1. Dynamics of Contagion: Comparing Agent-Based and Differential Equation Models
Hazhir Rahmandad and John Sterman
MIT-Albany Colloquium
April 30, 2004
11/3/2003

2. Motivation
Agent Based (AB) models are widespread: e.g. Santa Fe, Wolfram’s A New Kind of Science
Many exciting applications, but lots of hype, not enough understanding of when AB adds value and when it is inappropriate
Question is not ‘which type of model is right?’: All models are wrong.
Question is
Which type of model is best suited for different purposes?
How robust are policy conclusions to modeling methods?
How can best attributes of both modeling paradigms be integrated?
As a field we need to be open to when other approaches are helpful and when the assumptions are OK.
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3. DE vs. AB: What are the differences?
Differences in typical assumptions:
Level of aggregation of similar elements
Treatment of Time
Continuous (solved numerically, results (should be) insensitive to time step or numerical integration method)
Discrete (time periods often undefined, can’t easily be varied)
Differences in typical practice
Modeling problems vs. modeling systems
Emphasis on stochastic elements
Software and representation
Giver the example of world population
These differences have prohibited more cross-boundary collaboration. Clarifying them and bridging the gaps is one of the contributions.
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4. SEIR Epidemic Model: DE version
Explain the main loops.
Explain the main equations.
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5. Translating SEIR into AB
Explain the contact rate first and go from there for a few steps.
C[J,k]=IF(S[J]*CP[J,K]*IP[K], CR[J,K]>Rn[J,K],1,0) CP[J,K]=LCR[J,K]*DT
IP[J]= E[J]*IES+I[J]*IIS CR[K]=S[K]+CE/CS*E[K]+CI/CS*I[K]+CR/CS*R[K]
LCR[J,K]=f(NW[J,K], CS, K, a, TUL[J]*TUL[K])
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6. AB SEIR Overview # of States: N*4 vs. 4,
N=200: Total # of variables and parameters: over vs. 35
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7. Experimental Design AB SEIR Settings: 10 combinations (5*2) N=200
Network Structure
Uniform, Random, Scale-Free, Small-world, Lattice
Heterogeneity
Low and High
N=200
Simulating each setting 1000 times
Comparing with Base DE and Calibrated DE on 3 measures of Diffusion Fraction (F), Peak Time (TP) and Peak Value (IMAX)
What are the main reasons that AB assumptions can make a difference: NW and Het + Stochasticity
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8. Networks: Random & Uniform
Uniform: Everybody is connected to everybody else
Random: There is a random network structure (same chance for all possible links)
Poisson distribution for the number of links an individual has
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9. Networks: Scale Free The number of links has a power law distribution
A few hubs with lots of links and a lot of poorly connected individuals
Barabasi and Albert
Examples in WWW, biological structures, less for humans but we can think of scenarios.
Because people move, it becomes more interesting when there are physical hubs (e.g. public places)
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10. Networks: Small-world & Lattice
Small world, with k expected links:
Expected links to neighbors with distance up to k/2: k*p
Connected to k/2-far neighbors with probability p
Expected long distance links: k*(1-p)
Connected to others with k*(1-p)/(N-k)
Lattice: No long distance link
Watts and Strogatz
Has several realistic features.
Can change depending on p, but we choose a reasonable and stable value
Lattice is more like tree’s etc
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11. Heterogeneity Contact Rate[J,K]= Low High
More link for individual (N) =>Proportionally less contact per link (α=1)
Fixed individual tendencies to use links (TUL[J]=1)
High
Contact per link independent of individual connectivity (α=0)
Uniform distribution of TUL ~U( )
L is an appropriate constant so that the total contacts are kept same for all networks.
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12. Calibration Optimized DE is more realistic than base DE
Best fitting DE model matching MEAN Infected in AB simulation
Optimize over
Infectivity of Exposed and Infectious (0<CE,CI)
Average Incubation Time (0<ε<30)
Average Duration of Illness (5<δ<30)
Real world cases calibrate models based on data to find the parameters, specially for fast response, e.g. SARS
There is some understanding of physical constraints, so the parameters are not unbounded.
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13. A typical simulation Populations Imax Susceptible Exposed Infectious
200
150
100 50 30 60 90
120
180
210
240
270
300
Time (Day)
Susceptible
Exposed
Infectious
Recovered Tp S0
S0-S∞
F=( S0-S∞)/ S0
Imax
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14. Overview: Uniform & Random
Blue, Green, white for 95, 75, and 50% bounds
Highlight slightly faster in heterogenous
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15. Overview: Scale-Free and Small-world
Heterogeneity can become important in case of SF, because of huge number of links
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16. Overview: Lattice
This is like burning a circled structure
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17. Results: Diffusion Fraction
DE Dif Frac: 0.984
Highlight the generally decreasing trend, except that there are more hermits in scale-free
The only statistically significant difference is lattice: because standard deviation is increasing.
Note the effect of heterogeneity.
