An SEIDR Model with Saturated Behavioral-Feedback Incidence and Capacity-Constrained Optimal Control for Workplace Burnout Dynamics
David Olutunde Daniel *
Department of Mathematical Sciences, University of Texas at Dallas, Richardson, TX 75080, USA.
Magret Tolulope Daniel
University of Alabama at Birmingham, Birmingham, AL 35294, USA.
*Author to whom correspondence should be addressed.
Abstract
Aims: To build a population-level model of workplace burnout that captures how workers and workplaces respond to visible distress, to identify which prevention capacity is the binding bottleneck under realistic budget limits, and to express the answer in dollar terms that can guide prevention investment.
Study Design: A compartmental model with capacity-constrained optimal control, supported by a global sensitivity analysis and a study of a re-infection variant of the same model.
Place and Duration of Study: Department of Mathematical Sciences, University of Texas at Dallas, in collaboration with the University of Alabama at Birmingham, 2024 to 2026.
Methodology: A five-compartment model is constructed with susceptible, exposed, infected, distressed, and recovered groups, governed by ordinary differential equations. The exposure rate uses a Capasso-Serio form with two saturation terms, one on visible acute distress and one on visible chronic strain. The basic reproduction number is obtained by the next-generation method. A three-channel optimal-control problem (primary, secondary, and tertiary prevention) is then posed, with a pointwise cap and a total-spending cap on each channel. Existence follows from Cesari’s theorem and short-horizon uniqueness from a Lipschitz-Gronwall argument, with the optimal controls characterized by Pontryagin’s Minimum Principle.
Results: At the illustrative baseline, R0 = 1.9743. The capacity-constrained run returns an ROI of 12.02, against 19.94 with no budget caps. For this baseline, the shadow prices rank secondary prevention as the binding capacity at $130,769 per unit of budget, above tertiary ($26,978) and primary ($1,328). A re-infection variant shows a saddle-node bifurcation at R0 ≈ 0.76 and a bistable region below threshold.
Conclusion: The framework offers a model-based, regime-dependent way to identify which prevention capacity binds in a given workforce state. The saddle-node hysteresis explains, in formal terms, why a workforce already in chronic burnout does not recover once pressure is eased.
Keywords: Compartmental epidemiological model, saturated incidence, behavioral feedback, isoperimetric optimal control, Pontryagin’s Minimum Principle, shadow prices, allostatic load, LHS-PRCC sensitivity, saddle-node bifurcation, workplace burnout.