Dynamics and Optimal Control of the Growth of the Gut Microbiome with Varying Nutrient

Abstract

The human gut microbiome consists of all the microbes that make up the human intestinal tract, and many diseases are associated with certain microbial compositions in the gut. First, a mathematical model describing the growth of gut microbiome inside and on the wall of the gut is developed based on the chemostat model with wall growth. Both the concentration and flow rate of the nutrient input are time-dependent, which results in a system of non-autonomous differential equations. First the stability of each meaningful equilibrium is studied for the autonomous counterpart. Then the existence of pullback attractors and its detailed structures for the nonautonomous system are investigated using theory and techniques of nonautonomous dynamical systems. In particular, sufficient conditions under which the microbiome vanishes or persists are constructed. Numerical simulations are provided to illustrate the theoretical results. Then a second model is developed describing the growth of one beneficial bacterial population with time-varying controlled rate of the input flow of the nutrient. First the stability of each meaningful equilibrium is studied for a constant input case. Then schemes are developed using optimal control theory to find an optimal time-varying rate of input flow. Numerical comparison simulations are provided.