The 2014 Ebola virus (EBOV) outbreak in West Africa is the largest outbreak
of the genus Ebolavirus to date. To better understand the spread of infection
in the affected countries, it is crucial to know the number of secondary cases
generated by an infected index case in the absence and presence of control
measures, i.e., the basic and effective reproduction number. In this study, I
describe the EBOV epidemic using an SEIR
(susceptible-exposed-infectious-recovered) model and fit the model to the most
recent reported data of infected cases and deaths in Guinea, Sierra Leone and
Liberia. The maximum likelihood estimates of the basic reproduction number are
1.51 (95% confidence interval [CI]: 1.50-1.52) for Guinea, 2.53 (95% CI:
2.41-2.67) for Sierra Leone and 1.59 (95% CI: 1.57-1.60) for Liberia. The model
indicates that in Guinea and Sierra Leone the effective reproduction number
might have dropped to around unity by the end of May and July 2014,
respectively. In Liberia, however, the model estimates no decline in the
effective reproduction number by end-August 2014. This suggests that control
efforts in Liberia need to be improved substantially in order to stop the
current outbreak.Comment: Published version, PLOS Currents Outbreaks. 2014 Sep 

Althaus, Christian L.

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Christian L. Althaus

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PLoS Currents

Estimating the Reproduction Number of Ebola Virus (EBOV) During the 2014 Outbreak in West Africa

The 2014 Ebola virus (EBOV) outbreak in West Africa is the largest outbreak of the genus Ebolavirus to date. To better understand the spread of infection in the affected countries, it is crucial to know the number of secondary cases generated by an infected index case in the absence and presence of control measures, i.e., the basic and effective reproduction number. In this study, I describe the EBOV epidemic using an SEIR (susceptible-exposed-infectious-recovered) model and fit the model to the most recent reported data of infected cases and deaths in Guinea, Sierra Leone and Liberia. The maximum likelihood estimates of the basic reproduction number are 1.51 (95% confidence interval [CI]: 1.50-1.52) for Guinea, 2.53 (95% CI: 2.41-2.67) for Sierra Leone and 1.59 (95% CI: 1.57-1.60) for Liberia. The model indicates that in Guinea and Sierra Leone the effective reproduction number might have dropped to around unity by the end of May and July 2014, respectively. In Liberia, however, the model estimates no decline in the effective reproduction number by end-August 2014. This suggests that control efforts in Liberia need to be improved substantially in order to stop the current outbreak

Althaus, Christian L

Bern Open Repository and Information System (BORIS)

