Using Differential Equations in the Analysis of Dental Caries Growth and Development: A Mathematical Modeling Approach

Abstract

Dental caries represents one of the most pervasive chronic oral pathologies globally, characterized by the metabolic degradation of dietary carbohydrates by acidogenic bacteria, leading to the demineralization of the tooth enamel matrix. This study establishes a mathematical framework to model the coupled dynamics of oral bacterial proliferation and enamel degradation over time. Bacterial population growth within the dental biofilm is formulated using the non-linear logistic differential equation. Concurrently, a rate-of-decay differential equation is derived to track continuous enamel loss as a function of the bacterial load. Complete analytical derivations, explicit integration steps, and exact parameterizations are provided, and all reported numerical results have been independently re-verified. A comparative numerical analysis of ten distinct patient risk profiles evaluated at t = 10 days is presented, with the underlying computations corrected to be fully consistent with the derived closed-form solution. The findings demonstrate that mathematical modeling yields quantifiable, precise, and predictive insight into disease progression, offering an evidence-based foundation for personalized preventive dentistry and clinical risk stratification.