X,Y,Z,Q,R)=Nap1ˉXˉR(XˉXˉXlnXˉX)+Nap1ˉXˉR(YˉYˉYlnYˉY)+1βˉY(ZˉZˉZlnZˉZ)+Nap2ˉQ(ˉY+ˉZ)(QˉQˉQlnQˉQ)+Nap2ˉQ(ˉY+ˉZ)(RˉRˉRlnRˉR), (4.12)

    When L2 is C1, a proper positive definite function, and E1 is the global minimum of L2 on Ω, we obtain

    L2(ˉX,ˉY,ˉZ,ˉQ,ˉR)=0.

    The time derivative of L2 computed along solutions of Eqs (2.1)-(2.5) is

    dL2dt=Nap1ˉXˉR(1ˉXX)dXdt+Nap1ˉXˉR(1ˉYY)dYdt+1βˉY(1ˉZZ)dZdt+Nap2ˉQ(ˉY+ˉZ)(1ˉQQ)dQdt+Nap2ˉQ(ˉY+ˉZ)(1ˉRR)dRdt. (4.13)

    Under Eqs (2.1)-(2.5), if the system meets E1,

    r(1NK)=ap1ˉRN+h,(β+h)ˉY=r(1NK)p3ˉY+ap1ˉXNˉR,(γ+h)ˉZ=βˉY,μˉQ=Λap2ˉQˉY+ˉZN,μˉR=ap2ˉQˉY+ˉZN.

    We rewrite this to obtain

    dL2dt=Nap1ˉXˉR(1ˉXX)(ap1XNˉRap1XNR)+Nap1ˉXˉR(1ˉYY)(ap1XNRap1ˉRˉXNYˉY)+1βˉY(1ˉZZ)(βYβˉYZˉZ)+Nap2ˉQ(ˉY+ˉZ)(1ˉQQ)(Λap2QY+ZNQˉQ(Λap2ˉQˉY+ˉZN))+Nap2ˉQ(ˉY+ˉZ)(1ˉRR)(ap2QY+ZNap2ˉQˉY+ˉZNRˉR). (4.14)

    The derivative of L2 then becomes

    dL2dt=Nap2ˉQ(ˉY+ˉZ)Λ(ˉQQ)2QˉQ+1+XˉX+QˉQ+Y+ZˉY+ˉZZˉZYˉZˉYZXRˉYˉXˉRYQˉR(Y+Z)ˉQR(ˉY+ˉZ)0. (4.15)

    Using the arithmetic-geometric means inequality, we can see that dL2dt0. Note that, dL2dt=0 only if X=ˉX, Y=ˉY, Z=ˉZ, Q=ˉQ, and R=ˉR.

    Therefore, the largest compact invariant set in {(X,Y,Z,Q,R)ΩdL2dt=0} is the singleton E1=(ˉX,ˉY,ˉZ,ˉQ,ˉR). Thus, dL2dt0. This implies that E1 is globally-asymptotically-stable in Ω.

    We have established that the disease-free equilibrium point E0 is globally asymptotically stable and that the disease can be controlled as long as the threshold R01. If R0>1 the endemic equilibrium point E1 is globally-asymptotically-stable and the disease will persist. From an epidemiological point of view the goal of policy is to control CMD outbreaks by reducing R0 to below one and maximizing the uninfected population. However, an agricultural viewpoint would instead focus on maximizing the economic returns. We therefore need to identify the strategy that maximizes profits. To do this we first identify the parameter that plays the greatest role in the stability of R0.

    Sensitivity analysis is used to assess the relative impact of different factors on the stability of a model under data uncertainty. The analysis is also able to determine which parameters play critical roles. We perform the analysis by calculating the sensitivity indices of the basic reproduction number, R0 to the parameters in the model using both local and global methods.

    The local sensitivity analysis is based on the normalized forward sensitivity index R0. The sensitivity index of R0 with respect to the parameters in our model are derived as follows [35]:

    ΓR0P=R0P×PR0. (5.1)

    Here, P is a parameter value from Table 2, while the value of R0 is computed from Eq (4.2). We analyze the sensitivity of R0 by substituting the parameter values into Eq (5.1). For example, the sensitivity index R0 with respect to h is

    ΓR0h=R0h×hR0=0.0713.
    Table 2.  Sensitivity Indices of R0 (Definition from Table Table 1).
    Parameter Parameter Value Sensitivity Index
    h 0.003 0.0713
    β 0.008 0.2693
    γ 0.03 0.0881
    K 15,000 0.4965
    r 0.025 0.0677
    a 200 +0.9929
    Λ 0.2 +0.4965
    μ 0.0142 0.9929
    p1 0.0033 +0.4965
    p2 0.0033 +0.4965
    p3 0.1 +0.0071

     | Show Table
    DownLoad: CSV

    Table 2 lists the sensitivity indices of R0 obtained using Maple.