Pre-print version of manuscript published at PLOS Currents Outbreaks (2 September 2014)Estimating the reproduction number of Ebola virus(EBOV) during the 2014 outbreak in West AfricaChristian L. Althaus1*AbstractThe 2014 Ebola virus (EBOV) outbreak in West Africa is the largest outbreak of the genus Ebolavirus to date. Tobetter understand the spread of infection in the affected countries, it is crucial to know the number of secondarycases generated by an infected index case in the absence and presence of control measures, i.e., the basic andeffective reproduction number. In this study, I describe the EBOV epidemic using an SEIR (susceptible-exposed-infectious-recovered) model and fit the model to the most recent reported data of infected cases and deathsin Guinea, Sierra Leone and Liberia. The maximum likelihood estimates of the basic reproduction number are1.51 (95% confidence interval [CI]: 1.50-1.52) for Guinea, 2.53 (95% CI: 2.41-2.67) for Sierra Leone and 1.59(95% CI: 1.57-1.60) for Liberia. The model indicates that in Guinea and Sierra Leone the effective reproductionnumber might have dropped to around unity by the end of May and July 2014, respectively. In Liberia, however,the model estimates no decline in the effective reproduction number by end-August 2014. This suggests thatcontrol efforts in Liberia need to be improved substantially in order to stop the current outbreak.1Institute of Social and Preventive Medicine (ISPM), University of Bern, Bern, Switzerland*Corresponding author: christian.althaus@alumni.ethz.chIntroductionThe 2014 Ebola virus (EBOV) outbreak in West Africa is thelargest outbreak of the genus Ebolavirus to date. The outbreakbegan in Guinea in December 20131 and later spread to SierraLeone, Liberia and Nigeria. While public health interventionshave been introduced in all affected countries, the numbersof infected cases and deaths from EBOV continue to increaseand the effects of control measures remain to be determined.Real-time analysis of the numbers of infected cases and deathsdue to EBOV could provide helpful information for publichealth policy.Two key parameters describing the spread of an infectionare the basic and the effective reproduction numbers, R0 andRe, which are defined as the number of secondary infectionsgenerated by an infected index case in the absence and pres-ence of control interventions. If Re drops below unity, theepidemic eventually stops. Several studies have fitted mathe-matical models to data from previous outbreaks of the genusEbolavirus.2,3,4 Previous estimates of R0 from two outbreaksin Congo (1995) and Uganda (2000) range from 1.32 to 2.7.4It will be important to know the reproduction numbers of thecurrent EBOV outbreak and how it is affected by public healthinterventions. This will facilitate making projections of theepidemic during the next months and will allow comparisonsof the effects of control measures in each country.In this study, I describe the 2014 EBOV epidemic usingan SEIR (susceptible-exposed-infectious-recovered) model.Fitting the model to the most recent data about reported casesand deaths in Guinea, Sierra Leone and Liberia providedestimates of the reproduction numbers of EBOV in absenceand presence of control interventions.MethodsThe transmission of EBOV follows SEIR (susceptible-exposed-infectious-recovered) dynamics and can be described by thefollowing set of ordinary differential equations (ODEs):2dSdt=−β (t)SI/N, (1)dEdt= β (t)SI/N−σE, (2)dIdt= σE− γI, (3)dRdt= (1− f )γI. (4)After transmission of the virus, susceptible individuals Senter the exposed class E before they become infectious indi-viduals I that either recover and survive (R) or die. 1/σ and1/γ are the average durations of incubation and infectiousness.The case fatality rate is given by f . The transmission rate inabsence of control interventions is constant, i.e., β (t) = β .After control measures are introduced at time τ ≤ t, the trans-mission rate was assumed to decay exponentially at rate k:3β (t) = βe−k(t−τ), (5)i.e., the time until the transmission rate is at 50% of its initiallevel is t1/2 = ln(2)/k. Assuming the epidemics starts witha single infected case (I0 = 1 and C0 = 1), the