A model for the spread of an epidemic from local to global: A case study of COVID-19 in India
aa r X i v : . [ phy s i c s . s o c - ph ] J un A model for the spread of an epidemic from localto global: A case study of COVID-19 in India
Buddhananda Banerjee ∗ , Pradumn Kumar Pandey † , and Bibhas Adhikari ‡ Abstract.
In this paper we propose an epidemiological model for the spread of COVID-19.The dynamics of the spread is based on four fundamental categories of people in a population:Tested and infected, Non-Tested but infected, Tested but not infected, and non-Tested andnot infected. The model is based on two levels of dynamics of spread in the population: atlocal level and at the global level. The local level growth is described with data and param-eters which include testing statistics for COVID-19, preventive measures such as nationwidelockdown, and the migration of people across neighboring locations. In the context of India,the local locations are considered as districts and migration or traffic flow across districtsare defined by normalized edge weight of the metapopulation network of districts which areinfected with COVID-19. Based on this local growth, state level predictions for number ofpeople tested with COVID-19 positive are made. Further, considering the local locationsas states, prediction is made for the country level. The values of the model parameters aredetermined using grid search and minimizing an error function while training the model withreal data. The predictions are made based on the present statistics of testing, and certainlinear and log-linear growth of testing at state and country level. Finally, it is shown thatthe spread can be contained if number of testing can be increased linearly or log-linearly bycertain factors along with the preventive measures in near future. This is also necessary toprevent the sharp growth in the count of infected and to get rid of the second wave of pandemic.
Keywords.
COVID-19, metapopulation network, grid search
COVID-19 is a pandemic that is actively spreading in the whole world and is an unprecedentedchallenge for the human race. All the countries infected with COVID-19 are struggling tomitigate the spread through various strategies. This disease is spread by inhalation or contactwith infected droplets or fomites. It is observed that successful medical testing and as a result,detection of people infected with SARs-Cov-2 becomes one of the crucial control strategiesfor the spread of COVID-19 [1]. For instance, the epidemic curve in The Republic of SouthKorea suggests that this control strategy in South Korea has curtailed the epidemic. Besides,testing is also linked to tracing contact lists of the infected people and finally self-isolationof those people help against the spread. The success of containment of COVID-19 in theRepublic of Taiwan has also the influence of proactive testing [2].Given the fact that there in no effective antiviral vaccine or drug should coming soon,different prevention strategies are adapted by different countries that include voluntary orcompulsory quarantine, stopping of mass gatherings, closure of educational institutions orworkplaces, social distancing or even nationwide lockdown. However, these strategies may ∗ Department of Mathematics and Centre for Excellence in Artificial Intelligence, IIT Kharagpur, E-mail:[email protected] † Department of Computer Science and Engineering, IIT Roorkee, India Email: [email protected] ‡ Corresponding author, Department of Mathematics and Center for Theoretical Studies, IIT Kharagpur,India, E-mail: [email protected]
Introduction act less significant for the infected people who are at the pre-symptomatic stage, and inthat case they act as invisible spreaders for the disease [3]. Thus it becomes increasinglyimportant for mass medical testing for a country. Several researchers around the world areactively working on producing mathematical models of the spread of COVID-19. Here wequote that ‘model-based predictions can help policy makers make the right decisions in atimely way, even with the uncertainties about COVID-19 [4].The primary preventive steps adapted by the Government of India fall into five categorieswhich include social distancing, movement restrictions public health measures, social andeconomic measures, and nationwide lockdown. A few notable decisions by the Governmentof India are given in Table 7. It should be noted that a complete nationwide lockdown fromMarch 25 till May 13 helped to control the spread the disease at large distances but failedto prevent it in neighboring districts, as observed in [5]. For example, before lockdown,infected cases are reported from different districts across India which are at large distancesapart, however during the lockdown period it has been observed that new spread is reportedin districts which are neighbors of infected districts. Besides, due to lack of well plannedpolicy for migrant workers several of them have been travelling to their native districts duringlockdown. Unavailability of data of such a traffic flow across the districts can be crucial inorder to do a precise analysis of the spread. It can also be seen that the preventive measuresproposed by the Government of India are similar to those adapted in other counties.It is observed in various studies that COVID-19 exhibits significantly different epidemio-logical attributes than other well studied epidemics in past. Thus it is of paramount interestto develop mathematical models which can characterize the inherent dynamics of the spreadof COVID-19. Standard epidemic models such as SIR model considers human-to-humantransmission, and it describes the diffusion process through three mutually exclusive stages ofinfection: Susceptible, Infected and Recovered. These models are also called compartmentalmodels [6] which enables to compartmentalize different individuals based their states for theepidemic in a population. This model can help gain some insights about the growth of theinfection based on approximating the model parameters from the available data. However dueto a peculiar growth of COVID-19 in different countries, researchers have extended the SIRmodel and other existing models such as SIS model in order to acquire meaningful insightsabout spread of the COVID-19 [7]. It is very important to note that these studies can helpus to frame control strategies and policies that can mitigate the epidemic [8] [9].One of the first models for the spread of COVID-19 is proposed by Anastassopoulou et al.based on the data of confirmed cases reported at the Hubei province of China from the 11thof January until the 10th of February, 2020 [10]. They propose a discrete SIRD (Susceptible-Infected-Recovered-Dead) model and estimate the mean values of the corresponding epidemi-ological parameters such as basic reproduction number, the case fatality and case recoveryratio from the data. This model enables to forecast about the spread in near future. In ananother attempt, in [11] the authors study the datasets of transmission from within and out-side Wuhan, China to estimate how transmission in Wuhan varied between December 2019,and February 2020, and assess the potential for sustained human-to-human transmission tooccur in locations outside Wuhan through a stochastic transmission dynamic model. In [12],a mean-field epidemiological model is proposed for COVID-19 epidemic in Italy by extend-ing the classical SIR model. Here, in addition to susceptible (S) and infected (I), the otherstages of individuals are considered as diagnosed (D), ailing (A), recognized (R), threatened(T), healed (H) and extinct (E), collectively termed as SIDARTHE. In [6], an Age-stratifiedmodel of the COVID-19 is proposed to capture the age-dependent dynamics for nowcastingand forecasting for Switzerland. This model incorporates the compartments of symptomaticand asymptomatic infected individuals along with susceptible and exposed individuals. In[13], the authors propose a model of COVID-19 epidemic dynamics under quarantine condi-tions. They also develop methods to estimate quarantine effectiveness in a country or a regionwhich is infected with COVID-19. Besides, a few models are proposed for understanding andpredicting the spread of COVID-19 based on metapopulation network approach, see [14] [15] Introduction [16].Several mathematical models are also proposed based on the the available COVID-19 dataof India and fitting them into classical epidemic models incorporating other