Prabhakar fractional-order modeling of malaria transmission with vaccination and waning immunity

In this study, we develop a classical malaria transmission model by extending an existing framework to incorporate partial vaccination and waning immunity, introducing compartments for vaccinated susceptible, exposed, infected, and recovered individuals. To enhance biological realism and capture complex memory and hereditary effects, we fractionalize the model using the Caputo-type Prabhakar derivative. This generalized fractional operator, with its non-local Mittag–Leffler kernel, enables a more flexible representation of disease progression than classical derivatives. Furthermore, we show that by taking appropriate parameter limits in the Prabhakar operator, the model reduces to a special case involving the Atangana–Baleanu fractional derivative in the Caputo sense (ABC type), thereby connecting the generalized framework to a widely used non-singular fractional operator. We prove that the fractional model retains positivity and boundedness, and we determine the equilibrium points and basic reproduction number (R0) using the next-generation matrix method. Sensitivity analysis reveals the most influential parameters affecting R0, offering practical insights for targeted control strategies. Numerical simulations are performed via a fractional fourth-order Runge–Kutta method. Results show that decreasing the fractional order a1 delays infection peaks, smooths transitions between compartments, and slows disease spread, reflecting the influence of long-term memory in moderating epidemic dynamics. Phase portraits indicate that trajectories approach equilibrium more gradually for smaller a1 values, while 3D time-dependent plots illustrate the evolution of each compartment under varying memory effects. Additionally, 3D interaction surfaces between vaccinated and unvaccinated subgroups highlight the interplay between immunity and transmission. These findings confirm the versatility of Prabhakar fractional-order modeling for realistic malaria dynamics.