Prediction of Third Wave in India

Prediction of Third Wave in India

 

Dr Himanshu Shekhar

 

I dedicate this blog to my Math Professor Late Shri SB Singh of MIT, Muzaffarpur, Bihar who nurtured and honed my mathematical skills.

A predictive approach using sinusoidal variation is attempted in this post of the blog. This post was created due to absolute faith of my Friend TK Sadhana Singh in me and my calculations. The encouragement from each reader is my asset and I look forward for more input to give me correct direction for prediction.

 

Introduction

England has claimed to have achieved very nice progress in vaccination, which resulted in poor penetration of virus at the general health of people. Although England has attributed control, due to lockdown, but at the same time, now vaccination is stated to be the main reason for non-fatal next variant or wave in England. India is not able to progress rapid vaccination due to vast population and scarcity of vaccines. However, based on the available data till date, an attempt is made to predict possibility of third wave, using various mathematical technics. As number of data-points specifying waves is insufficient, some assumptions are made for the prediction. First an attempt is made to model both the waves in India, using single equation of damped vibration.

 

Nature of Waves

India has seen two waves, each attained a peak, followed by a decline. The reasons for advent of second peak was neither known nor investigated, properly. In absence of any correct social engineering parameters known, it is purely mathematical exercise, which is repeated. At first nature of the variation of daily confirmed cases are observed. Although normal distribution curve fits well, but this is not sufficient to predict the re-occurrences and wave nature. Normal distribution has only one peak at mean position with proper standard deviation representing span of each wave. To get rid of this, damped vibration equations are fitted, after assessing the nature of both the waves. First concern is peak confirmed cases in both the waves, which has a multiplication factor of 4. First wave has peak cases of 1 lakh, while second wave has 4 lakh peak daily cases. Second feature is span of the wave on time scale. Time span of both the waves has a factor of 4.

For modelling this, curves are generated incorporating these features, where peak value of each wave changes and so is the time-scale. First curve represents equation of the type y = a.[1+Sin(2pt/T)], where a is amplitude or peak value of the sinusoidal wave, T is time period of wave and t is time lapsed. The addition of unity is to make all the terms of the curve positive. The plotted curve is shown below.

 

 

The indicated curve has same amplitude and time-period for each oscillation. Exponential decay is introduced in peak by incorporating exponential term. The governing equation becomes y = a.e^(-bt). [1+Sin(2pt/T)], where b is exponential decay factor. A typical representative curve is indicated below for decaying peak of sinusoidal wave.

 

 

 

 

However, the actual curve is not decaying in peak but each subsequent wave has higher peak than the previous wave in current situation. This can be handled by taking a negative value of exponential decay factor. For time compression, the time-period is also assumed to be varying with time, linearly. The same is incorporated in equation and T is replaced by (T-td), where d is time-period decay factor. The produced curve is shown below.

 

The below represented exponential decay and varying time step. If now the mirror image of the curve is taken in y-axis, the designed curve can represent the actual situation of confirmed cases. The curve has reduced peak for each wave and the time period is high for first wave followed by sharp decline for the second wave and so on. This type of equation is used for simulating the curve for daily confirmed cases.

 

 

 

 

Integrated Equation for Waves

The process of equation development is followed for the prediction and the variation of daily confirmed cases is simulated using equation mentioned in previous section. The peak cases is amplified by a factor 4 and the time period is reduced by a factor of 4, as obvious from the used equation. The curve generated is superimposed over the actual number of daily confirmed cases.

 

 

It is clear that curve is representing to a fair degree of accuracy, the daily confirmed cases. As per this, the third wave of curve will follow immediately after end of second wave and probably it rise will be much faster. The peak may be of the order of around 4.74 lakh and the phase may start by 07 June 2021 and end by 29 June 2021. However, if somehow, the lockdown is extended up to 15 June 2021, all over India, the occurrences can be avoided and probably then no third wave will come.

