Synchronization and dynamics of modified fractional order Kawasaki disease model with chaos stability control

In this paper, fractional calculus has proven to be invaluable in disease transmission dynamics and the creation of control systems, among other real-world problems. To investigate vaccine and treatment dynamics for disease control, this work focuses on Kawasaki illness and uses a unique fractional operator called the modified Atangana-Baleanu-Caputo derivative. The stability analysis, positivity, boundedness, existence, and uniqueness, are treated for the proposed model with novel fractional operators. Additionally, it investigates the effects of different parameters on the reproductive number. It verifies the existence and uniqueness of the solutions to the suggested model using Banach fixed point and the Leray-Schauder nonlinear alternative theorem. Employs Lyapunov functions to determine the model equilibria analysis global stability. The numerical simulation and results utilized the two-step Lagrange interpolation approach at various fractional order values. The results are contrasted with those obtained using the widely recognized ABC method and comparisons are also made to show the effects of the proposed method for the epidemic system. This model advances beyond existing Kawasaki disease models by incorporating fractional-order dynamics, which capture memory effects and long-range dependencies in biological systems, offering more accurate representations of disease progression. The inclusion of chaos stability control provides novel insights into managing complex, nonlinear behaviors, enhancing both theoretical understanding and potential clinical applications.