cumulativenumber of infected cases C and deaths D are given by thesolutions to dC/dt = σE and dD/dt = f γI, respectively. TheODEs were solved numerically in the R software environmentfor statistical computing6 using the function ode from thepackage deSolve.Estimating the reproduction number of Ebola virus (EBOV) during the 2014 outbreak in West Africa — 2/5GuineaCumulative number of individuals01 Dec 2013 01 Mar 2014 01 Jun 2014 01 Sep 20140100200300400500600700CasesDeathsLiberiaCumulative number of individuals01 May 2014 01 Jul 2014 01 Sep 201402004006008001000CasesDeathsSierra LeoneCumulative number of individuals01 May 2014 01 Jul 2014 01 Sep 201402004006008001000CasesDeathsFigure 1. Dynamics of 2014 EBOV outbreaks in Guinea, Sierra Leone and Liberia. Data of the cumulative numbers ofinfected cases and deaths are shown as red circles and black squares, respectively. The lines represent the best-fit model to thedata. Note that the scale of the axes differ between countries.Table 1. Parameter estimates for the 2014 EBOV outbreak. The basic reproduction number is given by R0 = β/γ where1/γ = 5.61 days is the infectious duration from the study by Chowell et al.2 The primary case is defined as the index patientthat caused the subsequent outbreak. The date of appearance of the primary case in Guinea was set at 2 December 2013.1 95%confidence intervals (CI) are shown in brackets. *A Likelihood ratio test showed that treating k as a free parameter does notimprove the fit.Parameter Guinea Sierra Leone LiberiaBasic reproduction number, R0 1.51 (1.50-1.52) 2.53 (2.41-2.67) 1.59 (1.57-1.60)Transmission rate, β (per day) 0.27 (0.27-0.27) 0.45 (0.43-0.48) 0.28 (0.28-0.29)Case fatality rate, f 0.74 (0.72-0.75) 0.48 (0.47-0.50) 0.71 (0.69-0.74)Rate at which control measures 0.0023 0.0097 0*reduce transmission, k (per day) (0.0023-0.0024) (0.0085-0.0110)Date of appearance of primary case, T – 23 Apr 2014 14 April 2014(19-25 Apr 2014) (11-16 Apr 2014)Outbreak data for Guinea, Sierra Leone and Liberia (Table2) were based on the cumulative numbers of reported totalcases (confirmed, probable and suspected) and deaths fromthe World Health Organisation (WHO).5 The total populationsize N = S+E + I+R in each country was assumed to be106 individuals. Note that the exact population size does notneed to be known to estimate the model parameters as long asthe number of cases is small compared to the total populationsize.2 In particular, the basic reproduction number is simplygiven by R0 = β/γ . The effective reproduction number isgiven by Re = β (t)S/(γN)≈ β (t)/γ as long as the number ofcases remains much smaller than S.Maximum likelihood estimates of the parameters wereobtained by fitting the model to the data, assuming the cumu-lative numbers of cases and deaths are Poisson distributed. Iused the optimization algorithm by Nelder & Mead that isimplemented in the function optim. 95% confidence intervals(CI) were calculated from the likelihood profile.ResultsThe limited number of data items and the fact that the EBOVoutbreak is ongoing prevent the estimation of all model pa-rameters. Hence, I made three simplifying assumptions. First,the average duration of the incubation and infectious periodwere fixed to previous estimates from an outbreak of thesame EBOV subtype in Congo in 1995 (1/σ = 5.3 days and1/γ = 5.61 days).2 Second, I assumed that control measuresbegan after the appearance of the first infected case, i.e., τ = 0.Third, the date of appearance of the primary case of the out-break in Guinea was set to 2 December 2013.1 The remainingparameters to be estimated are the transmission rate β , thecase fatality rate f , the rate k at which control measures re-duce the transmission rate and the time of appearance of theprimary case that caused the subsequent outbreak T (for SierraLeone and Liberia only).The model fits the reported data of cases and deaths inGuinea, Sierra Leone and Liberia well (Figure 1). The maxi-mum likelihood estimates of the basic reproduction number,R0, are 1.51 (95% CI: 1.50-1.52) for Guinea, 2.53 (95% CI:2.41-2.67) for Sierra Leone and 1.59 (95% CI: 1.57-1.60) forEstimating the reproduction number of Ebola virus (EBOV) during the 2014 