The role of follow-up programs in weight-loss maintenance: an optimal control approach

Weight-loss maintenance remains a persistent public health challenge, partly because social influence and behavioral relapse can return individuals who have achieved a healthy weight to an overweight state. This study extends a previously developed social obesity model by introducing a time-dependent follow-up control κ(t) that reduces the effective relapse rate from the recovered class. An optimal control problem is formulated to minimize the prevalence of overweight and obesity together with the implementation cost of follow-up support. We prove the existence of an optimal control and use Pontryagin's Maximum Principle to derive the corresponding adjoint system and pointwise characterization of the optimal strategy. Under the baseline weighting, initial condition, and 30-year planning horizon, the optimal intervention lowers the combined overweight-and-obese prevalence at T=30 by approximately 6.9% relative to the uncontrolled trajectory and maintains a larger recovered population, although the total burden is not reversed over the intervention horizon. Over the 30-year horizon considered, the optimal strategy is front-loaded, with stronger support early and lower intensity when implementation is more costly. Post-intervention simulations show that the benefit persists for many years after support is withdrawn but gradually erodes as the system relaxes toward the uncontrolled endemic equilibrium.