Comparing Compartment and Agent-based Models Shannon Gallagher JSM - - PowerPoint PPT Presentation

Comparing Compartment and Agent-based Models

Shannon Gallagher JSM Baltimore, MD August 2, 2017

Thesis work with: William F. Eddy (Chair) Joel Greenhouse Howard Seltman Cosma Shalizi Samuel L. Ventura

Goal: Combine two good models into a better one

1

Studying infectious disease is important

2

Compartment vs. Agent-based Models

Compartment models (CMs) describe how individuals evolve over time

Assumptions (Anderson and May 1992) :

• 1. Homogeneity of individuals
• 2. Law of mass action

I t 1 I t

4

Compartment models (CMs) describe how individuals evolve over time

Assumptions (Anderson and May 1992) :

• 1. Homogeneity of individuals
• 2. Law of mass action

I(t + 1) ∝ I(t)

4

Agent-based models (AMs) simulate the spread of disease

Assumptions (Helbing 2002):

• 1. Heterogeneity of agents
• 2. Model adequately reflects reality

5

Agent-based models (AMs) simulate the spread of disease

Assumptions (Helbing 2002):

• 1. Heterogeneity of agents
• 2. Model adequately reflects reality

5

CMs and AMs: a side by side comparison

CMs

∙ Equation-based ∙ Computationally fast ∙ Homogeneous individuals ∙ No individual properties

AMs

∙ Simulation-based ∙ Computationally slow ∙ Heterogeneous individuals ∙ Individual properties

6

Combining the two together

(Bobashev 2007, Banos 2015, Wallentin 2017) ∙ ad hoc approaches ∙ perspective from non-statisticians

Goal: Create a statistically justified hybrid model

7

Combining the two together

(Bobashev 2007, Banos 2015, Wallentin 2017) ∙ ad hoc approaches ∙ perspective from non-statisticians

Goal: Create a statistically justified hybrid model

7

Current Work

There are two main avenues of improvement

• 1. Quantifying how similar CMs and AMs are
• 2. Speeding up AM run-time

9

The SIR model: a detailed look

(Kermack and McKendrick 1927)

    

dS dt

= − βSI

N dI dt

= βSI

N − γI dR dt

= γI ∙ β – rate of infection ∙ γ – rate of recovery ∙ N – total population size

10

The SIR model: a detailed look

(Kermack and McKendrick 1927)

    

∆S ∆t

= − βSI

N ∆I ∆t

= βSI

N − γI ∆R ∆t

= γI ∙ β – rate of infection ∙ γ – rate of recovery ∙ N – total population size

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Our stochastic CM approach

ˆ S(t + 1) = ˆ S(t) − st ˆ R(t + 1) = ˆ R(t) + rt ˆ I(t + 1) = N − ˆ S(t + 1) − ˆ R(t + 1), with st+1 ∼ Binomial ( ˆ S(t), βI(t) N ) rt+1 ∼ Binomial ( ˆ I(t), γ ) .

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Our stochastic AM approach

For an agent xn(t), n = 1, 2, . . . , N, the forward operator for t > 0 is xn(t + 1) =      xn(t) + Bernoulli (

βI(t) N

) if xn(t) = 1 xn(t) + Bernoulli (γ) if xn(t) = 2 xn(t)

• therwise

. where xn(t) = k, k ∈ {1, 2, 3} corresponds to state S, I, and R, respectively Let the aggregate total in each compartment be ˆ Xk(t) =

N

∑

n=1

I{xn(t) = k}

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The means overlap

25 50 75 100 25 50 75 100

Time % of Population

Type

S−CM I−CM R−CM S−AM I−AM R−AM 1000 agents; 5000 runs; β = 0.10; γ = 0.03

Mean Proportion of Compartment Values

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The distributions look the same

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These approaches are equivalent

Theorem Let the CM and AM be as previously described. Then for all t ∈ {1, 2, . . . , T}, ˆ S(t)

d

= ˆ XS(t) (1) ˆ I(t)

d

= ˆ XI(t) ˆ R(t)

d

= ˆ XR(t).

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These approaches are equivalent

Theorem Let the CM and AM be as previously described. Then for all t ∈ {1, 2, . . . , T}, ˆ S(t)

d

= ˆ XS(t) (1) ˆ I(t)

d

= ˆ XI(t) ˆ R(t)

d

= ˆ XR(t).

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We can compare CM/AM pairs and AM/AM pairs by fitting the underlying model

0.028 0.029 0.030 0.031 0.096 0.098 0.100 0.102

β γ

Simulation Type

CM AM 1000 agents; 5000 runs; β = 0.10; γ = 0.03

Fitted SIR parameters distribution

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AMs are appealing because they can be run multiple times

∙ Simulate an epidemic en masse! ∙ A run - same initial parameters, different random numbers ∙ Runs (L) are independent of one another = ⇒ parallelization ∙ Roughly, the variance of compartments ↓ when N, L ↑

Goal: Improve computation time without sacrificing statistical details

18

AMs are appealing because they can be run multiple times

∙ Simulate an epidemic en masse! ∙ A run - same initial parameters, different random numbers ∙ Runs (L) are independent of one another = ⇒ parallelization ∙ Roughly, the variance of compartments ↓ when N, L ↑

