Stage-structure model for the dynamics of whitefly transmitted plant viral disease: an optimal control approach

Abstract

In this article, a mathematical model is formulated to study the dynamics of whitefly transmitted viral diseases in plants. Here, the aim is to capture the effect of whitefly’s age-stages on the disease dynamics. The existence of the equilibria, basic reproductive number (${\mathcal {R}}_0$), and stability have been studied through qualitative analysis. It is found that the onset of oscillations may occur through Hopf bifurcation in the system. Forward bifurcation is also observed at ${\mathcal {R}}_0=1$. Finally, optimal control theory has been applied for the cost-effectiveness of disease management.

Appendices

Appendix A: Proof of part (iii) of Theorem 1

To prove the theorem, we use the following normal form representing of the system on the central manifold

$$\begin{aligned} \dot{\mathbf{u}}=L_1 \mathbf{u}^2+L_2\phi \mathbf{u} \end{aligned}$$

(25)

where

$$\begin{aligned} L_1=\frac{\mathbf{v}}{2}\cdot D_{xx}f(x_0,\phi _0)\mathbf{w}^2\equiv \frac{1}{2}\sum ^n_{k,i,j=1}v_kw_iw_j\frac{\partial ^2 f_k}{\partial x_i\partial x_j }(x_0,\phi _0) \end{aligned}$$

(26)

and

$$\begin{aligned} L_2=\mathbf{z}\cdot D_{x\varphi }f(x_0,\phi _0)\mathbf{w}\equiv \sum ^n_{k,j=1}v_kw_i\frac{\partial ^2 f_k}{\partial x_i\partial \varphi }(x_0,\phi _0) \end{aligned}$$

(27)

In (26) and (27), $\phi$ is a bifurcation parameter to be chosen, $\phi _0$ is the critical vale; $f_k$ denotes the right hand side of system (3), $\mathbf{x}$ denotes the state vector, $\mathbf{x}_0$ the disease-free equilibrium and $\mathbf{v}$ and $\mathbf{w}$ denote, respectively, the left and right eigenvectors corresponding to the null eigenvalue of the Jacobian matrix of a system, evaluated at the critical point.

Now, the system (3) is assumed at ${\mathcal {R}}_0=1$ that is $\varLambda b =a$. Any of the parameters in the expression of ${\mathcal {R}}_0$ can be assumed as the bifurcation parameter. At the steady state $E_0$, two eigenvalues of the characteristic equation are $-\rho <0$ and $-\mu <0$, and the remaining roots satisfy the cubic equation (14), that is, for ${\mathcal {R}}_0=1$, one eigenvalue is zero and other two satisfy

$$\begin{aligned} \xi ^2+(\rho a+2)\xi +(1+b+\varLambda b)=0, \end{aligned}$$

whose roots are real negative quantities. Thus, for ${\mathcal {R}}_0=1$ the disease-free equilibrium $E_0$ is a non-hyperbolic equilibrium.

The right eigenvectors w = $(w_1,w_2, w_3,w_4,w_5)^T$ satisfies $A(E_0)\mathbf{w }=0$, that is

$$\begin{aligned} A(E_0)=\left[ \begin{array}{ccccc} -\rho &{} ~~-\rho &{} 0 &{} 0&{}-\rho \\ \ \\ 0 &{} ~~-\rho a &{} 0 &{} 0&{} \rho \\ \ \\ 0 &{}0&{} ~~-\mu &{}~~~0&{} ~~~ 0\\ \ \\ 0 &{} ~-b&{} 1 &{} -1 &{} 0\\ \ \\ 0 &{} ~ \varLambda b &{} 0 &{}0 &{} -1\\ \end{array} \right] , \end{aligned}$$

This gives $\mathbf{w }=(-1-a, ~1, ~0, ~ -b,~a)^T$. Again, the left eigenvectors z = $(z_1, z_2, z_3,z_4,z_5)^T$ satisfy $A(E_0,\varLambda _0)^T\mathbf{v} = 0$, this yields z = $(0, ~1,~0,~0~\rho )^T.$

The coefficients $L_1$ and $L_2$ is now computed using (26) and (27). Considering the system (3) and considering only the non-zero components of the left eigenvector $\mathbf{z}$, it follows that:

$$\begin{aligned} L_1= & {} \frac{1}{2}\left[ z_2w_1w_5\frac{\partial ^2 f_2}{\partial x\partial w }(E_0,\varLambda _0)+z_5w_2w_4\frac{\partial ^2 f_5}{\partial y\partial w }(E_0,\varLambda _0)\right] \\= & {} -\left[ (1+a)+\rho ^2 b\right] <0,~ \text{ and }\\ L_2= & {} z_5w_2\frac{\partial ^2 f_5}{\partial y\partial \varLambda }(E_0,\varLambda _0)=\rho a > 0, \end{aligned}$$

and thus the bifurcation is forward.

