Applying Levitt’s Equation for Prediction of Pandemic Control

Applying Levitt’s Equation for Prediction of Pandemic Control

 

Dr Himanshu Shekhar

 

Shri Vinay Pandey, one of my friends forwarded details of Levitt’s Equation and its applicability for the prediction of time for the Control of Pandemic. This post is based on his input.

 

Introduction: Shri Vinay Pandey sent me details of a study related to prediction of control of Pandemic. This was based on Levitt’s equation, which is generating a parameter, similar to growth factor, but for the total confirmed cases, followed by fitting a straight line to extrapolate it a value unity. The date on which unity value is realized is taken as date of Pandemic control. The same exercise is undertaken for data up to 20.07.2020 for India and its various states in this post, to understand the date of turnaround or Pandemic control.

 

Calculation for India: The ratio of total number of confirmed cases for India for two consecutive days is taken and the values are plotted against days lapsed since 14.03.2020. The values are slightly unstable in the beginning, but it stabilized after 26.04.2020 (day 40) and the plot is curtailed from 40^th day since 14.03.2020 till 20.07.2020. Both uncensored and curtailed curves are given below for completeness. For the curtailed curve, the unity value of Levitt’s number is calculated on day 122, which is 14.07.2020. The date has already crossed and it is incorrect way of calculation.

 

 

The plot is curtailed from beginning till 26.04.2020 (day 40). A linear variation is fitted and the time to reach unity is calculated as 206, which is predicting a control on 06 October 2020.

 

 

Indian States: Similar analysis is repeated for some of the major contributing states of India. The major contributor is Maharashtra and the data assuming 09.03.2020 as day zero is plotted. Again there are certain oscillations and the data from day 40 (18.04.2020) is considered for calculation. The linear fit is applied and the unity value of Levitt’s number is predicted to be achieved on 07 August 2020.

Tamil Nadu has peculiar variation and it has a very high value in between as shown below. The data is curtailed before 65 days (10.05.2020) and the linear fit gave unit value of Levitt’s number on 27 September 2020.

The calculation for further states are carried out on similar lines and unit value of Levitt’s number is calculated. The nature of curves is given below, along with the value of control date, based on unity value of Levitt’s number.

 

Conclusion: The linear variation is criticised for Levitt’s number, as the variation slows down at the later stage. However, this method is considered the easier, handy, faster and quicker way to predict control of Pandemic. It is also obvious that linear fit has very poor R-square values and the curve fitting may not be a good and optimized. In addition to this extracting data from the entire set is also a concern and data is very stable in the beginning. The values are found to be very sensitive to extraction of data. For some states, specially, Karnataka and Bihar, the curve is found to be slopping upward, as these states are moving towards burst of cases and data is extracted to get a negative slope of the curve. The summary of entire calculation is tabulated below.

 

Region	R-Square	Trunaround
India	0.7581	06-Oct-20
Maharashtra	0.6201	07-Aug-20
Tamil Nadu	0.6145	27-Sep-20
Delhi	0.5122	31-Aug-20
Gujarat	0.6539	29-Sep-20
Karnataka	0.1153	11-Aug-20
Uttar Pradesh	0.353	07-Aug-20
Bihar	0.1083	23-Sep-20

  

Regards.

Dr Himanshu Shekhar

Statistical Analysis of Mumbai between 01.05.2020 and 16.07.2020

Statistical Analysis of Mumbai between 01.05.2020 and 16.07.2020

 

Dr Himanshu Shekhar

 

It was frustrating to see the rapid rise in number of confirmed cases in India at an exorbitant rate. It is not the question of correctness of prediction, but the basic ethics, economy, emotion and essential fibre of India has withered.  I stopped posting, but continued my analysis, without publicly posting it. Today, I got a message from Shri Guru Prakash, one of my friends, in Mumbai, who celebrated his Birthday, this year, under lot of stress, due to the Pandemic. This post is forwarded because of his insistence.

 

Introduction: Mumbai is economical capital of India and it was once upon a time, declared as the Corona Capital of the country, competing with Delhi, for first place. Equally disturbing is the fact that Maharashtra continued to supplement the largest number of confirmed cases in India, but Mumbai displayed a better control, now and Pune, another big city of Maharashtra has emerged as the largest contributor in Maharashtra. Nevertheless, Shri Guru Prakash, provided me with the data from 01.05.2020 to 16.07.2020 and the same is analysed in this post.

 

Confirmed Case Analysis: Confirmed cases in Mumbai is analysed and the trend is plotted below. It is clear that the variation is almost linear, since around 19.05.2020. The case is rising at a steady and probably uniform linear rate.

 

For simulation 30.04.2020 is taken as reference, as the data from 01.05.2020 is made available by my friend. This makes 01.05.2020 as day one for calculation. A power law is fitted and index of fitted curve by Chi-square optimization is obtained as 0.8845. With such closeness to 1 for the index, it is tried to fit a linear curve to the total number of confirmed cases of Mumbai. It is observed that on an average 1260 cases per day is the current rate of diagnosing confirmed cases in Mumbai and it is almost constant from 19.05.2020 till 16.07.2020.

