Optimal Multi-Drug Therapies for Antimicrobial Resistance with Horizontal Transfer

The emergence of antimicrobial resistance (AMR) is a global health threat that can be mitigated through the careful planning of multi-drug therapies. We analyse a model that captures the interplay between the host immune system and susceptible and resistant bacterial populations, accounting for the use of antibiotics and for treatment-induced mutations and horizontal transfer phenomena that favour the growth of resistant populations. We extend the model to consider multiple antibiotics and all the ensuing possible combinations of resistance. We formulate an optimal control problem aimed at minimising the overall bacterial population size as well as the use of antibiotics, and we rigorously prove that an optimal solution exists, which can be thus computed numerically. For the considered optimal control problem, we analyse necessary conditions for optimality based on the Pontryagin minimum principle, as well as singular controls. We also discuss an alternative approach for treatment design through a model predictive control (MPC) scheme. Our computational experiments, which illustrate both the direct solution to the optimal control problem and the MPC approach, with realistic parameters from the biological literature and initial conditions chosen according to several different scenarios, validate our theoretical results and demonstrate the flexibility of the framework, which adapts the therapy to the presence and numerosity of resistant populations, so that the optimal treatment strongly depends on both the cost functional weights and the initial conditions for the system dynamics.