 

Conclusion

As concluding remarks, I would like to reiterate that the daily confirmed cases may be simulated as sinusoidal variation with varying peak and time-period. If mathematics is to be believed, the third wave will be faster and not more severe than second wave. If lockdown is extended till 15 June 2021, the third wave can be avoided. This is a purely mathematical exercise and the assessment needs validation in course of time. I personally don’t see any chances of third wave in India, whether vaccination is complete or incomplete.

 

Stay safe, stay healthy, follow social distancing, and remain connected through social media. All the best.

Regards,

Dr Himanshu Shekhar 

Comparison of Confirmed cases with Prediction and an Analysis of Daily Deaths in India

Comparison of Confirmed cases with Prediction and an Analysis of Daily Deaths in India

 

Dr Himanshu Shekhar

 

Unlocking in Progress: A quick look at Situation in India by May 2021 End

 

Introduction

In my post on 09.05.2021, I have fitted normal distribution curve to the daily confirmed cases in India and tried to optimize the curve to predict a turnaround and further expectation of cases. The turnaround was said to be achieved by 07.05.2021 and continuous fall in daily number of confirmed cases was contemplated. The situation is analysed by end of May 2021 for the daily confirmed cases, with respect to the prediction. One data of concern is daily number of deceased, which is not reducing in proportion to the fall in daily confirmed cases. This article takes a look at this data, too.

 

Confirmed cases

The prediction on 09.05.2021 was made for a peak of around 4 lakh daily confirmed cases, in India. The situation is found to be valid and passage proved that the expectation matched with reality. In fact, the discrete data of daily confirmed cases, matched well with the normal distribution curve, which generally depicts any random event. The data collected from internet up to May-2021 end is plotted against prediction on 09.05.2021.

 

 

The curve indicates a good match. However, daily confirmed cases declined faster than prediction as per normal distribution curve. The mathematical analysis indicated that, a better fit will be obtained, if keeping all other data same, turnaround is fixed to 04 May 2021, rather than 07 May 2021. However, the prediction made on 09 May 2021 is still valid to a fair degree. A more valid picture will emerge with the data of cumulative confirmed cases, which is matching for the prediction and actual confirmed cases.

 

 

Deceased Cases

Generally deceased cases follow the same trend, as that of the confirmed cases with some scaling. It indirectly indicates that percentage deceased on daily basis is almost a fixed fraction of the daily confirmed cases. However, after turnaround, the daily confirmed cases are reducing drastically, with very insignificant change in daily deceased. This indirectly indicates a higher fatality and higher daily death percentage.

 

 

Daily deceased as percentage of daily confirmed cases in India has been plotted since 01 April 2021. 14 April 2021 is taken as reference date. It is clear that percentage is monotonically rising. The rate of rise is faster in recent days. With respect to lapsed days from the reference date, the percentage deceased is fitted into a mathematical equation and a quadratic fit is found to be suitable. The value of daily death = 0.0006[1000+x(10+x)]. This is a disturbing trend.

 

 

 

Concluding Remarks

As predicted for the daily confirmed cases, the figure is continuously declining. The decline is found to be faster than prediction. However, the cumulative confirmed cases has almost matching trend. The validity of prediction on 09.05.2021 is thus proven till 29.05.2021 and hopefully, the trend will continue to follow the predicted normal distribution curve and cases will be under control, as predicted. However, the trend is number of daily deceased is not declining and this is a matter of concern. Although this also can be modelled as normal distribution curve, but wasting time and energy on such negative activity is avoided. Currently death percentage has reached around 2% and although trend is stagnating in last 4-5 days, but the best fit curve predicts a deceased percentage of above 3% by 12 June 2021. This is a cumulative effect of reduction in daily confirmed cases and stagnation or relatively slow declining in daily deceased cases.

 

Hope that this analysis is as per the expectations of readers. Comments are always welcome. Regards. Stay Safe, Stay Healthy, Stay at a distance, Stay connected remotely.

 

Dr Himanshu Shekhar 

Analysis and Exploration of Doubling Period for Prediction of Cumulative Confirmed Cases

 

Analysis and Exploration of Doubling Period for Prediction of Cumulative Confirmed Cases

 

Dr Himanshu Shekhar

 

Hope that readers help me in improving my analysis.