factors such asnationwide lockdown, social distancing etc., see [17] [18] [19] and the references therein. In[20], a mathematical model of the spread of COVID-19 is proposed based on an age-structuredSIR model. However, the comparison of this model prediction with real data is criticized byDhar in [21]. In [22], the authors perform state-wise analysis of the data of infected populationin different states based three models: Exponential Model, Logistic Model and the SIS model.They also provide state-wise prediction for number of infected people for different states inrecent future. An elementary network-based model for geographical spread of COVID-19 inIndia is proposed in [23] . In [24], a model for the spread of COVID-19 in India is proposedemphasizing on migration of population based on the spatial network of cities, incorporatingthe growth-dynamics of SIR model at the city-level.In this paper, we propose an epidemiological model for the spread of a contagious epidemicin a region or country. The entire model is based on combining two growth processes of thespread at local and global level. By local, we mean at the level of city or town or districts orprovince, and global mean at the level of state or country. First we develop a new discretemodel for the growth-dynamics of infected people at local level as follows. We consider fourtype of individuals living at a location. These are individuals who are tested as infected ( X ),tested as non-infected ( X ), untested but infected (asymptomatic or pre-symptomatic, X ),and untested and non-infected ( X ) for the disease. Total number of such individuals equalsthe total population living at that location. Given the time series data of these numbers X i ( t ) , t = 1 , , ,
4, we define the growth statistic X i ( t + 1) − X i ( t ) utilizing X j ( t ) , j = i and four other parameters each one of them is related to the the spreading pattern of thevirus which causes the disease. Note that the different standard compartmental models existin literature based on susceptible, infected, recovered, and diseased, which do not preservethe effect of parameters in an epidemic like COVID-19. In our proposed model, the growth-dynamics at local level include the following parameters:(a) Spread due to infected but asymptomatic and pre-symptomatic individuals(b) Effect of preventative measures like lockdown or restricted movement of individualsacross locations(c) Daily testing statistics.Then we consider the metapopulation network of all the locations at local level in orderto incorporate the transmission dynamics of disease at the global level. Here we mentionthat the metapopulation network model is a standard and popular model for analyzing thespread of highly contagious diseases which include Zika virus [25]. Also see [26] and thereferences therein. In our proposed model, the vertices of the metapopulation network arethe locations infected with the disease and the links connecting them represent the possiblemode of transportation or spatial distance such as the great circle distance of the latitude andlongitude coordinates of the locations at local level. The weight of these links, that representthe rate or percentage of transmission of population per unit time such as a day. Then thefinal model is defined by combining the dynamics of the spread at local and global level. Thevalues of the model parameters are obtained by a learning technique based on training dataand an error minimization.In the case of COVID-19, we consider the model parameters at the local level as testingstatistic, social distance, and rate of infected people by an infected but untested individual(asymptomatic or pre-sympotatic) per unit time. In the context of India, the locations areconsidered as districts which constitutes the states and union territories of India. There are28 states and 8 union territories in India, and there are a total of 718 districts in India.Based on the proposed model we predict number of COVID-19 infected people both atstate level and the country (India) level. The prediction depends on the number of testing The proposed model COVID+ ve COVID − ve TotalTested X [ l ]1 ( t ) X [ l ]2 ( t ) T l ( t )Non-tested X [ l ]3 ( t ) X [ l ]4 ( t ) ¯ T l ( t )Total C + l ( t ) C − l ( t ) N l Tab. 1:
Distribution of population in location l at time t performed per day. The results show that the total number of infected people at India levelwill be approximately 0.46 Millions on July 7, 2020, 1.9 Millions on November 7, 2020, and4.6 Millions on May 7 2021 when the number of testing is approximately 1,00,000 per dayat the country level (which is the number of testing as on May 7, 2020 approximately). Ifthe number of testing grows linearly (with a certain rate see Section 3.2) then the number ofpeople tested positively with COVID-19 would be approximately 2 Millions on July 7, 2020,59 Millions on November 7, 2020, and 130 Millions on May 7, 2021. Finally, if the number oftesting grows log-linearly (with a certain rate, see Section 3.2) then the number COVID-19infected people in India would be approximately 1.3 Millions on July 7, 2020, 3.77 Millions onNovember 7, 2020, 8.5 Millions on May 7, 2021. Note that these above mentioned predictionsare made when there is no external measure is used to control the spread, for example, usingany cure like a vaccine or drug discovered in between. Further using numerical simulationwe show that the spread stops when daily number of testing increases linearly or log-linearly,however if the number of testing remains approximately the same as of May 7, 2020 the spreadneed not stop in recent future, say in the year 2021. Let V = { l | l is the index of a location } be the set of locations where persons infected withCOVID-19 are likely to stay in or move to on a day t . Suppose that N l is the populationsize in location l . Now we introduce the following notations to model the distribution anddynamics of pandemic. If T l ( t ) denotes the number of tested individuals in the location l then¯ T l ( t ) = N l − T l ( t ) stands for the number non-tested individuals up to time t . Let C + l ( t ) and C − l ( t ) be the total number of people infected and non-infected with COVID-19, respectivelyin a location l ∈ V . Here these temporal data varies with time ( t ) measured in days. In anylocation l for a given day t, we define a random vector X [ l ] ( t ) = h X [ l ]1 ( t ) X [ l ]2 ( t ) X [ l ]3 ( t ) X [ l ]4 ( t ) i T with four components for the distribution of population the N l . Based on the above discussion X [ l ] ( t ) can be represented in a 2 × X j =1 X [ l ] j ( t ) = N l , ∀ l ∈ V , the total population at the location l , though X [ l ]3 ( t ) & X [ l ]4 ( t ) are unobserved or latent randomvariables. Unlike the standard epidemic models, the asymptomatic infected people or whoare infected with COVID-19 but not tested, that is, X [ l ]3 ( t ) may influence the number X [ l ]1 ( t ′ )at a future date t ′ > t. Besides, C + l ( t ) highly depends on the contact networks of C + l ( t ′′ ) at aprevious date t ′′ < t. But only X [ l ]1 ( t ) is observed. Thus the number of people who are testedfor COVID-19 at a given day governs the dynamics of X [ l ] ( t ) at a location l over time. The proposed model Let e T l ( t + 1) be a strategic number which provides the target quantity of new tests forCOVID19 to be performed on day ( t + 1) in location l. Given the statistic X [ l ] ( t ) , new testsalso depends on the availability of test-kits. However, this also depends on ¯ T l ( t ), the numberof people not tested for the disease at the location l. Hence, we define the possible number oftests to be performed at l as e T ∗ l ( t + 1) = min { e T l ( t + 1) , ¯ T l ( t ) } . (1)In Table 2, we introduce some generic notations of model-parameters that are used todevelop the dynamics of the system and some more hyper-parameters that are involved intraining and updating of model-parameters. All the parameters modified with suffix/super-fixaccording to the time and locations accordingly.Parameters Interpretations λ Testing-coverage probability among the infected λ Infection spreading probability λ Probability of population migration among locations α Average family size θ Mobility of individuals ǫ Error parameterHyper-parameters Interpretations α Changing rate of λ α Changing rate of λ for future β Changing rate of λ r Rate of increment in testing under linear growth. r Rate of increment in testing under log-linear growth.
