Biological Attack Model(BAM)

Formal Progress ReportApril 5, 2007

Sponsor: Dr. Yifan Liu

Team Members:Richard BornhorstRobert GrilloDeepak JanardhananShubh KrishnaKathryn Poole

2

Agenda

• Project Status– Project Plan– Work Breakdown– Progress Tracking

• Model Discussion– Model Status– Model Diagram & ODEs– Model Implementation– Input Parameters

• Analysis Plan• Containment Strategies• Transmission Rate Decay• Effective Reproductive Number

3

Project Plan

Description WEEK 7 WEEK 8 WEEK 9 WEEK 10

WEEK 11

WEEK 12

WEEK 13

WEEK 14

WEEK 15

Detailed Design and Model DevelopmentProgress PresentationStatus Report # 2Progress DiscussionTesting, Evaluation, and RecommendationsFormal Progress PresentationFinal Report DraftingFinal Report DuePresentation PreparationFinal Presentation

4

Work Breakdown

Project Task Week 11 Week 12

Units----> RB RG DJ SK KP RB RG DJ SK KP

Project Management 1         1        

Configuration Mangement 1         1        

Group Meetings 1.5 1.5 1.5 1.5 1.5 1.5 1.5 1.5 1.5 1.5

Online Discussions 0.5 0.5 0.5 0.5 0.5 0.5 0.5 0.5 0.5 0.5

Status/Progress Brief Preparation 3 2 2

Develop a disease behavioral model

3 3 8 8 8

Develop containment model

Testing and Evaluation of models     8 8 8

Evaluate the effectiveness of various emergency response strategies.

·        Containment strategies, 2 5      

·        Emergency response procedures,       2 2

·        Check point recommendations, 2     2  

Final Report Drafting 5 5       5 5 2 2 2

TOTALS 12 10 10 10 10 16 14 14 16 14

5

Progress Tracking

• 650 man-hours of work completed

• On schedule– Initial draft of final report

completed

– Model 95% complete

– MATLAB model 75% complete

– Evaluation and analysis will begin this week

EV (Earned Value) – Technical/Schedule Performance

Project Tracking

0100200300400500600700800

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15

Weeks

Tota

l Man

hour

s

Planned Hours

Actual Hours

EV

6

Model Status

• Ordinary Differential Equations completed– Eight states, eight ODEs

• Evaluating numerical methods for solving ODEs– Initial implementation was done with Forward Euler method

• Simplest numerical method

• Prone to the most error

– Current implementation is with the Runge-Kutta (fourth-order) Numerical Method

• Least error compared with other numerical methods

• Error Analysis– Currently testing the stability, convergence and error properties

of the two numeric methods

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Model Diagram

E (exposed) S (susceptible)

I (infectious)

Q1 (quarantined non-symptomatic)

Q2 (quarantined symptomatic)

D (dead)

R (recovered)

Q=Q1+Q2

B(t)

QS(t)

QE(t)

QI(t)

DI(t)

MQ(t)

RQ2(t)

RI(t)

RQ1(t)

C(t)

QQ(t)

M (maimed)

MI(t)

DQ(t)

RS(t)

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Model ODEs

)()()( tRtBtQdt

dSSS

)()()( tQtCtBdt

dEE

)()()()()( tQtRtMtDtCdt

dIIIII

)()()()( 11 tQtRtQtQ

dt

dQQQES

)()()()()( 22 tQtRtMtDtQ

dt

dQQQQQI

)()( tDtDdt

dDQI

)()( tMtMdt

dMQI

)()()()( 21 tRtRtRtRdt

dRSQQI

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Model Implementation

• Model seeks solutions to ODEs as an Initial Value Problem • Parallel Implementation in Excel & Matlab

– Allows for a comparative study and sanity checks

• Excel implementation uses the Forward Euler method • Matlab uses the Runge-Kutta method

– ODE45 Solver (alternately ODE113 Solver)

• Work on Tolerances & Stability is in progress– To ascertain Local Formula Error and Round-off error and ultimately

estimate global error– To determine stiffness by varying time steps

• Next steps – Complete adjustments based on current results– Align with units of “Known” Input parameters– Code for “Controllable” Input parameters– Tracking and Cataloging of solver outputs for analysis & reporting

