The global stability analysis for the mathematical model of an infectious disease is discussed here. The endemic equilibrium is shown to be globally stable by using a modification of the Volterra–Lyapunov matrix method. The basis of the method is the combination of Lyapunov functions and the Volterra–Lyapunov matrices. By reducing the dimensions of the matrices and under some conditions, we can easily show the global stability of the endemic equilibrium. To prove the stability based on Volterra–Lyapunov matrices, we use matrices with the symmetry properties (symmetric positive definite). The results developed in this paper can be applied in more complex systems with nonlinear incidence rates. Numerical simulations are presented to illustrate the analytical results.

SUMMARY: This paper presents a novel robust hybrid fractional order proportional derivative sliding mode controller (HFOPDSMC) for 2-degree of freedom (2-DOF) robot manipulator based on extended grey wolf optimizer (EGWO). Sliding mode controller (SMC) is remarkably robust against the uncertainties and external disturbances and shows the valuable properties of accuracy. In this paper, a new fractional order sliding surface (FOSS) is defined. Integrating the fractional order proportional derivative controller (FOPDC) and a new sliding mode controller (FOSMC), a novel robust controller based on HFOPDSMC is proposed. The bounded model uncertainties are considered in the dynamics of the robot, and then the robustness of the controller is verified. The Lyapunov theory is utilized in order to show the stability of the proposed controller. In this paper, the EGWO is developed by adding the emphasis coefficients to the typical grey wolf optimizer (GWO). The GWO and EGWO, then, are applied to optimize the proposed control parameters which result in the optimized GWO-HFOPDSMC and EGWO-HFOPDSMC, respectively. The effectivenesses of the optimized controllers (GWO-HFOPDSMC and EGWO-HFOPDSMC) are completely verified by comparing the simulation results of the optimized controllers with the typical FOSMC and HFOPDSMC.