Uncertainty in the Spread of COVID-19: An Analysis in the Context of India

Objectives: Prevention measures play an important role in controlling infectious diseases. We eagerly want to know how to observe the impact of prevention measures, just by looking at the pandemic curve. To explain this impact, we observe that the graphical representation of an infectious disease on a logarithmic scale is more suitable compared to a linear scale. To achieve our result, we also verified that the curve of the cumulative confirmed cases of pandemic COVID-19 follows an almost exponential growth. Furthermore, we tested the flattening of the logarithmic curve, which indicates the effect of prevention measures are working well. Methods: We use the numerical and statistical method introduced by Baruh. We divided the cumulative confirmed COVID-19 data of 240 days into 12 equal parts (20 days per part) after the starting of the vaccination programme in India. We apply the exponential growth model to check the exponential growth of cumulative confirmed cases of COVID-19 on a linear scale and verify it by the comparison of the actual and the predicted values obtained by exponential model. Also, we compute the first difference of logarithmic cumulative confirmed cases and find its strong linear relationship with time ’t’. Furthermore, we apply the student t-test to confirm the linear relationship between them. We find the number of days require to flatten the logarithmic curve. Findings: Our results show that the uncertainty of the cumulative confirmed cases of COVID-19 spread pattern may continue in the upcoming days. The logarithmic curve would be flattened within 127 days from 23rd August 2021. The logarithmic scale explains the impact of the prevention measures better than the linear scale. Because the flattening of the logarithmic curve appears earlier than the flattening of the linear scale. Novelty: In the context of India, our study exhibits the importance of graphic presentation of COVID-19 data and compare between the logarithmic scales to the linear scale. As per our knowledge, this kind of study is new in the context of India.


Introduction 2 Data and Methodology
According to Baruah,[8][9][10][11] , the spread of COVID-19 in India is growing exponentially. We collect the required data of cumulative confirmed of COVID-19 in India from the Johns Hopkins University (25) . We select the data after starting the vaccination program in India. We analyze the consecutive 240 days from 15 th January to 11 th September 2021.
To check the exponential growth of epidemic curve, we define an exponential function k(t) by the equation: From equation (1), we get, This is linear in 't' . To use this model, we have to choose the value of 't' as a base, The value of the constant 'a' is already known. When the estimated values of 'b' are about the same, the trend has become nearly exponential. To compute 'b' at a time 't' , we use cumulative confirmed cases for some days. As ∆log C(t) becomes constant, then the structure of the total number of confirmed cases tends to be almost exponential.
If ∆log C (t) performs the reduction, the structure of the total confirmed cumulative cases becomes almost logarithmic. It may not reach exactly zero otherwise the total number of confirmed cases remains stable in an unexpected way, but the 'S' https://www.indjst.org/ shaped arch for the models does not permit it. Therefore, the transform from the almost exponential structure to the almost logarithmic structure has just taken place, before the ∆log C (t) becomes zero. We are enthusiastic to recognize the shift from exponential to logarithmic, by assessing the data from 15 th January 2021 to 11 th September 2021. We divided this period into 12 equal parts, and each part has 20 days.

