Did the lockdown curb the spread of COVID-19 infection rate in India: A data-driven analysis
DD ID THE LOCKDOWN CURB THE SPREAD OF
COVID-19
INFECTION RATE IN I NDIA : A
DATA - DRIVEN ANALYSIS D IPANKAR M ONDAL ∗ S IDDHARTHA
P. C
HAKRABARTY † Abstract
In order to analyze the effectiveness of three successive nationwide lockdown enforced in India, we presenta data-driven analysis of four key parameters, reducing the transmission rate, restraining the growth rate, flat-tening the epidemic curve and improving the health care system. These were quantified by the considerationof four different metrics, namely, reproduction rate, growth rate, doubling time and death to recovery ratio.The incidence data of the COVID-19 (during the period of 2nd March 2020 to 31st May 2020) outbreak inIndia was analyzed for the best fit to the epidemic curve, making use of the exponential growth, the maximumlikelihood estimation, sequential Bayesian method and estimation of time-dependent reproduction. The best fit(based on the data considered) was for the time-dependent approach. Accordingly, this approach was used toassess the impact on the effective reproduction rate. The period of pre-lockdown to the end of lockdown 3, sawa reduction in the rate of effective reproduction rate. During the same period the growth rate reduced from during the pre-lockdown to after lockdown 3, accompanied by the average doubling time increasingform - days to - days. Finally, the death-to-recovery ratio dropped from . (pre-lockdown) to . after lockdown 3. In conclusion, all the four metrics considered to assess the effectiveness of the lockdown,exhibited significant favourable changes, from the pre-lockdown period to the end of lockdown 3. Analysis ofthe data in the post-lockdown period with these metrics will provide greater clarity with regards to the extentof the success of the lockdown. Keywords: Lockdown; Reproduction Number; Estimation; COVID-19
NTRODUCTION
As of 5th June 2020, the coronavirus disease 2019 (COVID-19) with its epicenter in Wuhan, China [1], hasresulted in more than . million confirmed cases and , , causalities [2]. The global pandemic resultingfrom COVID-19 was preceded by two other outbreaks of human coronavirus, in the 21st century itself, namely,severe acute respiratory syndrome coronavirus (SARS-CoV) and Middle East respiratory syndrome coronavirus(MERS-CoV) infections [3]. The possibilities of the source of the transmission of COVID-19 outbreak includes(but is not limited to) animals, human-to-human and intermediate animal-vectors [3]. The index case for COVID-19 outbreak in India was reported on 30th January 2020, in case of an individual with a travel history fromWuhan, China [4]. The data available on [4], suggests that during the early stages, the COVID-19 positive casesin India, were limited to individuals with a travel history involving the global hotspots of the outbreak. However,subsequently, cases were detected in individuals who neither had a travel history involving the global hotspots,nor had any contact with individuals who were already infected, which indicated the possibility of communityoutbreak. This resulted in the Government of India announcing a lockdown across the country, driven by thenecessity of ensuring that the social distancing norms are strictly observed. While the lockdown was not theonly response to the pandemic, it was a very crucial step towards curbing the growth of COVID-19 in densely ∗ Department of Mathematics, Indian Institute of Technology Guwahati, Guwahati-781039, Assam, India, e-mail:[email protected] † Department of Mathematics, Indian Institute of Technology Guwahati, Guwahati-781039, Assam, India, e-mail: [email protected],Phone: +91-361-2582606, Fax: +91-361-2582649 a r X i v : . [ q - b i o . P E ] J un opulated countries, like India. Given the concurrent economic cost of the lockdown, it is even more criticalfrom the epidemiological as well as economic perspective, to assess its effectiveness. This paper presents a data-driven analysis to examine the effectiveness of the lockdown, with an emphasis on the question as to whetherthe lockdown succeeded in curbing the intensity of COVID-19 spread rate in India ? In order to answer this, weempirically analyze four different metrics, namely, reproduction number, growth rate, doubling time and deathto recovery ratio, which quantify the transmission rate, the growth rate, the curvature of epidemic curve and theimprovement of health care capacity, respectively.We now give a brief summary of some of the available literature on quantitative approaches to the modeling oftransmission of COVID-19 outbreak. A system of ordinary differential equation (ODE) driven model for phasictransmission of COVID-19, was analyzed for calculating the transmissibility of the virus, in [5]. Kucharskiet