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 50 th 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/wi ki/file:danielbernoulli.jpg)
Mathematical Models of Infectious Disease Platform for synthesis of best available data, so can: Estimate key parameters (e.g., R 0 ) 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 ds/dt = -bsi di/dt = +bsi-i/d di/dt = +bsi-i/d-mi dr/dt = I/D S I R m (mortality)
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: R 0 & 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 growth peak decelerating growth final size.
Single Equation Approaches (2) E.g. Richards model, but many other forms. Provide information on likely final size, turning points, can estimate R 0 via exponnentiation. Scaled in epidemic time (serial intervals) rather than calendar/clock time.
Serial Interval: Measles DAYS 1 SERIAL INTERVAL Case 1 Case 1 Latency Infectious Case 2 Case 2 Latency Infectious
Serial Intervals: Tuberculosis YEARS 1 SERIAL INTERVAL Case 1 Case 1 Latency Infectious Case 2 Case 2 Latency Infectious
Serial Intervals (2) For a given generation (t) and a given R 0 (say, 3) number (n) of incident infections in that generation is: Generation Cases (n t-1 ) Cases (n t ) 0 --- 1 1 1 3 2 3 9 n = R 0 t
Richards Model I (t)=ri[1-(i/k) a ] R 0 = e rt
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
Estimated Reproductive Number 20. 3 1 Richards Model (Respiratory Visits) Richards Model (ILI Visits) SEIR Model (Respiratory Visits) SEIR Model (ILI Visits) Southern Ontario Estimate 7.4 2 2.7 1 10 1 1.5 2 2.5 3 3.5 4 4.5 Serial Interval (Days)
Difficulties with Richards and Other Logistic Models Non-intuitive. Assumptions about exponent of deviation. R 0 estimates seem high?
IDEA Model In the absence of intervention or immunity: I(t) = R 0 t But: intervention occurs, people become immune. Growth decelerates in an accelerating fashion! IDEA Model (Incidence Decay and Exponential Adjustment): I(t) = [R 0 /(1+d) t ] t
Algebraically: IDEA Model (2)
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, RR t ).
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-R 0 ) Therefore R 0 = -[(1/N)-1]
MERS Co-V, May 2013 http://pandemicinformationnews.blogspot.ca/2013/05/promed-thoughts-ontransmissibility-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.
Incident Cases Cumulative Cases 120 100 Observed Incident Cases Observed Cumulative Cases Modeled Cumulative Cases Modeled Incident Cases 600 500 80 400 60 300 40 200 20 100 0 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 0
Estimated R0 13 11 9 7 5 3 1 Reported Cases Total Cases Total Cases-Background Total Cases x 2 (50% Under-reporting) 3 5 7 9 11 13 15 Serial Intervals Utilized
Estimated d 1 0.8 0.6 0.4 0.2 (Figures 3a and 3b) Reported Cases Total Cases Total Cases-Background Total Cases x 2 (50% Under-reporting) 0 3 5 7 9 11 13 15 Serial Intervals Utilized
Incident Cases 120 100 80 60 40 20 0 0 5 10 15 Generation/Weeks Since March 1, 2014 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).
R 0 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?