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Dynamics of Contagion: Comparing Agent-Based and Differential Equation Models

Hazhir Rahmandad and John Sterman

MIT-Albany Colloquium

April 30, 2004

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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?

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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

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SEIR Epidemic Model: DE version

SusceptiblePopulation S

B

ExposedPopulation E

Depletion

InfectiousPopulation I

EmergenceRate ER

RecoveredPopulation R

RecoveryRate RR

AverageIncubation

Time e

-

++

AverageDuration of

Illness d

TotalInfectiousContacts

CContact Ratefor Exposed

Infectivity ofExposed Contact Rate

for Infectious

Infectivity ofInfectious

+

++

+

+

+

R

Contagion

R

Contagion

InfectionRate IR

++

-

TotalPopulation

N

-

IR = C(S/N) ER = E/e

RR = I/dC = cEiEE + cIiII

dSdt

= – IR, dEdt

= IR – ER, dIdt

= ER – RR, dRdt

= RR

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Translating SEIR into AB

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])

Susceptible S

B

Exposed E

Depletion

Symptomatic IEmergence

Rate ER

Infectivity ofExposed IES

Infectivity ofInfectious IIS

Infection Rate IR

Total InfectiousContacts TIC

+

R Contacts Rn

Contact Frequencyfor Healthy Cs

Relative ContactRisk for Infectious

RCI

<TIME STEP>

Contact ProbabilityNetwork CP

InfectiousContacts C

Contact RiskCR

Infection Risk IP

ContactNetwork NW

+

<Exposed E><Susceptible S>

<Recovered R>

<Relative ContactRisk for Exposed

RCE>

R

Contagion

R

Contagion

<Noise Seed>

<TIME STEP>

Relative Contact forRecovered RCR

Observed LinkPer Person K

Link ContactRate LCR

Tendency to Use Linksfor Individual TUL

Eff Link Num onContact Coeficient a

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AB SEIR Overview

• # of States: N*4 vs. 4, – N=200: Total # of variables and parameters: over 300000 vs. 35

Susceptible S

B

Exposed E

Depletion

Symptomatic IEmergence

Rate ER

Recovered R

RecoveryRate RR

AverageIncubation Time e

-

+ +

ExpectedDuration of Illness

d

Infectivity ofExposed IES

Infectivity ofInfectious IIS

InfectionRate IR

-

Total InfectiousContacts TIC

+

R Contacts Rn

Contact Frequencyfor Healthy AC

Relative ContactRisk for Infectious

RCI

R Recovery

Probability ofRecovery

++

<TIME STEP>

R Emergence

Probability ofEmergence

+

+

<TIME STEP>

+

+<TIME STEP>

InitialExposed E

One Day

<One Day>

Noise Seed

Contact ProbabilityNetwork

InfectiousContacts C

Contact RiskCR

InfectionRisk IP

<ContactNetwork NW>

Switch IndivHeterogeneity

+

<Exposed E><Susceptible

S><Recovered

R>

<Relative ContactRisk for Exposed

RCE>

B

Depletion

B

Depletion

R

Contagion

R

Contagion <Noise Seed>

<Noise Seed>

<TIME STEP>

<Exposed E>

<SymptomaticI>

Relative Contact forRecovered RCR

<Observed LinkPer Person>

Link ContactRate LCR

<Total ObservedNumber of Links>

ExpectedContact per DT

Tendency to UseLinks for Individual

TUL

Total RelativeContact for Links

TCL

<Noise Seed>Effect of Link Numberon Contact Rate ELN

Eff Link Num onContact Coeficient

EffectiveIndividual Contact

Rate

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Experimental Design

• AB SEIR Settings: 10 combinations (5*2)– 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)

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Networks: Random & Uniform

• Uniform: Everybody is connected to everybody else

• Random: There is a random network structure (same chance for all possible links)

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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

individualsLogaritmic Graph of Number of Links per Node

0.00001

0.0001

0.001

0.01

0.1

1

1 10 100 1000

Number of Links

Ob

serv

ed P

rob

abli

ty o

f L

ink

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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

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Heterogeneity

• Contact Rate[J,K]= • Low

– 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(0.25-1.75)

])[*][(

][*][*

KNJN

KTULJTULL

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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)

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A typical simulationPopulations

200

150

100

50

0

0 30 60 90 120 150 180 210 240 270 300

Time (Day)

Susceptible

ExposedInfectious

Recovered

Tp

S0

S0-

S

∞

F=

( S0-S

∞)/ S0

I max

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Overview: Uniform & Random

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Overview: Scale-Free and Small-world

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Overview: Lattice

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Results: Diffusion Fraction

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Results: Peak Time & Peak Value

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Results: Calibration Insights

• Very good fit: 0.97<R2<1.00

• Calibrated parameters absorb networks and heterogeneity effects

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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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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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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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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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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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Agenda

• AB and DE Models

• SEIR Model: DE and AB

• Study Design: – Networks, Heterogeneity, and Calibration

• Results– Overview, Three Metrics

• Other Considerations

• Conclusions and Lessons

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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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AB vs. DE: A continuum, not an oppositionExample: 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

Highly disaggregated

Typical AB model

Typical DE models

Agent model still aggregates lower-level entities

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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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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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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

SusceptiblePopulation S

B

ExposedPopulation E

Depletion

InfectiousPopulation I

EmergenceRate ER

RecoveredPopulation R

RecoveryRate RR

AverageIncubation

Time e

-

++

AverageDuration of

Illness d

TotalInfectiousContacts

CContact Ratefor Exposed

Infectivity ofExposed Contact Rate

for Infectious

Infectivity ofInfectious

+

++

+

+

+

R

Contagion

R

Contagion

InfectionRate IR

++

-

TotalPopulation

N

-

IR = C(S/N)

ER = E/e

RR = I/d

C = cEiEE + cIiII

dSdt

= – IR, dEdt

= IR – ER, dIdt

= ER – RR, dRdt

= RR

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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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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 timeExample 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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Example: SARS Cumulative Probable Cases, Taiwan

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SARS: Reported Cases, Taiwan

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SARS: Geographical Dist.

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Crossing Boundaries: A Simple Model

• Probability of Recovery= 1-(1-1/d)^One Day/TIME STEP

• Symptomatic(t)=E( )

• Learning Lessons: Unit Consistency and Independence from TIME STEP

i

i tcISymptomati )(

Symptomatic I Recovered R

Recovery RateRR

ExpectedDuration of

Illness d

-

R Recovery Probability ofRecovery

+

+

<TIME STEP>

+One Day

Symptomatic

Recovery

+

<ExpectedDuration ofIllness d>

-

Noise Seed

SD Model

Agent Based Model

DE Model

• Recovery= S/d