Do read and share my book “Pandemic Survivor Diary”, available on AMAZON Kindle.

 

 

Introduction

Doubling Period is defined as number of days in which cumulative confirmed cases become double. Although mathematically, it may be a fractional number but it is expressed as whole number. This is one of the performance parameter of a country to express the control of Pandemic. Higher doubling period indicates a better control, as rise in case is taking longer time. Currently, although it seems that turnaround has been achieved in India on 07.05.2021 (as predicted in post on 09.05.2021), the doubling period if continuously declining. The control of Pandemic as a function of doubling period is explored in this post.

 

Earlier Turnaround

As India is witnessing second wave of Pandemic, it has already undergone one cycle of turnaround. It has attained a peak in September 2020 and after that continuous decrease in daily confirmed cases is observed. When doubling period data is observed, it was continuously and linearly rising at a constant rate during 15 October 2020 to 25 March 2021. After that the rise has slowed down and in April 2021, it has declined sharply, leading to second wave. Currently the value of doubling period of India is reduced to 46 days, after witnessing a peak (control) value of 187 days on 13.04.2021 (One month back).

 

 

 

When variation of doubling period during 15 October 2020 to 15 March 2021 is assumed to be a straight line rise, the slope of the curve is around 0.87 days of doubling period rise every day. If the slope becomes 1, then cumulative confirmed cases will be under control. In the current situation of second wave, the same slope is to be attained at the earliest occasion to ensure control of Pandemic.

 

 

Simulation and Prediction

Although not very realistic assumption, but it is assumed that now onwards, the doubling period will rise at a rate of 0.87 days per day. Current value of doubling period of 46 days, will become 46.87 days tomorrow, 47.74 days day after tomorrow, and so on. By end December 2021, the doubling period at this rate can reach 250 days. Under this assumption, the value of cumulative confirmed cases are also plotted till 31 December 2021.

 

 

 

It is clear that the current slope of cumulative confirmed cases under the assumption of control, as observed during end of first wave, results in slower rise. Definitely, this slower rise is difficult to achieve in practice. It is also not known that doubling period will be reducing to what level. Overall, an assumption of end of first wave is simulated to predict similar fate of second wave in terms of doubling period. It leads to cumulative confirmed cases of around 3.71 Crores by the end of 2021.

To have better appreciation, prediction for cumulative confirmed cases made from optimized normal distribution curve is superimposed on this doubling period based curve. It is observed that cumulative distribution curve based on both the assumption show almost same value on 31.10.2021. However, variation of both are different. It is clear that calculation based on normal distribution follow the current slope, and the slope reduces to bring the curve to horizontal position end by June 2021. However, the doubling period based approach gives a reduction in slope of cumulative confirmed cases, but the rate of rise is slower, which picks up slowly. The cumulative confirmed cases are shown to be rising continuously. The nature of both the curve are quite different.

 

 

 

Conclusion

It is clear that the end of first wave gave a constantly rising doubling period for India. The value of 0.87 days of doubling period rise per day is assumed for the current second wave also and from 13.05.2021, onwards, this rise in doubling period is assumed. However the cumulative confirmed cases as per this simulation has a kink, a sudden reduction in slope and a continuously rising value. This trend is not acceptable, as such. However, the calculation gives almost similar numerical value, as obtained by prediction based on optimization of normal distribution curve on 31.10.2021. Of course doubling period may not be representing the actual variation, but a control may be possible in this manner only.

 

 

Do read and share. Keep a positive mind in negative body. Take care and be safe. Regards,

 

Dr Himanshu Shekhar

Number of Testing, Vaccination and Deaths: Vital Demographic and Statistical Input

Number of Testing, Vaccination and Deaths: Vital Demographic and Statistical Input

 

Dr Himanshu Shekhar

 

During a healthy discussion in one of the WhatsApp groups, my name and posts were referred again and again. This encouraged me to continue my analysis further. Hope that readers help me in improving myself. 

Do read my book “Pandemic Survivor Diary”, available on AMAZON Kindle.

