Dr. Fisman is an infectious disease physician and epidemiologist at the University of Toronto’s Dalla Lana School of Public Health, where he is Professor of Epidemiology. He is also a practicing Infectious Diseases clinician at the Toronto Western Hospital. He trained in clinical infectious diseases at Beth Israel Deaconess Medical Centre in Boston and from 1999 to 2001 was an AHRQ Fellow in Health Policy at Harvard Centre for Risk Analysis. He has held prior faculty appointments at Drexel, McMaster and Princeton Universities. Current research interests focus on the mathematical epidemiology of infectious diseases, and health economic evaluation of communicable disease control programs.

Ebola virus is a zoonotic filovirus that causes a hemorrhagic fever syndrome in humans, with a high case-fatality rate

In addition to a case-fatality rate greater than 50%, the current West African outbreak has proven difficult to control, has resulted in international travel advisories, flight cancellations and border closures ^{,}

Mathematical models of infectious disease outbreaks and epidemics can be useful tools for synthesizing available information on an infectious disease process, transforming data into useable knowledge, and defining and quantifying uncertainty about infectious diseases^{,}^{,}

Case data, including cumulative incidence, and cumulative deaths, by date of report, for Liberia, Sierra Leone, Guinea, and Nigeria were obtained from a public data repository maintained by Caitlin Rivers of Virginia Polytechnic Institute (https://github.com/cmrivers/ebola). These data are derived from official case counts from the World Health Organization, but have been aggregated and organized, making them an efficient resource for model fitting. Total case counts from this source do not distinguish between suspect, probable and confirmed case counts, and consequently cumulative incidence may decrease between measurements reflecting suspect cases who have been excluded through testing or other means. As only a fraction of cases are subject to virological confirmation, we also obtained virologically-confirmed case counts, by date, from the Virology Down Under blog, maintained by Dr. Ian Mackay (http://virologydownunder.blogspot.com.au/). These estimates are, again, derived from World Health Organization reports. Dr. Mackay's graphs are created in the Tableau application, and numerical data can be obtained using Tableau (http://www.tableausoftware.com).

We utilized the previously described "incidence decay with exponential adjustment" (IDEA) model to evaluate epidemic dynamics _{0}) and in terms of simultaneous decay, brought about by behavioral change, public health interventions, increased immunity in the population, or any other dynamic change that slows disease transmission; the model is descriptive and cannot distinguish between putative controlling mechanisms, but has the advantage of allowing epidemic growth to slow even before the critical fraction of susceptibles in the population is exhausted ^{,}_{0} is low or moderate _{0} is low or moderate. Furthermore, comparison with simulations suggests that the model can identify multi-wave epidemics or abrupt changes in control based on sudden changes in the value of the control parameter

The model utilizes the following functional form: _{0 }the basic reproductive number, and _{t }represents incident cases in a given generation. In the absence of control, incident case counts grow to the power of ^{2}

We utilized prior estimates of incubation period for Ebola virus infection (mean approximately 13 days) _{0} for Ebola virus have ranged from 1.5 to 2.7 ^{,}

In our base case analyses, we fit our model to time series data iteratively, using a progressively increasing number of outbreak generations. Fits were performed using the Berkeley Madonna software package (Berkeley, California, http://www.berkeleymadonna.com/), both using the built-in "curve fit" function, and also by evaluating root-mean-squared distances between model estimates and observations for varying combinations of R_{0} and

Model fits utilized epidemic time series available as of August 22, 2014. In addition to fitting models to overall epidemic cumulative incidence curves, we fit models to country-specific data from Guinea, Liberia, and Sierra Leone. A separate model was not fitted to data from Nigeria due to the low case count (N = 15) at the time of writing. For the purposes of fitting country-level models, we used the same estimated start date for the Guinea epidemic as was used for the outbreak overall, while the first generation of Liberia's outbreak was dated to March 27, 2014, and Sierra Leone's to May 27, 2014.

As increasing numbers of outbreak generations were used, best fit R_{0} estimates and estimates of _{0 }and d that provided approximate fits to observed case counts, but RMSD was lowest, by an order of magnitude, for R_{0 }values close to 1.8, and d values close to 0.01 (Figure 2). Our best fit model identified Ro as 1.78, and d as 0.009. Cumulative model case counts were projected to be 2435 as compared to 2473 observed cases (Figure 3).

Based on these parameter values, and in the absence of increase in

We also fit separate models to epidemic curves derived from reported deaths, rather than cases, curves based only on virologically confirmed cases, as well as curves based on varying assumptions about case-under-reporting, epidemic duration prior to first reporting in March 2014, and generation times. None of these analyses provided estimates of R_{0} and

In our base case we fitted IDEA models to overall cumulative epidemic curves, but fitting curves to individual country-level epidemics (Figure 5), and summing these curves, also reproduced the overall epidemic curve well (Figure 6). However, the epidemic dynamics of individual countries were quite distinct from one another. In Liberia, a low R_{0 }with no control (_{0} and high control (

The graph plots best fit values of R_{0} (blue curve) and

Figure plots root mean squared distance (RMSD) of model projected case counts from observed case counts in models utilizing varying combinations of R0 (X-axis) and

Best fit model (dark curve) (R0 = 1.78, d = 0.009) to observed cumulative incidence for West Africa by generation (gray bars). A 15 day serial interval is assumed, and first reported cases are assumed to have been reported in generation 5.

