Forecasting the transmission of Covid-19 in India using a data driven SEIRD model
FForecasting the transmission of Covid-19 in India using a data driven SEIRD model
Vishwajeet Jha , Nuclear Physics Division, Bhabha Atomic Research Centre, Mumbai-400085, India and Homi Bhabha National Institute, Anushaktinagar, Mumbai-400094, India
The infections and fatalities due to SARS-CoV-2 virus for cases specific to India have been studiedusing a deterministic susceptible-exposed-infected-recovered-dead (SEIRD) compartmental model.One of the most significant epidemiological parameter, namely the effective reproduction number ofthe infection is extracted from the daily growth rate data of reported infections and it is includedin the model with a time variation. We evaluate the effect of control interventions implemented tillnow and estimate the case numbers for infections and deaths averted by these restrictive measures.We further provide a forecast on the extent of the future Covid-19 transmission in India and predictthe probable numbers of infections and fatalities under various potential scenarios.
I. INTRODUCTION
Almost every continent of the planet is grappling witha large number of infections arising due to virus calledCoronavirus 2, SARS-CoV-2 [1]. These infections thatmay result in a mild to severe symptomatic disease calledCoronavirus disease 2019 or COVID-19 were first de-tected in Wuhan, a city in central China [2, 3]. Laterthe infections spread across the globe and it has forcednations to undertake drastic measures to minimize theloss of precious human lives [4, 5]. For a populous coun-try like India, which has a dense and large population ( ≈ th March when the number of cumu-lative SARS-CoV-2 infections were around 650. Thesestrict measures prevented any large scale disaster andslowed down the rate of infections in the initial stagesand helped in geographical containment of the epidemic.However, recent days have seen no real decrease perhapsdue to gradual weakening of restrictive measures owingto pragmatic social and economic reasons. From June1 st India continues to have a complete lock-down only inthe defined containment zones where the infection ratesare high. These steps of gradual easing of lock-downhave been necessitated as the balance between life andlivelihoods are intertwined, which calls for invoking moreintelligent strategies because a complete extended lock-down cannot be sustained for very long time withoutother competing collateral losses to the most vulnerablesections of society. Alternative steps based on isolation ofinfected patients through the lock-down in the contain-ment zones and more widespread testing and contact-tracing are being followed for controlling the rate of in-fection. This represents the transition from suppressionto mitigation strategy for the resolution of any potentialoutbreak but efficacy of these steps remains to be seen as the execution of these policies on ground level are chal-lenging.The transmission dynamics of viral epidemics in anypopulation is an interplay of various factors related toviral, immunologic, environmental and sociological con-ditions. A number of mathematical and physical modelshave been proposed in general to understand the evo-lution of epidemics, aiming to make reliable predictionsso that to help governments to formulate proper policiesand response plans for effective control of the disease [6–8]. Simple deterministic mathematical models based onthe formulation of differential equations have been exten-sively used to provide information on the transmissionmechanism of various viral epidemics. The SIR modelis a one of the simplest epidemiological models that isbased on dividing the population among three compart-ments, the susceptible, the infected and the recovered (ordeceased) populations and determining their time evolu-tion [9]. The SEIR [10] model is a simple extension of theSIR model, where an additional compartment of exposedpopulation with a latency period is introduced which ismore appropriate for COVID-19 like epidemic which hasan inherent latency and asymptomatic transmission [11].Extended models have been employed that use severalseparate compartments for various sub-populations suchas, asymptomatic, quarantined, hospitalized or compo-nents based on the variations for example, according toage, gender etc. [12, 13]. However, this entails incorpo-ration of many unknown parameters and uncertain ini-tial conditions about which the information is either notavailable or there are large associated uncertainties.In the present article, we employ a dynamic SEIRDmodel with the inclusion of population of deaths as aseparate compartment in the SEIR model. Several workshave been already performed in the Indian context to ex-plain the COVID-19 dynamics in the initial phase of itstransmission [14–18]. We incorporate the crucial param-eter of contact period with a time variation connectingits value at the beginning of the epidemic to the currentreduced value. The reduction in the values of contactrate has been achieved due to many isolation measures,primarily the imposed nation-wide lock-down. The timevariation in the contact rate β ( t ) is determined through a r X i v : . [ q - b i o . P E ] J un the effective reproduction number R ( t ) that is in turn re-lated to the doubling time of the rate of infection growth[19–21]. We integrate this parameter in the SEIRD modelcalculations and estimate the role of interventions in