 

Introduction

One of the avid readers of my post, Shri S.A. Joshi, a dedicated and workaholic scientist once suggested me to look into the number of deaths, when this pandemic was not there. Whether any rise in deaths before advent of COVID-19 with data of current situation can be correlated. It was an eye-opener and number of deaths, due to Pandemic is found to be lower than what was coming earlier. My friend Priyamvad has a positive YouTube post, which all must see to get positive energy. On further scrutiny, as post-mortem of death due to COVID is not conducted, many deaths are assigned to be pandemic-deaths, including, murders, gang-wars, dowry-deaths, rivalries, and so on. The moral and ethical fibre of our society has withered in acquiring personal properties and India, which boasts of its rich heritage has failed miserably to act in humanitarian way to combat the Pandemic. Nevertheless, in this post, some vital statistics are looked into.

 

India in General

India has a population of around 138 Crores, written as 1380000000. Considering the area of India, the population density is around 450. The high population density is affecting the faster spread of this pandemic and social distancing has been a major concern due to this reason, in India.  Cardiovascular death rate around 282 and around more than 10 % population has diabetes. The percentage of smokers in India is also above 20 %. This amounts to high co-morbidity issues too. The life expectancy in India is around 70 years and around 6% population is above 65 years of age. People above 70 years is around 3.5%. Poverty is another factor, which gives poor affordability to people. Extremely poor population in India is around 21%.

Overall, the spread of Pandemic in India is due to bigger population and more than that a very high population density. Extreme poverty is also an issue to consider. Comorbidity factor is also high. Acquired co-morbidity is also high, due to smoking and drinking. The long queues at wine-shops, clearly indicate the susceptibility of Indians to acquire co-morbidity symptoms faster. The life expectancy concerns and percentage population above average life expectancy is also one of the factors to be considered, while situation in India is reviewed. So, overall India has seemed to have touched the peak of daily confirmed cases, but extreme situations still prevails.

 

Confirmed Cases

Daily and cumulative confirmed cases are plotted on same graph with different scales. Primary axis is for the cumulative confirmed cases (Shown in blue on left). The value is fast trying to reach a value of 2.5 Crores, in a monotonically rising way. The daily confirmed cases are shown on right side secondary axis (Brown) and it has a reduction in value. The purpose of this article is not on confirmed cases, but to understand the other aspects in the country related to Pandemic.

 

 

 

Number of Testing

Number of testing is considered to be influencing the daily confirmed cases. Higher number of testing is resulting in higher number of confirmed cases. This is observed in many cases. To assess the situation correctly, first daily and cumulative testing are considered. It is clear that daily testing is oscillating and probably the weekly variation in confirmed cases is a function of weekly variation in number of testing, too. The daily testing is oscillating and currently it is above 15 lakhs per day. The cumulative number of testing is touching 32 crores.

 

 

One peculiar feature to be observed is that daily number of testing has dipped during January-February 2021, as depicted by the daily testing curve. A doubt is obvious that it was intentional to get less number of cases in those periods. The cumulative confirmed cases and cumulative testing is plotted on same graph. It is clear that there is variation in slope of both the curves. However, rather than commenting on this, it is better to look at the positivity rate, which is expressed in percentage.

 

 

 

 

Positivity rate is defined in percentage as daily confirmed cases per unit daily testing, expressed in percentage. This is plotted and during the so called end of second wave, the value has gone to as low as 1.78%. However, it is now rising and currently the positivity rate is exceeding 20%. It indirectly says that if 100 persons are tested, 20 will be positive or it is 1 in 5. In other words, currently 5 persons are to be tested to get a positive person. If it drops to 2%, to get 1 positive person 50 people will be tested. So, a lower value of positivity rate is desired, at the earliest.

 

 

 

The main criticism of the data of number of testing is people are not getting themselves tested for fear of quarantine and isolation. In addition to this, for marriages and family gatherings and religious functions, one person with symptom gets himself tested and others follow the medicine, prescribed to the affected person. So, in a herd of 100-200 people, only 2-3 are tested, despite others are also infected. The actual number of cases are said to be higher than reported numbers. Another aspect is now this testing has become mandatory for any travel and surgery. These people have no symptoms and they get the test done to just get a negative certificate. This gives a false reduction in positivity rate. This leads to reduction in accuracy of number of testing data.  

