The Well-Mixed Micro-Simulation – Canada Mosquitoes Data Portal

Simulating Pandemics in a Digital Petri Dish: The Well-Mixed Micro-Simulation Agent-Based Model

An accessible deep dive for the scientifically curious

🌍 Introduction

Imagine a city where every person is a digital avatar—going to work, meeting friends, riding buses, and occasionally falling ill. Now imagine a virus slipping into this virtual world, hopping from one avatar to another. How fast will it spread? Who gets infected first? Can a mask mandate or school closure slow it down?

These are the questions tackled by agent-based epidemic models (ABMs)—computational frameworks that simulate disease transmission by tracking individuals (agents) and their interactions. Among the simplest yet surprisingly powerful ABMs is the Well-Mixed Micro-Simulation model. Despite its name, “well-mixed” doesn’t mean chaotic; it refers to a foundational assumption: within certain social settings (like a classroom or office), every susceptible person has an equal chance of contacting every infectious person. This mirrors the classic “mass-action” principle from traditional compartmental models (like SIR), but with a crucial twist: individuals are explicitly represented, allowing for rich heterogeneity in behavior, immunity, and contact patterns.

In this article, we unpack the mechanics of the Well-Mixed Micro-Simulation model—its rules, assumptions, strengths, and modern extensions—making it a cornerstone of the Agent-based Epidemic Models Toolbox. Whether you’re a public health student, a data scientist, or just a curious reader with a science background, you’ll come away understanding how digital societies help us prepare for real-world outbreaks.

🧪 Model Description

At its core, the Well-Mixed Micro-Simulation model simulates an epidemic by advancing time in discrete steps (usually daily) and updating the health state of each agent based on probabilistic rules. Agents progress through disease stages following a SEIR-like structure:

S: Susceptible
E: Exposed (infected but not yet infectious)
I: Infectious
R: Recovered (or removed)

Unlike differential-equation-based SEIR models that track fractions of populations, this model tracks individuals. Each agent carries personal timers for incubation and infectious periods, drawn from statistical distributions to reflect biological reality.

🔢 Core Transmission Mechanism

Each day, every susceptible agent j interacts with a set of contacts Cⱼ(t)—a union of contacts from home, work/school, and community settings. Among these, only those who are infectious (I) can transmit the disease.

For each infectious contact i ∈ Cⱼ(t) ∩ I, the per-contact transmission probability is:

pᵢⱼ(t) = 1 − exp(−β · g(τᵢ))

Where:

β (beta) is the baseline per-contact transmissibility scale (unitless, typically 0.01–0.3).
τᵢ is the time since agent i became infectious (in days).
g(τ) is the infectiousness curve—a function describing how contagious an individual is over time. Often modeled as a piecewise bell-shaped curve (e.g., gamma or log-normal distribution), peaking around symptom onset.

The total daily infection probability for susceptible j is then:

Pⱼ(t) = 1 − ∏ᵢ∈Cⱼ(t)∩I (1 − pᵢⱼ(t))

This equation captures the idea that infection can occur via any infectious contact—probabilities combine multiplicatively for independent events. If Pⱼ(t) exceeds a random draw from a uniform distribution, agent j transitions to the Exposed (E) state.

⏳ Disease Progression

Once exposed, each agent i draws two key durations from empirical or assumed distributions:

Tᴱᵢ: Incubation period (time from exposure to becoming infectious)
Tᴵᵢ: Infectious period (time during which they can transmit the virus)

These are often modeled as log-normal or gamma distributions to reflect real-world variability. For example, for SARS-CoV-2:

Mean Tᴱ ≈ 5.0 days (SD ≈ 1.5)
Mean Tᴵ ≈ 7.0 days (SD ≈ 2.0)

Agents automatically progress:
S → E (after exposure) → I (after Tᴱ) → R (after Tᴵ)

No recovery is assumed during the infectious period—once R, agents are immune for the simulation duration.

📊 Key Parameter Definitions & Typical Values


N	Population size	Total number of agents	1,000 – 10⁷
β	Transmissibility scale	Baseline infection risk per contact	0.01 – 0.3
g(τ)	Infectiousness curve	Relative infectiousness at time τ post-infection	Peaks at τ ≈ 2–3 days
⟨Tᴱ⟩	Mean incubation period	Average time from exposure to infectiousness	4–7 days (e.g., flu: 2d, measles: 10d)
σᵀᴱ	Incubation SD	Variability in incubation	1–2 days
⟨Tᴵ⟩	Mean infectious period	Duration of infectiousness	3–10 days
σᵀᴵ	Infectious SD	Variability in infectious duration	1–3 days
cₕ, cᵥ, c꜀	Daily contacts	Avg. contacts per day in home (ₕ), work/school (ᵥ), community (꜀)	Home: 2–5; Work: 5–20; Community: 5–15

💡 Note: Contact numbers are setting-specific. A teacher may have cᵥ = 30, while a remote worker has cᵥ = 2. This heterogeneity is a key strength of micro-simulations.

