1. 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

2. 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. 3 Project Plan

4. 4 Work Breakdown Project TaskWeek 11Week 12 Units---->RBRGDJSKKPRBRGDJSKKP Project Management1 1 Configuration Mangement1 1 Group Meetings1.5 Online Discussions0.5 Status/Progress Brief Preparation322 Develop a disease behavioral model 33888 Develop containment model Testing and Evaluation of models 888 Evaluate the effectiveness of various emergency response strategies. · Containment strategies, 25 · Emergency response procedures, 22 · Check point recommendations,2 2 Final Report Drafting55 55222 TOTALS1210 1614 1614

5. 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

6. 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

7. 7 Model Diagram

8. 8 Model ODEs

9. 9 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

10. 10 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

11. 11 Input Parameters ParameterDefinition Small Pox Value Ebola Value βtransmission rate20.025 αclose contacts identification rate50.8 dmortality rate of the disease0.050.4 - 0.9 mdisability rate of the disease0.050.1 φtreatment rate 4 % 0.0 % (No effective treatment) γquarantine rate3% μ1μ1 incubation period102 - 20 μ2μ2 infectious period128 - 12 Initial Values (deterministic model)

12. 12 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

13. 13 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

14. 14 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

15. 15 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, β 0 After time t*, β(t) will start to decay down to β 1, at rate q, (β 1 < β 0 ) β 0 = transmission rate prior to precautionary measures β 1 = transmission rate after precautionary measures are in full effect t* = time of the onset of the precautionary measures q = decay rate

16. 16 Effective Reproductive Number The effective reproductive number, R eff (t), measures the average number of secondary cases per infectious case t time units after the introduction of the initial infections –R eff (t) is a common comparative parameter used in epidemic modeling –N = population size R eff (t) = β(t)*μ 2 *(S(t)/N) In a closed population, R eff (t) is non-increasing as the size of the susceptible population, S(t), decreases When R eff (t) ≤ 1, the threshold to eventual control of the outbreak is crossed

17. 17 Questions ?