    We first consider the pathogen parameters p1,p2 and p3. Our model limits the severity of an outbreak by controlling three infection terms:

    (1) Transmission by whitefly, ap1XNR.

    (2) Infection of whitefly by CMD virus from infected cassava, ap2QY+ZN.

    (3) CMD spread by replanting of infected cuttings, r(1NK)p3Y.

    Whitefly-cassava transmission (p1) and cassava-whitefly transmission (p2) were given the same probability, and these parameters had the highest sensitivity index magnitude of +0.4965 (Table 2). This suggested that the most important determinants of the severity of a CMD outbreak are transmission by whitefly to cassava and transmission to whitefly from infected plants.

    The model also assigned a probability to CMD spread by planting of infected cuttings (p3). As can be seen from Table 2, this was the parameter with the lowest sensitivity index magnitude of + 0.0071. Replanting of infected cuttings does increase the number of plants exposed to the virus, but contributes less to the spread than does whitefly transmission. Control of the whitefly population is therefore the key policy goal.

    We next analyzed the parameters that determine the whitefly population. The whitefly death rate (μ=0.9929) and the average number of cassava plants visited (a=+0.9929) turn out to be significant. The birth rate of whitefly also affects the stability of the system (Λ=+0.4965). As μ is negative, the whitefly population will decrease as μ increases. However, the sensitivity index values of a and Λ are positive, so that R0 will increase as these parameters increase. The significance of μ, a and Λ in determining R0 (Table 2) suggests that whitefly elimination is the optimal approach.

    Global sensitivity analysis is used to examine the model which responses to parameter variation within a wider range in the parameter space. The parameter values and ranges are listed in Table 1. Partial rank correlation coefficient (PRCC) between the R0 and each parameter for the model are derived as in references [36,37,38]. We computed the PRCC by setting of input parameter values sampled using the Latin Hypercube Sampling (LHS) method. Table 3 shows the results of PRCC and P-value from 1000 independent simulations.

    Table 3.  PRCC between R0 and each parameter.
    Parameter PRCCs P-values
    h 0.0075 0.8136
    β 0.1415 0.0000
    γ 0.0055 0.8624
    K 0.1298 0.0000
    r 0.0012 0.9704
    a +0.4684 0.0000
    Λ +0.1449 0.0000
    μ 0.4312 0.0000
    p1 +0.4049 0.0000
    p2 +0.4358 0.0000
    p3 +0.0286 0.3671

     | Show Table
    DownLoad: CSV

    Parameters with absolute maximum PRCC values as well as corresponding lowest P-values are the most critical factor in disease spread. As shown in Table 3, we noticed that the parameter a has the most effect on R0, followed in decreasing order by p2 (cassava-whitefly transmission), μ (death rate of whitefly), and p1 (whitefly-cassava transmission). The most important parameters for R0 from the global sensitivity analysis match those from the local sensitivity analysis, i.e., a, p2, μ, and p1. These results allow us to considerably reduce R0 by decreasing the whitefly population.

    Local and global sensitivity analysis are used to examine the impact of parameters on R0. We found that killing whitefly is the optimal policy. In practice, whitefly density in plantation area is also a factor that affects to the severity of CMD outbreaks. We therefore investigated the sensitivity of R0 in Eq (4.2) to vary the different sizes of the plantation area (K).

    Figure 1a shows that the sensitivity of R0 to a increases as a increases and K decreases. Figure 1b shows that the sensitivity of R0 to Λ increases as Λ increases and K decreases, which means that R0 is more sensitive to the whitefly birth rate if the plantation area is small. In contrast, the sensitivity of R0 to μ increases as μ and K decrease. This means that CMD outbreaks are more severe when K is small, and a whitefly elimination policy will fail to eliminate the disease. A cost-effective control strategy must therefore take account of the plantation area. To determine the most cost-effective policy, we conducted numerical simulations.

