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Wednesday, April 29, 2015
Ro and the History of the Ebola Epidemic in West Africa
A chart of the Basic Reproduction Number (Ro) shows the history of the Ebola epidemic in West Africa. Ro measures the rate of transmission of disease, and can be calculated from the history of daily new cases. Ro can be considered analogous to acceleration in physics – cumulative cases represent position; daily new cases represent velocity; and Ro is the rate of change in daily new cases.
Ro rose rapidly in May and June of 2014, then fell steadily thanks to the implementation of public health measures. Ro fell below the critical level of 1.0 at the end of October, reaching a minimum of 0.9 by January, 2015. Since then, progress against the epidemic has slowed. Ro has climbed back towards 1.0, indicating the epidemic may reach a steady state or resume growth in the rate of disease transmission.
The current disease transmission rate of about 40 new cases per day is still very dangerous, and tragic for every victim. More stringent efforts are necessary to contain the disease.
The Basic Reproduction Number, Ro
The basic reproduction ratio (Ro) is the measure of how quickly a disease spreads. Ro is simply the number of new infections caused by each case of the disease. Highly contagious diseases such as measles can have Ro values up to 15. Many common diseases, including seasonal flu, have Ro values of approximately 1.3. This means that on average, each patient with the flu will give the bug to 1.3 additional people. As long as Ro is greater than 1, an epidemic is growing. When Ro is less than 1, the disease is shrinking and the outbreak will eventually end.
Case numbers from the Ebola epidemic in West Africa allow calculation of Ro for the history of the outbreak.
I have tracked the cumulative case numbers for the West Africa Ebola outbreak since early August, 2014. The cumulative case number includes all suspected, probable and confirmed cases reported by the World Health Organization (WHO). Considering the known issue of under-reporting of cases, I believe that this number is the most accurate representation of the epidemic from the available data, as illnesses misdiagnosed as Ebola may be offset by unreported cases of the actual disease.
The first case of Ebola in the current outbreak is believed to be a two-year old boy, who contracted the disease after playing under a tree frequented by bats. The boy died December 6, 2013. Doctors Without Borders began international efforts to combat the disease, in March, 2014, followed soon afterward by the World Health Organization. By early May, transmission of the disease declined to nearly zero. However, the disease was not successfully contained, and transmission rates increased. By late May, the cumulative case count was growing exponentially. Exponential continued until about October, 2014, leading to considerable anxiety about the future course of the epidemic. It appears that the most effective procedure in fighting the epidemic was simple public education about patient isolation, sanitation, safe care, and safe burials. It is my belief that public health education was the essential factor in combatting the epidemic. The rate of transmission declined below the critical level of Ro = 1 before the arrival of most international aid and the construction of new Ebola treatment centers. If the disease had continued to grow exponentially, the new international treatment facilities would have been completely inadequate.
The cumulative case chart follows an S-shaped pattern on a linear scale. This chart represents the total number of confirmed, probable and suspected cases reported periodically by the World Health Organization (WHO). I interpolated the figures to arrive at common reporting dates for all of the countries involved in the epidemic. This S-shaped pattern has been described as the characteristic pattern for cumulative cases in previous, smaller Ebola outbreaks.
On a logarithmic chart, cumulative cases form a straight line from late May until October, 2014. This pattern of exponential growth was broken in mid-October, and shows a progressive flattening of the cumulative case chart until January, 2015.
The chart of daily new cases is derived from the history of cumulative cases. I interpolated the cumulative case numbers to a daily count, and then took a 9-day rolling average to smooth the variations due to irregular reporting periods. Daily new cases rose sharply from 20 cases per day in late July, 2014 to approach 140 cases per day in September. The exact peak of the epidemic is hard to determine, due to chaotic reporting and significant data revisions during October, 2014. Daily new cases began to decline in early December, 2014. The rate of decline flattened significantly in mid-January, 2015, and progress since that date has been slow.
In order to see the history of Ro for the epidemic, I used a 6th order polynomial regression to smooth the daily new cases data further. The function has “edge effects” with unnecessary high-frequency waves at the tails of the function.
I spent a ridiculous amount of time trying to remove these artifacts, and finally chose a simple eye-ball fit to the data, using an arbitrary smoothing function on the end-tail of the polynomial expression. The following chart shows the modified fit to the data.
To calculate Ro, I assumed an eight-day average interval between an initial case and subsequent infections resulting from contact with the infected patient. The chart of Ro using the original polynomial fit shows a sharp rise in June 2014 and reaches a peak in July. Although the number of cumulative cases continued to rise sharply, Ro began falling in late summer of last year. Ro can be considered analogous to acceleration in physics – cumulative cases represent position; daily new cases represent velocity; and Ro is the rate of change in daily new cases. Ro crossed the critical value of 1 in late October, 2014. At that point (according to the smoothed function), the number of daily new cases began to fall.
The number of daily new cases fell rapidly after Ro dropped below 1.0. The rate of decline changed in mid-January, seen by flattening of the curve on the daily new cases chart. At that point, Ro reached a minimum of about 0.9.
Since mid-January , progress in the epidemic has been slow. Although the number of daily new cases continues to fall, the rate has been progressively slower. Ro has climbed back toward 1.0.
The next few weeks will be critical in determining whether the Ebola epidemic in West Africa will be contained, reach a steady-state, or resume growing. The current level of disease transmission, at about 40 cases per day, is still very dangerous. And tragic for every victim.
World Health Organization Ebola Situation Reports
Colin Freeman, November 6, 2014, Magic Formula That Will Determine if Ebola is Beaten, The Telegraph, UK.