Result and discussion
We inspired by Baruh's work [24]. He used his own statistical method based on exponential model to study the pandemic COVID-19 situation in India, USA and world. Though our study durations different, yet our results are almost same. Previously, Semra Sevi et al. (26) studied how different ways of visually presenting COVID-19 data affect Canadian citizens' views, attitudes, and support for public policy.
To examine the trend of ∆log C(t), we split the data into twelve similar parts, as shown in Table 1D. The total confirmed values for each part is following an exponential growth that be verify by applying the exponential growth model N = Ae Bt where A > 0, B > 0, t ≥ 0. by the strong exponential regression correlation coefficient in table 1. For each part, the calculation is arranged in a tabular form. See table 2 to table 13. Next, we arrange all the predicted values of ∆log C(t) in tabular form (table 14 and table 15) for the better comparison.
As we can observe that the average value of ∆log C(t) is decreasing from the first part to the second part, but after that, it is increasing continuously till the part 6. From the part 6, it decreases continuously till the part 11 and then slightly increase in the part 12.
Now we tabulated the linear equation and correlation coefficient of each part in the following table 16. The linear relationship between ∆log C(t) and 't' can easily be detected by the above table 16. The value of the correlation coefficient shows that the linearity becomes stronger as time increases. But in the part 6, this linear relationship becomes weak. If we put ∆log C(t) = 0 in the 12 th part linear equation, we get t = 127 days approximately, which means the logarithmic curve flatten within 127 days from the date 23 th August 2021.
As ∆log C(t) reduce continuously, the pattern of growth would stay exponential. As mentioned in the table 16 , the slopes increase part by part. Even the first part has a negative slope, but after that, the slope becomes positive and more positive. This means that the exponential trend has continued over time due to the strong positive linear trend between ∆log C(t) and 't' . The negative slope of this linear relationship indicates that the exponential pattern is beginning to convert in the log pattern .
The exponential growth of each part can be easily recognize in the following figures 1 to 12. In the following figures, we compared the actual and predicted value of total confirmed cases through the exponential growth model, y = Ae Bt , where y is the cumulative confirmed cases, t is the time in days and A,B are positive constable. Both the curve of actual and predicted values almost coincide that indicate the cumulative confirmed cases follow exponentially growth. Now, we compare the actual and the predicted values of ∆log C(t) for each part in the figure 13. Our actual and predicted values of ∆log C(t) follow a strong trend in the figure 13 (iv), 13 (v), 13 (vi), 13 (vii), 13 (viii). Figure 13 (i) shows that a decreasing linear trend between these two values, if it tends to zero (it cannot be exactly zero), then we can say that the exponential curve started to turn into a logarithmic curve. https://www.indjst.org/    https://www.indjst.org/ Figure 13 (ii) indicates that there is a suddenly increasing linear trend between these two values. It means the exponential growth curve remained exponential. Figure 13 (iii) demonstrates that this increasing linear trend remains increasing and has not started to decrease again. Figure 13 (iv) implies that there is more strong, increasing linear trend between these two values, we conclude that the exponential growth curve of total confirmed cases being continued in this phase also, and maybe remain continue in the next phase. Figure 13 (v) reflects that there is a more powerful increasing linear trend between these two values. It implies that these values are almost coinciding with each other. In this part, the exponential curve of total confirmed cases continued to be exponential and there is no chance to convert it into a logarithmic curve for some time. Figure 13 (vi) , 13 (vii) and 13 (viii) reflect that there is a powerful decreasing linear trend between these two values. This part indicates that the change from exponential to logarithmic is about to start. Figure 13 (ix) depicts that this powerful decreasing in ∆log C(t) is slightly become weak. Figure 13 (x) and 13 (xii) depicts that there is no linear relationship between t and ∆log C(t). Now, We apply the statistical tests at the 5 % significance with the degree of freedom 18, to the null assumption that there is no significance linear relationship between ∆log C(t) and 't' , against the two-tailed alternative assumption that there is a significant linear correlation between the variables ∆log C(t) and 't' referred to the equations in table 16 of each part separately. The twotailed hypothetical value of 't' is 2.101. Now, we apply the student t-test in each part to check the validity of our hypothesis, we summarize the values in the table 17. Student 't' value for each part is calculated by the formula, Where n = 20 and r represent the linear correlation coefficient between ∆log C(t) and t. In part 10 and part 12 the calculated value of t are 0.003 and 1.781 respectively, which is less than the hypothetical value of 2.101. Surely more than 5%, the null assumption is true. Thus, we deduce that the linearity between ∆log C(t) and 't' is not significant. This is identical to deciding that the coefficients of the regression (-0.0000008 and -0.00001 respectively) do not significantly differ from zero. It means the exponential growth may continue in upcoming days.
For part 1-9 and part 11, the observed value of t are more than the hypothetical value of t = 2.101. Thus, we deduce that the null assumption is to be dismissed for these parts, and there is a meaningful linear relationship exists between ∆log C(t) and 't. The regression coefficients in these parts reasonably differ from zero. Figure 14 (i) and 14 (ii) is a graphical representation of COVID-19 total confirmed cases till the 12 th September 2021 in the linear and the logarithmic scale respectively. Both the figures represent the same data, but flattened of logarithmic curve display the clearest picture of COVID-19 cases with respect to different prevention measures. At the flatten point of a logarithmic scale the public health measures begin to produce the desired effect and result. The logarithmic chart will highlight any substantial changes in the trend-whether it is up or down. Because of the way the scale is compressed, the logarithmic diagram shows the lines flattened earlier as compared to the linear scale. using logarithmic scale is better than as compare to linear scale. The interpretation of working of different prevention measures and vaccination can be seen in the epidemic curve on the logarithmic scale easily. https://www.indjst.org/

Conclusion
In this paper, we analyzed the total confirmed COVID-19 data for 240 days from 15 th January to 11 th September 2021, by dividing it into 12 equal parts. We have shown that during these periods the total confirmed cases followed an almost exponential growth in India. We have shown that the exponential growth curve changes into logarithmic or not, as the value of ∆log C(t) tend to zero or not. We performed the statistical significance test and concluded that there is a strong positive linear relationship between ∆log C(t) and time 't' . As per current government guidelines on COVID-19 (Janata curfew, social distancing, wearing a mask, vaccination, etc.), It will take about 127 days from August 23, 2021 to change the cumulative confirmed case pattern from exponential to logarithmic, and has begun to flatten. In the context of India, our research demonstrates the importance of graphical presentation of COVID-19 data and compares between the logarithmic and linear scale. The Flattening of the logarithmic curve indicates that the prevention measures are working well to stop the spread of infectious diseases.