pre-venting the number of probable infections and death tillnow. Further, we consider different potential scenariosfor the rate of growth of infections for making projectionsof SARS-CoV-2 transmission. We make a forecast for theprobable numbers of infections and fatalities in the com-ing times. The projections provide information for theextent of suppression and containment strategies thatneed to be employed to mitigate the impact of Covid-19 in coming times. It is to be mentioned that resultsobtained in this work are to be used for the research pur-poses only. SEIRD MODEL AND EFFECTIVEREPRODUCTION NUMBER dSdt = − β I ( t ) NdEdt = β I ( t ) N − φEdIdt = φE − γI (1) dRdt = γIdDdt = − δI where, S ( t ) is the susceptible population, E ( t ) is the ex-posed population, I ( t ) is the infectious population, R ( t )is the recovered population and D ( t ) is the number ofdeaths at any instant t and N = S + E + I + R + D is the total population. We have not included separatecompartments for the number of asymptomatic, quaran-tined, hospitalized populations or the variations accord-ing to age or gender, as these lead to increase in numberof unknown parameters and therefore lead to large un-certainties in the predictions. In any case, these numberscan be estimated in an average way with their relationsto populations that have been considered. In addition,assumption about the no re-infection of the recoveredpopulation is made as there is no evidence to the con-trary.The parameters of the above set of equations are thelatent period of being exposed A = 1/ φ that is relatedto the incubation period of the virus, the contact period B of infection = 1/ β , the period of being free from beinginfected G = 1/ γ commonly known as the recovery time , the parameter corresponding to death D = 1/ δ of the in-fected population. These parameters determine the tran-sitions that occur across the compartments as the timeevolves. Here, the parameters A , G and D are specificto characteristics of SARS-CoV-2 and only weakly corre-lated to the health responses of the country and thereforeexpected to have similar values across countries. The pa-rameter B represents the strength (speed) of the virustransmission which is intimately related to the prevailingconditions of containment measures undertaken by spe-cific countries. Apart from these parameters, the fractionof the susceptible population at the beginning to the totalpopulation α = S (0) /N is a very important parameter.Taking total population of the region as S (0) may lead togross overestimation of case numbers, because the partof population may be inherently immune or less affectedby the virus or live in isolated conditions. Furthermore,the extent of initial exposed latent population definedby (cid:15) = E (0) /I (0), parameter, may also be an importantparameter that indicates the presence of a number ofundetected or asymptomatic exposed individuals at thebeginning.One of the most significant parameters that describesthe pandemic is the basic reproduction number of infec-tion R , which is defined as the number of individualsthat are infected from the uninfected, susceptible pop-ulation by one infected individual under normal condi-tions [6, 19]. There are challenges in determining R interms of the parameters of deterministic model as onerequires estimates of included parameters that are un-certain [23]. During the spread of the epidemic one candefine an effective reproduction number R i ( t ), which isa time dependent quantity that changes because of con-trol measures and depletion of susceptible population.It provides the dynamic information on the strength ofthe epidemic transmission as the time evolves. In gen-eral, the infection continues to expand if R i ( t ) has valuesgreater than 1, while the epidemic stops eventually if R i ( t ) is persistently less than 1. The estimation of theeffective reproduction number is complicated and manymodels have been proposed for its determination [24, 25]from the data. Here we use a simple method based onfitting the incidence data growth rate by a distributionwith gaussian shape to determine the behaviour of R i ( t ).It must be mentioned however, that reported data hasan inherent delay as compared to the instantaneous pop-ulation numbers that are required for the estimation ofits actual value. In the SIR - type models or their sim-ple extensions, such as one described above R i ( t ) can beexpressed as R i ( t ) = β.S ( t )( γ + δ ) .N (2)In the initial stages of the infection, S ( t ) ∼ N and R i = βγ , since ( γ (cid:29) δ ). The R i ( t ) value can be estimated usingthe initial doubling time T d of the number of infections[20] R i ( t ) = 1 + G ln (2) T d (3)The T d value can be determined by fitting the reportedgrowth in the cases of infection, which shows an expo-nential growth at the beginning of the epidemic, I ( t ) = I (0) exp ( β − γ ) t (4) T d = ln (2)( β − γ ) = ln (2) ln (1 + r ( t )) (5)where, the daily growth rate r ( t ) is determined from thedata of reported cases of infections. At smaller values of r ( t ) it has a simple relationship R i ( t ) = 1 + G.r ( t ). Thevalues of r ( t ) are extracted from the reported data ofdaily growth rate of infections starting from 14 th Marchto 28 th May (day 76) with a 9-day moving average. It isfitted with a function in the following form r ( t ) = a [ e − ( t − t σ + b ] (6)where a , b , σ and t are fit parameters. These parame-ters are determined from the best fit approach throughthe local minimization of the sum of squares of the error.The resulting fit to the daily growth rate is shown in Fig.1a along with the band with standard error on fit pa-rameters. In addition, the projections for next days after28 th May are also shown for various probable scenariosby the straight lines that are used for the extrapolationsof infection growth rate. It is seen from the figure thatIndia had a peak daily growth rate of ∼