Tab. 2:
Model parameters and hyper-parametersNow we define the dynamics of change of X [ l ] ( t ) for any location l.X [ l ]1 ( t + 1) − X [ l ]1 ( t ) = ∆ t X [ l ]1 = bin (cid:16) min { e T ∗ l ( t + 1) , X [ l ]3 ( t ) } , λ [ l ]1 ( t + 1) (cid:17) (2) X [ l ]2 ( t + 1) − X [ l ]2 ( t ) = ∆ t X [ l ]2 = min { e T ∗ l ( t + 1) − ∆ t X [ l ]1 , X [ l ]4 ( t ) } , (3) X [ l ]3 ( t + 1) − X [ l ]3 ( t ) = ∆ t X [ l ]3 = max {− ∆ t X [ l ]1 + min { a [ l ] ( t + 1) , X [ l ]4 ( t ) } , − X [ l ]3 ( t ) } (4) X [ l ]4 ( t + 1) − X [ l ]4 ( t ) = ∆ t X [ l ]4 = max {− ∆ t X [ l ]2 − min { a [ l ] ( t + 1) , X [ l ]4 ( t ) } , − X [ l ]4 ( t ) } (5)where a [ l ] ( t + 1) = bin( α ∆ t X [ l ]1 , λ [ l ]2 ( t + 1)) + Pois λ [ l ]3 ( t + 1) N X k =1 m kl ( t ) X [ l ]3 ( t ) ! + Pois( ǫ ) .λ [ l ]1 ( t + 1) ∈ (0 ,
1) is testing-coverage probability among the infected in location l attime ( t + 1). Hence, only a fraction of X [ l ]3 ( t ) will be will be identified as ∆ t X [ l ]1 . So it ismodelled with binomial distribution. λ [ l ]2 ( t + 1) ∈ (0 ,
1) is a probability indicating the averagespread of infection among near by people of a group of infected individuals. So, new spreadidentified-infected people is also modelled with binomial random variable bin( α ∆ t X [ l ]1 , λ [ l ]2 ( t +1)). Now λ [ l ]3 ( t + 1) ∈ (0 ,
1) is a probability closed to zero indicating the influence fromadjacent locations. As a consequence it is modelled with Pois (cid:16) λ [ l ]3 ( t + 1) P Nk =1 m kl ( t ) X [ l ]3 ( t ) (cid:17) .Parameter ǫ > X [ l ]3 ( t ) and X [ l ]4 ( t ) are latent variables at a given day t. The parameters λ j ( t + 1) , j = 1 , , The proposed model Now we consider the meta-population network G ( t ) with vertex set V of locations in orderto incorporate the effect of transmission of COVID-19 across the locations. Let A ( t ) = [ a kl ( t )]denote the adjacency matrix associated with G ( t ) . Let d kl denote the distance between k and l. Then define the weights of the edges of G ( t ) as w kl ( t ) ∝ exp (cid:26) − d kl θ ( t ) (cid:27) (6)where θ ( t ) is the mobility parameter. Here w kl denotes diffusion weight for the human trafficflows per day between the neighboring locations k and l. The value of θ ( t ) > θ ( t ) may be considered as a small value. Now we define the matrix M ( t ) = [ m kl ( t )]where m kl = w kl ( t ) P |V| l =1 w kl ( t )which is a row-stochastic matrix. Finally we propose the following predictive model at thelevel of state and country for the number of COVID-19 infected people.Note that the traffic flow between locations influences the value of X [ l ]4 ( t + 1) as followedby Eq. (5) which contribute to X [ l ]3 ( t + 1) and finally to the number of infected people X [ l ]1 ( t + 1) . Besides, the number of nodes in the metapopulation network G ( t ) varies withtime. At time t, the nodes of G ( t ) represented the districts which are affected by the diaeaseat time t. Thus at the level of state S which consists of some locations, X [ S ]1 = P l ∈ S X [ l ]1 atanytime t. Further, the number of infected people at the country level is calculated based onthe proposed dynamics of X [ l ] i , i ∈ { , , , } where l is a state. This is done presumably dueto the traffic flow between neighboring districts may be different from the traffic flow betweenneighboring states. Hence, at the country level, say India, denoted by I, X [ I ]1 = P S ∈ I X [ S ]1 at anytime (day) t. In this section we discuss how to determine the values of the parameters involved in theproposed epidemiological model. Note that the initial values can be assumed wisely based onits characteristics observed from data and then as the time passes the model can update thevalues of the parameters from observed and simulated data. Let [ t , t ] be the learning periodthroughout which the real data is available and the model can learn the data for estimating thevalues of the parameters. Consequently, the growth-dynamics of parameters can be definedwhich can update the values of the parameters when the real data is not available in future.First we consider the parameter λ [ l ]1 ( t ) . Then define λ [ l ]1 ( t + 1) = λ [ l ]1 ( t ) + α e X [ l ]1 ( t ) − X [ l ]1 ( t ) m [ l ] , (7)for some α ≥ m [ l ] = max t ′ ≤ t | e X [ l ]1 ( t ′ ) − X [ l ] ( t ′ ) | , where, e X [ l ]1 ( t ) is the reported number of tested-positive cases in location l at time t , and X [ l ]1 ( t ) is the value of tested-positive cases obtained from simulation. In Eq. (7), infectionspreading rate λ [ l ]1 ( t ) is updated in such a way that if the number of tested and infected casesare more than the simulated values then infection spreading rate would be more as comparedto current rate of of infection and vice versa. The value of α represents the slope of the linealong which λ [ l ]1 ( t ) increases with time linearly.Eq. (7) is explained pictorially, in Figure 1, consider that points connected via black linesare corresponding to real data points, and points connected via green lines are corresponding The proposed model to simulated points using λ [ l ]1 ( t ) = λ [ l ]1 ( t + 1). In such scenario error e X [ l ]1 ( t ) − X [ l ]1 ( t ) increases.For better fit of the model we need to update the parameter λ [ l ]1 ( t ) in such a way that X [ l ]1 ( t +1)can come closer to X ′ [ l ]1 ( t + 1). In figure, e X [ l ]1 ( t ) > X [ l ]1 ( t ). X [ l ]1 ( t + 1) is the number of testedpositive cases, if we increase the rate of infection spread λ [ l ]1 ( t ) then we can get X [ l ]1 ( t + 1)closer to e X [ l ]1 ( t + 1), point X [ l ]1 ( t + 1) connected to point X [ l ]1 ( t ) via blue line.For the growth of λ [ l ]2 ( t ) over time which represents the probability of the spread of thedisease at a location l. Thus we define λ [ l ]2 ( t + 1) = λ [ l ]2 ( t ) − β e T l ( t + 1) − e T l ( t ) P Nj =1 e T j ( t + 1) ! , (8)where β ≥ e T l ( t ) denotes the number of tests performed at the location l at time t . Hereobserve that, the intuition behind Eq. (8) is that the probability of spread of the diseasedepends on the number of testings done at the location l. We consider constant values of λ [ l ]3 ( t ) and ǫ in current version. Fig. 1:
Figure explains the way to update infection spreading rate.
Recall that individuals who are at the asymptomatic and pre-symptomatic stages of infection,act as invisible spreaders for the disease. Hence, detection of individuals who are infectedwith the virus plays an important role into the growth-dynamics of the number of infectedindividuals at a particular location. Thus one of the control strategies to prevent the spreadis to conduct enough number of tests per day and separate-out the infected people. In acountry like India, where approximately 1.4 billion people live, conducting enough tests perday could be a difficult exercise. Besides, due to lack of huge number of test-kits and medicalfacilities, India is facing a lot of challenges to perform enough tests per day. The testing datain India is plotted in Fig. 2 which is obtained from [27]. It may be observed that the datais not available for three consecutive days after the 30th day. Besides the testing data is notavailable before March 19, 2020.Note that testing for COVID-19 for random sampling of individuals is not desired due toscarcity of enough testing kits for a large population and medical support facilities. Indeed,targeted testing by tracing social contacts of newly detected individuals with COVID-19can be more efficient for identifying asymptomatic and pre-symptomatic individuals who areinfected with the virus. Hence the increment of number of testing per day should depend onthe testing-coverage probability among the infected individuals at a particular location, thatis, λ [ l ]1 . The proposed model Fig. 2:
Testing performed daily in India, from March 19,2020 to May 20,2020 [27].In this model we incorporate two possible growth of testing data over time at a location:linear and log-linear. The parameters which we call rate of gain in the number of tests forCOVID-19, are denoted by r and r for the following linear and log-linear growth equationsrespectively. From the real data it can be observed that the number of tested positive caseshas positive correlation (0 . λ [ l ]1 ( t )has positive dependency over the number of tested positive cases. Thus, the number of testsperformed has positive relation with λ [ l ]1 .Let e T l ( t ) denote the number of tests performed at a location l on a day t. Then definelinear increment of testing: e T l ( t + 1) = l e T l ( t ) + r λ [ l ]1 ( t ) m . (9)and log-linear increment of testing: e T l ( t + 1) = l(cid:16) r λ [ l ]1 ( t ) (cid:17) e T l ( t ) m , (10)Thus assigning small values of λ [ l ]1 ( t ) , λ [ l ]2 ( t ) in the beginning of the simulation of the model, λ [ l ]1 ( t ), λ [ l ]2 ( t ), and e T l ( t ) are updated according to Eqs. (7), (8), and (10) or (9) respectively.Further, α , β , and r can be selected from the interior of the unit cube given by (0 , × (0 , × (0 , , whereas r can be larger than 1. The searching method is well-known as as threedimensional grid search . Indeed, mapping the growth given by Eqs. (7), (8), and (10) withreal data, the values of α , λ [ l ]1 ( t ), β , λ [ l ]2 ( t ), r (or r ), and e T l ( t ) can be learned and estimatedsuch that the total testing (cid:16)P l e T l ( t ′ ) (cid:17) , and total tested and infected cases (cid:16)P l X [ l ]1 ( t ′ ) (cid:17) attime t ′ ≤ t that are close to real data. It is discussed in details in the next subsection. Theseestimated values can be used for the training of the model.Suppose that α , λ [ l ]1 ( t ′ ), β , λ [ l ]2 ( t ′ ), r ′ (or r ′ ), and e T l ( t ′ ) are the learned values from thegiven data. However, for any t > t ′ when the real data are not available, the trained modelcan be used for prediction. Thus we define the update of λ [ l ]1 ( t ) as follows: λ [ l ]1 ( t + 1) = λ [ l ]1 ( t ) + α X [ l ]1 ( t ) − X [ l ]1 ( t −
1) + X [ l ]1 ( t − P Nl =1 (cid:16) X [ l ]1 ( t ) − X [ l ]1 ( t − (cid:17) , (11) The proposed model Let X [ l ]1 ( t ) and e X [ l ]1 ( t ) be the simulated and observed numbers of detected after test as infectedwith COVID-19 respectively at a location l at the time (day) t. Consider the time series ofreal data e X [ l ]1 ( t ) where t ≤ t ≤ t , for a particular location l ∈ V which is the vertex set ofthe metapopulation network. Then the complete observed data-set is given by X = { e X [ l ]1 ( t ) : l ∈ V , t ≤ t ≤ t } . Then the data X is divided into two sets which we call the training setand validation set as follows for estimating the model parameters which define X [ l ]1 ( t ) . Let t ′ ∈ ( t , t ) . Define X T = { e X [ l ]1 ( t ) : l ∈ V , t ≤ t ′ } (Training set) (12) X V = { e X [ l ]1 ( t ) : l ∈ V , t ′ < t ≤ t } . (Validation set) (13)The model parameters are calculated which minimize the error function e = wT e + (1 − w ) V e (14)where w = | X V || X T | + | X V | (15) T e = 1 | X T | X l ∈V , e X [ l ]1 ( t ) ∈ X T | e X [ l ]1 ( t ) − X [ l ]1 ( t ) | (16) V e = 1 | X V | X l ∈V , e X [ l ]1 ( t ) ∈ X V | e X [ l ]1 ( t ) − X [ l ]1 ( t ) | . (17)Note that the weight w is defined such that T e and L e are computed over two differentsets X V and X T respectively to avoid the imbalances in the data. Now we discuss how the values of the model parameters estimated by real data at locallocation can be used to predict the number of infected people at a global level such as stateand country level in near future. We propose to train the model based on two methodologiesat the the state level and country level. Recall that a state in India consists of several districts(locations denoted by l ), and in India there are 28 states and 8 union territories. In this paperwe adapt two-step approach for the prediction. The global level parameters include the socialmobility parameter θ , and the traffic flow across the local level locations, given by the edgeweight of the metapopulation network.First, we make state level prediction, that is, X [ S ]1 = P l ∈ S X [ l ]1 ( t ) when X [ l ]1 is consideredat district level l , where S is a state of India. The metapopulation network for a state S isformed by the vertices which are districts belong to the state S, and the traffic flow which isrepresented by weights w kl defined by Eq. (6). The distance d kl between two districts k, l isdefined by the great circle distance between the longitude and latitude coordinates of k and l. Next, once the estimates for X S are obtained for all states S in India, the prediction at thethe nation level is obtained by applying the proposed model treating the location as states.Thus model parameters are further estimated comparing with the real data at the level ofstates, as described above. Further, the metapopulation network of states is constructed,and the traffic flow is calculated using the wight formula w kl where the distance between twostates is considered as the great circle distance between the longitude and latitude coordinatesof states k and l. Prediction with model and real data: a case study of India Data (Learning)ModelData (Validation)