 

Vaccination

Daily and cumulative vaccination is also plotted on a graph to understand the behaviour in India. Daily vaccination started in January 2021 with around 2 lakh on 15 January 2021. It quickly rose to 10 Lakh on 21 January 2021. It scored 30 Lakh on 15 March 2021 and 34 Lakh on 22 March 2021. After that it declined and now it is at around 21 lakh stage on 13.05.2021. One side, a rapid vaccination drive is desired and other side this poor progress is a concern. The cumulative vaccination has reached to a value of around 17.6 Crores, which is around 12.75% of our total population. This is one-eighths value in around 4 months. At this pace, it will take 8x period leading to 32 month, equivalent to around 3 years. This data seems to be a concern. The reluctance of public for vaccination, negative propaganda, administrative failure, centre verses state deadlocks, etc are reasons, but now India should unitedly try to consume all the vaccine available at the fastest pace to achieve the needed target of 100% vaccination. It may be noted that two dozes of vaccines are to be taken and this data is for total number of vaccination. Two dozes are administered to only around 1% population of India

 

 

 

Deaths

Pandemic related deaths are also concocted and many deaths, which are from other reasons are also included in the data, due to various reasons like malicious intensions, avoiding post-mortem, etc. Cases of dowry deaths, gang-war mortalities, illegal activities, personal grudges etc are culminating into death and they are also intentionally included in the death toll. However, the daily deceased cases are a good major of control of pandemic. Even if data on daily testing and daily confirmed cases are concocted, the daily deceased figures cannot be false. If daily deaths are rising, it is clear that daily confirmed cases are also rising. Hiding a death is more difficult. To consider this, daily confirmed cases and daily death are plotted on same graph at different scale. Both plots have similar nature. However, in last 15 days, the number of death as percentage of reported confirmed cases is rising. This clearly indicates that confirmed cases are not correctly reflected or more deaths are taking place or getting reported. Whatever may be the reason, the data floating on internet has to be believed.

 

 

 

Conclusion

This has become a long post, but many peripheral aspects are covered in this post. The demographic condition in India is reviewed in light of current wave of Pandemic, excluding the healthcare facilitates, medical emergencies, chaos for management of bed, oxygen and medicine, etc. Number of testing conducted is correlated with number of confirmed cases and positivity rate is plotted for better understanding of the situation. The status of vaccination is also explored and number of deceased is also analysed. Overall, India is bouncing back correctly, but rather than democracy, where every head counts, it is mandatory to have dictatorship to make only relevant heads count. Hope that this post will have the needed readership.

 

Do read and share. Keep a positive mind in negative body. Take care and be safe. Regards,

 

Dr Himanshu Shekhar 

Salient Parameters one Week after Turnaround of Second Wave

Salient Parameters one Week after Turnaround of Second Wave

 

Dr Himanshu Shekhar

 

Thanks you readers for giving more than 200 views to my previous post on prediction of turnaround. This weekend, I am taking stock of situation one week after the predicted turnaround.

 

Introduction

When number of daily confirmed cases were almost oscillating around 400000, during 30.04.2021 to 09.05.2021, optimizer is run again on 09.05.2021 and it was found that turnaround has already happened on 07.05.2021, 2 days before the date of prediction. Although it is a known fact that daily confirm cases are higher on Thursdays and Fridays, but it is but natural to rejoice and satisfy ourselves with the available information. Although, as expected, there is a rise in daily confirmed cases on Thursday 13.05.2021 and daily confirmed cases rose from 329942 on 11.05.2021 (Technology Day) to 348421 on 12.05.2021 to 362727 on 13.05.2021. The daily confirmed cases are bound to rise today, i.e. on 14.05.2021. We have to keep our fingers crossed to see that further rise due to festivals are not taking place, in daily confirmed cases on weekends and thereafter in next week. Current post discusses current situation in India, for different parameters.

