Mapping the Invisible Web of Disease: The Static Network Agent-Based Model
How fixed social ties shape epidemics—one connection at a time
🌐 Introduction
Picture your social life as a web: you’re connected to your partner, your coworkers, your gym buddies, your book club. These ties don’t vanish overnight—they persist, day after day. Now imagine a virus entering this web. It doesn’t spread randomly through a crowd; it travels along the threads of your real relationships.
This is the intuition behind the Static Network Agent-Based Model (ABM)—an epidemiological simulation where disease spreads across a fixed contact graph. Unlike models that assume everyone mixes equally (like a bubbling stew), this approach treats society as a network of enduring connections, much like a subway map of human interaction.
Static network ABMs have become indispensable tools for understanding outbreaks of diseases like influenza, measles, and even SARS-CoV-2 in settings where relationships matter more than random encounters—schools, offices, households, or close-knit communities [1–3]. In this article, we unpack how this model works, why it’s powerful, and how scientists extend it to tackle real-world complexity—all while keeping the math clear and the ideas vivid.
Welcome to the digital sociology of contagion.
🔗 Model Description
The Static Network ABM represents a population as an undirected graph G = (V, E), where:
- V is the set of N agents (|V| = N),
- E is the set of persistent edges representing regular contacts (e.g., household members, desk neighbors, classroom peers).
Crucially, the network does not change during the simulation—hence “static.” This captures stable social structures where people repeatedly interact with the same individuals.
Each agent occupies one of four health states in a SEIR framework:
- S: Susceptible
- E: Exposed (infected, not yet infectious)
- I: Infectious
- R: Recovered (immune)
Time advances in discrete daily steps. Transmission and recovery follow probabilistic rules grounded in individual biology and network topology.
🦠 Transmission Rule
On each day t, for every edge (i, j) where agent i is infectious (I) and agent j is susceptible (S), infection may occur with probability:
pᵢⱼ(t) = 1 − exp(−β · g(τᵢ))
Where:
- β (beta) is the baseline transmissibility scale (unitless, 0 < β ≤ 1),
- τᵢ is the time since agent i became infectious (in days),
- g(τ) is the infectiousness profile—a function that typically peaks early in infection (e.g., 1–3 days post-symptom onset).
This formulation ensures that transmission risk depends on both biology (g(τ)) and behavior (fixed contact). If a random number u ~ Uniform(0,1) satisfies u < pᵢⱼ(t), then agent j transitions to Exposed (E).
💡 Why exponential? The expression 1 − exp(−β·g(τ)) arises from modeling infection as a Poisson process over a contact interval—standard in micro-simulation epidemiology [4].
⏳ Natural History Progression
Once exposed, each agent i independently draws:
- Tᴱᵢ ~ Distribution of incubation periods (time from exposure to infectiousness),
- Tᴵᵢ ~ Distribution of infectious durations (time from infectiousness to recovery).
Common choices: log-normal or gamma distributions, reflecting biological variability. For example, for seasonal flu:
- Mean Tᴱ ≈ 2 days, SD ≈ 0.8
- Mean Tᴵ ≈ 5 days, SD ≈ 1.5
Agents automatically progress:
S → E (after exposure) → I (after Tᴱᵢ days) → R (after Tᴵᵢ days)
No reinfection occurs during the simulation unless explicitly modeled (see extensions).
📏 Key Parameter Definitions & Typical Values
| N | Population size | Number of agents (nodes) | 100 – 1,000,000 |
| k̄ | Mean degree | Average number of contacts per agent | 5 – 50 (e.g., school: ~15; village: ~8) |
| β | Transmissibility | Per-contact infection risk scale | 0.05 – 0.4 |
| g(τ) | Infectiousness curve | Relative transmission potential at time τ | Peaks at τ = 1–3 days |
| ⟨Tᴱ⟩ | Mean incubation | Avg. time to become infectious | 1–10 days (disease-dependent) |
| ⟨Tᴵ⟩ | Mean infectious period | Avg. duration of infectiousness | 3–14 days |
| σₖ | Degree heterogeneity | Standard deviation of node degrees | Often modeled via power-law or Poisson |
🌟 Network structure matters: A network with k̄ = 10 but high variance (e.g., a few “super-connectors”) spreads disease faster than a uniform network with the same average degree [5].
⚖️ Assumptions and Applicability
The Static Network ABM rests on clear, testable assumptions:
✅ Fixed contacts: Social ties do not form or break during the outbreak (reasonable for short epidemics or stable settings like schools).
✅ Undirected edges: If i contacts j, then j contacts i (symmetric interaction).
✅ Homogeneous mixing within edges: Each edge represents a regular, bidirectional contact opportunity.
✅ No external seeding: All cases arise from internal transmission (unless modified).
🎯 When should you use this model?
- Closed or semi-closed populations: Schools, military barracks, nursing homes, cruise ships [6].
