Carlos Castillo-Chavez is a Regents Professor and a Joaquin Bustoz Jr. Professor of Mathematical Biology at Arizona State University, the founding director of the Simon A. Levin Mathematical, Computational and Modeling Sciences Center, and has co-authored over 200 publications at the interface of the life, social and mathematical sciences. He has had 33 Ph.D. students, a group that includes 19 individuals from underrepresented groups. Recognitions to his work include: three White House Awards (1992,1997, and 2011), the 2010 American Mathematical Society Distinguished Public Service Award and the 2007 AAAS Mentor award. He is a fellow of the AAAS (American Association for the Advancement of Science), SIAM (Society for Industrial and Applied Mathematics), AMS (American Mathematical Society), and ACE (American College of Epidemiology). He has held honorary Professorships from Xi’an Jiatong University in China and Universidad de Belgrano in Argentina. Past appointments include a Stanislaw M. Ulam Distinguished Scholar at Los Alamos National Laboratory, a Cátedra Patrimonial at UNAM in México, and a Martin Luther King Jr. Professorship at MIT. He is a member of the Board of Higher Education at the National Academy of Sciences (2009-2016) and serves in President Barack Obama Committee on the National Medal of Science (2010-2015). He is a Fellow of the American Association for the Advancement of Science, the American Mathematical Society, the American College of Epidemiology and the Society for Industrial and Applied Mathematics.

Ebola virus disease (EVD) is a severe, often fatal viral infection in humans, with a case fatality risk (CFR) of up to 90% in previous outbreaks ^{,}^{,}^{,}^{,}^{,}^{,}^{,}^{,}

The spread of EVD requires direct contact with infected blood, tissue, or body fluids of either the living ill or the recently deceased, and the disease is particularly prone to transmission in unprotected homecare settings and during traditional burials ^{,}^{,}^{,}

Assessing the net positive or negative impact of such attempted intervention measures involves assessment of a complex adaptive system, where the system dynamics are intimately connected to the impact of multiple feedback mechanisms that evolve over time

In this analysis, we examine current outbreak incidence data for Guinea, Sierra Leone, and Liberia up to September 8, 2014, as estimated by the World Health Organization (WHO) ^{,}

As part of the HealthMap project

The estimated daily average of the new case incidence data for the three countries are shown in Figure 1. Updates to the WHO case counts occur irregularly in time (situation updates have occurred every 2 to 9 days since July 1st ), thus to estimate the daily average incidence data between two updates, we take the new incidence accumulated between updates, and divide by the time between updates.

Cumulative incidence data are highly inter-correlated from point to point, and thus are inappropriate for fitting with standard least squares or likelihood fitting methods, which have as an underlying assumption that the observations are independent ^{,}

Using a Negative Binomial likelihood fit to account for over-dispersion in the data

which simplifies to

where the

Because of the irregular updates in the data, the time spanned by ten contiguous points ranges from 21 to 34 days. Sensitivity analyses indicate the conclusions of this analysis do not depend on the exact number of contiguous points used for the fits.

We also fit an exponential curve to the entire time series for each country, from July 1st onwards (after July 1st the WHO EVD case count updates for each country became more regular, occurring on average every three days).

The basic reproduction number,

Given these limitations of

In an epidemic, given a short enough period of time, the value of

where

With estimates of

Typically, if the transmissibility and contact rates in the population remain constant, one expects to see the effective reproduction number decrease in time as the fraction of susceptible individuals in the population decreases. However, if

While we use an SEIR model and Equation 2 in this analysis to estimate

In Figure 1 we show the time series of the recorded daily average incidence data for Guinea, Sierra Leone, and Liberia. Overlaid in green, we show a selection of the piecewise exponential fits along the incidence time series, in which ten contiguous points are fit at a time, and the local exponential rate of rise,

In Figure 2 we show the time series of the estimates of the local exponential rise

Note that the rates of

Time series of recorded average number of new EVD cases per day during the initial phase of the 2014 West African outbreak, for Guinea, Sierra Leone, and Liberia (dots). The green lines show a selection of the piecewise exponential fits to the data (not all fits are shown to clarify the presentation); a moving window of groups of 10 contiguous points are taken at a time, and the rate of exponential rise estimated for those 10 points. The results for the estimations of the exponential rise for the full set of piecewise fits are shown in Figure 2. Shown in red is the fitted exponential rise from July 1st onwards.

Estimated rates of exponential rise from piecewise exponential fits to the average daily EVD incidence data, as shown in Figure 1; a moving window of groups of 10 contiguous incidence data time series points are taken at a time, and the rate of exponential rise estimated for those 10 points. The dates shown on the x axis are last date in each contiguous set of 10 points, and the vertical error bars denote the 95% confidence interval. The horizontal black line shows the estimated rate of rise of an exponential fit to the incidence time series from July 1st to Sep 8th, with the black dotted lines indicating the 95% confidence interval.