 

Confirmed Cases

The optimized normal distribution curve is superimposed over actual daily confirmed cases and one week after prediction, the values are still following the nature of prediction.

 

 

Better appreciation will come, if cumulative confirmed cases are plotted for prediction and actuals. The cumulative confirmed cases will have a continuously rising vales with time, without display of any daily oscillations. The curve is smoother, as shown.

 

 

It is clear from the curve that the cumulative confirmed cases are rising monotonically in almost a straight line fashion. The curve is expected to curl down and the point of inflection is already achieved for the curve. However, we have to wait for one more week to see that actual reduction in rate of rise of cumulative confirmed cases are taking place. If by 20.05.2021, we are getting less than 300000 daily confirmed cases, we, the Indians collectively are on a right track for recovery.

 

Control Parameters

Doubling period and daily growth rate are considered as control parameters. As far as doubling period is concerned, it is continuously dipping and the current value as reached 46 days. The continuously dipping curve of doubling period must start rising up, so that proper and conclusive statement of turnaround can be made. Currently, the reducing trend of doubling period is a concern.

 

 

Daily growth rate is also one of the parameters for study of trend. It is calculated using compound interest formula (high school) for seven days duration. One week duration is considered, as daily confirmed cases has a weekly oscillating trend. The value of daily confirmed cases exceeded 2.00% on 28.04.2021 and remained there for 3-4 days. After that, it is reducing at a slow pace. The reduction in daily growth rate is a favorable sign, indicating a control. Current value of daily growth rate is 1.66%.

 

Conclusion

The current status of daily confirmed cases is on the decline and is currently following the turnaround curve predicted on 09.05.2021. The cumulative confirmed cases are also following the predicted trend. However, a reduction in the slope of cumulative confirmed cases is still to come. Once that point of inflection is achieved then conclusive evidence of reduction can be assumed. The daily growth rate is also reducing, giving a positive indication for the turnaround. The doubling period is still reducing and is a point of concern. Next post will concentrate on trend of doubling period and future expectation from doubling period, for total control of daily confirmed cases in the country.

 

Do read and share. Keep a positive mind in negative body. Take care and be safe. Regards,

 

Dr Himanshu Shekhar

COVID under Control in India

 

COVID under Control in India

 

Dr Himanshu Shekhar

 

 

Introduction

The prediction made on 30.04.2021 about variation in daily confirmed cases needs an update, now, in light of daily confirmed cases stagnating at around 400000. It was predicted that daily confirmed cases will have a peak on 29.05.2021 and at that time daily confirmed cases will reach around 700000. The posted plot is reproduced below, as ready reference.

 

 

 

Current Situation

Although reasons are unknown, but daily confirmed cases has stagnated at around 400000 during last 7 days. This stagnation means rise of cases have slowed down and probably, India can have a faster than expected turnaround with less number of peak daily confirmed cases. With that intension in mind, the actual data up to 09.05.2021 is taken and optimization of normal distribution curve is again attempted for the second wave. The actual and predicted curves superimposed over each other is reproduced below.

 

 

It indicates that turnaround has already come and the peak value of around 400000 is expected.

 

Other Control Curves

However, Doubling period and Daily growth rate curve is to be seen in correct perspective. Nature of doubling period curve is disturbing as it is showing continuous fall in value. This indirectly says that now cases will be becoming double in less time. Current value of doubling period is 68 days. For the current cumulative number of cases in India, the value must be higher than 200 days.

 

 

 

Daily growth rate based on weekly variation has a rising trend and the value is around 2.0%. Again this is a high value for the current situation in India, in light of its value being less than 0.50 % in the beginning of April 2021. So both Doubling Period and daily growth rate are not very favourable.

 

 

 

Concluding Remarks

The stagnation of value of daily confirmed cases around 400000 in India for last 9 days has been a good sign of control. Because of this, mathematics is bound to predict that turnaround has already happened in India on 07.05.2021 (2 days before) and now cases with decline. However, doubling period and daily growth rate negates any such incident. I am hopeful that cases will reduce and am happy to get this result of already achieved turnaround in India. Of course, it must be seen in light of two aspects (i) number of testing conducted (ii) weekly low confirmed cases during weekends. We have to keep our fingers crossed to see that on Next Thursday (13.05.2021), the cases are not rising.