- Diseases spread via close, repeated contact: Influenza, RSV, tuberculosis, measles.
- Studying “superspreading”: Networks naturally generate high-degree nodes that can ignite outbreaks.
- Evaluating targeted interventions: Vaccinate high-degree nodes? Isolate clusters?
It is less suitable for:
- Diseases with environmental transmission (e.g., cholera),
- Highly dynamic populations (e.g., mass gatherings, urban commuting),
- Long-term epidemics where social networks evolve (e.g., multi-year HIV spread).
In such cases, dynamic or adaptive network models are preferred [7].
🔧 Model Extensions and Variants
While the base static network model is elegant, real epidemics demand richer features. Here are key enhancements used in modern toolboxes:
1. Weighted Edges
Assign a weight wᵢⱼ to each edge representing contact intensity (e.g., hours per day). Then:
pᵢⱼ(t) = 1 − exp(−β · wᵢⱼ · g(τᵢ))
Weights can be derived from surveys or mobility data [8].
2. Layered (Multiplex) Networks
Model multiple contact contexts as separate network layers:
- Gₕ: Household layer
- Gᵥ: Workplace/school layer
- G꜀: Community layer
Transmission occurs across the union of layers, with layer-specific β values (e.g., βₕ > β꜀ due to closer contact) [9].
3. Asymptomatic Carriers
Introduce a probability fₐ that an infection is asymptomatic. Asymptomatic agents have modified infectiousness:
gₐ(τ) = α · g(τ), with α < 1 (e.g., α = 0.6 for SARS-CoV-2 [1])
They may also have different Tᴵ distributions.
4. Vaccination & Immunity
Pre-assign immune agents (R at t=0) based on coverage v. Or model vaccine efficacy ε by reducing susceptibility:
pᵢⱼ(t) = (1 − ε) · [1 − exp(−β · g(τᵢ))]
for vaccinated j [10].
5. Stochastic Seeding & Multiple Introductions
Instead of one index case, introduce infections at random nodes with probability λ per day to mimic importation.
These extensions preserve the static network’s core while adapting to policy questions like “Should we vaccinate teachers or students first?” or “How many index cases trigger an outbreak in this dorm?”
🎉 Conclusion
The Static Network ABM transforms an abstract idea—“social structure matters”—into a computational reality. By freezing human connections into a graph, it reveals how the architecture of our relationships shapes the fate of outbreaks. A single well-connected individual can ignite an epidemic; a sparse network can halt it in its tracks.
This model is not just a theoretical toy—it has guided real interventions, from school closures during H1N1 to contact tracing strategies for Ebola [6,9]. And while it assumes fixed ties, that very simplicity makes it fast, interpretable, and ideal for scenario planning in stable communities.
As we collect better data on human contact patterns—from wearable sensors to digital diaries—static network models will only grow more precise. But their core insight remains timeless: epidemics don’t spread through populations; they spread through people, and the threads that bind them.
So next time you shake hands, hug a friend, or sit next to a coworker, remember: you’re not just sharing a moment. You’re part of a network—and in the eyes of an epidemiologist, that network is the battlefield where pandemics are won or lost.
📚 Key References
- Lloyd-Smith, J.O. et al. (2005). Superspreading and the effect of individual variation on disease emergence. Nature. https://doi.org/10.1038/nature04153
- Keeling, M.J. & Eames, K.T.D. (2005). Networks and epidemic models. Journal of the Royal Society Interface. https://doi.org/10.1098/rsif.2005.0051
- Mossong, J. et al. (2008). Social contacts and mixing patterns relevant to the spread of infectious diseases. PLOS Medicine. https://doi.org/10.1371/journal.pmed.0050074
- Ferguson, N.M. et al. (2005). Strategies for containing an emerging influenza pandemic in Southeast Asia. Nature. https://doi.org/10.1038/nature04017
- Pastor-Satorras, R. et al. (2015). Epidemic processes in complex networks. Reviews of Modern Physics. https://doi.org/10.1103/RevModPhys.87.925
- Cauchemez, S. et al. (2009). Closure of schools during an influenza pandemic. The Lancet Infectious Diseases.
- Funk, S. et al. (2009). Spread of epidemic disease on networks. Physical Review E. https://doi.org/10.1103/PhysRevE.79.026101
- Prem, K. et al. (2017). Projecting social contact matrices in 152 countries. PLOS Computational Biology. https://doi.org/10.1371/journal.pcbi.1005697
- Aleta, A. et al. (2020). Modelling the impact of testing, contact tracing and household quarantine on second waves of COVID-19. Nature Human Behaviour. https://doi.org/10.1038/s41562-020-0931-9
- Glass, K. et al. (2022). Agent-based models for infectious disease policy. Philosophical Transactions B. https://doi.org/10.1098/rstb.2021.0030