Sequential Monte Carlo algorithms for agent-based models of disease transmission

Sequential Monte Carlo algorithms for
agent-based models of disease transmission
Jeremy Heng
ESSEC Business School
Joint work with Phyllis Ju and Pierre Jacob (Harvard)
Probability and Statistics Seminar
University of Kansas, Department of Mathematics
7 April 2021
JH Agent-based models 1/ 39

Outline
1 Agent-based models
2 Agent-based SIS model
3 Sequential Monte Carlo

Agent-based models
• Agent-based models specify how a population of agents
interact and evolve over time
• Flexible, interpretable and widely employed in many fields
• Can render realistic macroscopic phenomena from simple
microscopic rules
Figure: SimCity by Electronic Arts

Software for agent-based models

Calibration of agent-based models
• These models are typically calibrated by matching key
features of simulated and actual data
• Can be computationally intensive and difficult to calibrate
126 CHAPTER 5
Figure 5.3. Simulated and historical settlement patterns, in red, for Long House
Valley in A.D. 1125. North is to the top of the page.
of the 1270–1450 period could have supported a reduced but substantial
population in small settlements dispersed across suitable farming habitats
located primarily in areas of high potential crop production in the
Figure: Simulated and historical settlement patterns in long house valley

Statistical inference for agent-based models
• Given occasional noisy measurements of the population, we
could consider statistical inference for such models
• Few works have addressed this important topic as
likelihood-based inference is computationally challenging
• We propose sequential Monte Carlo algorithms for some
classical agent-based models in epidemiology
• The general principle is to ‘open the black box’ nature of
these models and exploit its inherent structure

Compartmental models in epidemiology
• A population-level approach assigns the population to
compartments and models the number of people in each
compartment over time
SIR model
Susceptible
Infected
Recovered
λ
γ
SIS model
Susceptible
Infected
λ
γ

Agent-based models in epidemiology
• The agent-based approach assumes agents can take these
states and models the state of each agent n over time
SIR model
Susceptible
Infected
Recovered
λn
γn
SIS model
Susceptible
Infected
λn
γn

Why agent-based models?
• May be unrealistic to assume all agents interact
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Figure: Fully connected network versus small world network

Why agent-based models?
• May be unrealistic to assume agents are interchangeable
0.0
0.1
0.2
0.3
0.4
0.5
0 10 20 30
Incubation period
Density
(days)
Gender Men Women
0.0
0.1
0.2
0.3
0 10 20 30
Incubation period
Density
(days) Age <50 >=50
Figure: Gender-specific (left) and age-specific (right) distributions of
COVID-19 incubation period (Zhao et al. 2020, AoAS)

Outline

Agent-based SIS model
• We consider the agent-based SIS model and encode
Susceptible = 0 and Infected = 1
• Let Xt = (Xn
t )n∈[1:N] ∈ {0, 1}N denote the state of a closed
population of N agents at time t ∈ [0 : T]
• Initialization X0 ∼ µθ given by
Xn
0 ∼ Ber(αn
0), independently for n ∈ [1 : N]
• Markov transition Xt ∼ fθ(·|Xt−1) at time t ∈ [1 : T] is
given by
Xn
t ∼ Ber(αn
(Xt−1)), independently for n ∈ [1 : N]

• Transition probability specified as
αn
(Xt−1) =
(
λnD(n)−1
P
m∈N(n) Xm
t−1, if Xn
t−1 = 0
1 − γn, if Xn
t−1 = 1
• Interactions specified by an undirected network: D(n) and
N(n) denote the degree and neighbours of agent n
• Infection and recovery rates are modelled using
agent-specific attributes
λn
= (1 + exp(−β>
λ wn
))−1
, γn
= (1 + exp(−β>
γ wn
))−1
,
where βλ, βγ ∈ Rd are parameters and wn ∈ Rd are the
covariates of agent n (similarly αn
0 depends on β0)

• If the network is fully connected D(n) = N, N(n) = [1 : N]
and the agents are homogeneous λn = λ, γn = γ
• We recover the classical SIS model of Kermack and
McKendrick (1927), which has a deterministic limit as
N → ∞
• These simpler models offer dimension reduction which
facilitates inference
• However, one cannot incorporate network information and
agent attributes
• We will use these simplifications to construct efficient SMC
proposal distributions for the agent-based model

