# A hybrid Daubechies wavelet collocation approach for a fractional-order SIR epidemic model with delay effects

## TL;DR

This paper introduces a new wavelet-based method to solve a fractional-order epidemic model with delays, improving accuracy and efficiency in simulating disease spread.

## Contribution

A novel hybrid Daubechies wavelet collocation method is proposed for fractional-order epidemic models with delay effects.

## Key findings

- The Daubechies wavelet-based method converges more stably and tracks memory and delay effects better than existing methods.
- The method offers higher approximation accuracy with lower computational overhead compared to traditional numerical techniques.
- The approach assumes fixed transmission rates, which may limit its applicability to heterogeneous real-world scenarios.

## Abstract

This paper studies the transmission dynamics of influenza by using a fractional SIR (Susceptible-Infected-Removed) epidemic model with discrete delay to describe the short-term dynamics. The model includes history-dependent effects through Caputo fractional derivative and maturity delays, which are biologically motivated as the incubation periods or delayed immune responses. In this paper, we will solve this model by introducing a hybrid collocation method with the Daubechies wavelet basis that can be used to efficiently take into account the fractional-order system and the delay system. The reliability and efficiency of the presented algorithm are investigated by means of comparison with some well-known numerical methods, such as the classical Runge-Kutta method (RK4), the Rational Polynomial Spectral Method of order 7 (RPSM7), the Generalized Wavelet Collocation Method (GWCM), and the Genocchi wavelet method. Numerical simulations demonstrate. Our Daubechies wavelet-based method is reported to converge more. Stably and better track both memory and delay effects in the context of the numerical simulations. Nonetheless, the technique presumes fixed parameters (specifically transmission rates), simplifying the situation of multiple unknown parameter values, which are often encountered. Such heterogeneity, particularly in transmission rates, is likely to impact the model’s predictions and should be accounted to have a better and realistic epidemic model. In addition, the method’s efficiency may increase in the case of systems on a large scale or real-time simulations. However, it offers a higher approximation accuracy with lower computational overhead when compared to the known methods.