rsmodel

A qualitative, two-state dynamical model of the interaction between depressive symptoms and psychological resilience under external adversity — the resilience–symptom (RS) model:

dr/dt = (-s + (1-s)*r) * (1-r) * r
ds/dt = (e*(1+s-r) - s*r) * (1-s) * s

where r ∈ [0,1] is the resilience level (0 = depleted, 1 = full), s ∈ [0,1] the depressive symptom level (0 = healthy, 1 = severe), and e ∈ [0,1] an external adverse input. The unit square is positively invariant.

This is the reference implementation for

Björn S. Rüffer & Michael Schönlein, A mathematical model for depression and resilience.

Preprint: arXiv:2608.22073 — under review.

Interactive playground, no installation required: https://rsmodel.org

Warning Neither the model nor its time scale is calibrated against clinical data. This is a qualitative model, not a quantitative one, and it does not represent the effects of medication or of any other intervention. It is not a diagnostic or clinical tool.

Install

uv add rsmodel

Usage

from rsmodel import RSModel, Patient
from rsmodel.utils import get_predefined_scenarios

model = RSModel()
model.rhs(_r=0.84, _s=0.67, _e=0.3)         # -> (dr/dt, ds/dt)
model.plot_equilibria()                      # equilibrium manifold over e ∈ [0,1]
model.plot_streamlines(external_input=0.2)   # phase portrait at constant e

patient = Patient(s0=0.67, r0=0.84)
patient.get_stimulus_response(get_predefined_scenarios()["multiple adverse events"])

Symbolic analysis (sympy):

from rsmodel.analysis import compute_jacobian, analyze_corner_equilibria

J = compute_jacobian(RSModel())                     # symbolic 2x2 Jacobian
analyze_corner_equilibria(RSModel(), e_value=0.5)   # stability of the four corners

RS2Model is the variant with the 3*s*r coupling term used for comparison in the paper.

Interactive notebooks

The repository ships three marimo notebooks — an interactive playground (also published at https://rsmodel.org), the full symbolic analysis, and a script that regenerates every figure in the manuscript. See https://github.com/bjoseru/rsmodel for details.

Citation

The manuscript is under review; until it is accepted, cite the arXiv preprint alongside the archived software:

@misc{rueffer_schoenlein_rsmodel_preprint,
  author        = {R{\"u}ffer, Bj{\"o}rn S. and Sch{\"o}nlein, Michael},
  title         = {A mathematical model for depression and resilience},
  year          = {2026},
  eprint        = {2608.22073},
  archivePrefix = {arXiv},
  url           = {https://arxiv.org/abs/2608.22073}
}

@misc{rueffer_schoenlein_rsmodel_software,
  author    = {R{\"u}ffer, Bj{\"o}rn S. and Sch{\"o}nlein, Michael},
  title     = {{rsmodel}: a resilience--symptom dynamical model of depression},
  year      = {2026},
  publisher = {Zenodo},
  doi       = {10.5281/zenodo.22018288},
  url       = {https://rsmodel.org}
}

License

MIT