 

Be safe, Take care, Keep positive mind in a negative body. Regards.

 

Dr Himanshu Shekhar

COVID Wave of Daily Confirmed Cases In India

COVID Wave of Daily Confirmed Cases In India

 

Dr Himanshu Shekhar

 

Introduction

The daily confirmed cases are appearing in different countries as waves. The rise and fall of daily confirmed cases are clearly visible in many countries of the world and some of them are covered in previous posts of this blog. Analysis of multiple waves in Russia, Australia, Israel, Italy, Japan, Ireland, UK, USA and Canada were presented and it was concluded that it is not mandatory that next wave is always having higher peak as compared to previous waves, like in Italy. However, in general higher peak is found for next wave always. To model the daily confirmed cases in India, as waves, equation for damped vibration is used.

 

Damped Vibration

When we are riding a car, it has springs, which react to the road surface undulations and moves car up and down, when it traverses a road. The car is nothing but the curve of daily confirmed cases. The up and down movement is called oscillations and it leads to loss of wheel-grip on road. Chances of skidding are also envisaged. For that the oscillations must stop after traversing certain distance. This is accomplished by a unit called damper. The purpose of damper is to damp or reduce the amplitude of oscillation with time and finally impart stability to running car. The governing equation for a damped vibration is given by d²x/dt² + (b/m) dx/dt + (k/m) x = 0. Here ‘m’is mass involved, ‘k’is stiffness or spring constant, ‘b’is damping coefficient. A typical damped vibration curve is shown below, which is obtained using solution as x = A x exp(-ct) x Cos(wt). Here, ‘A’ is amplitude or peak value, ‘c’ is decay constant, representing ratio of two subsequent peaks, ‘w’ is angular velocity representing frequency or occurrence of event in a given time.

 

 

The horizontal axis is time and vertical axis is amplitude or daily confirmed cases. The curve is plotted from a peak. It decays with a decaying factor of 7 in the graph, so that next peak appears as one seventh of the magnitude of first. And third peak appears as 1/49^th of the first peak, and so on. Finally, stability of curve indicates that vehicle is stable and daily confirmed cases are under control.

 

Modifications in Curve

Although generation of curve and derivation of equation for the same is a cumbersome exercise, this can make the post interesting. So, straightway the manipulation of curve is carried out to negotiate negative number of daily confirmed cases and employing inverse of damping in the plots. In the plot below, green curve represents the curve when it is shifted up appropriately using an exponential function, such that all values are positive. The red curve is obtained by mirroring or reversing the shifted curve. The red curve has lower amplitude in the beginning and higher amplitudes for subsequent waves. A rise factor is 7, in this case.

 

Multiple Waves in India

To represent multiple waves in India, the data of daily confirmed cases till 07.05.2021 is considered. The shifted and reversed damped oscillation curve is adjusted such that it matches the available data. For this, following numerical parameters are taken. A = 2800000, c = 7, w = 20, Data plotted from 30 January 2020, 1 day is equivalent to 0.001 time unit. The superimposed curve is shown below.

 

 

It is clear that the first two waves of daily confirmed cases can be represented by the shifted reverse damped vibration curve. The second wave has faster growth as compared to the prediction. However, the prediction says a peak value of around 700000 daily confirmed cases, on 01.08.2021 and end of second wave by 03 Dec 2021. A very high third peak is expected by 22 May 2022.

 

Conclusion

This is a purely mathematical exercise to simulate variation of daily confirmed cases with the equation of damped vibration with decay factor of 7 and angular velocity of 20. 1 day is taken as 0.001 time unit and the plot indicates a peak number of daily confirmed cases as 7 lakhs on 01 Aug 2021. Although peak may be matching to earlier prediction, but the time for peak was calculated delayed by 2 months. Earlier predication of peak was 29 May 2021, but this calculation indicates only 4.50 lakhs cases by 29 May 2021. Future is unknown, but analysis of daily confirmed cases as damped vibration is needed. The correctness of data is not claimed, but mathematical exercise is just highlighted. The actual values are much higher than expectation, even on 07.05.2021.