20 % at thebeginning of the epidemic which reduced to ∼ th Marchhas been continually relaxed in phased manner and existsnow only in the containment zones from 1 st June. How-ever, after the decrease in growth rate in infections in theinitial phase following the lock-down, the cases of infec-tion have continued to grow at somewhat constant ratefor a while. The extrapolations for next 30 days thatdefine various probable scenarios are approximated as alinear reduction or increase from the present value of in-fection growth rate. The quantity r ( t ) determined fromthe data is also used to study the evolution of R i ( t ) intime as shown in Fig. 1b. It must be noted that R i ( t ) alsodepends on the period of infection for which, we presentthe result for values G = 12 . G = 20 days. The R i ( t ) values have been extracted from the r ( t ) of the re-ported cases and also obtained through fitted value of r ( t ). These values are seen to decrease from a peak valueof ∼ ∼ G = 12 . T d ( t ) value that is directly extracted from the data andalso from the fit shows a constant value of ∼
16 days.
Lock-down P e r c e n t a g e G r o w t h o f I n f e c t e d c a s e s Time (days since hundred cases)
Reported rate of infectionCurrent rateScenario 1 : The best caseScenario 2 : OptimisticScenario 3 : Most likelyScenario 4 : ProblematicScenario 5 : Alarming
Lock-downLock-down E ff e c t i v e r e p r o d u c t i o n nu m b e r R i ( t ) D o u b li n g r a t e t d Time (days since hundred cases) R i (t) from reported rate, G = 12.7 days R i (t) from fit G = 12.7 days R i (t) Scenario 1: Optimistic CaseR i (t) from fit G = 20 dayst d from reported ratet d from fit t d Scenario 1: Optimistic Case
FIG. 1. a) The variation of the percentage rate of growthof infections with time is shown with a 9-day moving aver-age. The data is shown by red points. The solid blue lineis the fit to data with the standard error shaded band as de-scribed in text. The values of rate r(t) for the next 30 dayscorresponding to different scenarios are plotted as green dot-ted line, pink dashed double dotted line, orange dashed line,brown long dashed line and red dashed-dotted line respec-tively. b) The time variation of R i ( t ) for G = 12.7 days isshown by red points as extracted from data with G = 12.7days and blue line as determined from the fit. R i ( t ) for G =20 days is shown by from the fit is shown by orange dottedline. The R i ( t ) projection for the optimistic case scenario isshown by pink dashed double dotted line. The time variationof T d ( t ) is shown on second y-axis by red points as extractedfrom data and brown dashed line as determined from the fit.The T d ( t ) projection for the most optimistic case is shown bypink dashed double dotted line. The period of nation-widelock-down is indicated by the horizontal line. The value of R i ( t ) and T d values are also shown for oneprobable scenario where the rate reduces by one-half ofthe present value in a linear manner. This shows a mod-erate reduction in the value of R i ( t ). In addition, therate decrease leads to a significant increase in T d values. RESULTS
The SEIRD model calculations using eqn. 1 have beenperformed to make comparisons with the data aggregatedfor India using the reported cases of infected, recoveredand dead populations up to 28 th May and to make fore-casts about the future scenarios. The contact rate pa-rameter β ( t ) is taken to be time dependent with the pa-rameters β and β c fixed in accordance of equation 6.The parameter A is taken as 5.1 days, which is the meanincubation period and bit larger than the latency period.The value of parameter δ = 0.025 is taken, which de-termines the death population and very weakly affectspopulations in other compartments. The parameter γ ( t )is taken in the following form γ ( t ) = γ [1 + exp ( − κt )] (7)The time variation in this parameter with γ = 0.079 cor-responding to period of 12.7 days and κ =0.01 takes intoaccount the larger value of G ≈
26 days that is needed toexplain the behaviour of data in the initial stages. As thetime elapses, a reduction in the recovery period is seenand γ approaches γ value.The model was applied from the day of the epidemicwhen cumulative number of infections were ∼
100 as on14 th March. The fraction of the population at day 1 incompartments are set as follows : I = 88 / . e R =10 / . e D = 2 / . e α and (cid:15) are theunknowns in the model. We take α =0.1 which is similarto value of α =0.08 extracted for European countries inRef. [26]. The parameter (cid:15) = 3.2 A , is important for theinitial description of data but it does not affect long timedynamics of the epidemic as predicted by the model.The results of calculations with these parameters thatuse the time varying β ( t ) parameter as determined aboveprovide a good description of the evolution in the casenumbers of reported infected, recovered and death pop-ulation as shown in Fig. 2a. In addition, the calculationshave also been performed for constant β = 0 .