For each country in Figure 2 the solid and dotted black lines denote the central value and 95% CI, respectively, of the rate of exponential rise of case incidence data between July 1st and September 8th . The estimates of the rates of exponential rise between July 1st and September 8th are ρ=0. 052 ± 0. 008, ρ=0. 015 ± 0. 008, and ρ=0. 046 ± 0. 005 for Guinea, Sierra Leone, and Liberia, respectively. The value estimated from the combined data samples is ρ=0. 037 ± 0. 006. The values are statistically consistent for the Guinea and Liberia data (one-sided Z test p = 0. 28). However, the estimate for the Sierra Leone data is significantly lower than the estimates obtained from the Guinea and Liberia data (one-sided Z test p = 0. 0005, and p = 0. 0003, respectively). In Table 1 we show the values of the effective reproduction number, ^{,}^{,}^{,}

Estimates of the effective reproduction number for the 2014 West African EBV outbreak data, for an SEIR model as obtained from Equation 2 under several hypotheses for the incubation and infectious periods. The estimates and the 95% confidence intervals are derived from the rate of rise of an exponential fit to the new case incidence data for the various countries between July 1st and September 8th (the beginning of July is when the World Health Organization began frequent updates of the new case information for all three countries).

Hypothesized incubation |
Guinea | Liberia | Sierra Leone | All |
---|---|---|---|---|

5 and 5 | 1.6 [1.4,1.8] | 1.5 [1.4,1.6] | 1.2 [1.0,1.3] | 1.4 [1.3,1.5] |

7 and 5 | 1.7 [1.5,2.0] | 1.6 [1.5,1.8] | 1.2 [1.0,1.4] | 1.5 [1.3,1.7] |

10 and 5 | 1.9 [1.6,2.3] | 1.8 [1.6,2.0] | 1.2 [1.0,1.5] | 1.6 [1.4,1.8] |

5 and 7 | 1.7 [1.5,2.0] | 1.6 [1.5,1.8] | 1.2 [1.0,1.4] | 1.5 [1.3,1.7] |

7 and 7 | 1.9 [1.6,2.2] | 1.8 [1.6,1.9] | 1.2 [1.0,1.5] | 1.6 [1.4,1.8] |

10 and 7 | 2.1 [1.7,2.5] | 1.9 [1.7,2.2] | 1.3 [1.0,1.6] | 1.7 [1.5,2.0] |

5 and 10 | 1.9 [1.6,2.3] | 1.8 [1.6,2.0] | 1.2 [1.0,1.5] | 1.6 [1.4,1.8] |

7 and 10 | 2.1 [1.7,2.5] | 1.9 [1.7,2.2] | 1.3 [1.0,1.6] | 1.7 [1.5,2.0] |

10 and 10 | 2.3 [1.8,2.8] | 2.1 [1.9,2.4] | 1.3 [1.0,1.7] | 1.9 [1.6,2.2] |

Two recent papers have attempted to estimate the effect of control measures on the spread of EBV during the West African outbreak. The first (Althaus, 2014) employs a Poisson likelihood fit to the cumulative incidence under the assumption of an SEIR model with control that begins the day the first case appears in the population, and causes the transmission rate to exponentially decay

We stress again that our analysis makes no assumptions about whether or not control measures have been employed, and/or when they were employed. We simply examine the local rates of exponential rise to estimate how the effective reproduction number appears to be changing in time. We note that up until mid-August, the values of

It is unclear why the transmission rate of the disease apparently rose for both Guinea and Liberia between mid July to mid August, and why the transmission rate in Sierra Leone is systematically lower, although is important to note that the WHO data are obtained from rudimentary surveillance systems in under-developed countries, under the stress of a rapidly evolving outbreak situation. The temporal patterns we observe may thus partly be due to variations in surveillance during the outbreak, under-reporting, and/or reporting delays. In addition, serial passage of the disease as the outbreak progresses may be leading to increased pathogenicity, and a subsequent increasingly larger rate of increase in case counts . However, it also must be considered that otherwise well intentioned attempts at intervention may in fact be making the situation worse, at least in some regions; in a joint meeting of officials from Guinea, Liberia, and Sierra Leone on Aug 1st , it was announced that

Based on our estimates of the exponential rise in cases between July 1st to the beginning of September, if this rise is to continue unabated, there will be approximately 4400 new EVD cases in West Africa during the last half of the month of September (95% CI [3000, 6800]), 500, 900, and 3000 of which will be in Sierra Leone, Guinea, and Liberia, respectively.

We have presented an analysis where piecewise exponential fits are employed to estimate the local rate of exponential rise in cases in the 2014 West Africa Ebola outbreak. We have shown that these local rates of exponential rise can be used in conjunction with a mathematical model to estimate the temporal changes in the effective reproduction number (in the case of this analysis, we employed an SEIR model). Our analysis indicates that the spread of the disease to densely populated cities, and/or the imposition of

The authors have declared that no competing interests exist.

The authors wish to acknowledge useful discussions with Gerardo Chowell regarding these studies.

<strong>Correction</strong> The confidence intervals for expected number of cases have been corrected following the discovery of an error by the authors. The correct 95% confidence interval on the expected number of cases by the end of the month is [3000,6800]. This has been updated in the text of the manuscript.