al. [1] considered a stochastic transmission model on the data for cases in Wuhan, China (including casesthat originated there) to estimate the likelihood of the outbreak taking place in other geographical locations. Aliterature survey by Liu et al. [6], summarized that the reproductive number (and hence the infectivity) in case ofCOVID-19, exceeded that of SARS. A Monte-Carlo simulation approach to assess the impact of the COVID-19pandemic in India, was carried out in [7]. In carrying out the mathematical and statistical modeling of COVID-19,it would be helpful to refer to the quantitative models analyzed in case of the two preceding outbreaks of humancoronavirus, namely SARS and MERS. In [8], a network model was analyzed to identify localized hotbeds, aswell as super-spreaders for SARS. Constrained by somewhat limited availability of data, a simple compartmentmodel was used in [9], for in-silico predictive analysis of SARS outbreak in Beijing, China. Yan and Zou [10],determined the optimal and sub-optimal strategies for quarantine and isolation in case of SARS. A predictivemodel in [11], on imported cases of MERS, was used to ascertain the likelihood of a MERS diagnosis, during thetime window between immigration and onset of the disease. The trajectory of MERS outbreak was calibrated toa dynamic model in [12], with the goal of studying the role of time, in implementing the control measures.A key identifier for the transmissibility of epidemiological diseases such as COVID-19 is the basic reproduc-tion number R , which is defined as the average number of secondary infections resulting from an infected case,in a population whose all members are susceptible. Accordingly, we seek to estimate the data-driven value of R , for the outbreak of COVID-19 in India. Further, we also seek to determine the time-dependent reproductionnumber R t , for better clarity on the time-variability of the reproduction number, particularly in the paradigm ofits dynamics during the phases of the nationwide lockdown in India. In addition, we also estimate and analyze thestatistical performance of growth rate, doubling time and death to recovery ratio.The paper is organized as follows. In Section 2, we detail the source of the data as well as the statisticalapproaches used for the estimation of R and R t . This will be followed by the discussion of the results for theoutbreak in India, in Section 3. In Section 4, we present the data driven analysis of the impact of the lockdown.And finally, in the concluding remarks in Section 5, we highlight the main takeaways for this analysis.2 M ETHODOLOGY FOR ESTIMATING REPRODUCTION RATE
The data of incidences used for the analysis reported in this paper was obtained from the website of IndiaCOVID 19 Tracker [4], and used for the purpose of estimation of R . This estimation was carried out makinguse of the R0 package [13] of the statistical package R . The standardized approach included in the R0 packageincludes the implementation of the Exponential Growth (EG), Maximum Likelihood (ML) estimation, SequentialBayesian (SB) method and estimation of time dependent reproduction (TD) numbers, used during the H1N1pandemic of 2009. The package is designed for the estimation of both the “initial” reproduction number, as wellas the “time-dependent” reproduction number. Accordingly, we present a brief summary of the four approachesused in the paper.1. Exponential Growth (EG):
As observed in [14], the reproduction number can be indirectly estimated fromthe rate of the exponential growth. In order to address the disparity in the different differential equationmodels, the authors observe that this disparity can be attributed to the assumptions made about the shapeof the generation interval distribution. Accordingly, the choice of the model, used for the estimation of thereproduction number, is driven by the shape of the generation interval distribution. Based on the assumption2hat the mean is equal to he generation intervals, the authors obtain the important result of determining anupper bound on the possible range of values of the reproduction number for an observed rate of exponentialgrowth, which manifests into the worst case scenario for the reproductive number. Let the function g ( a ) be representative of the generation interval distribution. If the moment generating function M ( z ) of g ( a ) is given by M ( z ) = ∞ (cid:90) e za g ( a ) da , then the reproduction number is given by R = 1 M ( − r ) subject to thecondition that M ( − r ) exists. In particular, the Poisson distribution can be used in the analysis of theinteger valued incidence data [15, 16], for (discretized) generation time distribution. An important caveat isthat this approach is applicable to the time window in which the incidence data is observed to be exponential[13].2. Maximum Likelihood (ML) estimation:
The maximum likelihood model as proposed in [17] is based on theavailability of incidence data N , N , . . . , N T , with the notation N t , t = 0 , , , . . . , T denoting the countof new cases at time t . In practice, we take the index t in days, while noting that this indexing is applicablefor other lengths of time intervals. This approach is driven by the assumption that the Poisson distribution,models the number of secondary infections from an index case, with the average providing the estimatefor the basic reproduction number. If we denote the number of observed incidences for consecutive timeintervals by n , n , . . . , n T and let p i denote the probability of the serial interval of a case in i days (whichcan be estimated apriori), then the likelihood function is the thinned Poisson: L ( R , p ) = T (cid:89) t =1 e − µ t µ n t t n t ! .Note that here µ t = R k,t ) (cid:88) i =1 n t − i p i and p = ( p , p , . . . , p k ) . The absence of data from the index casecan lead to an overestimation of the initial reproduction number, and accordingly a correction needs to beimplemented [13].3. Sequential Bayesian (SB) method:
A SIR model driven sequential estimation of the initial reproductionnumber was carried out by the sequential Bayesian method in [18]. It is based on the Poisson distri-bution driven estimate of incidence n t +1 at time t + 1 with the mean of n t e γ ( R − . In particular, theprobability distribution for the reproduction number R , based on the observed temporal data is givenby P [ R | n , n , . . . , n t +1 ] = P [ n , n , . . . , n t +1 | R ] P [ R ] P [ n , n , . . . , n t +1 | R ] , where P [ R ] is the prior distribution of R and P [ n , n , . . . , n t +1 ] is independent of R .4. Estimation of time dependent reproduction (TD):
The TD method is amenable to the computation of thereproduction numbers through the averaging over all networks of transmission, based on the observed data[19]. Let i and j be two cases, with the respective times of onset of symptoms being t i and t j . Further, let p ij denote the probability of i being infected by j . If g ( a ) denotes the distribution of the generation interval,then p ij = g ( t i − t j ) (cid:80) i (cid:54) = k w ( t i − t k ) . Accordingly, the effective reproduction number is given by R j = (cid:88) i p ij , whoseaverage is then given by R t = 1 n t (cid:88) t j = t R j . In absence of observed secondary cases, a correction can be madeto the time dependent estimation [20]3 E STIMATING THE REPRODUCTION NUMBERS AND FITTING THE EPIDEMIC CURVE
In this section, we undertake the fitting of the epidemic curve and the estimation of the reproduction numbersusing the approaches enumerated in Section 2. We have obtained the daily incidence data, for the period of 2nd3arch 2020 to 31st May 2020 [4]. The epidemic curve based on the data, for this period, is depicted in Figure 1,which indicates that the number of COVID-19 positive cases, were growing in an almost exponential manner.
Date D a il y c a s e s Figure 1: Epidemic curve for the period of 2nd March 2020 to 31st May 2020The initial reproduction number R according to the EG is . , with the confidence interval (CI) forthis estimation being [1 . , . . For the case of the ML method, the R is determine to be . and thecorresponding CI is [1 . , . .Method R /R t [1 . , . ML 1.26 [1 . , . SB 1.591 [1 . , . , ] TD 1.68 [1 . , . Table 1: Initial and time varying reproduction numbers using the four methodsFor the estimation of time-varying reproduction numbers or the effective reproduction numbers R t , the gen-eration time distribution is required. Accordingly, we use gamma distribution with mean of . days and thestandard deviation of . days as reported from China [21]. Now, the average R t , using the SB and TD methods,are . and . , respectively. The R values using EG and ML, and the R t values using SB and TD, along withthe corresponding confidence intervals are tabulated in Table 1. Further, the seven-day rolling R t , obtainedfor the cases of SB and TD, are plotted in Figure 2 and Figure 3, respectively.4 .00.51.01.52.02.53.03.54.04.55.0Mar Apr May Date R ( t ) Figure 2: Seven-day rolling R t using SB Method Date R ( t ) Figure 3: Seven-day rolling R t using TD methodBesides estimating the reproduction rate, we fit the epidemic curve, making use of the four models, namelyEG, ML, SB and TD. Accordingly, the predicted incidence (based on the fitted model parameters in each case)and the observed incidence for each method, are illustrated in Figure 4.5 Date D a il y c a s e s Exponential Model (a) EG
Date D a il y c a s e s Maximum Likelihood Model (b) ML
Date D a il y c a s e s (c) SB Date D a il y c a s e s Time−dependent Model (d) TD
