Analysis and Control of Probiotic Dynamic Models

Probiotic therapy involves using live microorganisms, primarily bacteria and yeasts, to improve or restore the balance of beneficial bacteria in the body, particularly in the gut. These microorganisms, when administered in adequate amounts, can offer health benefits to the host. Probiotic therapy is used for various conditions, including diarrhea, irritable bowel syndrome (IBS), and even to support immune function. The dynamics of probiotic therapy is extremely nonlinear. Bifurcation analysis is a powerful mathematical tool used to deal with the nonlinear dynamics of any process. Several factors must be considered, and multiple objectives must be met simultaneously. Bifurcation analysis and multi-objective nonlinear model predictive control (MNLMPC) calculations are performed on two dynamic models of probiotic therapy. The MATLAB program Matcont was used to perform the bifurcation analysis. The MNLMPC calculations were performed using the optimization language Pyomo in conjunction with the state-of-the-art global optimization solvers IPOPT and BARON. The bifurcation analysis revealed the existence of branch points in both models. The branch points (which cause multiple steady-state solutions from a singular point) are very beneficial because they enable the Multiobjective nonlinear model predictive control calculations to converge to the Utopia point (the best possible solution) in the models. It is proved (with computational validation) that the branch points were caused because of the existence of two distinct separable functions in one of the equations in each dynamic model. A theorem was developed to demonstrate this fact for any dynamic model.