Modelling the optimal control of fungi infection on the legumes plant in the presence of rhizobia bacteria

A non –linear mathematical model is formulated and analysed to study the Modelling Optimal Control of Fungi infection on the Legume plant in the presence of Rhizobia bacteria. The proposed model describes three types of interacting Legumes plant, Rhizobia bacteria and fungi populations. The population is divided into three namely; Legumes plant, Rhizobia bacteria and Fungi interaction. Optimal control model was formulated and analysed. A qualitative analysis of the model will be done in order to determine the boundedness and stability of equilibrium points. In qualitative analysis has been employed to analyse the model into five steady state (equilibrium points) that are trivial and four non-trivial equilibrium points. Both trivial and non-trivial steady states were found to be locally asymptotically stable and unstable. The existence and stability of the equilibrium points were analysed. It is found that linearization method by next generation matrix approach, as well as eigenvalues of Jacobian matrix, the equilibrium point is locally asymptotically stable if 0 λ < and unstable when 0 λ > . The Lyapunov function technique will be used to determine the global stability of the model. We estimate the model state initial conditions and parameters values from legumes plant, Rhizobia bacteria and fungi interaction data of other literature. We use Pontryagin’s maximum principle to derive the optimality system and solve the system numerically. We also discuss the characterization of the optimal control via adjoint variables. Matlab will be used for Numerical simulation of the model. It is finally concluded that numerical results shows that application of optimal control problems leads to decrease in the use of fungicide application hence reduce the fungi infection on the legumes plant and legume plant grow logistically.