Computational Methods
1 Parameter Estimation
Model parameters (\(\mathbf{\theta}\)) must be estimated from experimental data. For ME/CFS models, data sources include metabolomics (Naviaux et al. 2016) (Germain et al. 2020), cytokine panels (Hornig et al. 2015) (Montoya et al. 2017), two-day cardiopulmonary exercise testing (CPET) (Keller et al. 2024), and autonomic function assessments (Newton et al. 2007). Parameter estimation proceeds by minimizing the discrepancy between model predictions \(\hat{\mathbf{y}}(\mathbf{\theta})\) and observed data \(\mathbf{y}\):
$ = arg min_{} _{i=1}^{N} ( y_i - _i () )^2 / _i^2 $ {#eq-least-squares}
where \(\sigma_i\) is the measurement uncertainty for observation \(i\). For nonlinear ODE models, this optimization problem is typically non-convex, requiring global optimization methods (e.g., differential evolution, simulated annealing) or Bayesian approaches.
2 Bayesian Inference
Bayesian methods are preferred when parameter uncertainty must be quantified—a critical requirement for ME/CFS models given the limited and heterogeneous data available. Bayes’ theorem gives the posterior distribution over parameters:
$ p( | ) = $ {#eq-bayes}
where \(p(\mathbf{y} | \mathbf{\theta})\) is the likelihood, \(p(\mathbf{\theta})\) is the prior (encoding existing biological knowledge), and \(p(\mathbf{y})\) is the marginal likelihood (model evidence). Posterior distributions are typically computed via Markov chain Monte Carlo (MCMC) sampling. The resulting credible intervals on parameters propagate directly to credible intervals on model predictions, providing honest uncertainty quantification.
3 Sensitivity Analysis
Sensitivity analysis determines which parameters most strongly influence model outputs, guiding both experimental priorities and intervention design. Local sensitivity coefficients are defined as:
$ S_{i j} = dot $ {#eq-sensitivity}
where \(S_{i j}\) is the normalized sensitivity of state variable \(x_i\) to parameter \(\theta_j\). Global sensitivity analysis (e.g., Sobol indices) partitions output variance across parameters and their interactions, accounting for nonlinearities and parameter correlations. Parameters with high sensitivity indices are candidate intervention targets; parameters with low sensitivity indices indicate robustness of the model output to biological variability.