The Ebola Threat
by Seunghyun
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Introduction
Introduction
With a 50% average case fatality rate, the Ebola Virus Disease, or EVD, is a viral hemorrhagic fever that has affected more than 15,200 individuals over the past 45 years since 1976. Ebola spreads through physical contact with infected bodily fluids, causing internal bleeding. Identifying infectious diseases’ growth kinetics can be useful when understanding transmission, especially when the early ascending phase shows exponential growth.
In exponential growth, the number of Ebola cases increases over the number of days since the outbreak, proportional to the initially afflicted individuals. Considering there are no external influences like population growth, the function generally grows constantly. This works alongside the central idea of R0, or R naught, which indicates the average number of individuals who catch the disease from one infected person contingent on the infectious diseases’ basic epidemiology. If R0 equals one, the disease only stays stably alive. If R0 is larger than one, spread causes epidemics with compounded cases. As Ebola’s R0 typically ranges from 1.5 to 2.0, EVD shows an exponential pattern.
Hence, Ebola’s spread should complement this exponential growth function where:
If x increases by 1 when b>1, regardless of a, as b is the constant ratio, y would be the product of the previous output and base.
Sample data was compiled from The New York Times and the Centers for Disease Control and Prevention, or CDC, regarding historical Ebola outbreaks to fathom exponential growth. The 2014-2016 EVD outbreak in West Africa was selected as history’s largest outbreak, considering cases and geography, attributing unprecedented transmissions. Analyzing this data could aid in containing crowded urban outbreaks to later apply principles for smaller areas. Referring to Table 1, the article’s 24-day doubling time helped to condense data points and find unequivocal trends.
Is An Exponential Growth Model Appropriate?
Is An Exponential Growth Model Appropriate?
Table 2 shows first and second differences, calculated by subtracting the previous value from the next, alongside the ratio of successive y-values. Table 1’s y values were utilized for the “number of cases every 24 days” column to scrutinize data.
Table 2 highlights Table 1’s exponential relationship as for equal steps of x, while the ratio of successive y-values was constant, the first and second differences were not. As x is not the basis of Table 1’s change rate, it is illogical to model a quadratic or linear function.
As the successive ratio of y-values, calculated by dividing y-values by the previous one, emphasizes a ratio of two where a number greater than one was repeatedly multiplied, a characteristic of exponential growth functions is highlighted. Ratios stayed constant at two as the y-values doubled, corresponding to Ebola’s R0 range.
Developing The 2014 EVD Graph:
Developing The 2014 EVD Graph:
Using Table 1, the x-values were set as the “# of days since the outbreak” as it is the independent variable and the recorded cases as the y-values as it is the dependent variable for visualization.
As x, or the increasing number of days, approaches infinity, Figure 1’s upward curve meets the y-axis at (0,0), further alluding an exponential pattern. To develop Figure 1’s function, Table 1’s ordered pairs (24, 86) and (48, 172) were substituted into f(x) = ab^x:
The function shows that EVD’s growth multiplier is 1.03, meaning everyday, spread doubles by 1.03. Here, a represents the initially infected 41.6 individuals, and b shows a 3% growth rate as b=1+r. x varies based on time in days, where 48 was used to find a as (48,172) was substituted.
Is b Consistent?
Is b Consistent?
To show uniformity with various real-life scenarios, Congo’s EVD outbreak in 2018-2020 will be used as a comparison. Table 2’s method was utilized with data from the Humanitarian Data Exchange Organization, or HDEO, to ensure an exponential relationship, emphasizing that only the successive y-values’ ratio stayed constant at approximately 1.23 with a seven-day doubling period from 3 August to 28 September of 2018. Therefore, Figure 2 anticipates an exponential growth curve. Although the doubling period differed, b stayed the same because the growth occurring over each individual doubling period stayed at 3%.
Using the HDEO, on October 5 and October 12, respectively, there were 88 cases and 110 cases. The actual and estimated values seemingly have less than a 15% error, showing strong correlation.
Exactly How Accurate?
Exactly How Accurate?
As the measurements are approximates, upper and lower bounds were calculated with 5 of fourth significant figure of the smallest increment using Table 1’s point (24,86) as an example, making a=41.65 and b=1.035. Hence, the greatest number of cases in Figure 1:
The solution only has an error of 11.7%, akin to previous predictions. However, to further verify the solution’s accuracy, significant figures’ percentage errors were calculated.
Table 3 evaluates why three significant figures were utilized. Significant figures are pivotal to maintain accurate growth rates as both b<2 with a r of 3%, which is 0.03 in decimal form. A 0.01 difference changes r to 4%, shaping long-term ravagements as the exponential curve, or the y, increases as x increases. Thus, measuring against the expected value of 44032, 13% error for three significant figures compared to 99.9% for two shows increasing accuracy with larger significant figures.
Although percentage error could be ameliorated with four or more significant figures, innumerable decimal places are not necessary for short-term predictions and do not match the initial values’ precision. As x is small, b multiplies itself less, lowering the likelihood of major calculation errors as three significant figures only changes values by +-0.004.
Image source: blogs.unicef.org/blog/5-ways-unicef-is-fighting-ebolav1/
So What?
So What?
The solution of b=1.03 is valid as Figure 1 and Figure 2 have no anomalies with strong correlation, illustrating congruent early spread. Impoverished local health services and lack of external help may explain the continuous exponential curve. Perhaps outbreaks can become stagnant with a constant growth multiplier after identifying a to alert for care necessities.
However, the model disintegrates as a varies geographically, inducing an unspecified generalization. The possible time difference with official diagnosis’ is another limitation, where driving this down is critical in containing Ebola’s spread. Additionally, the growth rate may decrease after a few weeks, demonstrating linear growth due to precautions. Practicing careful hygiene alone moderates the exponential increase. Consequently, logarithmic growth could be utilized to model EVD outbreaks as it curves inversely with exponential growth and has no limiting value, supporting long-term data with a comprehensible complete growth curve. Perceiving infectious diseases’ growth paradigms aids in preventing exponential spread.
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