Ebola: Nonparametric Survival Analysis Without Life Data October 23, 2014 P[Time from case report to death > k weeks] P[Death in week k|Survive for k-1 weeks] Regression analyses of case reports Case Fatality Ratios by country standard deviations Caseload Forecasts Why? • News reports: 70% die, need 262 beds in Guinea by Dec. 1, 1000 new cases per week by end of year, 200,000 –250,000 cases by Jan. 20,… – How long to death? CFR and distribution – How long to release? Empirical distribution [NEJM] – How many cases in treatment each week? • Forecast case reports and caseloads – e.g., Harris and Rattner, Jewell et al., Majumder • Compare countries? Treatments? SIR? Ro? Data • http://www.who.int/csr/disease/ebola/en – Cumulative case reports = confirmed + probable + suspected and death counts – “The total number of cases is subject to change due to reclassification, retrospective investigation, consolidation of cases and laboratory data, and enhanced surveillance. ” • Used weekly counts to smooth corrections • Days from hospitalization to release [http://www.nejm.org/doi/pdf/10.1056/NEJMoa 1411100 appendix] Methods • Regress case reports on time: linear, logarithmic, and piecewise linear • Nonparametric ccdf (survivor function) estimates of time from case report to death from WHO counts – Maximum likelihood assuming nonstationary Poisson case reports [George and Agrawal] – Constrained least squares [Harris and Rattner, Gang, George] • minS|observed weekly deaths  estimated weekly deaths|2 [Gang] • Subject to one or both of… – S observed total deaths = S estimated total deaths – P[Time to death ≤ Now  first case date]  deaths/cases – Constrained maximum entropy Sp(t)ln(p(t)) [Tribus] • Length-of-Stay (in hospital) empirical cdf conditional on recovery, from NEJM article appendix Case Reports • Linear and logarithmic regression case reports Y on T, days – Notice R2 values for alternative models? Y=bmT Y = mT+b m, R2 b, seY m, R2 b, seY Guinea 5.68 -111 1.0120 104.25 R2, seY 0.83 164 0.93 0.21 Liberia 19.09 -1057 1.042 1.355 R2, seY 0.68 851 0.964 0.31 22.91 -514 1.0294 78.38 0.86 410 0.98 0.20 Sierra Leone R2, seY Piecewise Linear Regression • Cumulative case reports • Rate increases after knot points – Not splined, chose knot points for min SSE – Linear slope coefficients = case reports/day! Country Before\After Slope per day R2 Guinea 7/20/2014 2.95 0.97 Guinea 7/27/2014 13.22 0.97 Liberia 7/20/2014 1.11 0.56 Liberia 7/27/2014 56.78 0.98 Sierra Leone 8/3/2014 8.56 0.96 Sierra Leone 8/4/2014 41.26 0.95 Cumulative Case Reports and Forecasts 18,000 16,000 14,000 12,000 10,000 Guinea 8,000 Liberia Sierra Leone 6,000 Total Case reports 4,000 2,000 0 12/11/2014 11/11/2014 10/12/2014 9/12/2014 8/13/2014 7/14/2014 6/14/2014 5/15/2014 4/15/2014 3/16/2014 “Survivor” Function Estimates: CCDF of Weeks from Case Report to Death 1.00 0.90 0.80 0.70 0.60 Guinea 0.50 Liberia 0.40 Sierra Leone 0.30 0.20 0.10 0.00 0 4 8 12 16 20 Weeks from case report to death 24 28 32 Actuarial Weekly Death Rates Conditional on Survival 0.50 0.45 0.40 0.35 0.30 Guinea 0.25 Liberia 0.20 Sierra Leone 0.15 0.10 0.05 0.00 0 4 8 12 16 20 Weeks from case report to death 24 28 32 CFR = P[Death ≤ Time since first case] Country Probability of death Std. Dev. Guinea 62.52% ~1.2% Liberia 67.61% ~5.6% Sierra Leone 59.29% ~14.2% Nigeria 62% (mle) 69% (lse) na 43% (max.En.) • Survivor function estimates for Nigeria disagree: time, cases, and deaths are too few, August 2014 only Deaths/Cases is Biased Low • P[Time to death ≤ 32 weeks] > Deaths/Case Reports! – Because some haven’t died yet • Empirical standard deviation estimates are from March-April, March-May, March-June, March-July, March-August, March-September and total cohorts P[Death time ≤ 4 weeks] and P[LoS ≤ 4 weeks|Survive] • Death time = Death – Case Report • LoS = Release – hospitalization (Length of Stay) Country Probability of death in 4 weeks Std. Dev. LoS Guinea 48.1% 6.5% 100% Liberia 47.1% 14.4% 92.5% Sierra Leone 22.7% 14.0% 100% Nigeria 33.33% Length-of-Stay: P[LoS > t|Survive] • New England Journal of Medicine article appendix fit gamma distributions to data • Mark II eyeballs read hospitalization-to-release data from NEJM graphs. Used empirical cdf of LoS – Sample sizes are small • Use actuarial death rates and empirical cdf of LoS to forecast caseloads – Assume case report => case load in first week after report Caseloads vs. Required ETU Beds • WHO Dec. 1, 2014 required ETU beds – http://apps.who.int/iris/bitstream/10665/137091 /1/roadmapsitrep22Oct2014_eng.pdf?ua=1 • Forecast is from distribution of time from case report to either release or death Existing ETU beds Guinea 160 Liberia 620 Sierra Leone 346 WHO Required ETU beds 260 2690 1198 WHO Forecast Forecast Ratio % 61% 23% 29% Case-loads 838 2,880 2,737 Ratio % 19.09% 21.53% 12.64% Caseload Estimates and Forecasts (NOT cumulative) 3,500 3,000 2,500 2,000 Guinea 1,500 Liberia Sierra Leone 1,000 500 0 12/14/2014 11/14/2014 10/15/2014 9/15/2014 8/16/2014 7/17/2014 6/17/2014 5/18/2014 4/18/2014 3/19/2014 Analyses • Guinea http://pstlarry.home.comcast.net/EbolaGna.xlsm • Liberia http://pstlarry.home.comcast.net/EbolaLib.xlsm • Sierra Leone http://pstlarry.home.comcast.net/EbolaSL.xlsm • Regression and summary http://pstlarry.home.comcast.net/EbolaSIR.xlsx Workbook Spreadsheets • *.xlsm contain workbook tabs {Data, npmle. nplseSummary, MaxEntropy, Recovery, nplse of subsets} – Npmle didn’t fit well, partly due to data revisions – Nplse and MaxEntropy agreed tolerably – Macro is VBA convolution for actuarial forecasts • EbolaSIR.xlsx contains tabs (Guinea, Liberia, Sierra Leone, Survival Analysis, CaseLoad} – The country tabs contain regression analyses of case reports Conclusions: If survive week 1,… • Survivors may need care to prevent subsequent death due to secondary causes: liver damage and ??? • Case fatality ratio estimates in first week: 46.9% in Guinea, 40.5% in Liberia, ???% in Sierra Leone – More deaths in 4th or 5th week in Guinea and Sierra Leone, in 8th week in Liberia. Accounting phenomena? – More deaths in 12th week in Guinea. Accounting phenomenon? – Sierra Leone recently reported deaths affect estimates • Standard deviations of first weekly death rates: 6.1% in Guinea, 17.1% in Liberia, 16.7% in Sierra Leone, based on six monthly estimates • Exponential increases in cumulative case reports (=> Ro > 1) seems to have reverted to linear increases. Let’s hope so. • Caseloads ~7200 by end of year, based on The WHO case reports and deaths and NEJM LoS. Can’t treat unreported cases. Questions • Case confirmations and adjustments created problems with case reports and deaths – Why do survivor function estimates differ? – Standard deviation estimates could be reduced with more work, time, and data • Use case forecasts, caseloads, and standard errors for planning, allocation, and service levels? • Would WHO and Imperial College please share case data? – Should countries support WHO and West Africa without data, estimates, and uncertainty? Next Steps • Forecast tolerance limits on caseloads for specified service levels – Infectious time distribution for those who recover? – SIR by country or county? – Contact [email protected] if you would like more analyses – Updated weekly or whenever I get data • Get case data from WHO, Imperial College, and CDC References • • • • • • • • George, L. L. and Avinash C. Agrawal, “Estimation of a hidden service distribution of an M/G/∞ system,” Naval Research Logistics, 20: 549–555. doi: 10.1002/nav.3800200314 , http://pstlarry.home.comcast.net/MGinfi1.docx George, L. L., “Field Reliability Without Life Data,” SPES/QP News, vol. 5, no. 2, Dec. 1999, pp. 13-14, http://www.amstat-online.org/sections/qp/1299newsletter.pdf Harris, Carl M. and Edward Rattner and Clifton Sutton, “Forecasting the extent of the HIV/AIDS epidemic,” Socio-Economic Planning Sciences, 1992, vol. 26, issue 3, pages 149-168 Ibid. “Estimating and Projecting Regional HIV/AIDS Cases and Costs. 1990-2000: A Case Study,” Interfaces, Vol. 29, No. 5, Sept.-Oct. 1997, pp. 38-53 Gang Cheng, “The nonparametric least-squares method for estimating monotone functions with interval censored observations,” PhD thesis, University of Iowa, 2012, http://ir.uiowa.edu/etd/2839 Jewell, Nicholas et al., “Estimation of the Case Fatality Ratio with Competing Risks Data: An Application to Severe Acute Respiratory Syndome (SARS),” UC Berkeley Div. of Biostat. Working paper series Number 176, 2005 WHO Ebola Response Team, “Ebola Virus Disease in West Africa—the First 9 Months of the Epidemic and Forward Projections,” N Engl J Med. DOI: 10.1056/NEJMoa1411100, appendix Maimuna Majumder, “Mathematical Modeling of the 2014 Ebola Outbreak,” MIT Sept. 26, 2014, http://maimunamajumder.wordpress.com/2014/09/26/mathematical-modeling-ofthe-2014-ebola-outbreak/