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18. Results: Peak Time & Peak Value
DE Peak: 55 people, peak time: 49 days
Effect of heterogeneity is highlighted
What is significant: small-world and lattice
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19. Results: Calibration Insights
Very good fit: 0.97<R2<1.00
Calibrated parameters absorb networks and heterogeneity effects
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20. Results Summary Effect of Network small except lattice
Some Numerical, Little Behavioral Sensitivity
Clustering increases AB-DE gap
Network size decrease AB-DE gap
No gap with calibrated DE
Effect of heterogeneity small
Extreme: Disintegration into social and hermit (Scale-Free shows best)
The AIDS example
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21. AB vs. DE: Other Considerations
Data Availability
Extra Levers in AB Models
Complexity vs. Analyzability
Simulation Cost
Limits to Understanding
Purpose of Modeling and Cost of Error
More Feedback vs. Disaggregation
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22. Conclusions: Upsides of AB vs. DE
AB models offer additional insights when:
Sparse and locally connected networks
Capture “Non/low Diffusion” modes of behavior (important when low “contact number” (c*i*d) for epidemic)
Better tackle questions about effect of individual differences on overall behavior
Possibility of misleading parameter values in fitting curves to DE models
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23. Conclusions: Downsides of AB vs. DE
Data are rarely available to the detail needed for an AB model
Marginal precision improvement on complexity is usually low, expanding the boundaries may pay back better.
Analysis is very hard:
Structure-behavior connection hard to explain
Simulation cost can get prohibitive fast
Hard to make sense of so much data
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24. Process Insights
It is possible to build agent based models keeping up with good SD practice guidelines
Dimensional consistency
Independence from DT
Vensim software needs improvement to be used for AB models
Dealing with stochastic elements is not trivial!
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25. 11/3/2003

26. Agenda AB and DE Models SEIR Model: DE and AB Study Design: Results
Networks, Heterogeneity, and Calibration
Results
Overview, Three Metrics
Other Considerations
Conclusions and Lessons
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27. Policy recommendations might be affected by model type.
Example: Reducing risk of smallpox bioterror attack: What is the right vaccination strategy?
Kaplan, Craft & Wein (2002) use a differential equation model; conclude Mass Vaccination is superior
Halloran et al. (2002) use agent model, conclude Targeted Vaccination is superior
What accounts for difference? AB vs. DE method, or other assumptions?
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28. AB vs. DE: A continuum, not an opposition Example: modeling world population
Single stock
Disaggregated by age
Disaggregated by region, age
Disaggregated by country, age, gender, etc. …
Each person represented
People disaggregated into organs
Organs disaggregated into cells
Atoms
Quarks
Highly
aggregated
disaggregated
Typical DE models
Typical AB model
Agent model still
aggregates lower-
level entities
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29. Goals
What are the differences between AB and DE methods? When might it matter?
Modeling discipline: Learning across boundaries
Challenges of crossing the boundary
Learning opportunities for both communities
Example: The diffusion of an epidemic
AB: Value added under what conditions?
DE: What might it miss?
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30. Nonlinear differential equation paradigm:
dx/dt = f(x,u)
x vector of states; u, vector of exogenous inputs, including stochastic shocks; f() typically nonlinear
Typically in continuous time but difference equations also common
Finite number of compartments (elements of x)
No heterogeneity within a compartment. Heterogeneity added by enlarging number of compartments, e.g.:
Disaggregation by spatial structure:
World population P becomes population by country Pi
Disaggregation by attribute
People P become Pijk…, where, e.g., i, j, k = sex, age, health status, behavior, etc.).
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31. Example: SEIR Epidemic Model
4 compartments (S, E, I, R)
Perfect mixing within compartments
No heterogeneity in infectivity (within E, I) or in network structure of social contacts
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32. Agent-based paradigm:
Set A = {a1, … an} of agents, each agent has states xa
x can be e.g. health status, location, wealth, beliefs, decision rules, etc.
States xa change according to rules of interaction, e.g.,
Nearest neighbor (on lattice, torus, etc.) or other network structure;
Stochastic or deterministic.
Discrete time: xa(t) = Rule[xa(t-1)] for all a in {A}]
Heterogeneity across agents. Often, distribution of states across agents (often assigned randomly)
Aggregation:
Population is sum of agents; Number of people in each category (e.g., health status, gender) is sum of agents with those attributes each period.
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33. Example: Agent-Based Epidemic Model
Each person in one of 4 states (S, E, I, R)
Each person interacts (deterministically or stochastically) according to a specified network structure of social contacts (e.g., some people highly, others weakly, connected)
Probability of infection given contact can differ for each person (heterogeneous attributes of each agent)
Discrete time
Example Decision Rules:
If S, then become E if any of your contacts this period are in E or I state and if those contacts result in infection
If E, then become I e days after exposure
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34. Example: SARS Cumulative Probable Cases, Taiwan
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35. SARS: Reported Cases, Taiwan
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36. SARS: Geographical Dist.
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37. Crossing Boundaries: A Simple Model
DE Model
Recovery= S/d
Probability of Recovery= 1-(1-1/d)^One Day/TIME STEP
Symptomatic(t)=E( )
Learning Lessons: Unit Consistency and Independence from TIME STEP
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