outbreak in West Africa — 3/5GuineaEffective reproduction number,  Re01 Dec 2013 01 Mar 2014 01 Jun 2014 01 Sep 20140.00.51.01.52.02.5LiberiaEffective reproduction number,  Re01 May 2014 01 Jul 2014 01 Sep 20140.00.51.01.52.02.5Sierra LeoneEffective reproduction number,  Re01 May 2014 01 Jul 2014 01 Sep 20140.00.51.01.52.02.5Figure 2. Effective reproduction number of EBOV in Guinea, Sierra Leone and Liberia. The model assumes that thetransmission rate decays exponentially due to the introduction of control measures. In Guinea and Sierra Leone, the effectivereproduction number has dropped to around unity by the end of May and July 2014, respectively (dashed lines). In Liberia, theeffective reproduction number remains unchanged by end-August 2014. Note that the scale of the x-axis differs betweencountries.Liberia (Table 1). The case fatality rate f is estimated at 74%(95% CI: 72%-75%) for Guinea, 48% (95% CI: 47%-50%) forSierra Leone and 71% (95% CI: 69%-74%) for Liberia. Theestimates of the parameter k, which describes how the controlmeasures reduce the transmission rate, vary between countries(Table 1). This results in a different decrease of the effectivereproduction number, Re, after the outbreaks started in eachcountry. While Re seems to have dropped to levels aroundunity in Guinea and Sierra Leone by end-August 2014, themodel suggests that control interventions were not successfulin reducing Re in Liberia (Fig. 2).DiscussionThis study uses mathematical modeling to estimate the basicand effective reproduction numbers of EBOV during the 2014outbreak in West Africa. The maximum likelihood esitmatesof R0 are 1.51 for Guinea, 2.53 for Sierra Leone and 1.59 forLiberia and lie within the same range as previous estimates foran EBOV outbreak in Congo (1995) and an outbreak of Sudanvirus (SUDV) in Uganda (2000).2,3,4 The basic reproductionnumber in Sierra Leone seems to be significantly higher thanin Guinea and Liberia. This could be a result of differences inthe population structure or human mobility.This study provides real-time estimates of EBOV transmis-sion parameters during an ongoing outbreak. The mathemati-cal model of transmission is informed by previous studies2,3,4and provides a good description of the current epidemic. In-terestingly, the estimated date of appearance of the primarycase that caused the subsequent outbreak in Sierra Leone is inagreement with the time of the most recent common ancestorof a genome analysis of the EBOV outbreak.7The simplifying assumptions of the model mean that theresults of this study need to be interpreted with caution. Amajor limitation of the model is that the transmission ratedecays exponentially due to control measures after the ap-pearance of the first infectious case. Thus, the model cannotaccount for fluctuations in the number of new cases as seen inGuinea. These fluctuations could be a result of varying effectsof control interventions in different areas, suggesting that Rein some parts such as urban areas might well be above unity.As more data of the EBOV outbreak is becoming available,this modeling aspect might fit better to data from specificareas within a country. The effectiveness of current controlmeasures on the transmission rate is unknown at present andthe size and duration of the outbreak suggest that they need toimprove. As more data accumulate over the next weeks andmonths a more thorough analysis will allow more accurateestimates of R0 and Re together with epidemic projections andthe uncertainty around them.Real-time estimates of epidemic parameters and modelprojections are important for predicting the evolution of thisEBOV epidemic and investigating the effects of ongoing andnew interventions. Hygiene measures and social distancingintervention are already being implemented and WHO hasgiven permission for the use and evaluation of currently un-registered new drugs and vaccines. The results of this studysuggest that control measures might have been sufficient todecrease the effective reproduction number to around unity inGuinea and Sierra Leone by the end of May and July 2014, re-spectively. In Liberia, however, control efforts still need to beimproved substantially in order to