• Observations (Yt)t∈[0:T] are the number of infections
reported over time
• Modelled as conditionally independent given (Xt)t∈[0:T], and
Yt ∼ gθ(·|Xt) = Bin(I(Xt), ρ)
• I(Xt) =
PN
n=1 Xn
t is the number of infections and ρ ∈ (0, 1) is
the reporting rate
• Parameters to be inferred θ = (β0, βλ, βγ, ρ)

Graphical model representation
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3
1
3
2
3
3
3
4
1
4
2
4
3
4
ρ
βγ
βλ
β0
Figure: T = 4 time steps, a fully connected network with N = 3 agents

Likelihood of agent-based SIS model
• We have a standard hidden Markov model
pθ(x0:T , y0:T ) = µθ(x0)
T
Y
t=1
fθ(xt|xt−1)
T
Y
t=0
gθ(yt|xt)
• The marginal likelihood is
pθ(y0:T ) =
X
x0:T ∈{0,1}N×(T+1)
pθ(x0:T , y0:T ),
• Maximum likelihood estimation computes
arg max
θ
pθ(y0:T )
• Bayesian inference samples from
p(θ|y0:T ) ∝ p(θ)pθ(y0:T )

Likelihood of agent-based SIS model
• We have a hidden Markov model on a discrete state-space
• We can employ forward algorithm to compute the marginal
likelihood exactly
• The cost is of order
(no. of states)2
× (no. of observations) = O(22N
T)
• For large N, we have to rely on sequential Monte Carlo
(SMC) methods to approximate the marginal likelihood

Outline

Sequential Monte Carlo
• Sequential Monte Carlo (SMC) methods, aka particle
filters, are now quite advanced and well-understood since its
introduction in the 90s
• The idea is to recursively simulate an interacting particle
system of size P
• For time t ∈ [0 : T], we have P states and ancestor indexes
(X
(1)
t , . . . , X
(P)
t ), (A
(1)
t , . . . , A
(P)
t )

…
X0 X1 XT
Y0 Y1 YT
…
✓
For time t = 0 and particle p ∈ [1 : P]
sample X
(p)
0 ∼ q0(x0|θ)

…
X0 X1 XT
Y0 Y1 YT
…
✓
weight W
(p)
0 ∝ w0(X
(p)
0 )

…
X0 X1 XT
Y0 Y1 YT
…
✓
X
X
X
sample ancestor A
(p)
0 ∼ R

W
(1)
0 , . . . , W
(P)
0

, resampled particle: X
A
(p)
0
0

…
X0 X1 XT
Y0 Y1 YT
…
✓
X
X
X
sample X
(p)
1 ∼ q1(x1|X
A
(p)
0
0 , θ)

…
X0 X1 XT
Y0 Y1 YT
…
✓
X
X
X
weight W n
1 ∝ w1(X
A
(p)
0
0 , X
(p)
1 )

…
X0 X1 XT
Y0 Y1 YT
…
✓
X
X
X X
sample ancestor A
(p)
1 ∼ R

W
(1)
1 , . . . , W
(P)
1

, resampled particle: X
A
(p)
1
1

…
X0 X1 XT
Y0 Y1 YT
…
✓
X
X
X X
X
X
X
Repeat for time t ∈ [2 : T].

…
X0 X1 XT
Y0 Y1 YT
…
✓
X
X
X X
X
X
X
Repeat for time t ∈ [2 : T]. Note this is for a given θ!

Likelihood estimation
• Weight functions (wt)t∈[0:T] and proposals distributions
(qt)t∈[0:T] have to satisfy
w0(x0)
T
Y
t=1
wt(xt−1, xt) =
pθ(x0:T , y0:T )
q(x0:T |θ)
where q(x0:T |θ) = q0(x0|θ)
QT
t=1 qt(xt|xt−1, θ)
• We can compute a marginal likelihood estimator
p̂θ(y0:T ) =



1
P
P
X
p=1
w0(X
(p)
0 )






T
Y
t=1
1
P
P
X
p=1
wt(X
(A
(p)
t−1)
t−1 , X
(p)
t )



• Unbiasedness and consistency as P → ∞ follow from Del
Moral (2004)