Figure 4: Epidemic curve using the four methodsThe prediction provided by the EG, ML and TD, are reasonably close to the actual cases. However, it is clearlyobserved that the most poorly fitted model is the SB model. The SB model overestimates the epidemic curve, andthus the predictions according to this model are much higher than the actual incidences. Therefore, in order tofind the best-fitted model, the root mean squared errors,
RM SE := (cid:118)(cid:117)(cid:117)(cid:116) n (cid:88) i =1 ( ˆ y i − y i ) n , for all the models werecalculated. As expected, the RMSE for the SB model is the highest. On the other hand, the TD model has thelowest RMSE. The RMSE values for all the four models are tabulated in Table 2, from where we can conclude(based on the data set considered) that the best model for the estimation of the COVID-19 epidemic in India, isthe TD model. EG ML TD SB430 368 289 15377Table 2: RMSE for the four methods4 I MPACT OF LOCKDOWN
The nationwide lockdown was imposed, on 25th March, 2020, with the goal of arresting the spread of infec-tion, through strict restrictions on mass movement and encouraging social distancing, and it was expected thatthe spread rate would come down, along with the reduction in the possibility of community transmission. This inturn would result in curbing the number of cases from rising dramatically, thereby enabling the healthcare systemwith more time to make necessary arrangements for the better preparedness of the medical infrastructure. Thus,6he first phase of lockdown until 14th April, 2020, was extended to another two phases of lockdown, with slightlyrelaxed restrictions, and were enforced from 15th April to 3rd May, 2020 and from 4th May to 31st May, 2020.This section discusses the impact of the entire lockdown on COVID-19 spread, by analyzing various metrics,namely, the effective reproduction rate, the growth rate, the doubling time and the death to recovery ratio.4.1 I
MPACT ON EFFECTIVE REPRODUCTION RATE
One of the key mathematical indicator relied upon, in the paradigm of the spread of COVID-19 pandemic andconsequent policy decisions is the effective reproduction rate (ERR) or the time-varying reproduction number.As ERR provides the information of time varying transmission rate, it would be a natural choice to measure theimpact of the entire lockdown, as well as different phases of the lockdown. In the preceding Section 3, we haveshown that, amongst all the models, the TD is the best fitted model, for the Indian epidemic curve. Hence, wediscuss the impact of lock-down in the context of the TD-based R t . Date R ( t ) (a) R t using TD L0 L1 L2 L3
Time period A v g − da y R ( t ) (b) Average seven-day R t Figure 5: Impact of lockdown on ERRFigure 5a depicts the seven-day rolling ERR. It is clearly observed that, before the lockdown, the R t wasunsteady, but it started dipping downward after the commencement of the lockdown. In the pre-lockdown period,the average seven-day ERR was 2.23. Therefore, before the lockdown, if individuals had COVID-19, theywould have infected people on an average. In the first lock-down period, the average ERR came down to . , a 22% drop. Thus, at this rate, carriers would infect others on an average. In the second and thirdlockdown periods, the ERR furthers dipped to . and . , respectively. Therefore, from the pre-lockdown tothe end of lockdown 3, the overall rate of reduction of ERR was nearly . Figure 5b displays the phase-wise average R t . The descriptive statistics of R t and the corresponding confidence intervals are described in Table3. From these results, we can clearly infer that, so far, the lockdown has by and large succeeded, in reducing theERR. However, this observation come with the caveat that the three successive lockdowns did not drive the R t below , which is suggestive that the epidemic may exhibit a surge once all the restrictions are lifted.Periods 7-day ERR 95%-CIMin Max AverageL0 1.73 3.466 2.23 [1 . , . L1 1.30 2.317 1.73 [1 . , . L2 1.23 1.42 1.31 [1 . , . L3 1.07 1.29 1.22 [1 . , . Table 3: Reproduction rate and -CI in different periods L0, L1, L2 and L3 imply pre-lockdown, lockdown 1, lockdown 2 and lockdown 3, respectively
MPACT ON GROWTH RATE
The reduction of ERR should further reduce the growth rate of daily incidences. In order to see the growthrate, in a particular time period, we calculate the seven-day rolling growth rate in that period, and then take theaverage. Suppose that we have daily incidence numbers, D ( t ) , t = 1 , , , . . . , , for a period of days. Wefirst compute the seven-day rolling growth rates, D ( i + 7) − D ( i ) D ( i ) , where i = 1 , , , . . . , , and we get a datasetof points. Finally, the simple mean of the dataset is calculated. If the seven-day average growth is in amonth, then the average weekly number of positive cases would have increased from to in that month.