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Input Parameters

• “Known” input parameters – determined via research– Incubation period (generally given as a range)

• Deterministic model will use the mean

– Infection period (generally given as a range)• Deterministic model will use the mean

– Mortality rate– Disability rate

• Not readily available

• “Controllable” input parameters – modified as part of the containment analysis– Transmission rate

• Modify to represent the various containment strategies

– Close contacts identification rate– Quarantine rate– Treatment rate

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Input Parameters

Parameter DefinitionSmall Pox

ValueEbola Value

β transmission rate 2 0.025

α close contacts identification rate 5 0.8

d mortality rate of the disease 0.05 0.4 - 0.9

m disability rate of the disease 0.05 0.1

φ treatment rate 4 %

0.0 %

(No effective treatment)

γ quarantine rate 3% 3 %

μ1 incubation period 10 2 - 20

μ2 infectious period 12 8 - 12

Initial Values (deterministic model)

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Analysis Plan – Overview

• Initial analysis will focus on one disease (will expand to others if time permits)– Objective is to select a disease with ample, readily-available data

• Smallpox & possibly Ebola

– Research to determine realistic values for the input parameters is nearly complete

• Sensitivity analysis and parametric studies– Initial model is deterministic – sensitivity analysis will be used to

determine the impact of variations in the “known” input parameters• Incubation period, infectious period, mortality rate, and disability rate• Objective is to evaluate whether a deterministic approach is appropriate

– Sensitivity analysis may indicate that some parameters should be stochastic

– Parametric studies will be performed on the “controllable” parameters• Transmission rate, close contacts identification rate, quarantine rate, and

treatment rate• The variations in these parameters represent the control strategies that

BAM is going to evaluate

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Analysis Plan – Details

• Sensitivity analysis on the “known” input parameters– Incubation period, infectious period and mortality rate

• Available data provides a large range for some values– For example: Smallpox incubation period is 7-17 days with an average of 12-14

days• Will run multiple cases to determine how much the end result is impacted if

these parameters are varied from their mean– If the impact is “minimal” the mean will be used for the rest of the simulations

(will periodically review sensitivity analysis as the “controlled” parameters are varied)

– Disability rate – not a readily available number• Want to determine how much this parameter impacts the end results

• Parametric studies on the “controllable” input parameters– Parameter values will be selected to represent the control strategies– Objectives:

• Evaluate how modifications in the quarantine rate affect the total deaths and disabilities from the outbreak

• Compare the outcomes for mass vaccination vs. targeted vaccination strategies

– The results will be evaluated to determine the feasibility of quarantine and vaccination rates based on available resources

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Containment Strategies

• Six epidemic control strategies are being considered for analysis within BAM: – Quarantine/isolation– Voluntary confinement and movement restrictions– Ring vaccination– Targeted vaccination– Mass vaccination– Prophylactic vaccination

• Additional analysis is required to determine how to modify the input parameters to simulate the various control strategies– Research has provided initial values for the parameters of

interest

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Transmission Rate Decay

• Addresses the precautionary measures not accounted for in the state model– Includes voluntary confinement, use of protective equipment, and other

behavioral changes– β(t) = disease transmission rate as a function of time

• Prior to when precautionary measures are taken, the transmission rate is constant, β0  

• After time t*, β(t) will start to decay down to β1, at rate q, (β1< β0)

*)(

*)(

*)(101

0

tte

ttt

ttq

β0 = transmission rate prior to precautionary measures β1 = transmission rate after precautionary measures are in full effectt* = time of the onset of the precautionary measures q = decay rate

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Effective Reproductive Number• The effective reproductive number, Reff(t), measures the

average number of secondary cases per infectious case t time units after the introduction of the initial infections– Reff(t) is a common comparative parameter used in epidemic

modeling– N = population size

Reff(t) = β(t)*μ2*(S(t)/N)

• In a closed population, Reff(t) is non-increasing as the size of the susceptible population, S(t), decreases

• When Reff(t) ≤ 1, the threshold to eventual control of the outbreak is crossed

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Questions

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