A Wavelet Framework for Fractional Epidemic Models with Delay

A fractional-order epidemic model with a nonlinear incidence rate and a biologically motivated time delay is proposed using a new fractional derivative operator. The nonlinear incidence accounts for behavioral and psychological effects in disease transmission, while the delay represents latency and incubation periods. An efficient numerical scheme based on Euler wavelet expansion is developed to obtain approximate solutions of the resulting system. Fundamental analytical properties, including existence, uniqueness, and positivity of solutions, are established, and the stability of equilibrium points together with the basic reproduction number is analyzed. Numerical simulations demonstrate the influence of the fractional order, time delay, and nonlinear incidence on the qualitative dynamics of the epidemic model. The proposed framework generalizes several existing models and provides a unified approach for incorporating memory and delay effects in epidemic dynamics.