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.
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Acknowledgements
Sagar Adhurya acknowledges the University Grant Commission, Government of India for funding this research under NET-JRF scheme Sl no. 2061530673.
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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
where
and
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
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
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:
and thus the bifurcation is forward.
Appendix B: Proof of Theorem 2
Proof
The characteristic equation at the endemic equilibrium \(E^*\) is
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:
Now, we discuss the existence of Hopf bifurcation.
Using the conditions (15), the characteristic equation (28) can be rewritten as follows
Thus two roots of this equation are
and the remaining two roots, \(\xi _3\) and \(\xi _4\) satisfy the equation
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
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
where
Solving the (31) for \(\zeta '(\zeta ^*)\) and using the condition in (15) we have
Therefore, the transversality condition is satisfies. This confirms the occurrence of Hopf bifurcation at the critical value \(\zeta =\zeta ^*\). \(\square \)
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Adhurya, S., Basir, F.A. & Ray, S. Stage-structure model for the dynamics of whitefly transmitted plant viral disease: an optimal control approach. Comp. Appl. Math. 41, 154 (2022). https://doi.org/10.1007/s40314-022-01864-9
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DOI: https://doi.org/10.1007/s40314-022-01864-9
Keywords
- Mathematical model
- Basic reproduction ratio
- Bemisia
- Forward and Hopf bifurcation
- Optimal control
- Numerical simulation