A Stochastic Hybrid Model for Bipolar II Disorder with Lithium Maintenance Treatment

Bipolar II disorder (BD-II) is characterized by recurrent hypomanic and depressive episodes interspersed with periods of euthymia. This study proposes a stochastic hybrid framework that integrates multistable nonlinear mood dynamics, Gaussian white noise, Poisson jumps, and a compartmental pharmacokinetic model of lithium maintenance treatment. Unlike approaches based on intrinsic mood oscillations, the proposed model represents affective episodes as stochastic transitions between coexisting stable states. The deterministic system exhibits saddle-node bifurcations and, within a restricted parameter regime, four stable equilibria associated with hypomanic, depressive, and two euthymic states. Numerical simulations characterize how continuous and abrupt stochastic perturbations influence episode durations and recurrence frequencies, revealing asymmetric responses to perturbations acting on the two mood variables. Coupling the mood dynamics to periodic lithium administration generates periodic orbits whose stability is assessed using Floquet multipliers. Numerical continuation shows that pharmacological coupling can shift these orbits toward the euthymic region and, for suitable parameter values, produce stable periodic dynamics confined within it. The analysis of imperfect treatment adherence further shows that randomly omitted doses can alter episode occurrence and duration, while higher adherence generally favors euthymic dynamics under the investigated conditions. These findings establish a mathematical framework for examining the interplay between multistability, stochastic mood transitions, and lithium maintenance treatment, while providing a basis for future calibration and validation using longitudinal clinical data.