The Incidence Decay and Exponential

Adjustment (IDEA) model: a new single

equation model to describe epidemics

and their simultaneous control.

David N. Fisman, MD MPH FRCP(C) Professor, Dalla Lana School of Public Health, University of Toronto

McGill University Department of Epidemiology, Biostatistics and

Occupational Health 50th Anniversary Seminar Series

Montreal, PQ

October 27, 2014

Outline

• Single equation models to describe epidemic growth.

– Richards model (logistic growth).

– IDEA model.

• Case studies:

– MERS and Ebola

What Does Math Have to Do with Infectious Diseases?

• Communicable diseases: fundamental property is “transmission” (current cases produce future cases). – Contrast with non-communicable

diseases (e.g., cancer or diabetes). – When current cases produce on

average > 1 new case, have an exponential increase in case numbers, a.k.a., an “epidemic”.

• Daniel Bernoulli (1700-1782) applies math to smallpox control in Paris.

Daniel Bernoulli, Wikimedia Commons (http://en.wikipedia.org/wiki/File:Danielbernoulli.jpg)

Mathematical Models of Infectious Disease

• Platform for synthesis of best available data, so can: – Estimate key parameters (e.g., R0) by fitting

models to data.

– Manage uncertainty related to possible outcomes (but generally not predict the future).

– Perform stepwise “experiments” by varying parameters (e.g., identify importance of duration of immunity in pertussis).

– Platform for realistic CEA.

A Simple Schematic Model of an Infectious Disease

S I R

m (mortality)

dS/dt = -bSI

dI/dt = +bSI-I/D

dR/dt = I/D dI/dt = +bSI-I/D-mI

Figure 4. Effect of timing of epidemic peak on preferred vaccination strategy.

Tuite AR, Fisman DN, Kwong JC, Greer AL (2010) Optimal Pandemic Influenza Vaccine Allocation Strategies for the Canadian Population. PLoS ONE 5(5): e10520. doi:10.1371/journal.pone.0010520 http://www.plosone.org/article/info:doi/10.1371/journal.pone.0010520

Models in Acute Outbreak Settings

• Often want to characterize and model disease when novel pathogen emerges: – R0 & serial interval useful for disease control policy.

– When will it peak? When will it end? How big will it be?

– What are optimal intervention strategies?

• Recent examples: SARS, H1N1, MERS, Ebola, chikungunya, EV D68, others. – Though lots of missed opportunities (Lloyd Smith,

Science 2009).

“Low-Hanging Fruit” Modeling Human-Animal Interface

Lloyd-Smith J. et al., Science 2009

Challenges to Modeling Acute Outbreaks/Emergence

• Final size formula “incorrect” (because behavior changes and people intervene).

• Data hugging, non-transparency. • Poor quality data, aggregate data, missing data,

reporting delays. • Little information on subclinical cases, baseline

immunity/cross-immunity (e.g., H1N1). • Hard to parameterize compartmental models or

ABM without numerous assumptions re: immunity, mixing, biology, etc.

Single Equation Approaches

• Descriptive rather than mechanistic.

– Recognize epidemics as stereotyped logistic growth processes (Cum Inc): accelerating growthpeakdecelerating growthfinal size.

Single Equation Approaches (2)

• E.g. Richards model, but many other forms.

– Provide information on likely final size, turning points, can estimate R0 via exponnentiation.

– Scaled in “epidemic time” (serial intervals) rather than calendar/clock time.

Serial Interval: Measles

Case 1

Latency

Case 2

Infectious

Latency Infectious

Case 2

Case 1

1 SERIAL INTERVAL

DAYS

Serial Intervals: Tuberculosis

Case 1

Latency

Case 2

Infectious

Latency Infectious

Case 2

Case 1

1 SERIAL INTERVAL

YEARS

Serial Intervals (2)

• For a given generation (t) and a given R0 (say, 3) number (n) of incident infections in that generation is:

Generation Cases (nt-1) Cases (nt)

0 --- 1

1 1 3

2 3 9

n = R0t

Richards Model

I’(t)=rI[1-(I/K)a]

R0 = erT

Estimated impact of aggressive empirical antiviral treatment in containing an outbreak of pandemic influenza H1N1 in an isolated First Nations community

Influenza and Other Respiratory Viruses Volume 7, Issue 6, pages 1409-1415, 23 JUL 2013 DOI: 10.1111/irv.12141 http://onlinelibrary.wiley.com/doi/10.1111/irv.12141/full#irv12141-fig-0004

0

10

20

30

40

50

1 2 3 4 5

Est

ima

ted

Re

pro

du

ctiv

e N

um

be

r

Serial Interval (Days)

All Respiratory Visits

Influenza-Like IllnessVisits

Southern OntarioEstimate

0

1

2

3

1 1.5 2 2.5 3 3.5 4 4.5

Est

ima

ted

Re

pro

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ctiv

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be

r

Serial Interval (Days)

Richards Model (Respiratory Visits)

Richards Model (ILI Visits)

SEIR Model (Respiratory Visits)

SEIR Model (ILI Visits)

Southern Ontario Estimate

1

2.7

7.4

20.1

Difficulties with Richards and Other Logistic Models

• Non-intuitive.

• Assumptions about exponent of deviation.

• R0 estimates seem high?

IDEA Model

• In the absence of intervention or immunity:

I(t) = R0t

• But: intervention occurs, people become

immune. Growth decelerates in an

accelerating fashion!