The figure plots model-projected incidence (per 15-day generation) (solid red curve, scale on left Y-axis) and cumulative incidence (solid black curve, scale on right Y-axis) against time (X-axis). Dashed curves show the potential impact of intervention in September 2014 on incidence (dashed red curve) and cumulative incidence (dashed black curve), if intervention resulted in an increase of d by 0.005.

Graphs demonstrate good model fits (dark curves) to observed generation by generation cumulative incidence of infection in Guinea (top panel), Liberia (middle panel), and Sierra Leone (bottom panel).

Graphs show good agreement between the base-case model, fit to overall cumulative incidence data (all countries combined, solid gray curve) vs. summed outputs (solid black curve) from models fit to country-level data from Guinea (dashed black curve), Liberia (thin black curve), and Sierra Leone (dashed gray curve).

Alternate Assumption
_{0}
Base case
1.78
0.009
12 day generation time
1.68
0.009
18 day generation time
1.94
0.013
Outbreak recognized generation 3
2.19
0.022
Outbreak recognized generation 7
1.70
0.011
Outbreak 50% under-reported
1.92
0.013
Outbreak 100% under-reported
2.02
0.015
Virologically confirmed cases only
1.74
0.011
Deaths only
1.66
0.008
Guinea cases only
2.46
0.050
Liberia cases only
1.72
0
Sierra Leone cases only
8.33
0.22

The 2014 Ebola epidemic now stands as the largest ever recorded, and threatens not only health and healthcare institutions, but civil institutions, in affected countries. Based on models fit to available cumulative incidence data from August 2014, we project that in the absence of more effective control interventions, this epidemic will increase to affect tens, and possibly hundreds, of thousands of individuals. Given the high case fatality ratio associated with Ebola virus infection, such an occurrence would be nothing short of catastrophic. Based on data currently available to us, it appears that this threat is currently centered on the Liberian component of the epidemic, which can be characterized as a simple exponential growth process, with little evidence for slowing of transmission. This contrasts with outbreaks in Guinea and Sierra Leone.

The IDEA model is descriptive, and consequently it is not possible to attribute mechanisms to the "decay" parameter (d) which defines slowing of growth. In more complex and explicit models, the effects that occur via ^{,}

As with any mathematical model, ours is limited by the quality of data used for model calibration. Numerous factors, including limited resources, understandable concerns for personal safety among healthcare and public health personnel, civil unrest, and limited virological resources, are likely to combine to make accurate enumeration of cases difficult. We performed numerous sensitivity analyses, and found that use of deaths or virologically confirmed case numbers, variation in plausible starting date or generation time, and varying assumptions about (constant) under-reporting resulted in very little change in best-fit model parameters. That said, factors such as abrupt decreases or surges in case reporting (as opposed to occurrence) would be likely to result in distortion of model-based estimates.

For almost all scenarios evaluated, and for all countries except for Sierra Leone evaluated by single-country models, we found estimates of R_{0 }similar to those that have been reported previously for Ebola outbreaks ^{,}

Using a simple, two-parameter mathematical model, we find that the initial growth characteristics of the 2014 West African Ebola epidemic to be similar to those associated with prior Ebola outbreaks. Concerning is the lack of control evident, with epidemic processes growing in an essentially uncontrolled exponential manner, particularly in Liberia. While further data will permit model validation or re-calibration in the coming months, our findings indicate that this epidemic represents a public health emergency which has the potential to grow to extraordinarily destructive dimensions. We hope our model will add support to those voices already calling for an extraordinary international cooperative effort to control this epidemic.

The authors have declared that no competing interests exist.

The authors thank Maimuna S. Majumdar, Stephane Helleringer, John Brownstein, Ian Mackay, Caitlin Rivers, and Christian Althus for comments, suggestions, and guidance regarding data sources. In particular, Caitlin Rivers's data repository (https://github.com/cmrivers/ebola), and Ian Mackay's Virology Down Under blog (http://virologydownunder.blogspot.com.au/), were invaluable as sources for Ebola epidemic data.

Dr. Fisman is an infectious disease physician and epidemiologist at the University of Toronto’s Dalla Lana School of Public Health, where he is Professor of Epidemiology. He is also a practicing Infectious Diseases clinician at the Toronto Western Hospital. He trained in clinical infectious diseases at Beth Israel Deaconess Medical Centre in Boston and from 1999 to 2001 was an AHRQ Fellow in Health Policy at Harvard Centre for Risk Analysis. He has held prior faculty appointments at Drexel, McMaster and Princeton Universities. Current research interests focus on the mathematical epidemiology of infectious diseases, and health economic evaluation of communicable disease control programs.

Were you looking for this paper? http://currents.plos.org/outbreaks/article/assessing-the-international-spreading-risk-associated-with-the-2014-west-african-ebola-outbreak/

Where is the part that indicates the top 16 countries Ebola may spread to by percentage?

Good stuff, this is what CNN needs to show.