key: cord-0799143-tcs3ftiv
authors: Viswanath, N. C.
title: Analysis of the Second COVID-19 Wave in India and the United Kingdom Using a Birth-Death Model
date: 2021-06-18
journal: nan
DOI: 10.1101/2021.06.16.21259009
sha: ee706dd6263be55f24fb35cbb095d09a05d4fafd
doc_id: 799143
cord_uid: tcs3ftiv

Several countries have witnessed multiple waves of the COVID-19 pandemic between 2020 and 21. The method in [8] is applied here to analyze the COVID-19 waves in India and the UK. For this, a birth-death model is fitted to the active and total cases data for 30 days periods called windows starting from 16th March 2020 up to 10th May 2021. Peculiarities of the parameters suggested a classification of the above windows into three categories: (i) whose fitted parameters predicted a rise in the number of active cases before a fall to zero, (ii) which predicted a decrease, without rising, in the active cases to zero and (iii) which predicted an increase in the active cases until the entire susceptible population gets infected. It follows that some of the type (iii) windows are of the same or lesser concern when compared to some type (i) windows. Further analysis of the type (iii) windows leads to the identification of those which could be indicators of the start of a new wave of the pandemic. The study thus proposes a method for using the present data for identifying pandemic waves in the near future.

applied a generalized birth-death model [2] for modeling the total and actively infected COVID-19 cases in several countries. This model turned out to be a special case of the standard SIR model [3] , in which the susceptible cases variable has no explicit role. [8] discussed fitting of the model in [7] when a rise in the number of active cases occurs after a fall. The active and total cases curves of India [9] and the UK [10] are analyzed here to identify the parts of the curve which indicate the occurrence of a new wave.

More precisely, the active and total cases curves are observed through 30 days period windows starting from 16 th March 2020 up to 10 th May 2021. The method in [8] is applied to fit the data observed through a window to a generalized birth-death model [2] . The parameters thus obtained for a window are used to characterize whether it is an indicator of a new wave. 

... (2) According to [7] , a point of inflection on the curve (1) of actively infected cases where its decline starts is given by

[8] discussed application of the above model when a rise in the number of cases occurs after a fall. There the idea was to fit equations (1) and (2) to a dataset after adjusting the total cases so that the total and active cases remain the same at the starting epoch.

COVID-19 data of active and total cases for India [9] and the UK [10] is analyzed through 30 days period windows starting from 16 th March 2020 up to 10 th May 2021. For this equations (1) and (2) are fitted to the windowed data, following the method in [8] . Fitting is done by applying the Levenberg-Marquardt nonlinear least-squares algorithm [6] using MATLAB R2019b [5] software.

Depending on the nature of the parameters obtained, each window is classified into one of the following three types: predicted total cases at the end of the (i+1) th window, using the parameters for the i th window. The predicted total number of new cases during the (i+1) th window will then be given by

We denote this number by Next_Month_New_Cases.

The characterization of 15 windows for India, where the first one ranges from 16 th March till 14 th April 2020 and the final one ranges from 10 th May till 8 th June 2021, is given in Figure 1 . The parameters for the fit are given in Table 1 . Figure 1 identifies the windows starting from the dates 14 th June 2020, 13 th August 2020, 12 th October 2020, 9 th February 2021, and 11 th March 2021 as those falling in the category of Continuously Rise. It follows from Figure 2 that the Next_Month_New_Cases value for the 14 th June window is close to the 14 th July Rise & Fall category window, indicating that both windows raise similar concerns. The Next_Month_New_Cases value for the Continuously Rising 12 th October window suggests that this window is of lesser concern when compared to the Rise & Fall 12 th September window. Thus, . CC-BY-NC-ND 4.0 International license It is made available under a is the author/funder, who has granted medRxiv a license to display the preprint in perpetuity.

The copyright holder for this preprint this version posted June 18, 2021. ; https://doi.org/10.1101/2021.06.16.21259009 doi: medRxiv preprint the Continuously Rise status alone doesn't characterize a window as a cause for concern. Notice that the huge Next_Month_New_Cases value for the continuously rising 13 th August window raises the concern that the peak value of active cases may increase. Figure 3 shows that there is an increase in the Active_Peak value for the 12 th September window, compared to the 14 th July window. A reverse nature of the Peak_Time for the above windows, as observed in Figure 4 , suggests an increase in the active cases in a short period. This can also be observed in Figure 5 , which shows the real data of active cases. Hence the concern raised by the 13 th August window is worth noting. Thus the Continuously Rise status together with the Next_Month_New_Cases value characterizes the severity of a window. Notice that the two windows starting from 11 th December 2020 and 10 th January 2021 suggest a continuous decline in the active cases. After these, the 9 th February 2021 window raises concern against a new wave. Notice that this window spans from 9 th February to 10 th March 2021 and the number of active cases in India was around 200000 in the second week of March 2021, which reached a peak of around 3.7 million in the second week of May 2021 before started declining [9] . Hence a type (iii) window preceded by a type (ii) window needs special attention; especially when it is followed by another type (iii) window. However, these are not the only windows that raise concern. The Rise & Fall 15 th April 2020 window has a huge Active_Peak (please see Figure 3 ) as well as a long Peak_Time (please see Figure 4 ). Compared to this, the 10 th April 2021 window has a short Peak_Time but almost double Active_Peak. Hence both these windows raise severe concerns.

