A multi-type branching process model for the transmission of Ebola - PowerPoint PPT Presentation

A multi-type branching process model for the transmission of Ebola virus John M. Drake

What is required for Ebola Questions containment? What is required for Ebola containment? 1

Theory of Ebola transmission in human populations Objectives of the study • Provide better understanding of poorly understood aspects of epidemiology ( e.g. , under-reporting, compliance with policy, time-varying properties of the epidemic) • Enable counterfactual (“what if...”) investigation of alternative intervention scenarios • Forecast hospital demand until 31 December

Criteria for model specification • Emphasize the aspects of transmission we know to be important • Express the model as much as possible in terms of causal concepts and observable or measurable quantities • Capture key heterogeneities • Accommodate time-varying forces and interventions • Admit formulas for R 0 and R eff • Maximize “narrative parameterization” • Separate transmission from reporting

Proposal: Branching process model A branching process is a Markov process in which every individual in a population independently and probabilistically reproduces according to a specified probability distribution. Our model is: • Discrete-time (infection generations) • Multi-type

Model structure Key concepts: • Our model was constructed to represent the possible infection paths . Two equivalent interpretations differ with respect to whether individuals are classified by their source-of-infection or location-of-treatment . • Probability distributions are built from mixtures and convolutions. • In mathematical analysis, the location-of-treatment interpretation is preferred because it is two-dimensional. In numerical analysis, the source-of-infection interpretation is preferred because it better reflects intuition about disease progression and transmission.

Infection paths in a branching process model Key: C–Community, H–Hospital, CM–Community nursing, FNR–Funeral, VIS–Visitor, HCW–Health care worker

Realization HCW Community−nursing Hospital leakage 60 Funereal New cases 40 20 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 Infection generation

Basic reproductive ratio ( R 0 ) The mean process may be expressed in terms of a 2 × 2 or 4 × 4 mean matrix R 0 is the dominant of two eigenvalues of this matrix: 1 Λ = ( Nq ( 1 − h + αβ ) θ + ( g − 1 )( h − 1 ) φ + h λ h 2 � (( Nq φ ( h + αβ − 1 ) − ( g − 1 )( h − 1 ) φ ) 2 + h λ h ( 2 Nq φ ( 1 − h + αβ ) + 2 φ ( g − 1 )( h − 1 ) + h λ h ))) ± 1.0 1 1 0.8 2 Secure burial rate 0.6 0.4 3 0.2 0.0 0.0 0.2 0.4 0.6 0.8 1.0 Hospitalization rate

Model details Key model assumptions: • Outbreak is not self-limiting • Infection and removal do not substantially change the structure of local contact networks Obstacles to model fitting: • Model is a partially observed (latent variable) discrete time, time-dependent branching process – could be fit by MCMC, iterated filtering, etc. but these approaches take a lot of trouble-shooting, fine-tuning • Seek to avoid false impression of precision

Weekly cases/Hospital capacity 100 300 500 06/20/14 Cumulative cases Hospital capacity (beds) 06/27/14 07/04/14 07/11/14 07/18/14 07/25/14 08/01/14 08/08/14 08/15/14 08/22/14 08/29/14 09/05/14 09/12/14 Data 09/19/14 09/26/14 20 50 100 200 500 1000 2000 Cumulative number of cases

Method of plausible parameter sets We propose the method of plausible parameter sets . This method seeks only to find parameter combinations that could possibly have generated the observed data. Three pass tuning: • Fiddling • Least squares fitting initialized as the fiddled values • Latin hypercube search within a large neighborhood of the least squares fits

Fit during the tuning period Model predictions (reported cases) 5000 Model predictions (total cases) 2000 Cases (Liberia) 1000 500 200 100 50 20 Cumulative case reports 10 06/09/14 06/29/14 07/19/14 08/08/14 08/28/14 09/17/14 Day Conclusion: Fit model was consistent with observed data and a subtantial fraction of cases unreported.

Change in R 0 with interventions Effective reproduction number is evaluated at 15 day intervals (infection generations) over a set of plausible parameter values Effective reproduction number 4.0 3.0 2.0 1.0 2014−07−04 2014−07−19 2014−08−03 2014−08−18 2014−09−02 2014−09−17 2014−10−02 2014−10−17 These results allow that transmission may have become sub-critical as early as mid-August

Example Example solutions for the fit model (baseline scenario with no additional intervention) 1e+06 New cases 1e+04 1e+02 1e+00 Sep 02 Sep 17 Oct 02 Oct 17 Nov 01 Nov 16 Dec 01 Dec 16 Dec 31 Date

Projected epidemic size by end of 2014 Baseline 1 Baseline. Business-as-usual 2 Scenario A. DoD Scenario A commitment. 3 Scenario B. Improve Scenario B hospital capacity. 4 Scenario C. Improve hospital capacity and Scenario C compliance. 5 Scenario D. Improve hospital capacity Scenario D near-perfect compliance (best case). 1000 10000 100000 1000000 10000000

Forecasting hospital demand An interpolation scheme was devised to estimate daily hospital demand from infection generations that are projected at 15 day intervals Persons seeking hospitalization 180 150 median 10000 110 70 40 6000 x 2000 09/22/14 10/12/14 11/01/14 11/21/14 12/11/14 12/31/14 Day • This scenario – improved hospital capacity and 85% of cases seeking hospitalization – shows that epidemic response may be non-monotonic under realistic build out scenarios • The interaction between hospital capacity and transmission results in multiple epidemic “waves”

Update: fit hospitalization rate Persons seeking hospitalization 70 70 60 median 60 50 40 30 40 10 x 30 20 10 01/20/15 03/11/15 04/30/15 06/19/15 Day Drake, J.M. et al. 2015. Ebola cases and health system demand in Liberia. PLOS Biology 13:e1002056.

Conclusions Findings: • It mid-October it was still unclear whether or not the epidemic would be contained at planned level of intervention • Primary outcome of increased hospital capacity was to significantly reduce the change of a very massive epidemic • Containment achieved through the synergy of patient isolation, increased hospital capacity, and safe burial • Under present conditions, the epidemic appears to be on track for elimination in mid 2015 Needs for further research: • Probabilistic estimation and inference (maximum likelihood or Bayesian) • What role might be played by susceptible depletion, changing infection control practices, heterogeneity in population density, sexual transmission, and other factors?

Acknowledgements The UGA-MIDAS Ebola Modeling Working Group • Andrew Park • Reni Kaul • Matt Ferrari • Tom Pulliam • Drew Kramer • Suzanne O’Regan • Laura Alexander • JP Schmidt • Sarah Bowden • Pej Rohani • Sean Maher • David Hayman • David Yleah

Acknowledgements Paper: doi.org/10.1371/journal.pbio.1002056 Data & Code: doi.org/10.5061/dryad.17m5q Website: daphnia.ecology.uga.edu/midas Research supported by the National Institute Of General Medical Sciences of the National Institutes of Health under Award Number U01GM110744. The content is solely the responsibility of the authors and does not necessarily reflect the official views of the National Institutes of Health.