**By Tharosa Missaka Rajaratne –**

Doubling time is a mathematical concept which yields the amount of time for a particular population to double in its size, i.e. *“How long it would take N to be 2N”* Conversely, the number of a population at any given time could be obtained for a predetermined doubling time value, i.e. *“How much N would be at the day x if N doubled every y days”.* This concept has been in application since the days of Babylonian Mathematics in the field of calculating loan interests. The same concept is applicable to phenomena that are subjected to grow over time.

The theory of doubling time can be employed as an effective indicator to measure the severity of the Coronavirus outbreak and the overall effectiveness of the remediation process executed by a particular country. In this pandemic scenario, both of the aforementioned concepts [*“How long it would take N to be 2N” and “How much N would be at the day x if N doubled every y days”*] can be used as useful indicators. For the latter concept, a continuous graph could be plotted by using the parameters of active cases in a particular country and the number of days elapsed after the first confirmed case of that respective country. Several baselines that reflect the gradients of few selected doubling times are introduced to the graph to compare the behavior of the growth.

The semi-logarithmic method helps to assign the exponentially growing variable to a logarithmic scale (i.e. linearizing the variable), while retaining the other variable as linear by default. In the following graph, the semi-log relationship between the number of active cases (logarithmic) and the number of days elapsed after the first confirmed COVID-19 cases of China, Italy, South Korea, United States, and Sri Lanka are reported. It should be noted that due to the lack of diagnostic information during the initial period, the statistical timeline of China starts from December 31^{st} 2019 with 27 active cases. The United States and South Korea had positive COVID-19 cases by as early as 22^{nd} January 2020, and Italy by 31^{st} of the same month. Sri Lanka reported its first positive case on 11^{th} March.

The markers that are added onto the curves of Mainland China, Italy, South Korea, and United States indicate the respective nearest incidents that took place before that would have likely attributed to escalate the then-ongoing outbreak into an epidemic within the country.

The dinner involving 40,000 families in China, the Patient No.31 incident in South Korea, and the Lombardy clustered outbreak in Italy show the transition of the disease from an outbreak to an epidemic. The marker pinned on to timeline of The United States relates to identification of 36 COVID-19 positive individuals from the Diamond Princess Cruiser. Since then, until the 49th day, several state and counties have been reporting an alarming increase of COVID-19 cases. Yet a nationwide testing process was not introduced until 10th March. Consequently, the country rapidly stepped into an epidemic situation.

The shape of the curve and the gradient of it indicate the trend the epidemic would take in a constructive manner. Despite the breaching situation that was taken place in Mainland China and South Korea, it can be seen that both these countries have been able to quickly increase the doubling time of active cases. The increasing of the doubling time results in a decreasing gradient and eventually a negative gradient. Although the number of confirmed cases makes a considerable contribution to determining the death toll, the population of the active cases plays a critical role in determining the death toll; hence selecting the active case population instead of the confirmed cases would return a much straightforward graph indicating the degree of the severity of the epidemic.

Italy despite having maintained lower active cases for a longer time at the beginning compared to other countries, showed a similar spike in active cases similar to that of Mainland China and South Korea, but due to the shortcomings of the remediation process of Italy its gradient decreased at a slower rate. In the United States, on the other hand, after its spike, the number of active cases increased in a dreadful manner. Towards the tail of the curve, a little decrease in the gradient is observed. Unless an explicitly rigorous remediation process is executed in the United States, a catastrophe awaits to take place. Japan although indicates an increase of active cases, it is certain that the controlling is much effective than Italy and the United States. The following graph shows the efficacy of controlling carried out by selected countries in Asia.

Sri Lanka, despite having reported the first case as late as in mid-March (excluding the case of the Chinese citizen), experienced a steep increase in cases during the first two weeks. The remediation process has shown fruitful as for now however. The gradient of the active cases has decreased drastically and momentarily. However because of uncooperative individuals, several clustered cases have been reported. If these clustered cases pilot a surge of infection, a scenario similar to that took place in other countries could occur in Sri Lanka also. And given the economic and healthcare infrastructural differences between Sri Lanka and other countries, veraciously Sri Lanka is now in a crucial phase.

Logarithmic representation of data is sometimes effective in amplifying minute fluctuations while compressing values that are several powers of 10 (in this graph) apart. Moreover, utilizing the concept of doubling time as a baseline in comparison and prioritizing the population of active cases in analyzing diseases statistics shows a much accurate image of the direction of the epidemic over the conventional linear methods.

**To be continued.. *

## Amarasiri / April 13, 2020

Tharosa Missaka Rajaratne,

RE: The Coronavirus Pandemic: Doubling Time & Active Cases As Most Important Indicators

“Doubling time is a mathematical concept which yields the amount of time for a particular population to double in its size, i.e. “How long it would take N to be 2N” “

Thanks for your article. Can you mathematically estimate the doubling time of Para-Sinhala Para-“Buddhists” stupidity?

