To be actionable, calibrate the model with multi-source empirical data and quantify posterior uncertainty:
Data sources: polling/time-series for trust TT; CPI, unemployment, food prices for EE; event databases and social-media metrics for PP and HH; budget/execution data and security deployment logs for U,KU,\mathcal{K}.
Estimation methods:
State-space estimation: Extended Kalman Filter (EKF) or Unscented Kalman Filter (UKF) for online updating when noise is moderate and near-Gaussian.
Particle Filters: for strongly nonlinear, non-Gaussian regimes and for simultaneous state-parameter estimation.
Bayesian MCMC / Sequential Monte Carlo: to get full posterior distributions of parameters and predictive distributions for quantities of interest (e.g., Pr(P>Pc within 30 days) \Pr(P > P_c \text{ within } 30\text{ days})).
Robustness checks: run ensembles across parameter posteriors; compute credible intervals for forecast probabilities and MFPT estimates.
5. Sensitivity diagnostics that map to policy levers
Translate sensitivity results into policy priorities:
If Sobol indices show U\rho_U accounts for large variance in probability of escalation, prioritize accelerating relief programs and improving responsiveness.
If P\alpha_P (black-horse sensitivity to protest) is highly influential, invest in narrative/communication strategies and monitoring influencer networks.
If noise amplitude of media shocks (component of GG) is crucial, prioritize rapid counter-disinformation and transparency campaigns.
6. Early-warning metrics under stochasticity
Use combined deterministic and stochastic diagnostics:
Critical slowing down indicators: rising variance and lag-1 autocorrelation in PP or TT. Under stochastic forcing, these metrics increase before bifurcation but must be detrended and filtered for seasonality and exogenous cycles.
Skewness and kurtosis: heavy tails signal rising probability of extreme events.
Return time shortening: decreasing mean time between exceedances of moderate thresholds implies approach to criticality.
R-index trend: monitor estimated rt=^K(t)/^U(t)r_t=\hat\rho_{\mathcal{K}}(t)/\hat\rho_U(t) --- an upward trend is a red flag even if other indicators are stable.
7. Scenario generation and policy stress testing
Implement Monte Carlo scenario engines that sample:
Parameter vectors \theta from calibrated posteriors,
Noise realizations (both continuous and jump processes),
Alternative policy response rules (different \rho and cap values).
Outcomes to report:
Probability distributions (with credible intervals) of P(t)P(t), T(t)T(t), H(t)H(t) at 7/30/90/180 day horizons.
MFPTs to critical thresholds.
Bifurcation maps showing regions in parameter space associated with high risk (e.g., r>rcr > r_c and E<c\theta_E < \theta_c).