167 value,which is obtained from the best fit to the exponentialdistribution according to Eqn. 4. While the model re-sults as shown in Fig. 2b provide a good description ininitial days, it grossly over-predicts the case numbers ascompared to the reported cases. It is quite evident fromthe figure that the time dependence of β ( t ) is necessaryto understand the dynamics of infection spread for casesin India.The period of infection related to the recovery time ofthe infected individuals is also taken with a time vari-ation. This parameter is primarily the characteristic ofthe epidemic and it is only mildly dependent on the re-sponses of the health-care systems. In absence of anyeffective therapy or cure that may shorten the length ofthe infection it is relatively well known and it is taken as ∼ β -value [17] or evenwhen time variation in β -value is taken into account as
10 20 30 40 50 60 70 80 90 100
Time varying β C a s e s Time (days since hundred cases) infected (detected)recovereddead
10 20 30 40 50 60 70 80 90 100
Constant β = 0.167 C a s e s Time (days since hundred cases) infected (detected)recovereddead
FIG. 2. The time evolution of the reported cases of infection,recovered and dead is shown by blue filled circles, green opencircles and red squares, respectively. The results of calcula-tion for the infected, recovered and dead populations usingthe SEIRD model are shown by blue solid line, green dasheddotted line and red dashed line for the case when a) param-eter β is time varying b) parameter β =0.167 is constant intime. it has been found in the present study. The recoveryrates are continually improving, a feature also reflectedin the reported recovery data. Therefore, a time depen-dence in the parameter γ ( t ) is introduced to account forthis observation. We show the calculations in Fig. 3 withthe different values of infection period as parameter withconstant values ( G = 5 day, 8 day, 12 . β . Theinterventions have led to a decrease in the daily growthrate of infections which is intimately related to the β value. We use the constant values of β = 0.252 and 0.125,which correspond to the peak rate of growth and half its
50 100 150 200 250 5.0×10 I n f e c t e d c a s e s Time (days since hundred cases) γ = (5 day) -1 γ = (9 day) -1 γ = (12.7 day) -1 γ = (20 day) -1 γ = (27 day) -1 FIG. 3. The time evolution of the infected populations fromthe calculation using the SEIRD model for different values of γ parameter are shown by green dotted line (G = 5 days),pink dashed-dotted-dotted (G = 9 days), blue solid line (G =12.7 days), brown dashed line (G = 20 days) and red dasheddotted line (G = 27 days), respectively. value. The peak β -rate is expected to have prevailedin the early stages of infection spread in the absence ofany interventions such as, the lock-down or the condi-tions of no enhanced public awareness. In addition, wealso give results obtained from the β = 0.167, β = 0.01and time varying β -value. The infected, recovered anddeath populations for these β -values are shown in Fig.4a, Fig. 4b and Fig. 4c, respectively. From the com-parison it is evident that the lock-down and other in-terventions have prevented any large spread of infectionsand kept the death numbers particularly low. These in-terventions could have prevented around 4 million peakinfections and 200,000 deaths at the 100 day mark. Thelower growth rate also means that number of active infec-tions are low at any instant which helps to optimize theresponse of health care systems. The rate of infectionsin India have remained approximately constant after theinitial reduction for last several days. After an extendedlock-down slowly the restrictions have been loosened up.We extrapolate values of β (t) to predict the outcomes ofvarious probable scenarios. The β (t) values correspond-ing to growth rate value r pr as on 28 th May are variedso as to attain a given value at the end of next 30 daysassuming a time variation with constant slope. Thesescenarios are named as the best case, the optimistic case,the most likely, current, problematic and alarming sce-narios respectively. The time evolution of the epidemicis studied with these time variations for the future. Theresulting predictions for the populations of infected, re-covered and dead are shown in Fig. 5a, Fig. 5b and Fig.5c respectively. The growth rate same as r pr may leadto a high number of total infections ( ∼
50 100 150 200 250 2.0×10 I n f e c t e d c a s e s Time (days since hundred cases) β = 0.252 β = 0.167 β time varying β = 0.125 β = 0.01
