Fuzzy modeling of fractional-order dental caries system and bifurcation analysis

Dental caries is a widespread public health concern influenced by biological, behavioral, and environmental factors. To better understand its progression, this study formulated a fuzzy fractional-order SEIR model. The Caputo fractional derivative was used to capture memory effects. Fuzzy membership functions, dependent on viral load, incorporated uncertainty in transmission, recovery, and disease-induced mortality rates. Equilibrium points were calculated, and the basic reproduction number was derived as a function of viral load. Stability analysis showed the disease-free equilibrium is locally asymptotically stable when [Formula: see text], while an endemic equilibrium appears if [Formula: see text]. Bifurcation analysis with center manifold theory confirmed a forward bifurcation at the threshold [Formula: see text]. Existence and uniqueness of solutions were ensured through Lipschitz conditions. A fractional Adams–Bashforth scheme provided numerical approximations. Simulations illustrated how changes in fractional order and fuzzy parameters affect disease dynamics. The results showed fractional orders [Formula: see text] delay infection peaks and flatten epidemic curves. Higher viral loads reduce recovery rates and increase lethality. The proposed framework captures realistic dynamics under uncertainty, giving valuable insights for designing effective treatment strategies and public health interventions against dental caries.