Appendix B: Proof of Theorem 2

Proof

The characteristic equation at the endemic equilibrium $E^*$ is

$$\begin{aligned} H(\xi )=\xi ^4+\sigma _1\xi ^3+\sigma _2\xi ^2+\sigma _3\xi +\sigma _4=0. \end{aligned}$$

(28)

If the roots of the characteristic equation (28) have negative real parts, then $E^*$ is stable. Applying the Routh–Hurwitz criterion (Murray 2002) on the coefficients of (28), we can say that the (28) has roots with negative real parts if the following conditions are satisfied:

$$\begin{aligned} \sigma _1>0,~~\sigma _2> 0,~~\sigma _3> 0,~~\sigma _4>0,~~\sigma _1 \sigma _2 - \sigma _3>0,~~ \sigma _1 \sigma _2 \sigma _3 -\sigma _3^2-\sigma _4 \sigma _1^2 >0. \end{aligned}$$

(29)

Now, we discuss the existence of Hopf bifurcation.

Using the conditions (15), the characteristic equation (28) can be rewritten as follows

$$\begin{aligned} \left( \xi ^2+\frac{\sigma _3}{\sigma _1}\right) \left( \xi ^2+\sigma _1\xi +\frac{\sigma _1\sigma _4}{\sigma _3}\right) =0. \end{aligned}$$

(30)

Thus two roots of this equation are

$$\begin{aligned} \xi _{1,2}=\pm i\omega _{0},\quad \omega _0=\sqrt{\frac{\sigma _3}{\sigma _1}}, \end{aligned}$$

and the remaining two roots, $\xi _3$ and $\xi _4$ satisfy the equation

$$\begin{aligned} \xi ^2+\sigma _1\xi +\frac{\sigma _1\sigma _4}{\sigma _3}=0. \end{aligned}$$

Using (29) and applying Routh–Hurwitz criterion (Murray 2002), we can say that they both have negative real parts.

To verify the transversality condition, we first note that $\varPhi (\zeta ^*)$ is a continuous function of its argument, and hence, there exists an open interval $\zeta \in (\zeta ^*-\epsilon ,\zeta ^*+\epsilon )$, where $\xi _1$ and $\xi _2$ are complex conjugate roots of the characteristic equation, which can be written as

$$\begin{aligned} \xi _{1,2} (\zeta )= & {} \zeta (\zeta ) \pm i\nu (\zeta ), \end{aligned}$$

with $\xi _{1,2}(\zeta ^*)=\pm i\omega _0$.

Substituting $\xi _j (\zeta ) =\zeta (\zeta )\pm i\nu (\zeta )$ into the characteristic equation (28), differentiating with respect to $\zeta$, and separating real and imaginary parts gives

$$\begin{aligned} P(\zeta )\zeta '(\zeta ) -Q(\zeta )\nu '(\zeta )+R(\zeta ) =0,\nonumber \\ Q(\zeta )\zeta '(\zeta )+P(\zeta )\nu '(\zeta ) +S(\zeta ) =0, \end{aligned}$$

(31)

where

$$\begin{aligned} P(\zeta )= & {} 4\zeta ^3-12\zeta \nu ^2+3\sigma _1(\zeta ^2-\nu ^2)+2\sigma _2\zeta +\sigma _3, \\ Q(\zeta )= & {} 12\zeta ^2\nu +6\sigma _1\zeta \nu -4\zeta ^3+2\sigma _2\zeta , \\ R(\zeta )= & {} \sigma _1\zeta ^3-3\sigma _1'\zeta \nu ^2+\sigma _2'(\zeta ^2-\nu ^2)+\sigma _3'\zeta , \\ S(\zeta )= & {} 3\sigma _1'\zeta ^2\nu -\sigma _1'\nu ^3+2\sigma _2'\zeta \nu +\sigma _3'\zeta . \end{aligned}$$

Solving the (31) for $\zeta '(\zeta ^*)$ and using the condition in (15) we have

$$\begin{aligned} \left[ \frac{\text {d} \mathrm{Re}[\xi _j(\zeta )]}{d\zeta }\right] _{\zeta =\zeta ^*}= & {} \zeta '(\zeta ^*) =-\frac{Q(\zeta ^*)S(\zeta ^*)+P(\zeta ^*)R(\zeta ^*)}{P^2(\zeta ^*)+Q^2(\zeta ^*)} \\= & {} \frac{\sigma _1^3\sigma _2'\sigma _3(\sigma _1-3\sigma _3) -2(\sigma _2\sigma _1^2-2\sigma _3^2) (\sigma _3'\sigma _1^2-\sigma _1'\sigma _3^2)}{\sigma _1^4(\sigma _1-3\sigma _3)^2+4(\sigma _2\sigma _1^2-2\sigma _3^2)^2}\ne 0. \end{aligned}$$

Therefore, the transversality condition is satisfies. This confirms the occurrence of Hopf bifurcation at the critical value $\zeta =\zeta ^*$. $\square$