Bootstrap particle filter
• The bootstrap particle filter (BPF) of Gordon et al. (1993)
employs the proposal distributions
q0(x0|θ) = µθ(x0), qt(xt|xt−1, θ) = fθ(xt|xt−1)
and weight functions
wt(xt) = gθ(yt|xt)
• BPF can be readily implemented as simulating the latent
process is straightforward
• However, it suffers from curse of dimensionality for large N
– need large P to control variance of p̂θ(y0:T )
– p̂θ(y0:T ) can collapse to zero

Likelihood estimation
• Efficiency of SMC crucially relies on the choice of proposal
distributions
• Poor performance of BPF is not surprising, as it does not use
any information from the observations
• We show how to implement the fully adapted auxiliary
particle filter that accounts for the next observation
• We propose a novel controlled SMC method that takes the
entire observation sequence into account

Auxiliary particle filter
• The auxiliary particle filter (APF) was introduced in Pitt
and Shephard (1999) and Carpenter et al. (1999)
• It employs the proposal distributions
q0(x0|θ) = pθ(x0|y0), qt(xt|xt−1, θ) = pθ(xt|xt−1, yt)
and weight functions
wt(xt−1) = pθ(yt|xt−1)
• Sampling from these proposals and evaluating these weights
are not always tractable

• The predictive likelihood is
pθ(yt|xt−1) =
X
xt ∈{0,1}N
fθ(xt|xt−1)gθ(yt|xt)

pθ(yt|xt−1) =
X
xt ∈{0,1}N
=
X
xt ∈{0,1}N
N
Y
n=1
Ber(xn
t ; αn
(xt−1))Bin(yt; I(xt), ρ)

pθ(yt|xt−1) =
X
xt ∈{0,1}N
=
X
xt ∈{0,1}N
N
Y
n=1
Ber(xn
t ; αn
=
N
X
it =yt
PoiBin(it; αn
(xt−1))Bin(yt; it, ρ)
since the sum of independent Bernoulli with non-identical
success probabilities follows a Poisson binomial distribution

pθ(yt|xt−1) =
X
xt ∈{0,1}N
=
X
xt ∈{0,1}N
N
Y
n=1
Ber(xn
t ; αn
=
N
X
it =yt
PoiBin(it; αn
(xt−1))Bin(yt; it, ρ)
since the sum of independent Bernoulli with non-identical
success probabilities follows a Poisson binomial distribution
• Poisson binomial PMF costs O(N2) to compute (Chen and
Liu, 1997)

• To sample from pθ(xt|xt−1, yt), we augment It = I(Xt) as an
auxiliary variable
pθ(xt, it|xt−1, yt) = pθ(it|xt−1, yt)pθ(xt|xt−1, it)
• Conditional distribution of the number of infections is
pθ(it|xt−1, yt) =
PoiBin(it; αn(xt−1))Bin(yt; it, ρ)
pθ(yt|xt−1)
• Distribution of agent states conditioned on their sum is a
conditioned Bernoulli
pθ(xt|xt−1, it) = CondBer(xt; α(xt−1), it),
which costs O(N2) to sample (Chen and Liu, 1997)

• Hence the overall cost of APF is O(N2TP)
• We can reduce the cost to O(N log(N)TP) using two ideas
• Reduce cost of Poisson binomial PMF evaluation to O(N)
using translated Poisson approximation at a bias of
O(N−1/2) (Barbour and Ćekanavićius, 2002)
• Reduce cost of conditioned Bernoulli sampling to
O(N log(N)) using Markov chain Monte Carlo (Heng,
Jacob and Ju, 2020)

Controlled sequential Monte Carlo
• We introduce a novel implementation of the controlled SMC
(cSMC) proposed by Heng et al. (2020)
• The optimal proposal that gives a zero variance marginal
likelihood estimator is the smoothing distribution
pθ(x0:T |y0:T ) = pθ(x0|y0:T )
T
Y
t=1
pθ(xt|xt−1, yt:T )
• At time t ∈ [1 : T], the transition is
pθ(xt|xt−1, yt:T ) =
fθ(xt|xt−1)ψ?
t (xt)
fθ(ψ?
t |xt−1)
• ψ?
t (xt) = p(yt:T |xt) is the backward information filter (BIF)
and fθ(ψ?
t |xt−1) =
P
xt ∈{0,1}N fθ(xt|xt−1)ψ?
t (xt)