A comparison of the fitted and actual active and total cases for all the 15 windows for India is given in Appendix I of the supplementary information file (please refer to the odd-numbered figures in there). A comparison of the predicted and actual active and total cases for the nearest future window is also given (please refer to the even-numbered figures in Appendix I of the supplementary information file), where prediction is based on the parameters for the present window. The above figures show that even though the fits are reasonably good, the future predictions give much larger values than the actual counts when the parameters belong to a Continuously Rise window. Hence the Next_Month_New_Cases measure, which we used to characterize such a window, could be well above the actual count.

The characterization of 15 windows for the UK is given in Figure 6 and the fit parameters are given in Table 2 . The Continuously Rise windows are those starting from the dates 14 th July 2020, 13 th August 2020, 12 th September 2020, 11 th December 2020, 11 th March 2021, and 10 th May 2021. The 14 th July window is preceded by a continuously falling 14 th June window. It follows from Figure 7 that the Next_Month_New_Cases value for the 14 th July window is close to that for the 15 th April Rise & Fall category window. These together indicate the start of a new wave. This is further strengthened as the 13 th August window also belongs to the Continuously Rise category and its Next_Month_New_Cases value is . CC-BY-NC-ND 4.0 International license It is made available under a is the author/funder, who has granted medRxiv a license to display the preprint in perpetuity.

The copyright holder for this preprint this version posted June 18, 2021. ; https://doi.org/10.1101/2021.06.16.21259009 doi: medRxiv preprint higher than that for the 14 th July window. The actual data in Figure 10 shows that the second wave in the UK got started in September 2020. Hence, in the case of the UK also, a type (iii) window preceded by a type (ii) window needs special attention; especially when it is succeeded by another type (iii) window with a huge Next_Month_New_Cases value. Figure 9 shows that there is a sharp decrease in the Peak_Time for the Rise & Fall 11 th November window compared to the 12 th October window of the same type. There is also a reduction in the Active_Peak value as shown by Figure 8 . The above two facts suggest a decline in the active cases. However, the huge Next_Month_New_Cases value for the continuously rising 11 th December window raises the concern of a rise in the active cases (the concept of a new wave doesn't apply here as it comes in the middle of a wave). Though the succeeding 10 th January and 9 th February 2021 windows belong to the Rise & Fall category, their huge Active_Peak values confirm the concern raised by the 11 th December window. The Next_Month_New_Cases value for the Continuously Rise 11 th March 2021 window is similar to that for the same category 14 th July 2020 window. Notice that the 14 th July window was succeeded by the Continuously Rise 13 th August 2020 window, which raised the concern of a new wave. Compared to this, the 11 th March 2021 window is followed by a Continuously Fall 10 th April 2021 window, which dismisses the concern of another wave for a short period. However, the Continuously Rise status of the 10 th May 2021 window together with the huge Next_Month_New_Cases value, which is even higher than the 12 th September 2020 window renew the concern of a third wave in the UK.

A comparison of the fitted and the actual active and total cases for all the 15 windows for the UK is given in Appendix II of the supplementary information file (please refer to the odd-numbered figures in there).

A comparison of the predicted and actual active and total cases for the nearest future window is also given (please refer to the even-numbered figures in Appendix II of the supplementary information file), where prediction is based on the parameters for the present window. The above figures show that in the case of the UK also, the fits are reasonably good. However, the future predictions are worse than in the case of India in the sense that much larger values are obtained than the actual counts, especially when the parameters belong to a Continuously Rise window. Also, notice that the current model doesn't consider the effect of the ongoing vaccination program and other containment measures. Therefore the indication of the third wave by the 10 th May 2021 window needs further examination in light of the above facts. This is important since the epidemic forecasting of the COVID-19 pandemic had several setbacks [1].

The study proposes a method of windowing the data of active and total COVID-19 cases for using the present data for identifying pandemic waves in the near future. 30 days period windows are used here, . CC-BY-NC-ND 4.0 International license It is made available under a is the author/funder, who has granted medRxiv a license to display the preprint in perpetuity. (1) and (2) are fitted for different windowed data of active and total cases for the UK. is the author/funder, who has granted medRxiv a license to display the preprint in perpetuity. (which was not certified by peer review)

The copyright holder for this preprint this version posted June 18, 2021. 

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A Contribution to the mathematical theory of epidemics

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Update 3)

Nonlinear Regression

Analysis and Prediction of COVID-19 characteristics using a birth-and-death model

Prediction of COVID-19 Active and Total Cases After a Fall and Rise of Cases

Total Coronavirus Cases in India

Total Coronavirus Cases in India

 Figure 5 . Actual COVID-19 active cases data for India [9] .