50 100 150 200 250 2.5×10 R e c o v e r e d c a s e s Time (days since hundred cases) β = 0.252 β = 0.167 β time varying β = 0.125 β = 0.01
50 100 150 200 250 2.0×10 D e a t h s Time (days since hundred cases) β = 0.252 β = 0.167 β time varying β = 0.125 β = 0.01 FIG. 4. The results of calculation for the time evolution of thea) infected b)recovered and c) dead populations for differentvalues of β are shown for i) β = 0.252 by red dashed dottedline ii) β = 0.167 by brown dashed line iii) time varying β by blue solid line iv) β = 0.125 by pink dashed dotted-dottedline v) β = 0.01 by green dotted line peak of 450,000 active infections sometime in the monthof September. These numbers can be reduced if withcontainment measures the rate of growth can be broughtdown drastically so that we can see an early resolutionof the pandemic approximately in 3-4 months time. Inthis case, the death figures can be kept substantially lowin the range of 25,000-50,000. In contrast, if the rate ofgrowth were to increase from the present values due topre-mature lifting of the lock-down in the affected zonesand other lapses, the death numbers can be 500,000 witha rather alarming number of infected individuals in shorttime. Higher rates of growth also mean the large num-ber of active infected cases appearing early and that maystretch the health care systems to the brink. DISCUSSIONS
We have made detailed comparison of model predic-tions with the real data using the important parameterof contact rate and infection rate derived from the data it-self from the first principles. It must be noted that thereis an inherent delay in the reported rate and instanta-neous rate of the infection. In addition, the effect of anyrestrictive measures undertaken appears with a delay inthe reported rate, which is estimated by the fit parameter t ∼
15 days in the present case. The model calculationsare able to describe very well the case number of infectedand recovered populations of reported data till now. Theimposed restrictions have led to a reduction in the R i ( t )and an increase in the T d values as the time elapses. Thequantitative measure of the intensity of the imposed lock-down that reduced the growth rate r ( t ) to almost half itsvalue is given by the fit parameter σ = 14 days. It isseen from the calculations that a large number of infec-tions and fatalities have been averted due to impositionof the lock-down. Some part of this reduction may beascribed to the enhanced public awareness, and growingdisease monitoring and testing capabilities with the pas-sage of time. However, effect of complete lock-down inreducing the infection rate has been quite significant.After the initial period of 40 days following the com-plete lock-down, there has not been much gain in thereduction of infection spread rate in last 30 days. It isprobable that the gradual weakening of the lock-downdue to socio-economic reasons might have offset the gainsdue to restrictive measures. Nevertheless, the continuedrestrictions have prevented any rise in the rate of growthof infections, which in absence of any such measures isexpected to rise again. Even the growth rate of ∼ − th May suggest that more severe out-break may occur in coming times leading to high numberof infections. With the estimates from the most likelyscenario, over 450,000 would be clinically diagnosed atthe maximum resulting in ∼ D ( t ) to the cumulative num-ber of infections C ( t ). The CFR values have varied from ∼ ∼
50 100 150 200 250 5.0×10 I n f e c t e d c a s e s Time (days since hundred cases)
Scenario 5 : AlarmingScenario 4 : ProblematicScenario 3 : Most likelyScenario 2 : OptimisticScenario 1 : The best caseCurrent rate
50 100 150 200 250 2.0×10 R e c o v e r e d c a s e s Time (days since hundred cases)
Scenario 5 : AlarmingScenario 4 : ProblematicScenario 3 : Most likelyScenario 2 : OptimisticScenario 1 : the best caseCurrent rate
50 100 150 200 250 5.0×10 D e a t h s Time (days since hundred cases)
Scenario 5 : AlarmingScenario 4 : ProblematicScenario 3 : Most likelyScenario 2 : OptimisticScenario 1 : the best caseCurrent rate
