Numerical investigation of fractional order SEIR models with newborn immunization using Vieta–Fibonacci wavelets

In this article, we investigate the dynamics of a fractional-order SEIR epidemic model with special emphasis on the vaccination of newborns. By incorporating vaccination directly into the SEIR framework, newborns bypass the susceptible stage and enter the immune class directly, which enhances herd immunity and contributes to the overall reduction in disease spread. A novel operational matrix method based on Vieta–Fibonacci wavelets is developed to approximate the fractional-order SEIR model that includes newborn immunization, where the fractional derivative is taken in the Caputo sense. To begin with, the operational matrices of fractional-order integration are obtained via block-pulse functions. These matrices convert the underlying model into a system of algebraic equations that can solved using any classical method, such as Newton’s iterative method, Broyden’s method, or fsolve command in MATLAB software. The Haar wavelet method is also discussed to show its applicability and efficiency. The obtained results lucidly illustrate the dynamics of susceptible, exposed, infected, and recovered populations during an infectious outbreak. The decline in susceptible and infected individuals reflects the disease’s progression, while vaccination significantly reduces infection peaks. Variations in the fractional parameter $\alpha$ and transmission factor $\beta$ reveal the influence of these variables on the disease outbreak, with higher values of $\beta$ leading to rapid transmission. The chaotic attractors of the fractional-order SEIR epidemic model with newborn immunization are graphically represented using Vieta–Fibonacci wavelets.