The Wuhan China-9 virus, COVID -19 virus that originated from China, has brought forth the Para-Sinhala Para-“Buddhists” , and they have stared accusing the innocent Muslims, who are themselves victims like the others for the virus. In the meantime the Para-Sinhala Pata-“Buddhist” monks are chanting Pirith, to stop the viris, and virus ignores the monks.

//

The politicians and the imbeciles to prostrate to the monks should learn from the monks.

COVID-19 has called the BLUFF of the monks, Pope, priests, Ulama and politicians, and they all are STUMPED. The imbeciles, mean measured IQ 79, should make a note of this.

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## Amarasiri / April 13, 2020

Tharosa Missaka Rajaratne –.

RE: The Coronavirus Pandemic: Doubling Time & Active Cases As Most Important Indicators

https://en.wikipedia.org/wiki/2020_coronavirus_pandemic_in_Sri_Lanka

Looks like on the logarithmic scale, the curve seems to be flattening out.

March 24th 100 cases, April 11th, 200 cases. in 18 days. If the locations , clusters, can be identified and isolated, it be contained. Need to keep the foreigners out. They are going to be the new carriers.

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## DW / April 13, 2020

Your article attempts to capture the propagation characteristics of a natural phenomenon under unhindered conditions.

However, do note that the propagation of Covit-19 pandemic of different geographies cannot be mapped into one graph and the curves compared, since the pandemic management strategies adopted by different countries differ in a wide range.

Mathematically put, the pivot variables acting on each curve is localized and differ from each other. Thus the comparison of different country plots with each other makes no mathematical sense.

Also little is known about the natural inhibitors / accelarents such as temperature, RH, modes of spread, age and gender demographics, viral immunity of humans based on region and many such factors, all of which makes precise mathematical modeling of this phenomenon practically impossible.

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## Kathavarayan / April 13, 2020

Has SL flattened the curve like Taiwan, South Korea and China yet?

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1-Taiwan: 380 cases after 23rd day from 100th case.

Last 5 days of case rate= 0.005

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2-South Korea: 10450 cases after 50th day from 100th case.

Last 5 days of case rate= 0.003

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3-China: 83004 cases after 83rd day from 100th case.

Last 5 days of case rate= 0.001

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Sri Lanka: 197 cases after 17th day from 100th case.

Last 5 days of SL’s case rate= 0.023

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If we consider, first 3 countries have flattened the curve then official data taken from last 5 days can be fitted in a log-log plot with power rule using the following equation.

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Y=0.000728557 * X^(4.2031), R=0.99998

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If SL has already flattened the curve then expected COVID-19 cases (after 17th day from 100th case) should have been 108 (not 197).

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If SL is getting closer to flatten the curve then confirm cases (197) shouldn’t double in the next 5 days.

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## SJ / April 14, 2020

Do some of us wish the worst for this country when what is needed is responsible conduct on everybody’s part?

Sri Lanka seems near its peak if not passed already.

Most important today is not the infection but ensuring that all are fed and protected from illnesses more serious than the COVID-19.

The post-COVID-19 economy is a big worry as is the health of democratic institutions.

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## Kathavarayan / April 14, 2020

If you consider Taiwan, South Korea and China have flattened the curve then SL’s expected cases should have been 119 (not 210) for 19th day from 100th confirmed case. SL hasn’t reached its peak yet.

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By the way, regardless of wherever we all living now, we’re all indirectly or indirectly connected and COVID-19 is a thread for all. Many of us came from working class families and work to improve the well being of general population never misleads vulnerable people in crisis.

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Isn’t it every government’s responsibility to provide basic income for all in this situation?

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## DW / April 13, 2020

Your article attempts to capture the propagation characteristics of a natural phenomenon under unhindered conditions.

However, do note that the propagation of Covit-19 pandemic of different geographies cannot be mapped into one graph and the curves compared, since the pandemic management strategies adopted by different countries differ in a wide range.

Mathematically put, the pivot variables acting on each curve is localized and differ from each other. Thus the comparison of different country plots with each other makes no mathematical sense.

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## Prof Lasantha Pethiyagoda / April 13, 2020

Mathematical modelling should not be conducted as if all contributing environmental and demographic factors are equal. They are not. Furthermore the virus seems to behave differently in different geographical areas, and the rate of spread is probably influenced greatly by these diverse factors. As the author has observed, management strategies and their efficiency also help determine the behaviour of the “curve”. Sri Lanka’s case is not as rosy as the non-mathematically minded media seem to rejoice in depicting. Time will tell far more if the accurate picture can be determined without cover-ups or juggling with figures. Somehow given our history, I will not expect full transparency either.

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## JD / April 13, 2020

As the author says, it does not look like second order regression equation. It looks very much like a third order regression equation. Kathavarayan compares slopes of different parts of the graph.

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## Kathavarayan / April 13, 2020

Yes, regardless of the initial conditions of each curve as others have pointed out due to the rate of local dynamics of a heterogeneous system, only the slope of current trend of last n # of days was considered to check whether a local curve is getting closer to the apex or not.

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