FIG. 5. The results of calculation for the time evolution of thea) infected b)recovered and c) dead populations for differentfuture scenarios is shown by i) scenario 1: by green dotted lineii) scenario 2: pink dashed dotted-dotted line iii) scenario 3 :orange short dashed line iv) current model : solid blue line v)scenario 4 : brown long dashed line iii) scenario 5 : red shortdashed dotted line ble that both the number of fatalities and infections maybe underestimated. It is more likely that C ( t ) may beunderestimated more due to the presence of large num-ber of asymptomatic or non-critical infected cases whichleads to the overestimation of CFR, assuming reported D ( t ) cases to be true. CFR remains low as long as thehealth facilities are able to cope with the rate of patientsrequiring critical care. In the scenarios if the numberof active infected cases is large as predicted by multiplescenarios described above, requirements of hospitaliza-tions and critical care resources may increase sharply.In such a situation, the health care system is going to beseverely challenged in providing the critical care facilitiesfor prevention of fatalities. The CFR in these conditionsmay rise to higher values. Therefore, imposing strictermeasures inside the containment zones and more exten-sive testing and contact-tracing seems to be only viablelogical preventive option that can lead to a manageablereduction in infected cases and casualties in absence ofany therapies or large-scale immunity.There are limitations of the simplistic model employedhere and therefore the exact quantitative numbers pre-sented in the work are only indicative. In the presentmodel, the asymptomatic populations are taken only inan indirect way at the start of the epidemic through theintroduction of the parameter (cid:15) . Inclusion of this pop-ulation as a separate compartment however would leadto introduction of extra set of unknown parameters. Fur-ther, we have not considered the regional and age specificheterogeneities in the model. While we have made a rea-sonable assumption for the parameter α =0.1, implyinga 10% of the total population as the susceptible popu-lation, the overall numbers presented in this work maydiffer if it has a significantly different value. This numberis going to be affected as the country has seen large scalemigration from the infected areas to the other areas inrecent times which may increase the pool of susceptiblepopulation. Further, we have made forecasts in this workbased on probable daily growth rates. The determinationof contact rate parameter through the measured rate inthe simple way is uncertain due to stochastic fluctuationsin the early stages and inaccuracies and time delay of thereported data. Further, there are challenges on designingthe control mechanisms based on the basis of the num-bers of daily growth as discussed in Ref. [27]. However,this work shows the operational use of the R i ( t ) calcu-lated from the instantaneous infection rate to provide areasonable description of the transmission dynamics. CONCLUSIONS
In this article, we have presented results of SEIRDmodel calculations to study the role of interventions andmake future projections in the Covid-19 spread in In-dia. To make reliable forecast we have determined thetime dependent reproduction number R i ( t ) and contactrate parameter β ( t ) from the data for the daily rate ofincrease of infections. It is shown that timely imposi-tion of lock-down and other public health interventionshave led to a substantial reduction in the effective re-production number which decreased to a present valueof ≈ ≈ β ( t ) in the SEIRD model provide a gooddescription of the case numbers of infections, recoveredand deaths. We further make the projections for differ-ent probable scenarios. In the most likely scenario themodel predicts a peak of active infections around themonth of September with significant number of fatalitiesover the course of the epidemic around end of November.The results show the impending critical challenges forhealth care systems due to prospective high number ofpeople with infections. The salient feature of the simplemodel employed in this work is the use of minimal un-certain parameters and therefore in our opinion it makesreliable predictions of the infections and fatalities. Theprojections of peak infections suggest big challenges forthe available critical care health facilities in the manage-ment of pandemic. New innovative solutions have to becontinuously found and intelligent measures have to beeffectively implemented if the Covid-19 infections have tobe contained with a moderate number and the ensuingfatalities have to be minimized. The most important ex-tension of this study will be to incorporate the regionalvariability and apply this model by considering the statewise infection data and make predictions accordingly. ACKNOWLEDMENTS
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