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

Questions

1

What is required for Ebola containment?

What is required for Ebola containment?

Theory of Ebola transmission in human populations

Objectives of the study

• Provide better understanding of poorly understood aspects
• f 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 R0 and Reff
• 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

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16

Infection generation New cases

20 40 60

HCW Community−nursing Hospital leakage Funereal

Basic reproductive ratio (R0)

The mean process may be expressed in terms of a 2 × 2 or 4 × 4 mean matrix R0 is the dominant of two eigenvalues of this matrix:

Λ = 1 2 (Nq(1 − h + αβ)θ + (g − 1)(h − 1)φ + hλh ±

• ((Nqφ(h + αβ − 1) − (g − 1)(h − 1)φ)2 + hλh(2Nqφ(1 − h + αβ) + 2φ(g − 1)(h − 1) + hλh)))

0.0 0.2 0.4 0.6 0.8 1.0 0.0 0.2 0.4 0.6 0.8 1.0 Hospitalization rate Secure burial rate

1 2 3 1

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

Data

Weekly cases/Hospital capacity 100 300 500 06/20/14 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 09/19/14 09/26/14 20 50 100 200 500 1000 2000 Cumulative number of cases Hospital capacity (beds) Cumulative 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

Day Cases (Liberia) 06/09/14 06/29/14 07/19/14 08/08/14 08/28/14 09/17/14 10 20 50 100 200 500 1000 2000 5000 Model predictions (reported cases) Model predictions (total cases) Cumulative case reports

Conclusion: Fit model was consistent with observed data and a subtantial fraction of cases unreported.

Change in R0 with interventions

Effective reproduction number is evaluated at 15 day intervals (infection generations) over a set of plausible parameter values

Effective reproduction number 2014−07−04 2014−07−19 2014−08−03 2014−08−18 2014−09−02 2014−09−17 2014−10−02 2014−10−17 1.0 2.0 3.0 4.0

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)

Date New cases

Sep 02 Sep 17 Oct 02 Oct 17 Nov 01 Nov 16 Dec 01 Dec 16 Dec 31 1e+00 1e+02 1e+04 1e+06

1 Baseline.

Business-as-usual

2 Scenario A. DoD

commitment.

3 Scenario B. Improve

hospital capacity.

4 Scenario C. Improve

hospital capacity and compliance.

5 Scenario D. Improve

hospital capacity near-perfect compliance (best case).

Baseline

Projected epidemic size by end of 2014

Scenario A Scenario B Scenario C Scenario D 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

Day Persons seeking hospitalization

09/22/14 10/12/14 11/01/14 11/21/14 12/11/14 12/31/14

2000 6000 10000

x median

40 70 110 150 180

• 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

Day Persons seeking hospitalization

01/20/15 03/11/15 04/30/15 06/19/15

10 20 30 40 50 60 70

x median

10 30 40 60 70

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.