Robustness and resilience of dynamical networks in biology and epidemiology

TL;DR

This monograph provides a rigorous framework for robustness and resilience analysis of dynamical networks in biology and epidemiology, bridging control theory with systems biology. It centers on structural (parameter-free) and integrated structural-probabilistic methods to study how network topology enforces qualitative properties under uncertainty, and introduces formal resilience notions for stochastic perturbations around nominal attractors. The work unifies concepts across disciplines, presents the BDC-decomposition and sign-pattern tools for structural analysis, and extends robustness with probabilistic notions and resilience indicators (e.g., attraction time) to quantify the persistence of attractor behavior under noise. By connecting abstraction with data-driven and regime-shift perspectives, it aims to guide the design of biomolecular controllers, epidemic-management strategies, and detection of tipping points in complex biological networks. Overall, the book lays a formal, cross-disciplinary foundation for analyzing, predicting, and controlling robust and resilient behaviors in life-science networks.

Abstract

Natural systems are remarkably robust and resilient, maintaining essential functions despite variability, uncertainty, and hostile conditions. Understanding these nonlinear, dynamic behaviours is challenging because such systems involve many interacting parameters, yet it is crucial for explaining processes from cellular regulation to disease onset and epidemic spreading. Robustness and resilience describe a system's ability to preserve and recover desired behaviours in the presence of intrinsic and extrinsic fluctuations. This survey reviews how different disciplines define these concepts, examines methods for assessing whether key properties of uncertain, networked dynamical systems are structural (parameter-free) or robust (preserved for parameter variations within an uncertainty bounding set), and discusses integrated structural and probabilistic techniques for biological and epidemiological models. The text introduces formal definitions of resilience for families of systems obtained by adding stochastic perturbations to a nominal deterministic model, enabling a probabilistic characterisation of the ability to remain within or return to a prescribed attractor. These definitions generalise probabilistic robustness and shed new light on classical biological examples. In addition, the survey summarises resilience indicators and data-driven tools for detecting resilience loss and regime shifts, drawing on bifurcation analysis to anticipate qualitative changes in system behaviour. Together, these methodologies support the study and control of complex natural systems, guiding the design of biomolecular feedback architectures, the identification of therapeutic targets, the forecasting and management of epidemics, and the detection of tipping points in ecological and biological networks.

Figures (5)

• Figure 1: Flow graph representation of the chemical reaction networks associated with (a) the biomolecular network \ref{['eq:biomolec']}, (b) the chemical reaction network \ref{['eq:biomolec2']} and (c) the Lotka-Volterra system \ref{['eq:LV']}.
• Figure 2: (a) Schematic representation of a gene activation circuit, corresponding to Eq. \ref{['eq:gene-regulation2']}. TF stands for transcription factors, namely, proteins that regulate DNA transcription by binding promoters. (b) The family of activating $H^+(x)=\frac{(x/\beta)^h}{1+(x/\beta)^h}$ and inhibiting $H^-(x)=\frac{1}{1+(x/\beta)^h}$ Hill functions, for various values of $h$; $h = 1$ corresponds to Michelis-Menten functions, while the so-called "logic approximation" is $\lim_{h \to \infty} H^*(x) = \Theta^*(x - \beta)$, where $^*$ denotes either $^+$ or $^-$ and $\Theta^*$ is the (increasing or decreasing) Heaviside step function.
• Figure 3: Signal graph representation of the Incoherent Feed-Forward Loop activation-inhibition network associated with system \ref{['eq:IFFL']}. The negative self-loops for each node are not reported for simplicity.
• Figure 4: Graph representation of (a) the SIS model, Eq. \ref{['eq:sis']}, and (b) the SIR model, Eq. \ref{['eq:sir']}, akin to the flow graph representation of chemical reaction networks. Vertices represent compartments and edges represent flows of individuals governed by rate constants $\beta$ and $\gamma$.
• Figure 5: A multi-patch, or meta-population, networked model for epidemics capturing the heterogeneous disease transmission in different regions, or age classes (patches, associated with the blue nodes). The dynamics of each patch is described by a different SIR model with specific parameters, and patches are affecting one another through cross-coupling.

Theorems & Definitions (10)

• Example 1: Biomolecular network
• Example 2: Chemical reaction network
• Example 3: Lotka-Volterra system
• Example 4: IFFL
• Example 5: SIS model
• Example 6: SIR model
• Example 7: Extended SIRV model
• Example 8: Extended SEIRV model
• Example 9: Extended SIDARTHE-V model
• Remark 1: Epidemic models as chemical reaction systems