⚖️ Assumptions and Applicability

The Well-Mixed Micro-Simulation rests on several simplifying—but often reasonable—assumptions:

✅ Well-mixed within settings: In each setting (e.g., a classroom), every susceptible has equal contact probability with every infectious individual. This avoids modeling explicit networks but captures average mixing.

✅ Homogeneous mixing across time: Daily contacts are redrawn independently (no persistent friendships), though this can be relaxed in variants.

✅ No asymptomatic transmission: All infectious agents are symptomatic and equally likely to transmit (unless modified).

✅ Fixed population: No births, deaths (other than disease), or migration during simulation.

🎯 When is this model appropriate?

Early outbreak analysis where detailed contact tracing data is unavailable.
Policy screening: Testing interventions like school closures or mask mandates across heterogeneous populations.
Diseases with clear SEIR progression (e.g., influenza, measles, SARS-CoV-2).
Scenarios where individual variation matters (e.g., age-specific susceptibility, superspreading).

It is less suitable for diseases requiring explicit spatial structure (e.g., vector-borne illnesses) or where long-term immunity wanes significantly within the simulation window.

🔧 Model Extensions and Variants

While the base model is elegant, real-world complexity demands enhancements. Below are key variants integrated into modern toolboxes:

1. Asymptomatic & Presymptomatic Transmission

Introduce a fraction fₐ of infections that are asymptomatic (never show symptoms) but still infectious. Modify the infectiousness curve:

gₐ(τ) = α · g(τ)

where α (0 < α ≤ 1) scales infectiousness for asymptomatic cases (e.g., α = 0.75 for SARS-CoV-2 [1]).

2. Age-Structured Susceptibility

Assign age-specific susceptibility sₐ (e.g., children more susceptible to RSV). Then:

pᵢⱼ(t) = sₐ₍ⱼ₎ · [1 − exp(−β · g(τᵢ))]

3. Intervention Modules

Mask-wearing: Reduce β by factor m (e.g., m = 0.5 for cloth masks [2]).
Vaccination: Move agents directly from S → R with efficacy ε.
Case isolation: Remove infectious agents from community contacts after symptom onset.

4. Stochastic Re-seeding

To model imported cases, add external infection pressure:

Pⱼ⁽ᵉˣᵗ⁾(t) = 1 − exp(−λ(t))

where λ(t) is the daily importation rate (e.g., from travel data).

5. Waning Immunity

Allow recovered agents to return to S after time Tᵂ (drawn from distribution):

R → S after Tᵂ days

enabling modeling of seasonal reinfection (e.g., coronaviruses [3]).

These extensions preserve the micro-simulation’s granularity while adapting to specific pathogens or policy questions.

🎉 Conclusion

The Well-Mixed Micro-Simulation model strikes a sweet spot between simplicity and realism. By representing individuals—not just population averages—it captures the stochastic sparks that ignite epidemics and the heterogeneity that shapes their spread. While it assumes “well-mixed” settings, this is often a pragmatic and surprisingly accurate approximation for many respiratory diseases.

As computational power grows and data becomes richer (e.g., from mobile phones or surveys), these models evolve—incorporating mobility, behavior change, and genomic surveillance. Yet the core idea remains: to understand a pandemic, sometimes you need to simulate a society, one digital citizen at a time.

So next time you hear about “computer models” guiding public health decisions, remember: behind those curves and projections, there’s a virtual world where millions of tiny agents are living, interacting, and helping us stay one step ahead of the next outbreak.

📚 Key References

He, X. et al. (2020). Temporal dynamics in viral shedding and transmissibility of COVID-19. Nature Medicine. https://doi.org/10.1038/s41591-020-0869-5
Chu, D.K. et al. (2020). Physical distancing, face masks, and eye protection to prevent person-to-person transmission of SARS-CoV-2. The Lancet. https://doi.org/10.1016/S0140-6736(20)31142-9
Edridge, A.W.D. et al. (2020). Seasonal coronavirus protective immunity is short-lasting. Nature Medicine. https://doi.org/10.1038/s41591-020-1083-1
Ferguson, N.M. et al. (2005). Strategies for containing an emerging influenza pandemic in Southeast Asia. Nature. https://doi.org/10.1038/nature04017
Eubank, S. et al. (2004). Modelling disease outbreaks in realistic urban social networks. Nature. https://doi.org/10.1038/nature02541
Ajelli, M. et al. (2010). Comparing large-scale computational approaches to epidemic modeling. BMC Infectious Diseases. https://doi.org/10.1186/1471-2334-10-190
Kerr, C.C. et al. (2021). Controlling COVID-19 via test, trace, and isolate. Nature Communications.
Willem, L. et al. (2017). The impact of contact structure on infectious disease control. Epidemics. https://doi.org/10.1016/j.epidem.2017.02.012
Kretzschmar, M. et al. (2020). Impact of delays on effectiveness of contact tracing for COVID-19. The Lancet Public Health.
Glass, K. et al. (2022). Agent-based models for infectious disease policy. Philosophical Transactions B. https://doi.org/10.1098/rstb.2021.0030