Manuscrpit_Algorithm

A new computational algorithm to estimate case fatality rateKidane Desta Gebreyesus and Chuan-Hsiung ChangAbstract—Case fatality rate, an indicator of diseases estimate method, dividing number of deaths byseverity, is one of the most important parameters number of cases at specified time, is often used toin medical and epidemiological studies. Simple estimate the CFR. However, this method can leadnaive estimate method is often used to estimate to biased estimates due to a number of reasonscase fatality rate. However, the naive method can (e.g. incomplete data ,delay between case on setlead to biased estimates due to ascertainment and date of death).5,9,10 Overestimate (due toerror, incomplete data and other reasons. Here, we inaccurate number of deaths) and underestimatedeveloped a new algorithm called NELM (derived (due to inaccurate number of cases) can occur infrom Naive Estimate and Linear Model), which the estimation of CFR. Here, we developed a newincorporates both the naive method and the linear algorithm called NELM (derived from Naiveregression models. Where the regression model Estimate and Linear Model), which incorporatestakes Gaussian random error into account. We both the naive estimate and linear regressionillustrated the algorithm using Ebola epidemic model. Where the regression model takesdata of three West African countries. Our Gaussian random error into account. Wealgorithm provides insight into the ongoing Ebola constructed a diagram to show how the algorithmoutbreak and other emerging infectious diseases. can be implemented (Fig.1).Index Terms—parameter, case fatality rate, Ebola virus THE ALGORITHMdisease, computational model, naïve estimate The Computational algorithm we developed can beINTRODUCTION summarized in the following three main steps. Step 1: define a simple naïve estimate of CFR, at time t:THE Ebola Virus Disease (EVD) outbreak in three West African countries – Guinea, .Liberia, and Serra Leone – has increased Step 2: use linear regression model with Gaussian whitecontinuously since March 2014. A total of 27040 noise :cases, with 11140 deaths, have been reported as of26 May 2015.1 Computational models and (Considering the common assumptions of linear regressionalgorithms play an important role in the face of model, including linearity test). Its fitted model can besuch emerging epidemic diseases, as promising written asguides in medical and public health policies.Case fatality rate (CFR) is very important ,parameter, which helps to assess disease severity Using ordinary least square or maximum likelihood toand guides possible control strategies.2–5 estimate the parameters and .Accurate estimation of CFR is of great importance Step 3: using step and to obtain an updated estimate ofto the assessment of control intervention.6,7Addressing ascertainment bias is needed to :capturing disease burden data.2, 7–10 Simple naive OrManuscript received xxx; revised xxx. This work was supportedby . Go to step 1 whenever new data introduce.

epidemic dynamics: a novel computationalto estimate case fatality rate ta Gebreyesus1,2,3 and Chuan-Hsiung Chang3,4,* ormation Science, Academia Sinica, Taipei, 115, Taiwans Program, Taiwan International Graduate Program, Academia Sinica, Taipei, 115, Taiwanomedical Informatics, National Yang-Ming University, Taipei, 112, Taiwanstems and Synthetic Biology, National Yang-Ming University, Taipei, 112, Taiwan ce: [email protected] e, an indicator of diseases severity, is one of the most important parameters in medical and epidemiological e naive estimate method is often used to estimate case fatality rate. However, the naive method can leadmates due to ascertainment error, incomplete data and other reasons. Here, we developed a new model derived from Naive Estimate and Linear Model), which incorporates both the naive method and the linear