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 1/49th 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
Keep your mind positive and body negative. Very apt!!
ReplyDeleteThank you for picking the punchline.Regards.
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