393 191 47 32
L0 L1 L2 L3
Time period G r o w t h r a t e ( % ) Figure 6: Weekly growth rate of positive casesFigure 6 illustrates the average weekly growth rate in different time periods. In the pre-lockdown period (L0),the growth rate was 393%. It means that the weekly number of positive cases, increased drastically from to in the pre-lockdown period. The growth rate has decreased to in lockdown 1 (L1). It further reducedto and in lockdown 2 (L2) and lockdown 3 (L3), respectively. Therefore, we can conclude, that theimplementation of nationwide lockdown has resulted in slowing down the growth rate of COVID-19 positivecases. 4.3 I MPACT ON DOUBLING TIME
One of the key indicator to see the spread of any pandemic is the doubling time. It is referred to as the time(usually counted in number of days) it takes for the total number of cases to double. The doubling time of n daysmeans that if there were cases at day 0, then, on day n , the number of cases would be . The more thedoubling time is, the more the possibility of achieving a flattened epidemic curve.8 Time period N o o f da ys (a) Doubling time L0 L1 L2 L3
Time period N o o f da ys (b) Average doubling time in different periods Figure 7: Impact of lockdown on doubling timeFigure 7a displays the doubling rate for five-day moving averages. The escalation in doubling time is easilyseen from the figure. The doubling time during third lockdown period was about - days, up from - daysprior to the commencement of the lockdown. The phase-wise average doubling timings are shown in Figure 7b.The increment in doubling time is clearly visible from this figure. Therefore, from these results, we infer that thedoubling time has improved significantly after the enforcement of nationwide lockdown.4.4 I MPACT ON DEATH TO RECOVERY RATIO
In a pandemic, the performance of any nation’s health care system, is measured ultimately in terms of deathsand recoveries. This segment discuses the effect of lockdown on death to recovery ratio (DTR). The DTR isdefined as a ratio between total number of deaths and total number of recoveries:
DT R t = Total number of deaths upto time tTotal number of recoveries upto time t . The DTR stipulates the clinical management ability or the efficiency of health system. It is highly important tokeep the value of the DTR as low as possible. Mathematically, the closer this value is to zero, the better theefficiency of healthcare system, in dealing with the pandemic. For example,
DT R t = 0 . implies that, for every recoveries, infected patients would have died. The seven-day rolling DTR is plotted in Figure 8a. It isclearly seen that the DTR has declined significantly as time has progressed. The phase-wise bar chart also depictsthe reduction of DTR over the period of three months. In pre-lockdown (L0) and lockdown 1 (L1) periods together,the average DTR was 0.28. It reduced to . in lockdown 2 (L2) and further declined to . in lockdown 3 (L3),which shows that, in this short period, the Indian health care system has been improved significantly to tackle theCOVID-19 pandemic. 9 .000.050.100.150.200.250.300.350.40 Mar 15 Apr 01 Apr 15 May 01 May 15 Time period D T R (a) Death to recovery ratio L0+L1 L2 L3
Time period A v g . D T R (b) Average death to recovery ratio Figure 8: Impact of lockdown on death to recovery ratio5 C
ONCLUSION
In this paper, we have discussed the impact of lockdown on COVID-19 infection rate, in India. The aim wasto see whether the lockdown has really curbed intensity of spread. In order to do that, we empirically analyzeddifferent metrics that mainly measure the spread of infectious disease, like COVID-19. The metrics are effectivereproduction rate, growth rate, doubling time and death to recovery ratio (DTR). For case of ERR, it is seenthat the lockdown has reduced the reproduction rate by more than . The growth rate has also substantiallydecreased from the initial period to the end of lockdown. On the other hand, the doubling time has largelyimproved over the three month period. The rate of increment from pre-lockdown to lockdown 3 is nearly .Finally, we described the impact on DTR, which quantifies the number of death against the number of recoveries.We observed significant downfall of DTR from the month of April. On average, the initial DTR of . has dippeddownward to . at the third phase of lockdown. Therefore, despite rising cases of COVID-19 infection in India,the lockdown has managed to curb the spread to some extent. However, the caveat is that, despite the encouragingresults, the pandemic will persist, unless the ERR is driven below . It remains to be seen if there is a adversemovement of the metrics, after the relaxation of the restrictions. The behaviour of these metrics in post-lockdownperiod will provide a more accurate and complete information regarding the success or failure of lockdown.A CKNOWLEDGMENTS
This work was carried out under approved Grant No. MSC/2020/000049 from the Science and EngineeringResearch Board, Government of India. R
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