X: 78Y: 3.872e+04 (a) Data (Learning)ModelData (Validation)
X: 78Y: 1.509e+04 (b)
Data (Learning)ModelData (Validation)
X: 78Y: 1.245e+04 (c)
Data (Learning)ModelData (Validation)
X: 78Y: 5197 (d)
Fig. 3:
Plots are corresponding to data fitting (0-65 days), validation (66-72 days), and pre-diction of 78th day or May 20, 2020.Note that both predictions at state and country level incorporate the social mobilityparameter θ which preserves the effect of policies of the Government. For instance, duringlocklown the value of θ is considered as around 50 (more weights to local travel), and it willtake the value around 2000 (includes long distance travel) when there is no lockdown. Besidesthe metapopulation network between the locations l plays a crucial role into the prediction.The rate of traffic between two locations is considered as given by Eq. (6). Observed thatthe effect of social mobility of individuals is also incorporated with the traffic flow. In this section, the proposed model is trained with the data of infected population withCOVID-19 and number of testings performed in India from March 4, 200 to May 7, 2020[28]. Since there is nationwide lockdown during this period, the traffic flow across states isless. Therefore we simulate the model at an initial time t = 0 which is on March 4, 2020 bysetting λ [ l ]1 ( t ) = 1 × − , λ [ l ]2 ( t ) = 3 × − , r = 2 . × − and λ [ l ]3 ( t ) = 1 × − for all t ≥ t , for every location l. These values are assumed due to the following facts.(1) Testing-coverage probability is very small since the number of people infected withCOVID-19 in the beginning of the spread is small.(2) Since the average family size is 4, so approximately 3 people out of 10 may be exposedto get infected assuming that a person be in close and frequent contact with an groupof infected people.
Prediction with model and real data: a case study of India Data (Learning)ModelData (Validation)
X: 78Y: 5389 (a)
Data (Learning)ModelData (Validation)
X: 78Y: 5089 (b)
Data (Learning)ModelData (Validation)
X: 78Y: 3039 (c)
Data (Learning)ModelData (Validation)
X: 78Y: 2509 (d)
Fig. 4:
Plots are corresponding to data fitting (0-65 days), validation (66-72 days), and pre-diction of 78th day or May 20, 2020.(3) For simplicity of the model, we consider the value of r such that total number oftested individuals are close to real data. We obtained r = 0 .
28 during the optimization(training) using grid search. To match the number of testing performed each day,approximately 1 , ,
000 per day given after Eq. (10).(4) Due to nationwide lockdown, the traffic flow across the locations is less. Hence, θ ≤ λ ( t ) is very less. The values of θ and λ ( t ) [ l ] are obtained using grid search. Duringerror optimization, ( α , β , r , θ, λ [ l ]3 ) is selected from five dimensional grid search.After the initializing the model the parameter values are learned based on the real data.For instance, the number of testings throughout the period March 04, 2020 to May 20, 2020in India is approximately 10 per day [28, 27], and hence value of r is kept fixed during thetraining period of the model with real data. The value of number of testing at a location l isassumed as a random number between 1 to 5 when a first case of COVID-19 is reported.The model is trained with the real data collected from How India Lives [28, 27] for theperiod of March 04, 2020 to May 07, 2020 (65 days data). The remaining data, that is, thereal data for the period May 8 - 20, 2020 is validated based on estimated values of the modelparameters of the training dataset. A particular emphasis is given on the states where thereare more that 2000 cases of individuals infected with COVID-19 on May 7, 2020 (data usedfor training) which include Maharashtra (MH), Tamilnadu (TN), Gujrat (GUJ), Rajathan(RAJ), Madhya Pradesh (MP), Uttar Pradesh (UP), West Bengal (WB) and Andhra Pradesh(AP). The error function contains the absolute difference between data point (observed value)and corresponding value calculated using model. We define an error function which minimizes Prediction with model and real data: a case study of India Parameter MH TN GUJ RAJ MP UP WB AP INDIA α β θ
70 10 50 70 70 50 70 10 70 λ [ l ]3 ( t ) 1/100 1/100 1/100 1/20 1/10 1/1000 1/100 1/10 1/1000 error Tab. 3:
Trained values Model parametersthe absolute mean error of training and validation set, and trained model is used for predictionpurpose. The model parameters are learned by optimizing the error function as described inSection 2.3. On an average per location (district) error when the prediction is made at thestate level is given by 10.7050 (MH), 4.8018 (TN), 5.1519 (GUJ), 4.5936 (RAJ), 2.6368 (MP),2.3711 (UP), 2.5062 (WB) and 5.2155 (AP) when the number testing grows log-linearly as perthe rate r = 0 . . At the country level the error corresponding to real data and model basedprediction is calculated with error 37.2070 on average at each state when the testing growslog-linearly as above. Below we provide prediction at the level of state and India. Duringtraining the model, we consider log-linear gain in rate of testing, and error in fitting real datais reported in Table 3. After training and validation, two cases of gain in rate of testing areconsidered: linear and log-linear.
