Normalized Caputo–Fabrizio SVIR modeling and bifurcation analysis

This study presents a comprehensive formulation, theoretical analysis, and numerical investigation of a fractional-order SVIR epidemic model employing the normalized Caputo–Fabrizio (NCF) derivative. The NCF operator introduces memory effects through a normalized, non-singular kernel, ensuring physically consistent weighting of past states. Vaccination dynamics are explicitly incorporated, enabling realistic modeling of epidemic control strategies. We establish key mathematical properties of the model, including existence, uniqueness, positivity, boundedness, and conservation of the total population, and propose an efficient numerical scheme for its simulation. The influence of the fractional order and kernel normalization is explored through numerical experiments, highlighting their impact on epidemic peak magnitude, transient dynamics, and vaccination effectiveness. A rigorous equilibrium and bifurcation analysis shows that, for the closed SVIR structure with constant vaccination, neither backward bifurcation nor Hopf bifurcation can occur, even in the presence of saturated incidence. Any oscillatory behavior observed numerically is transient and induced by fractional memory effects rather than sustained periodic solutions. These results clarify the dynamical limitations of the closed NCF–SVIR framework and highlight the role of fractional memory in shaping epidemic trajectories. The proposed model provides a robust foundation for future extensions incorporating demographic turnover or imperfect vaccination, where richer bifurcation phenomena may arise.