stop the current outbreak.AcknowledgmentsI would like to thank Nicola Low and Sandro Gsteiger forhelpful comments on the manuscript.Estimating the reproduction number of Ebola virus (EBOV) during the 2014 outbreak in West Africa — 4/5Funding StatementChristian L. Althaus is funded by an Ambizione grant fromthe Swiss National Science Foundation (project 136737). Thefunders had no role in study design, data collection and analy-sis, decision to publish, or preparation of the manuscript.Competing InterestsThe author has declared that no competing interests exist.References[1] Baize, S., Pannetier, D., Oestereich, L., Rieger, T.,Koivogui, L., Magassouba, N., Soropogui, B., Sow, M. S.,Keı¨ta, S. et al. 2014 Emergence of Zaire Ebola VirusDisease in Guinea - Preliminary Report. N Engl J Meddoi:10.1056/NEJMoa1404505.[2] Chowell, G., Hengartner, N. W., Castillo-Chavez, C., Fen-imore, P. W. & Hyman, J. M. 2004 The basic reproductivenumber of Ebola and the effects of public health measures:the cases of Congo and Uganda. J Theor Biol 229, 119–26.doi:10.1016/j.jtbi.2004.03.006.[3] Lekone, P. E. & Finkensta¨dt, B. F. 2006 Statistical infer-ence in a stochastic epidemic SEIR model with controlintervention: Ebola as a case study. Biometrics 62, 1170–7.doi:10.1111/j.1541-0420.2006.00609.x.[4] Legrand, J., Grais, R. F., Boelle, P. Y., Valleron, A. J. &Flahault, A. 2007 Understanding the dynamics of Ebolaepidemics. Epidemiol Infect 135, 610–21. doi:10.1017/S0950268806007217.[5] Disease Outbreak News – WHO Re-gional Office for Africa: http://www.afro.who.int/en/clusters-a-programmes/dpc/epidemic-a-pandemic-alert-and-response/outbreak-news.html.[6] R Development Core Team 2014 R: A Language and En-vironment for Statistical Computing. R Foundation forStatistical Computing, Vienna, Austria.[7] Gire, S. K., Goba, A., Andersen, K. G., Sealfon, R. S.,Park, D. J., Kanneh, L., Jalloh, S., Momoh, M., Fullah, M.et al. 2014 Genomic surveillance elucidates Ebola virusorigin and transmission during the 2014 outbreak. Sciencedoi:10.1126/science.1259657.Estimating the reproduction number of Ebola virus (EBOV) during the 2014 outbreak in West Africa — 5/5Table 2. Reported cumulative numbers of cases and deaths of EBOV during the 2014 outbreak. Data were retrievedfrom the WHO website and based on the cumulative total numbers of clinical cases (confirmed, probable and suspected).5 Thefirst confirmed cases in Sierra Leone were reported on 27 May 2014. In Liberia, the cases that were reported on 16 June 2014were the first new cases since 6 April 2014.Date of report Guinea Sierra Leone LiberiaCases Deaths Cases Deaths Cases Deaths22 Mar 2014 49 2924 Mar 2014 86 5925 Mar 2014 86 6026 Mar 2014 86 6227 Mar 2014 103 6628 Mar 2014 112 7031 Mar 2014 122 8001 Apr 2014 127 8304 Apr 2014 143 8607 Apr 2014 151 9509 Apr 2014 158 10111 Apr 2014 159 10614 Apr 2014 168 10816 Apr 2014 197 12217 Apr 2014 203 12920 Apr 2014 208 13623 Apr 2014 218 14101 May 2014 226 14903 May 2014 231 15505 May 2014 235 15707 May 2014 236 15810 May 2014 233 15712 May 2014 248 17118 May 2014 253 17623 May 2014 258 17427 May 2014 281 186 16 528 May 2014 291 19329 May 2014 50 601 Jun 2014 328 208 79 603 Jun 2014 344 21505 Jun 2014 351 226 81 606 Jun 2014 89 715 Jun 2014 394 263 95 4616 Jun 2014 398 264 33 2417 Jun 2014 97 4920 Jun 2014 390 270 158 3422 Jun 2014 51 3430 Jun 2014 413 303 239 99 107 6502 Jul 2014 412 305 252 101 115 7506 Jul 2014 408 307 305 127 131 8408 Jul 2014 409 309 337 142 142 8812 Jul 2014 406 304 386 194 172 10514 Jul 2014 411 310 397 197 174 10617 Jul 2014 410 310 442 206 196 11620 Jul 2014 415 314 454 219 224 12723 Jul 2014 427 319 525 224 249 12927 Jul 2014 460 339 533 233 329 15630 Jul 2014 472 346 574 252 391 22701 Aug 2014 485 358 646 273 468 25504 Aug 2014 495 363 691 286 516 28206 Aug 2014 495 367 717 298 554 29409 Aug 2014 506 373 730 315 599 32311 Aug 2014 510 377 783 334 670 35513 Aug 2014 519 380 810 348 786 41315 Aug 2014 834 46616 Aug 2014 543 394 848 36518 Aug 2014 579 396 907 374 972 57620 Aug 2014 607 406 910 392 1082 624

Estimating the reproduction number of Ebola virus (EBOV) during the 2014 outbreak in West Africa

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