We consider 8 states which have highest number of tested positive cases. For each state,we learn a model and do the prediction of probable tested positive cases after 7 days of thelast day of validation data. We consider only those states which have sufficient data to trainthe model (at-least 2000 tested positive cases on May 7, 2020). Learned values of modelparameters corresponding to each state are given in Table 3.After training the models corresponding to data of each state, we do the prediction of totaltested positive cases in all the states on May 20, 2020. Predicted values and actual values arenoted in Table 4.In all the experiments performed in this work, we set ǫ = 2 , α = 4, and λ and θ areselected using grid search and values are given in Table 3. Day r MH TN GUJ RAJ MP UP WB APMay 20, 0 3 . × . × . × . × . × . × . × . × . × . × . × . × . × . × . × . × July 7, 0 1 . × . × . × . × . × . × . × . × . × . × . × . × . × . × . × . × (60 days) 5 × . × . × . × . × . × . × . × . × Nov 7, 0 3 . × . × . × . × . × . × . × . × . × . × . × . × . × . × . × . × (180 days) 5 × . × . × . × . × . × . × . × . × May 7, 0 7 . × . × . × . × . × . × . × . × . × . × . × . × . × . × . × . × (365 days) 5 × . × . × . × . × . × . × . × . × Tab. 4:
Prediction for states when the number of testing grows linearly at each district, basedon the training data up to May 7, 2020 and the validation data period is May 8 - 14,2020.
Prediction with model and real data: a case study of India Day r MH TN GUJ RAJ MP UP WB APJuly 7,2020 0 . . × . × . × . × . × . × . × . × (60 days) 0 . . × . × . × . × . × . × . × . × Nov 7,2020 0 . . × . × . × . × . × . × . × . × (180 days) 0 . . × . × . × . × . × . × . × . × May 7,2021 0 . . × . × . × . × . × . × . × . × (365 days) 0 . . × . × . × . × . × . × . × . × Tab. 5:
Prediction for states when number of testing grows log-linearly at each district basedon the training data up to May 7, 2020 and the validation data period is May 8 - 14,2020.
Maharashtra is the most affected state in India which has at-least 30% of total tested positivecases in the country on May 20, 2020. Apart from Maharashtra (MH), we consider TamilNadu (TN), Gujarat (GUJ), Rajasthan (RAJ), Madhya Pradesh (MP), Uttar Pradesh (UP),West Bengal (WB), and Andhra Pradesh (AP). Maharashra has 39297 tested positive casesas of May 20, 2020 and the proposed model predicts it as 38724 which is fairly close enoughto observed value. Similarly, TamilNadu, Gujarat, Rajasthan, Madhya Pradesh (MP), UttarPradesh (UP), West Bengal (WB), and Andhra Pradesh (AP) have total number of testedCOVID-19 infected cases as 13191, 12539, 6011, 5735, 5175, 3103, and 2560, and respectivepredicted values by the proposed model are 15092, 12453, 5197, 5389, 5089, 3039, and 2509as of May 20, 2020. Thus, we conclude that the model is able to learn and predict the totalnumber of tested positive cases in each of these states.In Figures 3 and 4, data and corresponding curve fitting is shown in which blue dotscorrespond to real data which are used as training data set, and the sky blue dots correspondto the validation data-set. The grey circles represent the trained and predicted values due tothe proposed model which are following the real data very well. Grey square in plots (X=78marked point in plots) indicates predicted value on May 20, 2020.Apart from next 7 days of prediction, we do a prediction for after 60 days, 180 days, and365 days from the date of validation (May 14, 2020) under different values of testing rates r and r . Predicted values are tabulated in Tables 4 and 5 for different states and in Table 6for India. Note that r = 0 means when the testing statistics remain same as per testing dataon May 07, 2020. However, if the number of total testing increases linearly or log-linearlyas defined by Eqs. (9) and (10) then the number of people detected with COVID-19 increasesignificantly. This will be discussed in the next subsection.The total estimate for the total number of COVID-19 infected people in India would beapproximately 4,60,000 on July 7, 2020; 19,00,000 on November 7, 2020; and 46,00,000 onMay 7, 2021, if the number of testing maintains the present statistics including lockdowncondition. Prediction with model and real data: a case study of India r =0r =5000r =50000 (a) r =0.0r =0.1r =0.4 (b) Fig. 5:
In this figure, we show the effect of testing rate in the expected number of testedpositive cases. In figures, horizontal axis represents the number of days and verticalaxis represents the total number of tested positive cases ( P l X [ l ]1 ). Gain in numberof testing samples is considered in two ways: ( a ) linear defined by Eq. (9), and ( b )non-linear defined by Eq. (10). In simulations, parameters are considered at statelevel. r =0.0r =0.1r =0.4 (a)
70 75 80 85 90 95 10001234567 10 r =0.0r =0.1r =0.4 (b) Fig. 6: ( a ) Magnified view of plots in Figure 5(b). When rate of testing increases underEq. (10) with r = 0 .