 

 

Daily confirmed cases: The daily confirmed cases is also plotted against time and the variation is almost constant between 1200 and 1400 cases per day from 19.05.2020 till 16.07.2020. The constant value of daily confirmed cases lead to problem with fitment of normal distribution curve for predicting a turnaround.  It leads to a degenerate case and mathematically turnaround, after chhi-square optimization is predicted beyond 2021. I feel that such degenerate mathematics must not be part of the post.

 

 

However, efforts are made to get the total confirmed cases from the optimized normal distribution and a short term prediction is shown below for Mumbai. It predicts attainment of 1 lakh cases in Mumbai by July 2020 and 2.0 lakh cases by mid-September.

 

 

Growth Factor: Growth factor of daily confirmed cases are calculated on daily basis for the data of Mumbai. The values are oscillating widely around 1.0. However, average value for the three months, including July 2020 till 16.07.2020 are all more than 1.0. This indirectly confirms that there is possibility of further rise and potential exists in Mumbai to supplement more and more confirmed cases. However, again this is a mathematical acitivity and is in no way a comment on the scenario in Mumbai.

 

 

Doubling Period: Doubling period is the parameter, which states the time taken for the confirmed cases to get to current level from the value which is half the current value. The doubling period in Mumbai is found to be rising continuously and currently, it is around 39 days.

 

 

However, the doubling period may not a correct estimate of control, so doubling period per lakh is calculated. It says that if confirmed cases become 4.0 lakh from 2.0 lakh in 50 days, then doubling period per lakh is only 25 days. Such rationalization helps in achievement of correct nature of variation.

 

 

It is clear that the doubling period per lakh confirmed cases has stabilized in Mumbai and it is almost constant for quite long period. Minor improvement in numerical value is also observed.

 

Growth Rate: Growth rate based on doubling period is accounted and calculated and it is found to reduce with passage of time. As sometimes, it is criticised that doubling period based growth rate has variation duration in consideration, so, it is better to calculate the growth rate on average of 7-days basis. Both the curves are generated for Mumbai and is reproduced below. The nature of both the curves are same but the 7-days average growth rate curve has lower value numerically.

 

Recovered, Active and Death Cases: The other forms of cases are also analysed. The recovered cases are continuously rising and active cases, after remaining stagnant for some times have declined. In fact, one of my senior colleague Shri JN Verma has given me ring to inform that number of active cases are declining and it is an indication of turnaround. In Mumbai, the active cases are reducing currently and probably, this may be an indication of some sort of control. The death cases are steadily rising and it is an irreversible process.

 

 

Rather than analysing absolute number of cases, the cases in Mumbai are also analysed for percentage values and the plot is shown below. The percentage death is plotted on right axis, which is 10 times enlarged. The value of percentage death out of total confirmed cases is 5.64%. in fact maximum value have touched around 5.86% on 30.05.2020. Percentage recovery is around 70% and percentage active cases is of the order of little less than 25%.

 

 

 

Concluding Remarks: This post and analysis is purely on insistence of my friend Shri Guru Prakash and I still feel that as a true citizen of India, we must demonstrate some social responsibility, to curtail and control further spread of Pandemic. 

In my office, some meeting took place, where, one of the participant got news that the person, whose farewell party, he attended 3 days back has become positive. He rushed to get himself tested and all those, who were there, in the meeting spent that nights under stress. Next day, his report came as negative, but he got news that his driver has become positive. Again he quarantined himself for 7 days to observe any sign. Later, he planned to come by his self-driven car, not chauffeur driven official car. Full office was sanitized and it was a great activity. This narrative is just to sensitize readers to please observe norms and reduce vulnerability of society towards for the Pandemic and Let India win this war, against Pandemic.

 

Thank you Guru Prakash for providing Excel Sheet for Mumbai.

Regards.

 

Dr Himanshu Shekhar

Analysis of Bihar for Pandemic

Analysis of Bihar for Pandemic

Dr Himanshu Shekhar

This is 50th Post of the Blog.

Thanks to all readers, seniors, friends, colleagues and viewers.

This post is created for my Friend Shi Arun Kumar Dubey, who has been regularly feeding me with data about number of testing, number of cases and number of deaths in Bihar, before it is published in media. This post is dedicated to Shri Arun Kumar Dubey and it analyzes the state of Pandemic in Bihar till 07.07.2020.

