Influenza is an important public health issue of worldwide concern due to its high transmission rate and seasonal mutations, and family physician surveillance data in Türkiye offer a vital understanding of its local dynamics. Although classical compartmental models provide a basic explanation of epidemics, they are sometimes inadequate to describe the non-linear, memory-dependent, and fractal characteristics of actual viral transmission. In this paper, we develop a new mathematical model by building a Hausdorff fractal model, with Susceptible, Exposed, Infected, Asymptomatic, and Recovered populations in a fractal-order derivative framework. The paper opens with an intensive mathematical analysis, including the proofs of the boundedness and positivity of solutions, to ensure the biological consistency of the model. We first calculate the equilibrium points and obtain the basic reproduction number using the next-generation matrix approach, and then conduct a detailed sensitivity analysis to determine which parameters have the greatest effect on the spread of the disease. To test our theoretical framework, we estimate parameters using real-world surveillance data from Türkiye (Week 48, 2025 - Week 13, 2026) with the Nelder-Mead optimization algorithm, calibrated to a fractal dimension of α=0.9. Simulations show that the epidemic dynamics associated with the fractal method are a better fit than those of classical models. Lastly, the paper wraps up with an analysis of how epidemiological surveillance systems can be improved in terms of predictive capabilities through fractal calculus.