• BIF satisfies the backward recursion ψ?
T (xT ) = gθ(yT |xT ),
ψ?
t (xt) = gθ(yt|xt)fθ(ψ?
t+1|xt), t ∈ [0 : T − 1]
• This costs O(22NT) to compute, so approximations are
necessary when N is large
• Our approach is based on dimensionality reduction by
coarse-graining the agent-based model
• We approximate the model with heterogenous agents by a
model with homogenous agents whose individual infection and
recovery rates given by their population averages, i.e.
λn ≈ λ̄ = N−1
PN
n=1 λn and γn ≈ γ̄ = N−1
PN
n=1 γn

• BIF of the approximate model ψt(I(xt)) can be computed
exactly in O(N3T) cost, and approximately in O(N2T)
• We then define the SMC proposal transition as
qt(xt|xt−1, θ) =
fθ(xt|xt−1)ψt(I(xt))
fθ(ψt|xt−1)
,
and employ the weight function
wt(xt) =
gθ(yt|xt)fθ(ψt+1|xt)
ψt(I(xt))
• Sampling and weighting can be done in a similar manner as
APF

• Quality of proposals depend on the coarse-graining
approximation
• We establish a bound on the Kullback–Leibler divergence from
q(x0:T |θ) and pθ(x0:T |y0:T ) (Chatterjee and Diaconis, 2018)
• Finer-grained approximations can be obtained using clustering
of the infection and recovery rates, at the expense of
increased cost

Better performance with more information, but higher cost
method ● ● ●
BPF APF cSMC
0:
O( )
O( 2 )
O( 2 ) + O( 3 )
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25
50
75
100
0 25 50 75
time
ESS%
method ● ● ●
BPF APF cSMC
0:
Figure: Effective sample size for N = 100 fully connected agents and
T = 90 time steps

Informative observations
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original
outliers
0 25 50 75 90
0
25
50
75
100
0
25
50
75
100
time
ESS%
method ● ● ●
BPF APF cSMC
∈ {25, 50, 75} 2
Figure: Bottom panel: observations at t ∈ {25, 50, 75} are replaced by
min(2yt, N)

Marginal likelihood estimation
ˆθ( 0: ) θ
= 100, = 90 = 2
Figure: Variance of log p̂θ(y0:T ) at data generating parameter

Marginal likelihood estimation
ˆθ( 0: ) βλ != β#
λ
β!
λ = (−1, 2) βλ = (−3
= 90 = 2
Figure: Variance of log p̂θ(y0:T ) at a less likely parameter

Numerical illustration
• Estimated log-likelihood function as the number of
observations increases
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T = 10 T = 30 T = 90
−4 0 4 −4 0 4 −4 0 4
−4
0
4
βλ
1
β
λ
2
−100 −20 −10−5 0
log−likelihood
Figure: MLE (black dot) and DGP (red dot)

Numerical illustration
• Estimated log-likelihood function as the number of
observations increases
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T = 10 T = 30 T = 90
−4 0 4 −4 0 4 −4 0 4
−4
0
4
βλ
2
β
γ
2
−100 −20 −10−5 0
log−likelihood
Figure: MLE (black dot) and DGP (red dot)

Concluding remarks
• SMC methods can be readily deployed within particle
MCMC for parameter and state inference (Andrieu, Doucet
and Holenstein, 2010)
• We considered APF and cSMC for the agent-based SIR model
• A general alternative to SMC methods is MCMC algorithms
to sample from the smoothing distribution
• Preprint https://arxiv.org/abs/2101.12156
• R package https://github.com/nianqiaoju/agents
• Slides https://sites.google.com/view/jeremyheng/

References
C. Andrieu, A. Doucet, and R. Holenstein. Particle Markov chain Monte Carlo
methods. Journal of the Royal Statistical Society: Series B (Statistical
Methodology), 72(3):269–342, 2010.
A. Barbour and V. Ćekanavićius. Total variation asymptotics for sums of
independent integer random variables. The Annals of Probability,
30(2):509–545, 2002.
J. Carpenter, P. Clifford, and P. Fearnhead. Improved particle filter for nonlinear
problems. IEE Proceedings-Radar, Sonar and Navigation, 146(1):2–7, 1999.
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