• IDEA Model (Incidence Decay and

Exponential Adjustment):

I(t) = [R0/(1+d)t]t

IDEA Model (2)

• Algebraically:

IDEA Model (3)

• Integrating (eek):

where

IDEA Model (4)

• Evaluated by comparing with outputs of difference (discrete time) SIR model where epidemic is subject to intervention.

– IDEA agrees well with SIR where intervention efficacy accelerates (first order, RRt).

Figure 1. Model fits and “order of control”.

Fisman DN, Hauck TS, Tuite AR, Greer AL (2013) An IDEA for Short Term Outbreak Projection: Nearcasting Using the Basic Reproduction Number. PLoS ONE 8(12): e83622. doi:10.1371/journal.pone.0083622 http://www.plosone.org/article/info:doi/10.1371/journal.pone.0083622

Figure 2. IDEA model fits for low R0 epidemics.

Fisman DN, Hauck TS, Tuite AR, Greer AL (2013) An IDEA for Short Term Outbreak Projection: Nearcasting Using the Basic Reproduction Number. PLoS ONE 8(12): e83622. doi:10.1371/journal.pone.0083622 http://www.plosone.org/article/info:doi/10.1371/journal.pone.0083622

Figure 3. IDEA model fits for higher R0 epidemics.

Fisman DN, Hauck TS, Tuite AR, Greer AL (2013) An IDEA for Short Term Outbreak Projection: Nearcasting Using the Basic Reproduction Number. PLoS ONE 8(12): e83622. doi:10.1371/journal.pone.0083622 http://www.plosone.org/article/info:doi/10.1371/journal.pone.0083622

Figure 5. Pandemic H1N1 case counts modeled with the IDEA Model.

Fisman DN, Hauck TS, Tuite AR, Greer AL (2013) An IDEA for Short Term Outbreak Projection: Nearcasting Using the Basic Reproduction Number. PLoS ONE 8(12): e83622. doi:10.1371/journal.pone.0083622 http://www.plosone.org/article/info:doi/10.1371/journal.pone.0083622

MERS Coronavirus

• Novel coronavirus, identified in Middle East in 2012.

• Sporadic cases of respiratory illness, retrospective identification of Jordanian hospital outbreak (13 cases).

• Presumed zoonosis (camels?), foodborne?

– More transmissible in healthcare setting.

– “Middle Eastern SARS”?

Source: http://www.cdc.gov/coronavirus/mers/

Branching Process

• When R0 < 1, average cluster size (including the index) is:

N = 1/(1-R0)

Therefore

R0 = -[(1/N)-1]

MERS Co-V, May 2013

http://pandemicinformationnews.blogspot.ca/2013/05/promed-thoughts-on-transmissibility-and.html

Breban et al., Lancet 2013

Fisman, Lipsitch and Leung, Lancet 2014

MERS, Saudi Arabia (June 6, 2014)

Healthcare focused outbreaks in Jeddah and Riyadh.

0

20

40

60

80

100

120

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20

Weeks Since March 1, 2014

Incid

en

t C

ases

0

100

200

300

400

500

600

Cu

mu

lati

ve C

ases

Observed Incident Cases

Observed Cumulative Cases

Modeled Cumulative Cases

Modeled Incident Cases

1

3

5

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11

13

3 5 7 9 11 13 15

Serial Intervals Utilized

Es

tim

ate

d R

0

Reported Cases

Total Cases

Total Cases-Background

Total Cases x 2 (50% Under-reporting)

0

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0.4

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0.8

1

3 5 7 9 11 13 15

Serial Intervals Utilized

Es

tim

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d d

Reported Cases

Total Cases

Total Cases-Background

Total Cases x 2 (50% Under-reporting)

(Figures 3a and 3b)

0

20

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0 5 10 15

Generation/Weeks Since March 1, 2014

Incid

en

t C

ases

Model (4 week fit)

Model (6 week fit)

Model (7 week fit)

Model (8 week fit)

Model (10 week fit)

Model (12 week fit)

Total Observed Cases

APPLICATION TO EBOLA, 2014 [PLOS CURRENTS OUTBREAKS, SEPTEMBER 8, 2014]

Model Fitting

• Used WHO time series (cumulative cases, cumulative deaths) available at http://virologydownunder.blogspot.com.au and https://github.com/cmrivers/ebola (Ian Mackay and Caitlin Rivers). – Publication based on cases to August 22, 2014. – Collapsed by generation, base case used data

aggregated across Liberia, SL, Guinea. – Assume initial recognition occurred in generation 5

(back estimated based on 40-80 cases in March 2014, with previously reported R0 ~ 1.5 or 2).

R0 and d, by Generation

[Source: Fisman et al, Plos Currents Outbreaks 2014]

Best-fit Parameters

[Source: Fisman et al, Plos Currents Outbreaks 2014]

[Source: Fisman et al, Plos Currents Outbreaks 2014]

[Source: Fisman et al, Plos Currents Outbreaks 2014]

Projection, and Effect of Intervention (d = 0.014)

[Source: Fisman et al, Plos Currents Outbreaks 2014]

Projections, October 2014

Hypothetical Vaccine (Re = 0.9)

Summary

• Modeling emerging pathogens in outbreak settings may provide important information for disease control.

• Publicly available data sources, aggregate data can still be utilized to identify plausible ranges of disease parameters.

– May be able to identify peak in near-real time with IDEA (though hard to project).

Summary (Ebola)

• IDEA appears to describe and predict Ebola epidemic behavior reasonably well.

– Fairly robust with varying assumptions about serial interval, start date, undercounting.

– Should identify shift in epidemic dynamics if one occurs (not seeing that with Ebola).

– Simple (can do this in a spreadsheet).

• Seems intuitive (per students).

• Use in the field?