Introduction: The Pandemic has knocked at the doorsteps of chief minister of Bihar on 06.07.2020, which triggered me to analyze the Pandemic for the state Bihar of India. Bihar reported first case on 22.03.2020, which reached 100 on 20.04.2020, which is a duration of 30 days. 1000 cases were reported on 14.05.2020, 24 days after 100^th case is reported. After that the rise has been very fast. 5000 cases were reported on 08.06.2020 and 10000 cases on 01.07.2020. One good thing about cases in Bihar is that number of deaths is limited and number of recoveries is good, resulting in very small and consistent number of active cases. Bihar reported 5000 recoveries by 19.06.2020 and active cases are most lingering around 3000. This post analyzes the trend of Pandemic in Bihar.

Types of Cases: The variation of confirmed cases, recovered cases and death cases are plotted for Bihar against time and unfortunately with a slow start, the number of confirmed cases is rising at a higher pace.

In the diagram, number of death is plotted on secondary axis and is 100 times smaller than other parameters, if scaled along primary y-axis. The confirmed cases and recoveries are moving almost parallel to each other. The death is also rising but pace is very slow. The active cases in second half of June was around 2000, but now it is picking up again. Death is touching 100 number. At present, daily number of cases is quite random in terms of recovery or death.

Derived Parameters: Growth factor is calculated for daily number of cases in Bihar and it has an average value slightly above 1 for June 2020. The doubling period has improved to 24 days, but it may not be a correct estimate. Rather than this, doubling rate per 1000 is another parameter devised, newly by me. This indicates equivalent doubling period, had the number of cases been 1000. So, current doubling period is rationalized to 1000 cases.

The rationalized doubling period takes into account, the initial value of confirmed cases. If 1000 to 2000 takes 15 days and 2000 to 4000 takes 30 days, then both have the same rationalized doubling period. The rationalized doubling period for Bihar is reducing continuously and currently it is around 1.98. As number of confirmed cases on 07.07.2020 is around 12000, the doubling period of 24 days is reduced accordingly.

The average daily growth rate, considering each last 7-days period is calculated. During second half of June 2020, it has come down to around 2.5, but now it has gone above 3.0%, indicating poor control and faster spread of Pandemic in Bihar.

Prediction: As Shri Arun Kumar Dubey was always worried about the rapid growth of confirmed cases, an attempt is made to predict the total number of confirmed cases. An exponential fit is attempted and Chi-Square optimization is applied to get the perfect curve matching to the other oscillating values of daily confirmed cases. 29.04.2020 is taken as reference date for calculation. The activation term is 0.023, which is less than that for India and probably rate of rise is slower in Bihar as compared to that in India (0.045).

From this optimized result, the value of total confirmed cases is calculated and current confirmed cases and prediction is superimposed on single plot. It is clear that the plot of prediction and actual if matching till 07.07.2020 and hopefully the rise is also captured by this plot.

From the calculation, following details are predicted for the number of confirmed cases. However, the number of deaths and number of recoveries are better at present. If the trend continues then some control is possible in future.

12 Jul	22 Jul	7 Aug	18 Aug	27 Aug	12 Sept	24 Sept
15000	20000	30000	40000	50000	75000	100000

Conclusion: On 07.07.2020, Bihar has 12570 confirmed cases, out of which 9284 cases have recovered and 104 deaths are reported. This makes number of active cases as 3182. Although a rise is suspected in Bihar, poor healthcare infrastructure may be detrimental to any such rise. Overall, the daily rise in confirmed cases in Bihar is checking the patience of people and current no control is a difficult situation. Hope that the predicted rise in confirmed cases are controlled and recoveries in Bihar must improve over period of time.

Dr Himanshu Shekhar

Prediction for India using Chi-Square Optimization

Prediction for India using Chi-Square Optimization

 

Dr Himanshu Shekhar

 

Introduction: Considering the tremendous daily confirmed cases in India, the chances of turnaround seems a remote possibility for India. It is clocking close to 18000 cases daily and now, I am finding many cases, in my vicinity and to someone I know. It is a disturbing situation, but my inquisitive mind continues to explore more from the data available till date for the confirmed cases. I have tried Chi-Square optimization of error, using normal distribution curve and prediction for future is made.

Chi-Square Minimization: The daily confirmed cases for India is taken from 15.03.2020 to 30.06.2020 and Normal distribution curve is plotted with chosen Mean, standard deviation and Amplitude. The Chi-square error value for each date is calculated and then the sum of error is minimized, using iterative scheme to optimize mean, standard deviation and amplitude. The optimized Normal distribution curve, superimposed over daily confirmed cases is given below.

 

Total Confirmed Cases: From optimized result, cumulative confirmed cases are calculated and superimposed over available data of total confirmed cases. It is extrapolated in future till September 2020.

The prediction is tabulated below.

Cases	Date
600000	02-07-2020
700000	07-07-2020
800000	12-07-2020
900000	16-07-2020
1000000	20-07-2020
1500000	08-08-2020
2000000	27-08-2020

 

Conclusion: This is a mathematical exercise and I am finding it difficult to understand, why these numbers for India is not reducing. Hope that we the people of India wake up with some sense to follow the norms, suitable for restriction of confirmed cases.

 

Dr Himanshu Shekhar