 

Do read, be safe, take care. Keep your mind positive and your body negative. Regards.

 

Dr Himanshu Shekhar

This is just damped vibration analysis. I have read it in theory of machine, one subject in BTech. Spring dashpot system has this type of behaviour. Whatever I have studied during BTech has remained idle for a very long time. It is refresher as well as refreshing. I have to get rid of negativity too, which is generated by watching useless debates on TV. Optimization is always used, as was suggested lasted year by Shri Gautam Sitesh. 

So, nothing extraordinary,

1. This is using equation of damped vibration.

2. It is using minimization of square of error for optimization.

Enjoy,

Regards.

Dr Himanshu Shekhar

Pandemic in Delhi: Ground Reality

 

Pandemic in Delhi: Ground Reality

 

Dr Himanshu Shekhar

 

This post is about analysis of Pandemic in DELHI, the capital of India.

 

Introduction

It is publicised that Delhi has been under fourth wave of Pandemic and scarcity of medicine, oxygen cylinder, hospital beds and infrastructure has been addressed with much publicity by various agencies. This post of the blog analyses the situation in Delhi with a quick look at various factors, relevant for understanding the spread.

 

Representing Fact

Delhi is witnessing fourth wave, each wave having higher number of daily peak confirmed cases. However, the data available on internet has the following variation of cumulative confirmed and recovered cases, over the entire time domain. Both the curves are showing same nature and recovery is closely following the confirmation of cases. It clearly indicates that in Delhi, intellectuals may look at the graphs and then comment positively to boost morale and reduce panic amongst masses.

 

 

Further data of cumulative number of deceased, also followed almost similar trend as cumulative confirmed cases, but the magnitude is many order less. However, attempts should be made to reduce this number, by any means. The cumulative number of testing is shown on right axis and is found to be independent of number of confirmed cases. The calculation of positive rate is shown under calculated outcome.

 

 

 

Calculation Outcome

The number of deceased per unit confirmed cases on daily basis is calculated and plotted against time. The value has attained 3.8% in June 2020, but after that it is decreasing continuously. Current value of percentage deceased in Delhi is of the order of 1.40%. Another data is positivity rate. It is expressed as percentage for daily confirmed cases per unit testing. The positivity rate reached a value of 16.5% and is reducing thereafter. The minimum value of 4.50% is attained in first week of April 2021. The current value of positivity percentage is around 6.98%.

 

 

 

 

The daily confirmed cases are extracted from the cumulative confirmed cases for Delhi and 4 waves are clearly visible. First wave has a peak of 3780 by End June 2020. The second wave in mid-September 2020 has a peak of 4470. The recovery of second wave was incomplete and third wave appeared in Delhi. It has third peak of 8590 by Mid-November 2020. However, the current 4^th wave has a very high peak of 28000 daily confirmed cases in April 2021. Hope that recovery will follow.

 

 

 

The doubling period has reflection of all the waves. The highest value of doubling period was 167 Days in the first week of April 2021. However after that control was poor. Doubling period has reduced to 142 days as of now.

 

 

 

The growth rate of total confirmed cases also represented all the waves adequately. It is clear that calculation on weekly basis, resulted in a low value for quite some time from November 2020 to February 2021. However, growth rate rose to above 3% in April 2021. Currently it is declining.

 

 

 

Conclusion

Delhi is capital of India and is given more attention than any other state or part of the country. The daily confirmed cases in Delhi was maintaining a low profile since November 2020 but the condition worsened in the first week of April 2021, suddenly. One peculiar and satisfying feature is closely following daily confirmed cases and recovery. It is the difference of both, only, which is responsible for scarcity in Delhi. However, the, Doubling period is rising and growth rate, although higher is also showing a declining trend. Hope that the daily confirmed cases attains lower values faster in Delhi.

 

Do read and share with all interested. Take care, be safe, be vigilant, be healthy. Regards.

 

Dr Himanshu Shekhar