    Figure 1.  (a) sensitivity of R0 to a as K increases from 10, 000 to 20, 000 and a increases from 150 to 250. (b) sensitivity of R0 to Λ as K increases from 10, 000 to 20, 000 and Λ increases from 0.1 to 0.3. (c) sensitivity of R0 to μ as K increases from 10, 000 to 20, 000 and μ increases from 0.0142 to 0.0166.

    To determine the effectiveness of a control policy, it is necessary to assess the number of whitefly in the plantation. Equations (2.4) and (2.5) are modified by introducing two control parameters: ϵ[0,1], the efficacy of pesticide spray in controlling whitefly, and g, the resulting whitefly death rate. Adding terms (μ+gϵ) to Eqs (2.4) and (2.5) produces Eqs (6.1) and (6.2). These take account of deaths of uninfected and infected whitefly from natural causes and from pesticide spraying.

    dQdt=Λap2QY+ZN(μ+gϵ)Q, (6.1)
    dRdt=ap2QY+ZN(μ+gϵ)R. (6.2)

    We used Eqs (2.1)-(2.3), (6.1) and (6.2) to represent the dynamics of a CMD outbreak when no control is applied (gϵ=0) and to determine the resulting cassava yield. We used two plantation capacity scenarios (K) to simulate the dynamics of such a CMD outbreak: (1) K=15,000 and (2) K=10,000. The other parameters were given the values in Table 1. Figures 2-5 assume a 300 d growing season, no control policy, and initial conditions X(0)=9,000, Y(0)=1,000, Z(0)=0, Q(0)=900, and R(0)=100.

    Figure 2.  Whitefly population without control, Scenario 1.
    Figure 3.  Cassava population without control, Scenario 1.
    Figure 4.  Whitefly population without control, Scenario 2.
    Figure 5.  Cassava population without control, Scenario 2.

    Figure 2 shows the number of infected whitefly increasing to a maximum at D 17, then decreasing. Figure 3 shows the numbers of healthy, latent, and symptomatic cassava. Latent and asymptomatic plants outnumbered healthy plants at D 19, or 2 d after the whitefly infection peak. Comparing Figure 2 with 3, the number of infected plants tracked the whitefly population. The impact was mainly on the rate of latent cassava, which exceeded the number of healthy cassava. The harvest of 1, 173 tubers from an initial planting of 9, 000 gave a daily rate of h=0.003.

    The numerical results in Figure 5 show that latent and symptomatic plants outnumbered healthy plants at D 17, lagging the trend in infected whitefly. Infected whitefly reached a maximum at D 17 (Figure 4), then decreased. By the time of harvest at D 300, 714 tubers out of 9, 000 remained healthy. In the smaller plantation, this increased whitefly density and CMD spread.

    Our sensitivity analysis showed that the rapidity of CMD spread was driven primarily by the whitefly population. A simple approach to CMD control is to spray pesticide. However, this may not be economical as more efficient pesticide is also more costly. It is therefore necessary to analyze the cost-effectiveness of spraying as well as its effectiveness.

    We simulated the three control strategies shown in Table 4, representing pesticide sprays with effectiveness of 5%, 10% and 15%. A whitefly control target (g) of 5% per day was set. We then determined the cost-effectiveness of using each spray. Table 4 shows the operating costs. Two kilograms of pesticide are sprayed once a week, requiring two laborers at a cost of 10eachper7,500 m^2 $. The operating costs are shown in the final column. The Cost-Effectiveness Ratio (CER) of [39] is then applied. CER is the ratio of total control costs to yield loss averted, and is used to measure the economic value of an intervention (Eq (6.3)).

    CER=total control costsyield loss averted. (6.3)
    Table 4.  Operating cost of spray pesticide policy.
    Unit per week Total cost per week
    Operating cost Cost per unit per 15, 000 m2 K=15,000 K=10,000
    (1) Labor $10/labor/7, 500 m2 2 labors $20 $13.333
    (2) Pesticide spray
    Type I ϵ=0.05 $10/kg. 2 kg. $20 $13.333
    Type II ϵ=0.10 $20/kg. 2 kg. $40 $26.667
    Type III ϵ=0.15 $50/kg. 2 kg. $100 $66.667

     | Show Table
    DownLoad: CSV

    The simulation used the parameter values given in Table 1. The numerical results in Figures 6 and 7 show that, when the three control strategies were applied, the increase in the healthy cassava population and decrease in the infected whitefly population tracked the efficiency of the pesticide spray. This gave a yield of 1173 tubers when no control strategy was applied. Infection of 518, 1221 and 1851 tubers were averted when Type I, II and III sprays were used. This represents a significant increase in profit. We then used CER to compare the cost-effectiveness of the three control strategies.

