Numerical and transcritical bifurcation analysis of Chikungunya dynamics within a fractional-fractal framework

Chikungunya is a mosquito-borne viral infection that remains a serious public health problem because of its rapid transmission and long-term clinical implications. Classical models based on integer-order derivatives are effective in describing the basic mechanisms of transmission, but do not consider the memory effects and temporal heterogeneity seen in actual epidemic processes. This study introduces a modified fractional-fractal mathematical model that combines fractional derivatives, representing nonlocal memory, and fractal time, representing scale-related temporal irregularities, to overcome these drawbacks. The fundamental properties, including the positivity and boundedness of solutions, are established. Equilibrium points are derived, and the basic reproduction number is calculated using the next-generation matrix approach. Local stability of the equilibrium states is carried out, and a formal transcritical bifurcation analysis is performed to study the stability switch at the threshold. A numerical scheme is developed based on the Adams-Bashforth-Moulton method to support the theoretical results. Numerical simulations including 3D phase portraits show the effect of the parameters of fractional and fractal models on the dynamics of the disease and distinctions from with classical models. The results suggest that fractional-fractal models are more flexible representations of Chikungunya transmission, and could help elucidate the complex dynamics of epidemics and inform control efforts.