Nonlinear Robust Sliding Mode Control for Measles with Parameter Uncertainties

Sailah Ar Rizka; Regina Wahyudyah Sonata Ayu; Hanna Hilyati Aulia; M. Ziaul Arif; Millatuz Zahroh

doi:10.20956/j.v22i2.47341

Authors

Sailah Ar Rizka Department of Mathematics, Universitas Jember, Jember, Indonesia https://orcid.org/0000-0002-8510-4823
Regina Wahyudyah Sonata Ayu Department of Mathematics, Universitas Palangka Raya, Palangka Raya, Indonesia
Hanna Hilyati Aulia Department of Sharia Economics, IAIN Metro, Metro, Indonesia
M. Ziaul Arif Department of Mathematics, Universitas Jember, Jember, Indonesia
Millatuz Zahroh Department of Mathematics, Universitas Jember, Jember, Indonesia

DOI:

https://doi.org/10.20956/j.v22i2.47341

Keywords:

adaptive gain, epidemic model, measles, nonlinear robust control, parameter uncertainties, sliding mode control

Abstract

Measles is a highly contagious viral disease that persists as a global threat following its resurgence in 2018–2019. The SVEIR epidemic model is considered to represent the dynamics of measles. Various factors affecting the spread of measles result in uncertainties and imprecision in the modelling of measles, thereby a robust control strategy is required to eradicate the disease. The aim of the study is to design a nonlinear robust sliding mode control to drive the number of individuals exposed to or infected with measles to zero through a targetted tracking scheme, despite uncertainties in measles dynamics. This is achieved by administering treatments to exposed and infected individuals while maintaining the existing levels of vaccination. The proposed control strategies have been proven to achieve the tracking objective analytically by employing Lyapunov's stability theorem and Barbalat's Lemma. Employing an adaptive switching gain that is updated online during the design process eliminates the necessity of prior information on the bounds of model uncertainties. Numerical simulations of different cases involving parameter uncertainty and diminishing rates were carried out to evaluate control performance. Using a saturation function and a tangent hyperbolic function instead of a sign function in the controllers and adjusting the rate of switching gain updating can reduce chattering incidents arising from the implementation of sliding mode control. Within the nine cases of parameter uncertainties considered, the robust sliding mode control strategy is effective in eradicating the disease, with average reductions in the number of exposed and infected individuals of 80.82–81.57% and 81.01–84.18%, respectively

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