A Caputo–Fabrizio fractional model for Escherichia coli dynamics in wastewater treatment systems

Using bacteriophage predation, bacterial resistance, and Monod-type growth kinetics, this study creates and evaluates a multi-stage mathematical model for the dynamics of Escherichia coli (E. coli) in wastewater treatment systems. The pre-primary, primary, secondary, and tertiary stages of wastewater treatment are represented by a seven-compartment deterministic model. Memory and hereditary effects in bacterial dynamics are captured by extending the traditional integer-order model to a Caputo–Fabrizio fractional-order framework, which includes both constant and variable fractional orders. Fixed-point theory is used to establish the existence and uniqueness of solutions. The three-step Adams-Bashforth approach is used to do numerical simulations that compare formulations of integer order, constant fractional order, and variable fractional order. Results indicate that the variable fractional-order model provides enhanced flexibility and improved representation of transient dynamics relative to classical approaches. The proposed framework offers a mathematically rigorous and adaptable tool for analysing E. coli control strategies in wastewater treatment, with potential relevance to public health and Sustainable Development Goals related to water and sanitation.