4, then spreading of COVID-19 under considered assumptionscan be controlled. ( b ) In this figure, we discuss that how under different testing rates,number of positive cases get changes and testing is able to trace all the branches ofinfection spreading after certain time limit. In simulations, parameters are consideredat state level. Currently, in India the number of testing conducted per day is approximately 1 , ,
000 sam-ples per day. However, training the proposed model over the real data provides the rate oftesting r = 0 . . The testing data is not available district level but at the state level. Afterlearning the data, the model uses different rates of gain in testing for validation. If r = 0then it means that per day testing is constant for the entire period of interest. In order toobserve the effect of statistics of testing on the count of total number of people infected withCOVID-19, we propose two types of growth in testing: linear and log-linear.First we perform this experiment when testing is increased linearly using Eq. (9), for Prediction with model and real data: a case study of India r = 5000 and r = 50000. As follows form Figure 5, we consider three cases: (I) Per daytesting is constant and corresponding curve is in grey color which is lowest among all three.The number of total cases of COVID-19 increasing slowly due to less number of testing overa huge population. (II) Increment of testings per day with r = 5000 results into a drasticchange in number of infected cases and it is increasing continuously and after certain time therate of infection goes down but does not contain. This implies that this rate of testing may notcontrol the spread of the disease. (III) Finally, When the increment of testing grows linearlywith r = 50000 then the simulation shows that the number of infected cases stabilizes andthe spread gets contained. During the simulation, we consider parameter values at state level.Now we discuss the effect of testing when it is increasing at log-linear rate under Eq. (10).We consider the following cases: (I) The number of testing is constant with the value as ofMay 7, 2020. (II) If the gain in the rate of testing is r = 0 . r = 0 . (a) λ ( t ) = 1 / (b) λ ( t ) = 1 / Fig. 7:
The plots shown the effect of mobility parameter θ , λ [ l ]3 ( t ), and testing r . There threeregions in both the sub-figures: (I) left narrow region corresponds to less numberof tested positive cases as compared to middle region (II) which is followed by widespread region (III). It signifies that after certain rate of testing r , infection spreadingcan be controlled before its pandemic like situation. ( a ) when λ [ l ]3 ( t )(= 1 / θ does not show its impact. Dark red patches are corresponding to points( r , θ ) which has large number of tested positive cases and these points are scatteredall over the middle (II) region (highest value of tested positive cases is almost 6 × ).While in ( b ) λ [ l ]3 ( t )(= 1 /
10) is significantly large and it shows the contribution of localmobility in spreading of infection. In plot, it is observe that dark red patches havemore concentration inside circle (lower values of θ and r ). Here in middle (red region)region, lower values of θ gives more weights to near by locations and less weights tolocations at distances while higher values of θ give almost equal weights to all regionsfor mobility. Now we study the effect of mobility of individuals ( θ ), probability of migration across locations( λ [ l ]3 ( t )) in the spread of the disease. Besides, we analyze the joint effect of the parameters θ , λ [ l ]3 ( t )) and the increment of number of testing r (see Eq. s (2) - (5)). For instance,during lockdown, the value of θ can be considered as a small value. Indeed, the mobilityparameter takes a smaller value as compared to the distance between districts which are far Prediction with model and real data: a case study of India Data (Learning)ModelData (Validation)
X: 78Y: 1.152e+05 (a) r =0Data (Learning)r =20000r =60000Data (Validation) (b) Fig. 8: ( a ) Blue dots are corresponding to data points used to train the model, grey circles arecorresponding to trained and predicted values, sky blue dots are data points used forvalidation of prediction. Model is trained for infection data of 65 days, and predictionhas been performed for next 7 days with high accuracy (validation set). Sky blue dots(real data) are close enough to corresponding predicted values (grey circles). Thereis shown a test point, corresponding to which, the values of tested positive cases is1 . × , and 1 . × is the actual value of it. ( b ) Cumulative number oftested positive cases under different values of testing parameter r . Plot correspondsto analysis of the status of the stabilization of spread of COVID-19.apart but belong to the same state, and it is also true for states within the country during thelockdown period. Besides, the probability of migration is very less corresponds to lockdowneffect. In simulation, we notice that the mobility parameter reaches the value up to 70 andthe migration probability is less than 0.1.In Figure 7, (in both the plots) we identify three regions: (I) The region in the middle,where number of tested positive (simulated) is highest, (II) A narrow region, left of (I) wherethe number of tested positive is less as compared to (I), and (III) stabilizing region, whichis right to (I) that has less number of tested positive cases and it signifies that after certainvalue of r , infection spreading can be controlled.We observe the following: ( a ) when λ [ l ]3 ( t ) = 0 .
01 no significant effect of θ in the numberof infected cases is found and it simulates the spread in neighboring locations of a locationwhich is infected (highest value of tested positive cases is almost 6 × ). On the other hand,in ( b ) when λ [ l ]3 ( t ) = 0 . θ and r . In the middle (red region), the number ofinfected cases is more as compared to the previous case.Moreover, note from Eq. (6) that if the value of social mobility parameter θ is muchlarger than max k,l ∈V ( t ) d kl then the traffic flow indicator w kl becomes almost uniform for all k and l. On the other hand, a small value of θ generates more traffic flow between neighboringlocations and induce less traffic flow across locations which are a large distant apart. If θ isconsidered comparatively larger value then the traffic flow are almost equal across locations.Besides, if θ is assigned in such a way that it induces higher traffic flow between thelocations which are significantly infected then the infected people in both the correspondinglocation significantly increase. Consequently, the dark red patches are wide spread over themiddle region in Figure 7(a). Therefore, it is reasonable to conclude that lower values of θ corresponds to dark red patches which signifies local transmission, and larger values of θ corresponds to long-distance transmission of COVID-19. Prediction with model and real data: a case study of India Day r INDIA r INDIAMay 20, 0 1 . × . × July 7, 0 4 . × × . × . . × (60 days) 6 × . × . . × Nov 7, 0 1 . × × . × . . × (180 days) 6 × . × . . × May 7, 0 4 . × × . × . . × (365 days) 6 × . × . . × Tab. 6:
Prediction for when number of testing grows linearly and log-linearly for certain valuesof r and r based on the training data up to May 7, 2020 and the validation dataperiod is May 8 - 14, 2020. The average error for the training dataset for the modelis 37 . r =0Data (Learning)r =5000r =9000Data (Validation) (a) r =0Data (Learning)r =1000r =5000Data (Validation) (b) r =0Data (Learning)r =1000r =5000Data (Validation) (c) r =0Data (Learning)r =1000r =5000Data (Validation) (d) Fig. 9:
Cumulative number of tested positive cases under different values of testing parameter r . Plots are corresponding to analysis of the status of the stabilization of spreade ofCOVID-19. In this section, we do the prediction at country level. X [ l ]1 ( t ) is defined at state level andmodel is trained using the infection diffusion data of covid-19 spreading from March 4, 2020 Prediction with model and real data: a case study of India r =0Data (Learning)r =1000r =5000Data (Validation) (a) r =0Data (Learning)r =1000r =5000Data (Validation) (b) r =0Data (Learning)r =1000r =5000Data (Validation) (c) r =0Data (Learning)r =1000r =5000Data (Validation) (d) Fig. 10:
Cumulative number of tested positive cases under different values of testing parameter r . Plots are corresponding to analysis of the status of the stabilization of spread ofCOVID-19.to May 14, 2020. In Figure 8(a), blue dots are corresponding to data points used to train themodel (from March 4, 2020 to May 7, 2020), grey circles are corresponding to trained andpredicted values, sky blue dots are data points used for validation of prediction (from May 8,2020 to May 14, 2020) [27]. In this plot, we have shown that model is trained for infectiondata of 65 days, validated using next 7 days data, and prediction has been performed fornext 7th days almost accurately. Sky blue dots (real data) almost coincide with the centresof corresponding to the predicted values (grey circles). Here training error is 37 . r and r . We consider r = 0 , , r = 0 , . , .
