Journal of Applied Nonlinear Dynamics
Modelling and Analysis of Pathogens Impact on the Plant Disease Transmission with Optimal Control
Journal of Applied Nonlinear Dynamics 11(3) (2022) 499521  DOI:10.5890/JAND.2022.09.001
Abdisa Shiferaw Melese$^{1}$, Oluwole Daniel Makinde$^{2}$, Legesse Lemecha Obsu$^{1}$
$^{1}$ Department of Applied Mathematics, Adama Science and Technology University, Adama, Ethiopia
$^{2}$ Faculty of Military Science, Stellenbosch University, Stellenbosch, South Africa
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Abstract
In this study, a nonlinear deterministic mathematical model for the impact of pathogens on plant disease transmission with optimal control is proposed. Mathematically, we analyzed the dynamics of the system properties including boundedness of the solutions, existence of pathogen free and coexistence equilibria, local and global stability of equilibrium points. Furthermore, we computed the basic reproduction number $R_{0}$ and studied its sensitivity with respect to model parameters to identify the most affecting parameters. The local stability of the equilibria is also established via the Jacobian matrix and Routh Hurwitz criteria while the global stability of the equilibria is proved by using an appropriate Lyapunov function. Using center manifold theory, we proved that the model exhibits forward bifurcation. An optimal control problem is proposed which minimizes costs incurred due to the disease and applied controls. The characterization of optimal paths is analytically derived using Pontryagin's Minimum Principle. We then numerically studied the optimal control problem. A comparative study is conducted by considering control strategies: implementation of only quarantine, chemical, execution of both quarantine and chemical. We observe that the inclusive use of quarantine and chemical controls can reduce the infected host population with minimum implementation cost.
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