Biological Attack Model
(BAM)
Formal Progress Report
April 5, 2007
Sponsor:
Dr. Yifan Liu
Team Members:
Richard Bornhorst
Robert Grillo
Deepak Janardhanan
Shubh Krishna
Kathryn Poole
Agenda
• Project Status
– Project Plan
– Work Breakdown
– Progress Tracking
• Model Discussion
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–
–
–
•
•
•
•
2
Model Status
Model Diagram & ODEs
Model Implementation
Input Parameters
Analysis Plan
Containment Strategies
Transmission Rate Decay
Effective Reproductive Number
Project Plan
Description
Detailed Design and Model
Development
Progress Presentation
Status Report # 2
Progress Discussion
Testing, Evaluation, and
Recommendations
Formal Progress Presentation
Final Report Drafting
Final Report Due
Presentation Preparation
Final Presentation
3
WEEK 7 WEEK 8 WEEK 9 WEEK WEEK WEEK WEEK WEEK WEEK
10
11
12
13
14
15
Work Breakdown
Project Task
Week 11
Units---->
RB
RG
DJ
Week 12
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
3
2
2
8
8
2
2
Status/Progress Brief Preparation
Develop a disease behavioral
model
3
3
8
8
8
Develop containment model
8
Testing and Evaluation of models
Evaluate the effectiveness of various emergency
response strategies.
·
2
Containment strategies,
5
·
Emergency response
procedures,
·
recommendations,
Final Report Drafting
4 TOTALS
Check
point
2
5
5
12
2
5
10
10
10
10
5
16
2
14
2
14
2
16
14
Progress Tracking
EV (Earned Value) – Technical/Schedule Performance
Total Manhours
Project Tracking
800
700
600
500
400
300
200
100
0
0
1
2
3
4
5
6
7
8
Weeks
5
Planned Hours
Actual Hours
EV
9
10 11 12 13 14 15
• 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
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
6
Model Diagram
B(t)
E (exposed)
S (susceptible)
RS(t)
QS(t)
QE(t)
C(t)
Q=Q1+Q2
Q1 (quarantined
non-symptomatic)
I (infectious)
QI(t)
DI(t)
MI(t)
RI(t)
DQ(t)
D (dead)
7
M (maimed)
QQ(t)
Q2 (quarantined
symptomatic)
MQ(t)
RQ1(t)
RQ2(t)
R (recovered)
Model ODEs
8
dS
 QS (t )  B (t )  RS (t )
dt
dE
 B (t )  C (t )  QE (t )
dt
dI
 C (t )  DI (t )  M I (t )  RI (t )  QI (t )
dt
dQ1
 QS (t )  QE (t )  RQ1 (t )  QQ (t )
dt
dQ2
 QI (t )  DQ (t )  M Q (t )  RQ 2 (t )  QQ (t )
dt
dD
 DI (t )  DQ (t )
dt
dM
 M I (t )  M Q(t )
dt
dR
 RI (t )  RQ1 (t )  R Q 2 (t )  RS (t )
dt
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
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–
–
–
9
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
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
10
Input Parameters
11
Parameter
Definition
Small Pox
Value
Ebola
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
4%
0.0 %
(No effective
treatment)
φ
treatment rate
γ
quarantine rate
3%
3%
μ1
incubation period
10
2 - 20
μ2
infectious period
12
8 - 12
Initial Values (deterministic model)
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
12
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
13
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
14
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)
t  t *
 0
 (t )  

 q ( t t *)
t  t *
  1  (  0   1 )e
β0 = transmission rate prior to precautionary measures
β1 = transmission rate after precautionary measures are in full effect
t* = time of the onset of the precautionary measures
q = decay rate
15
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
16
Questions
17