4; after training, r = 0 means testing will continuewith current volume (approximately 1,00,000 per day).If the number of testing increases linearly with r = 10 then the total number of peopleinfected with COVID-19 would be approximately 5.3 Millions on July 7, 2020; 88 Millionson November 7, 2020; and 220 Millions. For linear growth with r = 5 × in testingapproximates the total number of infected people in India as 2 Millions on July 7, 2020; 59Millions on November 7, 2020; and 130 Millions, see Table 6. Also the same for the log-linearincrease of testing per day is given in Table 6. In this section, we discuss about the stabilization of spreading of COVID-19 in future. Thismeans the number of newly affected gradually decrease, and the number of total number ofinfected people at country level becomes almost constant. From the analysis of the effect of
Conclusion mobility parameter θ and gain in testing rate r (log-linear) or r (linear), from Figure 7 itcan be concluded that higher testing rate is more effective as we know that presently availabledata is obtained under very less mobility rate. However, as the mobility will increase afterlifting the nationwide lockdown, the infection will presumably spread very fast.Here, we demonstrate time series analysis of infection spreading under different values oftesting parameter r for all the states and India. From Figures 8(b) and 6(a) for countrylevel, where as 9, and 10 for state level show the stabilization of tested positive cases withincreasing number testing of after certain threshold.
20 40 60 80 100 120 140 1600246810 10 r =0r =5000r =50000 (a) r =0.0r =0.1r =0.4 (b) Fig. 11:
Possibility of second wave of the spread of COVID-19. In both the figures, we observethat we have two plots (one from each sub-figure) in which once curves of totalinfected cases flattened and again rise very fast. It corresponds to the second waveof the spread of COVID-19.
The second wave of a pandemic is often observed in a region when interventions are effectivelyapplied to mitigate the spread of the disease and but are then lifted [29]. In the proposedmodel of this paper, the second wave can be examined under the following scenarios: (I) Ifthe number of testing performed daily is not enough, that is, it is at per with the cumulativenumber of all social contacts of previously detected people with COVID-19, then there wouldnot be any sign of second wave; (II) If the number of testing is large enough such that nextday available cases to be tested is decreasing continuously and spreading will get controlledsoon; and (III) If the number of testing performed daily is sufficient to detect the numberof cases at present state of the number of infections but somehow due to few events, (forexample, if number of tested positive daily decreases and also the number of testing to beperformed is below the required limit) then a second wave of diffusion may be observed. Thusit is important to track the last trail of infection diffusion completely to control it. Usingsimulation, we show that the second wave can be observed under different scenario whichinclude: Number of testings is increased ( a ) linearly ( b ) log-linearly, in Figure 11. We have proposed an epidemiological model for the spread of COVID-19. The model is basedon spread at local level which can be at the level of province, town, city or districts by com-bining a statistical approach and using the metapopulation network of infected locations. Themodel incorporates a few parameters which represent the effect of spread by asymptomaticor pre-symptomatic individuals, restricted mobility of individuals, and the testing statistics.Predictions of total number of tested with COVID-19 people are made at the level of state
Conclusion and the entire country, based on the data of testing as of May 7, 2020, and under linear andlog-linear growth of testing statistics. Finally it is shown that the spread can be contained invery near future if linear or log-linear growth of testing is adapted.The stabilization of infected cases primarily depends on the number of testing and theinter location transition of population or the strictness of lockdown. If the testing rate islow or moderate it may show less count of infected cases. But there is a chance of secondwave to hit-back. If the testing rate is sufficiently large and executed with proper samplingscheme then the count of positive will get stabilized much early. The proposed epidemiologicalmodel can be applied and generalized for prediction of total number of tested with COVID-19people at any country. Indeed, if metapopulation network is a network of countries then theprediction can be made at the world level based on the data of transmission of populationsacross the countries. Acknowledgement.
The authors thank Vaidik Dalal of How India Lives for his help withthe data.
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Appendix Date MeasuresJanuary 25 screening for travellers from 2019-nCoV affected countries (China)at points of entryFebruary 26 People coming from Republic of Korea, Iran and Italy or thosehaving history of travel to these countries may be quarantined for14 days on arrival to IndiaMarch 3 health scrrens at border crossingsMarch 5 Advisory against mass gatheringsMarch 16 closure of selected public institutions such as museums (incl. Taj Mahal)until March 31 and postponement of several local electionsMarch 17 Travel of passengers from Afghanistan, Philippines, Malaysia to Indiais prohibited with immediate effect. No flight shall take off from thesecountries to India after 1500 hours Indian Standard Time (IST) tillMarch 31 and will be reviewed subsequently.March 18 ban of entry for passengers from EU countries, EFTA countries,Turkey, UKMarch 22 No international flights to take off for India from foreign airportsafter 0001 hrs GMT of March 22, 2020 until 0001 hrs GMT March 29, 2020.20 hours maximum travel time. So no incoming international passengersallowed on Indian soil (foreigner or Indian) after 2001 hrs GMTof March 22, 2020.March 24 Complete lockdown of entire nation for 21 days. Agriculture-Farmingand allied activities exempted from LockdownApril 25 opening of certain categories of shops. In rural areas, all shops,except those in shopping malls are allowed to open. In urban areas,all standalone shops, neighborhood shops, shops in residential complexesare allowed to open. Shops in markets/market complexes andshopping malls are not allowed to open. It is clarified thatsale by e-commerce companies will continue to bepermitted for essential goods only.May 4 Extension of Lockdown for a further period of Two Weeks with effectfrom May 4, 2020
Tab. 7: