A mathematical model of multiple delayed feedback control system of the gut microbiota-Antibiotics injection controlled by measured metagenomic data

Recent biological studies on the gut microbiota have revealed associations between imbalance of microbial community composition and diverse types of disease development. Composition change or reduced diversity of the gut microbiota has been identified by metagenomic samples of the gut microbiota from patients of chronic inflammatory diseases. Not only artificial but also natural antibiotics potentially regulate abnormal microbial species composition. In this paper, we propose a novel mathematical model for a feedback control system of the gut microbiota: a moderate amount of antibiotics injection as negative feedback regulator to abundances of microbial populations measured via metagenomic data with time delay. A class of autonomous N-species Lotka-Volterra type competitive system with discrete delays, representing population dynamics of the gut microbiota with delayed negative feedback via antibiotics injection, is considered to quantitatively characterize effect of multiple delayed feedback regulations on the microbial populations. Explicit formula for the corresponding characteristic equation evaluated at the positive equilibrium of the main model is successfully derived. Analytic results as well as numerical simulations are given for the two species case to quantitatively evaluate allowable ranges of time delay for avoiding population oscillations.