Modeling Approaches
1 Ordinary Differential Equations
Ordinary differential equations (ODEs) describe the time evolution of state variables (concentrations, cell counts, hormone levels) under the assumption that spatial variation within compartments is negligible. For a state vector \(\mathbf{x}(t) \in RR^n\) and parameter vector \(\mathbf{\theta} \in RR^p\), a general ODE system takes the form:
$ = ((t), , t), (0) = _0 $ {#eq-general-ode}
ODE models are the primary framework used in Chapters Energy Metabolism Models through Neuroendocrine and Autonomic Models because they balance biological fidelity with mathematical tractability. The key assumptions are: (1) well-mixed compartments (e.g., plasma cytokine concentrations are uniform), (2) deterministic dynamics (stochastic fluctuations are negligible at the population level), and (3) continuous state variables (justified when molecule counts are large).
For ME/CFS-relevant systems, ODEs capture enzyme kinetics (Michaelis–Menten), receptor–ligand binding (Hill functions), and feedback regulation (inhibition/activation cascades). The Michaelis–Menten rate law, used extensively in energy metabolism models (Chapter Energy Metabolism Models), describes enzyme-catalyzed reactions:
$ v = $ {#eq-michaelis-menten}
where \(v\) is the reaction rate, \(V_\text{max}\) is the maximum velocity, \([S]\) is the substrate concentration, and \(K_m\) is the Michaelis constant (substrate concentration at half-maximum velocity).
2 Stochastic Models
When the number of molecules or cells is small—as in NK cell cytotoxicity assays or viral reactivation events—stochastic effects become important. The chemical master equation describes the probability \(P(\mathbf{x}, t)\) of the system being in state \(\mathbf{x}\) at time \(t\):
$ = _{j=1}^{M} [ a_j ( - _j) P( - _j, t) - a_j () P(, t) ] $ {#eq-master-equation}
where \(a_j (\mathbf{x})\) is the propensity function for reaction \(j\) and \(\mathbf{\nu}_j\) is the stoichiometry vector. In practice, the master equation is solved via the Gillespie algorithm (stochastic simulation algorithm), which generates exact sample trajectories. Stochastic models are used in Chapter Immune System Models for viral reactivation dynamics, where the transition from latent to lytic infection is a rare event governed by small numbers of viral genomes.
3 Network Models
Biological systems can be represented as networks where nodes are molecular species, genes, or cell types, and edges are interactions (activation, inhibition, binding). Network analysis provides structural insights—identifying hub nodes, feedback loops, and modular organization—without requiring kinetic parameters. Cytokine networks in ME/CFS (Chapter~Immune System Models) have been characterized using this approach, with studies identifying distinct cytokine signatures that correlate with disease duration (Hornig et al. 2015) (Montoya et al. 2017).
Formally, a biological network is a directed graph \(G = (V, E, w)\) where \(V\) is the set of nodes, \(E \subset.eq V \times V\) is the set of directed edges, and \(w: E -> RR\) assigns signed weights (positive for activation, negative for inhibition). Key network metrics include:
- Degree centrality: the number of edges incident on a node, identifying highly connected “hub” molecules
- Betweenness centrality: the fraction of shortest paths passing through a node, identifying bottleneck species
- Modularity: the degree to which the network decomposes into densely connected subnetworks, corresponding to functional modules (e.g., innate immunity, adaptive immunity)
4 Agent-Based Models
Agent-based models (ABMs) represent individual entities (cells, patients) as autonomous agents with rules governing their behavior and interactions. ABMs are particularly suited to modeling spatial heterogeneity and individual variation. In ME/CFS, ABMs could model immune cell trafficking across the blood–brain barrier (Chapter~Immune System Models) or patient-level variation in treatment responses (Chapter~Predictive Applications and Clinical Translation). However, ABMs require substantially more parameters than ODE models and are computationally expensive, limiting their current application to well-characterized subsystems.
5 Hybrid and Multi-Scale Models
No single modeling framework captures all relevant aspects of ME/CFS pathophysiology. Hybrid models combine frameworks across scales: ODEs for intracellular biochemistry, stochastic models for rare events (viral reactivation), and agent-based models for cell–cell interactions. Multi-scale coupling requires careful attention to timescale separation: fast dynamics (enzyme kinetics, milliseconds) can be approximated as quasi-steady-state relative to slow dynamics (immune cell proliferation, days), reducing the dimensionality of the coupled system. Chapter~Integrated Multi-System Models develops such multi-scale models for ME/CFS.