A Variable-Order Fractional Seir Model With Optimal Control: A Hybrid Wavelet Approach

Abstract

In this paper, a hybrid Chebyshev–Haar wavelet collocation method is developed for solving a variable-order fractional SEIR epidemic model with optimal control. The proposed model incorporates time-dependent memory effects through variable-order Caputo derivatives and includes vaccination and treatment strategies to control the spread of infection. The numerical scheme is constructed using an operational matrix of fractional integration derived via block pulse functions, which transforms the governing fractional differential system into a system of nonlinear algebraic equations. This transformation significantly reduces computational complexity while maintaining high accuracy. A comparative analysis with the Adams–Bashforth–Moulton (ABM) method demonstrates that the proposed hybrid wavelet method provides highly accurate and stable solutions with reduced computational time. Error and convergence analyses confirm the reliability and efficiency of the method.

Numerical simulations are performed to investigate the dynamics of susceptible, exposed, infected, and recovered populations under different fractional orders and control strategies. The results show that the variable-order fractional model effectively captures memory effects, leading to a more realistic description of epidemic behaviour.  It is also observed that decreasing the fractional order slows down the spread of infection and reduces the peak of infected individuals. Furthermore, the inclusion of optimal control significantly reduces the number of infected individuals and enhances recovery. The proposed hybrid Chebyshev–Haar wavelet method provides an efficient, accurate, and robust framework for solving variable-order fractional epidemic models and can be extended to a wide range of nonlinear problems in applied mathematics and biological systems,