    Figure 6.  Infected whitefly with control strategy in Scenario 1.
    Figure 7.  Healthy cassava with control strategy in Scenario 1.

    The CERs were calculated from Eq (6.3):

    CER(I)=1,600518,CER(II)=2,4001,221,andCER(III)=4,8001,851.

    Table 5 compares the cost-effectiveness. The sixth column gives the CER of each approach for K=15,000, where a smaller number represents a more desirable outcome. As can be seen, the Type II spray yielded the greatest overall economic benefit. This result provides policymakers with a useful tool for optimizing their control strategies.

    Table 5.  CER of control policies for K of 15, 000 m2 (Sce.1) and 10, 000 m2 (Sce.2).
    Yields infection averted Total cost CER
    Pesticide spray Sce.1 Sce.2 Sce.1 Sce.2 Sce.1 Sce.2
    Type I 518 154 $1600 $1067 3.088 6.942
    Type II 1, 221 449 $2400 $1600 1.965 3.567
    Type III 1851 777 $4800 $3200 2.593 4.121

     | Show Table
    DownLoad: CSV

    We next simulated the strategy with plantation capacity K=10,000, using the same parameters. The results are shown in Figures 8 and 9. The harvests of healthy tubers were 714 with no control, 868 with Type I spray, 1163 with Type II, and 1491 with Type III. Equation (6.3) was used to identify the most cost-effective policy.

    Figure 8.  Infected whitefly with control strategy in Scenario 2.
    Figure 9.  Healthy cassava with control strategy in Scenario 2.

    The final column of Table 5 confirms that Type II spraying yielded the greatest economic benefit. However, this strategy was less cost-effective when applied to the smaller plantation as the whitefly density was higher.

    In this study, we modeled control of a CMD outbreak driven both by whitefly transmission and planting of infected cuttings. From an epidemiological perspective, R0<1 is a sufficient condition for eradication of the disease from a cassava population. The Routh-Hurwitz criterion confirmed that E0 was locally-asymptotically-stable when R0<1, becoming unstable when R0>1. E0 was globally-asymptotically-stable if R01, while E1 was globally-asymptotically-stable if R0>1, as confirmed by application of Lyapunov's method and LaSalle's invariance principle.

    Our goal was to clarify the vectors that determine the severity of a CMD outbreak. To achieve this, we modeled increases in latent cassava due to whitefly transmission and due to planting of infected cuttings. By identifying the parameters that most affect the stability of the system, we hoped to optimize policy design. A sensitivity analysis determined that the most critical factor in disease spread was the presence of whitefly. We then sought to identify the most cost-effective pesticide for whitefly elimination. Numerical simulations showed that an increase in the death rate of whitefly through spraying was associated with a reduction in the CMD infection rate, and that the efficacy of the pesticide was key to control of the whitefly population. However, the increase in yield had to be offset against the cost of spraying.

    We compared the cost-effectiveness of three pesticide sprays to determine the optimal balance between input cost and yield. The CER showed that Type II spraying was the most cost-effective in both scenarios, and particularly in Scenario 2. Whitefly population control was more costly in a smaller plantation, where the whitefly density was higher and CMD spread more rapidly.

    The effects of rouging of infected plants should be investigated further, as should the effect of combining pesticide treatment with replanting. This may make spraying less efficient, thereby encouraging spread of CMD. Development of resistant strains is an alternative approach. However, it is typically costly and time consuming, and may also reduce the population of beneficial insects. Mathematical models may be used to compare such strategies to identify those that are most cost-effective in the real world.

    We developed a mathematical model to clarify the dynamics of CMD spread. The normalized forward sensitivity index of the reproduction number was used to compare the relative contribution of different factors to a CMD outbreak. We found that the two most important were the death rate of whitefly and the number of cassava plants visited. We also identified the most cost-effective policy for reducing the whitefly population using numerical simulations. CER was used to compare the economic effectiveness of three spraying strategies in farms of different sizes. The results may help stakeholders, including cassava farmers and government agencies, develop optimal policies for control of CMD outbreaks. This should increase yields, income, and profits.

    We would like to thank Mr. John Winward and Dr. Sopone Wongkaew for comments and suggestions.

    The authors declare that there is no conflict of interests regarding the publication of this paper. \newpage



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