Goal: Improve computation time without sacrificing statistical details

18

There is a tradeoff between the number of agents and number of runs

6 8 10 12 14 25 50 75 100

Time V(S1) V(S2)

5000 runs; β = 0.10; γ=0.03; Model 1−1000 agents, Model 2−100 agents

Ratio of Variance of # Susceptibles

6 8 10 12 14 25 50 75 100

Time V(I1) V(I2)

5000 runs; β = 0.10; γ=0.03; Model 1−1000 agents, Model 2−100 agents

Ratio of Variance of # Infected

6 8 10 12 14 25 50 75 100

Time V(R1) V(R2)

5000 runs; β = 0.10; γ=0.03; Model 1−1000 agents, Model 2−100 agents

Ratio of Variance of # Recovered

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The calculations show that the variance scales

∙ Note that for a given β and γ, if S1(0)

N1

= S2(0)

N2

= ⇒

S1(t) N1

= S2(t)

N2

∙ V S t 1 S t 1 pt pt 1 pt 2V S t ∙ V S2 t

N2 N1 V S1 t

V

1 L1 runs S1 t N1

V

1 L2 runs S2 t N2

L2N2

2

L1N2

1

V S1 t V S2 t L2N2 L1N1

We can replace agents with runs!

20

The calculations show that the variance scales

∙ Note that for a given β and γ, if S1(0)

N1

= S2(0)

N2

= ⇒

S1(t) N1

= S2(t)

N2

∙ V [ ˆ S(t + 1) ] = S(t)(1 − pt)pt + (1 − pt)2V [ ˆ S(t) ] ∙ V S2 t

N2 N1 V S1 t

V

1 L1 runs S1 t N1

V

1 L2 runs S2 t N2

L2N2

2

L1N2

1

V S1 t V S2 t L2N2 L1N1

We can replace agents with runs!

20

The calculations show that the variance scales

∙ Note that for a given β and γ, if S1(0)

N1

= S2(0)

N2

= ⇒

S1(t) N1

= S2(t)

N2

∙ V [ ˆ S(t + 1) ] = S(t)(1 − pt)pt + (1 − pt)2V [ ˆ S(t) ] ∙ V[ˆ S2(t)] = N2

N1 V[ˆ

S1(t)] V

1 L1 runs S1 t N1

V

1 L2 runs S2 t N2

L2N2

2

L1N2

1

V S1 t V S2 t L2N2 L1N1

We can replace agents with runs!

20

The calculations show that the variance scales

∙ Note that for a given β and γ, if S1(0)

N1

= S2(0)

N2

= ⇒

S1(t) N1

= S2(t)

N2

∙ V [ ˆ S(t + 1) ] = S(t)(1 − pt)pt + (1 − pt)2V [ ˆ S(t) ] ∙ V[ˆ S2(t)] = N2

N1 V[ˆ

S1(t)] V [

1 L1

∑

runs ℓ ˆ S1(t) N1

] V [

1 L2

∑

runs ℓ ˆ S2(t) N2

] = L2N2

2

L1N2

1

· V[ˆ S1(t)] V[ˆ S2(t)] = L2N2 L1N1 .

We can replace agents with runs!

20

The calculations show that the variance scales

∙ Note that for a given β and γ, if S1(0)

N1

= S2(0)

N2

= ⇒

S1(t) N1

= S2(t)

N2

∙ V [ ˆ S(t + 1) ] = S(t)(1 − pt)pt + (1 − pt)2V [ ˆ S(t) ] ∙ V[ˆ S2(t)] = N2

N1 V[ˆ

S1(t)] V [

1 L1

∑

runs ℓ ˆ S1(t) N1

] V [

1 L2

∑

runs ℓ ˆ S2(t) N2

] = L2N2

2

L1N2

1

· V[ˆ S1(t)] V[ˆ S2(t)] = L2N2 L1N1 .

We can replace agents with runs!

20

Through paralellization, we can get a speed-up without losing statistical information

0e+00 2e−04 4e−04 6e−04 25 50 75 100

Time Variance of ∑

l

S ^(t) (NL)

Simulation

100 agents, 4 cores 400 agents, 1 core S(t) − % susceptible averaged over # of runs

Variance of S(t)

0e+00 2e−04 4e−04 6e−04 25 50 75 100

Time Variance of ∑

l

I ^(t) (NL)

Simulation

100 agents, 4 cores 400 agents, 1 core I(t) − % infected averaged over # of runs

Variance of I(t)

0e+00 2e−04 4e−04 6e−04 25 50 75 100

Time Variance of ∑

l

R ^(t) (NL)

Simulation

100 agents, 4 cores 400 agents, 1 core R(t) − % recovered averaged over # of runs

Variance of R(t)

Simulation 1 (100 agents, 4 cores, 100 times): 3:30 minutes Simulation 2 (400 agents, 1 core, 100 times): 4:05 minutes

21

Future work

There is more work to be done: short-term

∙ Implementation of current methods in FRED

∙ FRED - an open source, supported, flexible AM ∙ Incorporate different levels of homogeneity

• 1. Independent agents
• 2. Agents go to one other activity (school, work, neighborhood)
• 3. Multiple activities

∙ Compare CM and AM parameters empirically

∙ Empirically determine when different regions can be combined

23

Thank you!

Questions?

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