els. Where the regression model takes Gaussian random error into account. We illustrated the model using data of three West African countries. Our model provides insight into the ongoing Ebola outbreak and other ous diseases.nDisease (EVD) outbreak in three West African countries – Guinea, Liberia, and Serra Leone – has increasednce March 2014. A total of 27040 cases, with 11140 deaths, have been reported as of 26 May 2015.1 Com-ls play an important role in the face of such emerging epidemic diseases, as promising guides in medical andlicies. Case fatality rate (CFR) is very important parameter, which helps to assess disease severity and guidesstrategies.2–5 Accurate estimation of CFFRigiusroef1g: rFelaotwimdpioargtraanmcettoohthoewatsoseismsmpleenmt eonf tctohnetrcoolminpteurtvaetinotinoanl.6a,l7gorithmrtainment bias is needed to capturing disease burden data.2, 7–10 Simple naive estimate method, dividing num-number of cases at specified time, is often used to estimate the CFR. However, this method can lead to biaseda number of reasons (e.g. incomplete data, delay between case onset and date of dotaecacbtuuhr)l.ia5n,r9t,fh1o1erOemsvtisemrteaosttiiopmnraeotsefentte number of deaths) and underestimate (due to inaccurate number of cases) can our results.RESULTS AND DISCUSSIONdeveloped a new model called NELM (derived from Naive Estimate and Linear Model), which incorporatesestcimomatpeuatantdiolWninaeleaarpurpsergeordaecshgsiraoannpdmhsiiomcdapelll.evWdieshruievaraeltiitozhnaetirineogtnhreetsmesiceohtnhnomidqosudseeelcsttaiaoknne.ds WGeaucsosniastnrurcatneddoamn error into account.an be implemented (see supplementary Fig.S1). algorithm to showd discussTioanble 1: Regression model fit for cumulative al visualizatinoun mtecbhenrisquoefs adnedattahbsul(aDr f)oramnsdtocparseessen(tCou).r results. Regression model fit for cumulative number of deaths (D) with cases (C), and its test of performance. Countries Model Fit Adjusted R2 P.value Dˆt = −38.1 + 0.66Ct Guniea Dˆt = −135.1 + 0.43Ct 0.994 < 0.01 Liberia Dˆt = −52.1 + 0.31Ct 0.985 < 0.01 Serialeon Dˆt = 204.8.1 + 0.4Ct 0.976 < 0.01 All countries 0.989 < 0.01e WHO Ebola data (2 July 2014 to 26 May 2015 reports), Figures 1 and 2 highlight the cumulative andnumber of cases and deaths trends for the three West African countries respectively. Different trends can bees and deaths in Guinea, Liberia and Serra Leone. The cumulative number of cases and deaths in the different Table 2. Ebola case fatality rate estimation using naive estimate (NE), linear model (LM, bˆ) and NELM method at three selected days of outbreak Country 25 August 2014 14 October 2014 29 March 2015 26 May 2015 Guinea NE LM NELM NE LM NELM NE LM NELM NE LM NELM Liberia Serra Leone 0.66 0.51 0.61 0.66 0.51 0.62 0.66 0.64 0.64 0.67 0.66 0.66 All countries 0.50 0.53 0.54 0.50 0.55 0.54 0.45 0.41 0.43 0.45 0.43 0.44 0.41 0.41 0.42 0.41 0.31 0.39 0.32 0.29 0.29 0.31 0.31 0.31 0.51 0.49 0.53 0.51 0.46 0.51 0.41 0.38 0.39 0.41 0.40 0.41 cBMTcturoaheamusniysnedutdscr2loiate0ohuts1rlindovs5uhetbghorehweeaoapenuWsdodtarstHthlrnsieegO)osh,tunwtEllFctyoiubogsvmofiuemlvrruataieillcrasdayartaili1nttvlrgaieenanie(nmd2sndsp(.uJta2hmucIentlbhytocweifog2ronch0tovolr1niaef4gtsrrtthoi,ctcltotaahslstet2herlia6ensntoeengscimescausemnoaeirudunknnleSadtdtfrieifoviafreeterrsra2net5nuhLrtmeAeersbuoecpeggnareiueoscso.netftssiT2.cv10ah1ae1senle4yWsdc.aauendndmDadep2aiupdf9tllfehaieMaesttrdihaevisrlnnecinsthhenGoa2utwr0rume1mein5ndboderedaeasres,fllpoucteLfoccatitcnubfiivaateetsritlbeohyieanse). acnudmudlaetiavtehnsumtrbeenr dofsdefaothrs wthieth ctahsrees.eThWeresist aAstfartiisctiacnally saingndificdaenat tlihnsearirnelathioenshdipifbfeetrweenetn tchoeucunmtruielastiveshnoumwbeedr of deaths (D) and cases (C), in all the three countries. The performance of the regression fitted linear model is evaluated using R2 statistics and P.value (Table 1). We showed that the coefficient parameter of the linear model (denoted by bˆ) can reasonably estimates CFR. We presented an estimates of Ebola CFR using naive estimate method, linear model and NELM at three selected time periods (August,October, March and May) (Table 2). We found that the three different methods have

slightly similar trends. In contrast, the slightly different estimate results at the early stage of the Ebola outbreak (e.g. as August). Using thenoncumulative number of cases and deaths current data (in May), however, we found quiteshowed a fluctuation trends throughout the two similar CFR estimate results. Using linear modelvertical lines (the two vertical lines marked at 25 and NELM, CFR in Guinea has estimated in the interval of 0.51-to-0.66 and 0.61-to-0.66August 2014 and 29 March 2015 respectively). respectively. It has consistently estimated at 0.66This could be as ainresduilftfeorfenvtarryeignigonism.1p1actWoef or 0.67 using naive estimate method, from Augustcontrol strategies to May. The NELM algorithm captures theapplied linear model to fit the cumulative number variation of these two methods and computesof deaths with cases. There is a statistically robust estimations. CFR decreased dramatically before March 2015, and slightly increased aftersignificant linear relationship between the January (Fig. 2, 3, and 4). In other words, the totalcumulative number of deaths (D) and cases (C), in number of cases reports gradually decreased fasterall the three countries. The performance of the than that of deaths after January 2015. The estimates of CFR vary slightly from country torRe2grsetsastiiosnticfsittaendd linear model is evaluated using P.value (Table 1). We showed country. The variations may possibly reflect the effect of different control strategies used inthat the coefficient parameter of the linear model(denoted by bˆ) can reasonably estimates CFR. different countries.We presented an estimates of Ebola CFR usingnaive estimate method, linearFigure 2: Trends of cumulative (a) and non-cumulative (b)number of deaths and cases of three countries, separately,and aver all.model and NELM at three selected time periods Figure 3: regression fitted of cumulative number of deaths(August, October, March and May) (Table 2). Wefound that the three different methods have

Figure 4: Graphical visualizations of CFR – trends and summary points. REFERENCES Striving for accuracy with emerging infectious diseases. NEJM J. Watch (2005).[1] WHO. Ebola data and statistics: hhttp://goo.gl/hejqvf. [9] Thompson CA, A. O. Selection bias modeling using[2] G., C. et al. Transmission dynamics and control of observed data augmented with imputed record-level ebola virus disease (evd): a review. BMC Medicine 12, probabilities. Annals of Epidemiology. 24(10), 747–753 196 (2014). (2014).[3] House, T. Epidemiological dynamics of ebola [10] . Althaus & L., C. Estimating the reproduction number outbreaks. Elife e03908 (2014). of ebola virus during the 2014 outbreak in west africa. PLOS Curr. Outbreaks. 1–9 (2014).[4] Legrand et al. Understanding the dynamics of Ebola epidemics. Epidemiology and infection. 135 (2007). [11] Siettos CI, R. L. Mathematical modeling of infectious disease dynamics. Virulence 4, 297–306 (2013).[5] Ghani,A.C.etal.Methods for estimating the case fatality ratio for a novel, emerging infectious disease.Am.J.Epidemiol. 162, 479–486 (2005).[6] Reich, G., N. et al. Estimating absolute and relative case fatality ratios from infectious disease surveillance data. Biomet- rics 68, 598–606 (2012).[7] Drake, J. M. Limits to forecasting precision for outbreaks of directly transmitted diseases. PLoS Med. 3, 57–62 (2006). 8. Team., W. E. R. Ebola virus disease in west africa—the first 9 months of the epidemic and forward projections. N Engl J[8] Med. (2014). 9. Mundy, L. M. The case fatality ratio: