Effective lockdown and role of hospital-based COVID-19 transmission in some Indian states: An outbreak risk analysis
EEffective lockdown and role of hospital-based COVID-19 transmissionin some Indian states: An outbreak risk analysis
Tridip Sardar , Sourav Rana a Department of Mathematics, Dinabandhu Andrews College, Kolkata, West Bengal, India b Department of Statistics, Visva-Bharati University, Santiniketan, West Bengal, India
Abstract
There are several reports in India that indicate hospitals and quarantined centers areCOVID-19 hotspots. In the absence of efficient contact tracing tools, Govt. and the poli-cymakers may not be paying attention to the risk of hospital-based transmission. To exploremore on this important route and its possible impact on lockdown effect, we developed a mech-anistic model with hospital-based transmission. Using daily notified COVID-19 cases from sixstates (Maharashtra, Delhi, Madhya Pradesh, Rajasthan, Gujarat, and Uttar Pradesh) andoverall India, we estimated several important parameters of the model. Moreover, we providedan estimation of the basic ( R ), the community ( R C ), and the hospital ( R H ) reproductionnumbers for those seven locations. To obtain a reliable forecast of future COVID-19 cases,a BMA post-processing technique is used to ensemble the mechanistic model with a hybridstatistical model. Using the ensemble model, we forecast COVID-19 notified cases (daily andcumulative) from May 3, 2020, till May 20, 2020, under five different lockdown scenarios inthe mentioned locations. Our analysis of the mechanistic model suggests that most of the newCOVID-19 cases are currently undetected in the mentioned seven locations. Furthermore, aglobal sensitivity analysis of four epidemiologically measurable & controllable parameters on R and as well on the lockdown effect, indicate that if appropriate preventive measures are nottaken immediately, a much larger COVID-19 outbreak may trigger from hospitals and quar-antined centers. In most of the locations, our ensemble model forecast indicates a substantialpercentage of increase in the COVID-19 notified cases in the coming weeks in India. Based onour results, we proposed a containment policy that may reduce the threat of a larger COVID-19outbreak in the coming days. Keywords:
COVID-19; Hospital-based transmission; Ensemble model forecast; Outbreak riskanalysis; Effective lockdown policy;
Introduction
Coronavirus disease of 2019 (COVID-19) was first observed in Wuhan, China and rapidlyspread across the globe in a short duration [1]. World Health Organization (WHO) declaresCOVID-19 as pandemic after assessing its various characteristics [2]. As of April 29, 2020–overthree million cases and over two hundred thousand deaths due to COVID-19 is reported acrossthe globe [3]. In India, the first confirmed case of COVID-19 was reported on 30 / / Email: [email protected] Corresponding author. Email: [email protected] a r X i v : . [ q - b i o . P E ] M a y student from Kerala studying in a university in Wuhan [5]. As of April 29, 2020–33065confirmed cases and 1079 deaths due to COVID-19 are reported in India [6].According to a daily monitoring report published by WHO, 22 ,
073 healthcare workersacross 52 countries are being tested positive to COVID-19 [7]. The report also noted that thenumber provided may be an underestimation as there is no systematic reporting of infectionsamong the healthcare workers [7]. In India, there are several reports that indicate hospitalsand quarantined centers are COVID-19 hotspots [8; 9; 10; 11; 12]. The doctors, nurses andother health workers are mainly vulnerable as they are in close proximity with the COVID-19 patients [9; 10; 11; 12; 13]. The close relatives of the notified COVID-19 patients in thequarantine centers may also be at risk of getting infection. In addition, those journalists whoare continuously visiting hospitals and quarantine centers to get updated reports on COVID-19, may also be at risk of getting infection [14; 15]. Therefore, a significant percentage ofsusceptible population in the community may be exposed to COVID-19 infection occurringfrom the contacts with patents in hospitals and quarantine centers. Due to unavailability ofefficient contact tracing tools, Govt. and the policy makers may be ignoring this importanttransmission route of COVID-19.Currently there is no vaccine and effective medicine available for COVID-19. Therefore, tobreak the transmission chain of COVID-19, Govt. had implemented a full nation-wise lockdown(home quarantined the community) staring from March 25, 2020 till April, 14, 2020. However,the large country like India with such diverse and huge population, lockdown all over the nationmay not be a very feasible and effective solution. In addition, lockdown already have a hugeimpact on the Indian economy specially on the short scale industries [16; 17]. To partiallyovercome this economic crisis as well reducing COVID-19 transmission, Govt. has proposedsome amendments (known as cluster containment strategy) on the lockdown rules from April20, 2020 [18; 19]. In these revised rules, Govt. has provided some relaxation in current rules bydividing different districts of the various states into three zones namely red (hotspot), orange(limited human movement), and green (economic activity) depending on the number of COVID-19 cases [18; 19; 20]. However, question remains whether this cluster containment strategy maybe successful in reducing COVID-19 transmission or not? If not then what may be otheralternative solutions to reduce COVID-19 transmission? These question can only be answeredby studying the dynamics and prediction of a mechanistic mathematical model for COVID-19transmission and testing the results in real situation.Mechanistic mathematical models based on system of ordinary differential equations (ODE)may provide useful information regarding transmission dynamics of COVID-19 and its control.Recently, there are several modeling studies that provide information on different effectivecontrol measures for reducing COVID-19 transmission [21; 22; 23]. In their study, Moghdas etal [21] developed an age-structure model on COVID-19 and projected required ICU beds fordifferent outbreak scenarios in USA. To improve the hospital capacity and possible containmentof COVID-19, they recommended self isolation and better hygiene practices in the community.Using a mechanistic model, Tang et al [22] estimated the control reproduction numbers andstudied the effect of various interventions on COVID-19 transmission in China. Recently,Sardar et al [23] provided an effective lockdown strategy to control COVID-19 transmissionin India. They recommended cluster specific lockdown policy in different parts of India foreffective reduction in transmission of COVID-19. There are some recent studies that usesdifferent statistical modeling techniques to provide reliable real-time forecast of COVID-19cases [24; 25; 26]. Therefore, a combination of mathematical and statistical models may be2ffective in determining the control measures as well providing a robust real-time forecast offuture COVID-19 cases and deaths in different parts of India and for other countries as well.In this paper, we formulated a mechanistic model with hospital-based transmission. Weassume that COVID-19 patients from the hospitals and quarantine centers can only be incontact with a small fraction of the susceptible population from the community. Furthermore,we assume different transmission rates for the community and the hospital-based infection. Inthe mechanistic model, we have incorporated the lockdown effect through home quarantine of acertain percentage of susceptible population from the community. Using daily notified COVID-19 cases from six states (Maharashtra, Delhi, Madhya Pradesh, Rajasthan, Gujarat, and UttarPradesh) and overall India, we estimated several important parameters of the mechanisticmodel. Furthermore, we estimated the basic ( R ), the community ( R C ), and the hospital ( R H )reproduction numbers for the seven locations under study. To obtain a reliable forecast of futureCOVID-19 notified cases in the above mentioned locations, we used a hybrid statistical modelthat can efficiently capture fluctuations in the daily time series data. BMA post-processingtechnique based on DRAM algorithm is used to ensemble our mechanistic mathematical modelwith the hybrid statistical model. Using the ensemble model, we forecast COVID-19 notifiedcases (daily and cumulative) from May 3, 2020 to May 20, 2020, under five different lockdownscenarios in the seven locations. To determine an effective lockdown policy, we carried out aglobal sensitivity analysis of four epidemiologically measurable & controllable parameters onthe lockdown effect (number of cases reduction) and as well on R . Method
We extend a previous mechanistic model [23] by considering hospital-based COVID-19 trans-mission (see Fig 1 and supplementary). We assume that hospitalized & notified infected pop-ulation can only be in contact with a small fraction ( ρ ) of the susceptible population from thecommunity (see Table 1 and supplementary). We assume different transmission rates ( β and β , respectively) for community and hospital-based infection. As it is very difficult to detectasymptomatic infected in the community therefore, we assume that only a fraction of symp-tomatic infected population being notified & hospitalized by COVID-19 testing at a rate, τ (see Fig 1 and Table 1). Following [23], the disease related deaths are considered only for thenotified & hospitalized population at a rate δ . We incorporated lockdown effect in our model(see Fig 1 and supplementary) by home quarantined a fraction of susceptible population ata rate l . We also assume that after the current lockdown period (cid:18) ω = 40 days (cid:19) the homequarantined individuals will returned to the general susceptible population (see Fig 1 and sup-plementary). Moreover, we assume that the home quarantined individuals do not mixed withthe general population (see Fig 1) i.e. this class of individuals do not contribute in the diseasetransmission. A flow diagram and the information on our mechanistic ODE model parametersare provided in Fig 1 and Table 1, respectively.Mechanistic ODE model (see Fig 1 and supplementary) we are using for this study maybe efficient in capturing overall trend of the time-series data and the transmission dynamics ofCOVID-19. However, as solution of the ODE model is always smooth therefore, our mechanisticmodel may not be able to capture fluctuations occurring in the daily time-series data. Severalcomplex factors like lockdown, symptomatic, asymptomatic, hospital transmission, awareness,rapid testing, preventive measures, etc may influence the variations in daily COVID-19 time-series data. Therefore, it is an extremely challenging job to fit and long term forecast using3his daily time-series data. To resolve this issue, we considered a Hybrid statistical modelwhich is a combination of five forecasting models namely, Auto-regressive Integrated MovingAverage model (ARIMA); Exponential smoothing state space model (ETS); Theta MethodModel (THETAM); Exponential smoothing state space model with Box-Cox transformation,ARMA errors, Trend and Seasonal components (TBATS); and Neural Network Time SeriesForecasts (NNETAR). Finally, the Hybrid statistical model and the mechanistic ODE model(see Fig 1 and supplementary) are combined together by a post-processing Bayesian ModelAveraging (BMA) technique which we discussed later in the manuscript.We used daily confirmed COVID-19 cases from Maharashtra (MH), Delhi (DL), MadhyaPradesh (MP), Rajasthan (RJ), Gujarat (GJ), Uttar Pradesh (UP) and overall India (IND)for the time period March 14, 2020 to April 14, 2020 (MH), March 14, 2020 to April 18, 2020(DL), March 20, 2020 to April 17, 2020 (MP), March 14, 2020 to April 18, 2020 (RJ), March19, 2020 to April 16, 2020 (GJ), March 14, 2020 to April 18, 2020 (UP), and March 2, 2020 toApril 19, 2020 (IND) for our study. As of April 29, 2020, these referred six states contributes of the total COVID-19 notified cases in India [6]. Confirmed daily COVID-19 cases formthese mentioned seven locations are collected from [6]. State-wise population data are takenfrom [27].We estimated several uninformative parameters (see Table 1) of our mechanistic model (seeFig 1 and supplementary) by calibrating our mathematical model (see Fig 1 and supplemen-tary) to the daily notified COVID-19 cases from the seven locations MH, DL, MP, RJ, GJ, UP,and IND respectively. As some initial conditions of our mathematical model (see Fig 1 andsupplementary) are also unknown therefore, we prefer to estimate these uninformative initialconditions from the data (see Table S1 in supplementary). In lockdown 1.0, the Indian Govt.has implemented a 21 days nationwide full lockdown (home quarantined the community) start-ing from March 25, 2020 to April 14, 2020 [28] and Govt. then extend the lockdown period up toMay 3, 2020 (lockdown 2.0) [29; 30]. Therefore, the daily COVID-19 time series data containsthe effect of with and without lockdown scenario, therefore, we prefer to use a combination oftwo mathematical models (without and with lockdown) for calibration. An elaboration on thecombination technique of the mathematical models without and with lockdown (see Fig 1 andsupplementary) is provided below: • We first use the mechanistic model without lockdown (see Eq S1 in supplementary) startingfrom the first date of the daily COVID-19 data up to end of March 24, 2020 for the sevenlocations MH, DL, MP, RJ, GJ, UP, and IND, respectively. • Using values of the state variables of the model without lockdown (see Model S1 in supple-mentary method) onset of March 24, 2020 as initial conditions, we run the mechanisticmodel with lockdown (see Fig 1 and Model S2 in supplementary) up to the end date ofthe daily COVID-19 data for the seven locations MH, DL, MP, RJ, GJ, UP, and IND,respectively.The nonlinear least square function (cid:48) lsqnonlin (cid:48) in the
MATLAB based optimization toolbox iscalled to fit the simulated and observed daily COVID-19 notified cases in those seven locationsmentioned earlier. Bayesian based (cid:48)
DRAM (cid:48) algorithm [31] is used to sample the uninformativeparameters and initial conditions (see Table S1 and Table S2 in supplementary) of the math-ematical models combination without and with lockdown (see Fig 1 and supplementary). Adetails on mechanistic model fitting is provided in [32].4alibration of the Hybrid statistical model for the mentioned seven states are done us-ing the R package (cid:48) f orecastHybrid (cid:48) . First, we fitted the individual models ARIMA, ETS,THETAM, TBATS, and NNETAR by calling the functions (cid:48) auto.arima (cid:48) , (cid:48) ets (cid:48) , (cid:48) thetam (cid:48) , (cid:48) tbats (cid:48) ,and (cid:48) nnetar (cid:48) respectively. The results generated from each of the above models are combinedwith equal weights to determine the Hybrid statistical model. Equal weight among the fiveindividual models are taken as it generates a robust result (see Table S4 in supplementary) forthe Hybrid statistical model [33].Post-processing BMA technique for combining the mechanistic model (see Fig 1 and supple-mentary) and the Hybrid statistical model is based on (cid:48) DRAM (cid:48) algorithm [31]. Let, Y ODE = { y ODEj } nj =1 and Y HBD = { y HBDj } nj =1 be n simulated observations from our mechanistic ODEmodel (see Fig 1 and supplementary) and the Hybrid statistical model respectively and let,ˆ Y = { y obsj } nj =1 be n observation from the data. Then Y E = w Y ODE + w Y HBD , (0.1)is our ensemble model, where, the weights w and w satisfies the constraints∆ = { w , w ≥ w + w = 1 } . We assume w and w follows Gaussian proposal distribution. Then the error sum of squarefunction [31] is defined as: SS (˜ θ ) = n (cid:88) i =1 (cid:16) ˆ Y − Y E (˜ θ ) (cid:17) . Posterior distribution of the weights ˜ θ = ( ˜ w , ˜ w ) for the ensemble model (0.1) are generatedusing Bayesian based (cid:48) DRAM (cid:48) algorithm [31] (see Table S3 and Fig S1 to S7 in supplementary).To save the countries short-scale industries and the agricultural sectors, Indian Govt. hasproposed some amendments on current lockdown rules from April 20, 2020 [18; 19]. In theserevised rules, Govt. has provided some relaxation in current rules by dividing different districtsof the various states into three red (hotspot), orange (limited human movement), and green(Economic activity) zones depending on the number of COVID-19 cases [18; 19; 20]. Implemen-tation of these new rules in our mechanistic models combination (see Fig 1 and supplementary)are based on the following assumptions: • Lockdown rule will be relaxed from April 20, 2020 in those states where the current estimateof the lockdown rate (see Table 2) is higher than a threshold value. This relaxation inlockdown is based on the fact that locations where lockdown are strictly implementedbefore April 20, 2020 are likely to have more impact on the economic growth. • Lockdown rule will be more intensive from April 20, 2020 in those states where the currentestimate of the lockdown rate (see Table 2) is below a threshold value. This assumptionis made because locations where lockdown are not implemented properly before April 20,2020 are likely to have more red (hotspot) zones. •
50% lockdown success is taken as the threshold value for our study. Here, 50% lockdownsuccess in Delhi means that 50% of the susceptible population in this state is successfullyhome-quarantined during the period March 25, 2020 till April 20, 2020.5
Insensitivity and relaxation in lockdown are measured in a same scale namely 10%, 20% and30% increment or decrement on the current estimate of lockdown rate (see Table 2).Following the above assumptions and using our ensemble model (0.1), we provided a forecastof notified COVID-19 cases (daily and cumulative) for MH, DL, MP, RJ, GJ, UP, and IND,respectively during May 3, 2020 till May 20, 2020. As COVID-19 notified cases are continuouslyrising in these seven locations therefore, it is more likely that lockdown period will be extendedbeyond May 3, 2020. Therefore, forecast using the ensemble model (0.1) during the mentionedtime duration in those seven locations are based on the following scenarios:( A1 ) Using our mechanistic models combination (see Fig 1 and supplementary) and the currentestimate of the lockdown rates (see Table 2), we forecast notified COVID-19 cases up to May20, 2020. This forecast is combined together with the results obtain from the hybrid statisticalmodel by using our ensemble model (0.1).( A2 ) Using our mechanistic models combination (see Fig 1 and supplementary) and thecurrent estimate of the lockdown rates (see Table 2), we forecast notified COVID-19 cases upto April 20, 2020. From April 21, 2020 till May 20, 2020, forecast are made using incre-ment or decrement (depending on the state) in the estimate of current lockdown rate. Thisforecast is combined together with the results obtain from the hybrid statistical model by usingour ensemble model (0.1).( A3 ) Using our mechanistic models combination (see Fig 1 and supplementary) and thecurrent estimate of the lockdown rates (see Table 2), we forecast notified COVID-19 cases upto April 20, 2020. From April 21, 2020 till May 20, 2020, forecast are made using incre-ment or decrement (depending on the state) in the estimate of current lockdown rate. Thisforecast is combined together with the results obtain from the hybrid statistical model by usingour ensemble model (0.1).( A4 ) Using our mechanistic models combination (see Fig 1 and supplementary) and thecurrent estimate of the lockdown rates (see Table 2), we forecast notified COVID-19 cases upto April 20, 2020. From April 21, 2020 till May 20, 2020, forecast are made using incre-ment or decrement (depending on the state) in the estimate of current lockdown rate. Thisforecast is combined together with the results obtain from the hybrid statistical model by usingour ensemble model (0.1).( A5 ) Using our mechanistic models combination (see Fig 1 and supplementary) and thecurrent estimate of the lockdown rates (see Table 2), we forecast notified COVID-19 cases upto May 2, 2020. From May 3, 2020 till May 20, 2020, forecast are made with no lockdown (seeEq. S1 in supplementary). This forecast is combined together with the results obtain from thehybrid statistical model by using our ensemble model (0.1).As, we assumed that lockdown individuals do not mixed with the general population there-fore, the basic reproduction number ( R ) with and without lockdown (see Fig 1 and supple-mentary) are equal [34]: R = β κσ ( µ + σ )( γ + µ + τ ) + β (1 − κ ) σ ( µ + γ )( µ + σ ) + β κρστ ( µ + σ )( δ + γ + µ )( γ + µ + τ ) .
6n the expression of R , sum of first two term indicate the community infection occur-ring from symptomatic and asymptomatic infected population. The third term in R specifythe hospital-based COVID-19 transmission. To distinguish the community and hospital-basedCOVID-19 transmission, we defined the community reproduction number ( R C ), and the hos-pital reproduction number ( R H ) as follows: R C = β κσ ( µ + σ )( γ + µ + τ ) + β (1 − κ ) σ ( µ + γ )( µ + σ ) , and R H = β κ ρ σ τ ( µ + σ )( δ + γ + µ )( γ + µ + τ ) . Using estimated values of epidemiologically uninformative parameters (see Table 2), we esti-mated R , R C , and R H for the seven locations MH, DL, MP, RJ, GJ, UP, and IND, respectively.Constructing an effective policy on future lockdown in a region will require some relationbetween effect of lockdown (number of COVID-19 case reduction) with some important epi-demiologically measurable & controllable parameters. Our mechanistic ODE model (see Fig 1and supplementary) has several important parameters and among them measurable and con-trollable parameters are β : average rate of transmission occurring from hospitalized & notifiedbased contacts (it can be controllable by following WHO guidelines); ρ : fraction of susceptiblepopulation from the community that are exposed to notified & hospitalized based contacts(it also can be minimized by following proper guidelines from WHO); κ : fraction of infectedthat are symptomatic (rapid COVID-19 testing can provide an accurate estimate); τ : notifi-cation & hospitalization rate of symptomatic infected population (it also depend on numberof COVID-19 testing). We perform a global sensitivity analysis [35] to determine the effectof these parameters on the lockdown effect and on the basic reproduction number ( R ), re-spectively. The effect of lockdown is measured in terms of differences in the total number ofCOVID-19 cases occurred during May 3, 2020 till May 20, 2020 under the lockdown scenarios( A1 ) and ( A5 ), respectively. We draw 500 samples from the biologically feasible ranges of thementioned four parameters (see Table 1) using Latin Hypercube Sampling (LHS) technique.Other informative and uninformative parameters during simulation of the mechanistic modelare taken from Table 1 and Table 2, respectively. Partial rank correlation coefficients (PRCC)and its corresponding p -value are evaluated to determine the effect of these mentioned fourparameters on the lockdown effect and the basic reproduction number ( R ), respectively. Results and Discussion
Testing of the three models (the mechanistic ODE model, the hybrid statistical model,and the ensemble model) on daily notified COVID-19 cases from Maharashtra (MH), Delhi(DL), Madhya Pradesh (MP), Rajasthan (RJ), Gujarat (GJ), Uttar Pradesh (UP), and India(IND) are presented in Fig. 2. Based on the performance on testing data from the mentionedseven locations, we estimated the weights ( w and w ) for our ensemble model (0.1) (see TableS2 in supplementary). Our result suggest that the mechanistic ODE model (see Fig 1 andsupplementary) displayed a better performance in RJ, UP, and IND compared to the hybridstatistical model (see Table S2 in supplementary). For rest of the locations (MH, DL, MP, andGJ), the hybrid statistical model has performed better than the mechanistic ODE model interms of capturing the trend of the time-series data (Table S2 and Table S3 in supplementary).The ensemble model (0.1), which is derived from a combination of the mechanistic ODE (see7ig 1 and supplementary) and the hybrid statistical model respectively, has provided a robustresult in all of these mentioned seven locations in terms of capturing time-series data trend (seeFig. 2).The estimates of uninformative parameters of the mechanistic ODE model (Fig 1 and sup-plementary) suggests that currently in the seven locations (MH, DL, MP, RJ, GJ, UP, andIND) the community infection is mainly dominated by contribution from the asymptomatic in-fected population (Table 2). Among the seven locations, the lowest percentage of symptomaticinfection in the community is found in RJ (about ) and the highest percentage is foundin IND (about ) (Table 2). Our estimates suggest that currently in the seven locations(MH, DL, MP, RJ, GJ, UP, and IND), the notification & hospitalization rate of symptomaticinfected population is about to
23 % (Table 2). Therefore, most of the COVID-19 in-fections in those mentioned seven locations are currently undetected. Our result agrees withthe recent report by the Indian Council of Medical Research (ICMR) [36]. Our estimates ofthe lockdown rate for MH, DL, MP, RJ, GJ, UP, and IND, respectively, suggest that lockdownis properly implemented in the two metro cities DL and MH. Also, in overall India (IND)lockdown is properly implemented. In these three locations (MH, DL, and IND), about to of the total susceptible population may be successfully home quarantined during thepresent lockdown period (Table 2). However, for rest of the locations, our results suggest thatlockdown may not be successful as about to of the total susceptible population maybe isolated (home quarantined) during the current lockdown period in MP, RJ, GJ, and UP,respectively (Table 2). In the seven locations, we found that about to of the total sus-ceptible populations may be exposed to hospital (notified & hospitalized population) relatedcontacts (Table 2). Considering the fact that estimates of the average hospital-based transmis-sion rates for the seven locations (MH, DL, MP, RJ, GJ, UP, and IND) are very high (Table 2),therefore, may be a significant amount of COVID-19 infection in these seven locations are cur-rently occurring due to notified & hospitalized infected related contacts. These findings canbe further justified by analyzing the estimates of the basic ( R ), the community ( R C ) and thehospitalized ( R H ) reproduction numbers for MH, DL, MP, RJ, GJ, UP, and IND, respectively(see Table 3). Except for the RJ , in the remaining six locations, we found that about to of the total COVID-19 transmission currently occurring from notified & hospitalizedinfected related contacts (see Table 3). These figures can be increased up to to ifproper measures are not taken in MH, DL, MP, GJ, UP, and IND, respectively, (see Table 3).This is a worrisome situation as higher value of the hospital-based transmission rate in MH,DL, MP, GJ, UP, and IND, respectively (Table 2), indicate that there may be super-spreadingincidents occurring from hospital-based contacts. In RJ, low contribution of R H on R (seeTable 3) may be due to existence of low percentage of the symptomatic infected population inthe community (Table 2) and that leads to low percentage of notified & hospitalized COVID-19cases. For further investigation on super-spreading events, we carried a global uncertainty andsensitivity analysis of some epidemiologically measurable and controllable parameters from ourmechanistic ODE model (Fig 1 and supplementary) namely, β : average rate of transmissionoccurring from notified & hospitalized based contact (it can be controllable following the WHOguidelines [37]), ρ : fraction of susceptible population that are exposed to hospital-based contact(it can be reduced by following proper guidelines from the WHO [37]), κ : fraction of the newlyinfected that are symptomatic (Rapid COVID-19 testing can provide an accurate estimate), τ : hospitalization & notification rate of symptomatic infected population (it also depend onnumber of COVID-19 testing) on the basic reproduction number ( R ). Partial rank correlation8oefficients (PRCC) and its corresponding p -value suggested that all these four parameters havesignificant positive correlation with R (see Fig. 4 and Fig. S7 to Fig.S12 in supplementary).Furthermore, high positive correlation of ρ on R indicate that small increase in the percentageof susceptible population from the community that are exposed hospital-based transmissionwill leads to significant increase in COVID-19 force of infection. Considering the fact that es-timated value of β (see Table 2) in MH, DL, MP, RJ, GJ, UP, and IND, respectively are veryhigh (much higher than community transmission rate), therefore, a small increase in ρ mayleads to a larger COVID-19 outbreak in those seven locations. Therefore, until and otherwiseany preventive measures are taken in these locations, a larger COVID-19 outbreak may triggerfrom hospitals and quarantine centers.Using the ensemble model (0.1), we forecast daily as well as total COVID-19 cases under fivedifferent lockdown scenarios in MH, DL, MP, RJ, GJ, UP, and IND, respectively, from May 3,2020 to May 20, 2020, (see Fig. 3, Table 4 and Fig. S1 to Fig. S6 in supplementary). Comparingthe projected total COVID-19 cases during May 3, 2020 to May 20, 2020, (see Table 4) withthe total observed cases [6] during March 2, 2020 till April 29, 2020, we found about two foldincrease in the total cases in MH, MP, GJ, UP, and IND, respectively. In summary, our forecastresult suggest that in the coming two weeks a significant increase in cases may be observed inmost of these locations.To determine which epidemiologically measurable and controllable parameters are mostinfluencing the effect of lockdown, we carried out a global uncertainty analysis of β , ρ , κ , and τ on the lockdown effect. The lockdown effect is measured in terms of differences in the totalnumber of COVID-19 cases during May 3, 2020 till May 20, 2020, in MH, DL, MP, RJ, GJ, UP,and IND, respectively, under the lockdown scenarios ( A1 ) and ( A5 ), respectively (see methodsection for details). For MH, PRCC and its corresponding p -value suggested that all these fourparameters have significant influence on the lockdown effect (see Fig. S7 in supplementary).Furthermore, significant negative correlation of β , and ρ with the lockdown effect (see Fig. S7in supplementary) suggested that only home quarantined the community may not be sufficientto reduce COVID-19 transmission in MH. Govt. and the policy makers may also have to focuson reducing the transmission occurring from hospital premises based on the guidelines fromthe WHO [37]. For DL, PRCC and its corresponding p -value suggested that β , ρ , and κ arethe main parameters that are influencing the lockdown effect (see Fig. S8 in supplementary).Moreover, significant negative correlation of β and ρ with the lockdown effect and as well assignificant positive correlation of κ with the lockdown effect (see Fig. S8 in supplementary)implies that an effective lockdown policy in DL may be a combination of lockdown in thecommunity, contact tracing of COVID-19 cases, and with some effort in reducing hospital-basedtransmission following WHO guidelines [37]. For MP, PRCC and its corresponding p -valuesuggested that κ and τ have high positive correlation with the lockdown effect (see Fig. S9in supplementary). Furthermore, ρ have significant negative correlation with the lockdowneffect (see Fig. S9 in supplementary). Therefore an effective lockdown policy in MP may be astrict implementation of lockdown in the red and orange zones, rapid COVID-19 testing in thecommunity and reducing hospital-based transmission by following guidelines from WHO [37].For RJ, PRCC and its corresponding p -value suggested that only κ have significant positivecorrelation with the lockdown effect (see Fig. S10 in supplementary). No significant correlationwith hospital-based parameters may be due to existence of low percentage of the symptomaticinfected population in the community (see Table 2) and that leads to low percentage of notified& hospitalized based COVID-19 transmission. Therefore, RJ Govt. may focused more on9ontact tracing in the community with relaxation may be given in the Green and Orange zonesto increase the percentage of symptomatic infected in the community. For GJ, PRCC and itscorresponding p -value suggested that all these four parameters have significant influence on thelockdown effect (see Fig. S11 in supplementary). Furthermore, significant negative correlationof β , and ρ with the lockdown effect (see Fig. S11 in supplementary) indicate that only homequarantined the community may not be sufficient to reduce COVID-19 transmission in GJ.Govt. of GJ and policy makers may also have to focus on reducing the transmission occurringfrom hospital premises based on the guidelines from the WHO [37]. For UP, PRCC and itscorresponding p -value suggested that β , ρ , and κ are the main parameters that are influencingthe lockdown effect (see Fig. S12 in supplementary). Moreover, significant negative correlationof β and ρ with the lockdown effect and as well as significant positive correlation of κ with thelockdown effect (see Fig. S12 in supplementary) implies that an effective lockdown policy in UPmay be a combination of lockdown (relaxation in Green zone), contact tracing in communitywith effort in reducing hospital-based transmission following the WHO guidelines [37]. Finallyfor IND, PRCC and its corresponding p -value suggested that β and ρ are the main parametersthat are most influencing the lockdown effect (see Fig. 4). Therefore, only home quarantinedthe community may not be sufficient to reduce COVID-19 transmission in IND. Govt. of INDand the policy makers may also have to focus on reducing the transmission occurring fromhospital premises based on the guidelines from the WHO [37]. Conclusion
Our analysis of the mechanistic model with hospital-based COVID-19 transmission suggestthat most of the new infections occurring in India as well most of the states are currentlyundetected. Furthermore, a global sensitivity analysis of two epidemiologically controllable pa-rameters from the hospital-based transmission on the basic reproduction nuber ( R ), indicatethat if appropriate preventive measures are not taken immediately, a much larger COVID-19outbreak may trigger due to the transmission occurring from the hospitalized & notified basedcontacts. Moreover, our ensemble forecast model (0.1) predicted a substantial percentage ofincrease in the COVID-19 notified cases during May 3, 2020 till May 20, 2020, (see Table 4) inmost of these locations. In Rajasthan, trend of the forecast data (see Fig S4 in supplementary)during May 3, 2020 till May 20, 2020, is showing a decreasing trend. This is may be due to lownumber of hospitalized and reported cases in this state (see Table 2). However, cases may risein Rajasthan if relaxation in lockdown is applied. Furthermore, trend of the forecast data inoverall India (see Fig 3) during May 3, 2020 till May 20, 2020, indicating the fact that reachingthe peak of the COVID-19 epidemic curve may be a long way ahead for India . Fi-nally, based on our results of global sensitivity analysis of the four important epidemiologicallymeasurable & controllable parameters on the lockdown effect, we are suggesting the followingpolicy that may reduce the threat of a larger COVID-19 outbreak in the coming days: Effective Lockdown Policy:
Dividing different states into three clusters (red, orange,and green) is well appreciated as it increases the percentage of symptomatic infection in thecommunity. However, more COVID-19 testing is needed as it increases the number of notified& hospitalized cases over the states. It is much easier to reduce hospital-based transmission incompare to community transmission. To reduce the hospitalized & notified based contacts, anefficient disaster management team is required. They will continuously monitor the situations10n different hospitals and quarantine centers across India. This team must ensure that propersafety measures are being followed based on the guidelines provided by ICMR and WHO [37].
Conflict of interests
The authors declare that they have no conflicts of interest.
Acknowledgments
Dr. Tridip Sardar acknowledges the Science & Engineering Research Board (SERB) majorproject grant (File No: EEQ/2019/000008 dt. 4/11/2019), Govt. of India.The Funder had no role in study design, data collection and analysis, decision to publish,or preparation of the manuscript. 11 eferences [1] C. Wang, P. W. Horby, F. G. Hayden, and G. F. Gao, “A novel coronavirus outbreak ofglobal health concern,”
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Figure 1: A Flow diagram of the mechanistic ODE model with hospital-based COVID-19 transmission and lockdown effect.Different class of population shown in this figure are S : Susceptible population, E : Exposed population, I S : COVID-19symptomatic infected population, I A : COVID-19 asymptomatic infected population, H : Notified & Hospital individualssuffering from COVID-19 infection, R : COVID-19 recovered population, and L : Home quarantined susceptible populationduring lockdown, respectively. Epidemiological information of the parameters shown in this figure are provided in Table 1. igure 2: Combination of mechanistic ODE models (Fig 1 and supplementary), Hybrid statistical model, and Ensemblemodel (0.1) fitting to the daily notified COVID-19 cases form six different states and overall India. Respective subscriptsare MH : Maharashtra, DL : Delhi, MP : Madhya Pradesh, RJ : Rajasthan, GJ : Gujarat, UP : Uttar Pradesh, and IND :India. Here, LD indicate the period after the lockdown implementation in overall India started at 25/03/2020 and NoLD specifies the period before lockdown. Lockdown effect is considered only for the mechanistic ODE model (Fig 1 andsupplementary) and consequently in the ensemble model (0.1). Shaded area specifies the 95% confidence region. igure 3: Ensemble model (0.1) forecast for the daily notified COVID-19 cases in India during May 3, 2020 till May 20,2020 under five different lockdown scenarios. Under lockdown scenarios ( A2 ) to ( A4 ), projections are made with 10%,20% and 30% decrement in the current estimate of the lockdown rate, respectively (see Table 2). In lockdown scenario( A1 ), projections are made with current estimated lockdown rate (see Table 2). Finally, in the lockdown scenario ( A5 ),projections are made with no lockdown during May 3, 2020 till May 20, 2020. igure 4: Effect of uncertainty of four epidemiologically measurable and controllable parameters of the mechanistic ODEmodel (see Fig 1, Table 1 and method section) on the effect of lockdown and the basic reproduction number ( R ). Lockdowneffect is measured in terms of the differences in total number of COVID-19 cases occurred during May 3, 2020 till May 20,2020 in India under the lockdown scenarios ( A1 ) and ( A5 ), respectively (see method section). Effect of Uncertainty ofthese four parameters on the two mentioned responses are measured using Partial Rank Correlation Coefficients (PRCC).500 samples for each parameters were drawn using Latin hypercube sampling techniques (LHS) from their respective rangesprovided in Table 1. ables Parameters Biological Meaning Value/Ranges Reference
Π = µ × N (0) Recruitment rate of human population Differs over states -1 µ Average life expectancy at birth Differs over states [38] β Average transmission rate of a symptomaticand asymptomatic COVID-19 infected (0 - 500) day − Estimated β Average transmission rate of a notified & hos-pitalized COVID-19 infected (0 - 500) day − Estimated ρ Fraction of the susceptible population that areexposed to hospital-based transmission 0 - 0.2 Estimated1 σ COVID-19 incubation period (1 - 14) days Estimated κ Fraction of the COVID-19 exposed populationthat become symptomatic infected 0 - 1 Estimated γ Average recovery rate of symptomatic infec-tion ( γ - 1) day − Estimated γ Average recovery rate of asymptomatic infec-tion ( γ - 1) day − Estimated τ Average hospitalization rate for the COVID-19 symptomatic individuals (0 - 1) day − Estimated δ Average death rate due to COVID-19 infec-tion in hospital Differs over states [6] γ Average recovery rate of the notified & hospi-talized populations Differs over states [6] l Average lockdown rate (0 - 0.9) day − Estimated1 ω Current lockdown period in India 40 days [29; 30]21able 2: Estimated uninformative parameters of the mechanistic ODE model combinations (see Fig 1and supplementary). Respective subscripts are MH : Maharashtra, DL : Delhi, MP : Madhya Pradesh, RJ : Rajasthan, GJ : Gujarat, UP : Uttar Pradesh, and IND : India. All data are given in the format
Es-timate (95% CI) . Region β β ρ × σ κ γ τ × γ l × MH . . − .
75) 21 . . − .
7) 7 . . − .
78) 0 . . − . . . − . . . − . . . − .
3) 0 . . − . . . − . DL . . − .
54) 220 . . − .
8) 7 . . − .
67) 0 . . − . . . − . . . − . . . − .
3) 0 . . − . . . − . MP . . − .
91) 42 . . − . . − .
72) 0 . . − . . . − . . . − . . − .
2) 0 . . − . . . − . RJ . . − .
74) 325 . . − . . − .
56) 0 . . − . . . − . . . − . . . − .
3) 0 . . − . . − . GJ . . − .
10) 186 . . − . . − .
20) 0 . . − .
94) 0 . . − . . . − .
98) 14(4 . − .
2) 0 . . − .
98) 11 . . − . UP . . − .
92) 242(8 . − . . − .
51) 0 . . − .
71) 0 . . − .
95) 0 . . − . . . −
5) 0 . . − . . . − . IND . . − .
24) 499(494 − .
8) 8 . . − .
70) 0 . . − .
15) 0 . . − .
96) 0 . . − .
99) 0 . . − .
53) 0 . . − . . . − . MH , DL , MP , RJ , GJ , UP , and IND are same as Table 2. All data are givenin the format
Estimate (95% CI) . Region R R C % R R H % R MH . . − . . . − . . . − .
93) 0 . . − . . . − . DL . . − .
52) 2 . . − .
49) 93 . . − .
95) 0 . . − . . . − . MP . . − .
11) 2 . . − .
63) 98 . . − .
81) 0 . . − . . . − . RJ . . − .
78) 1 . . − .
76) 99 . . − . − . . − . GJ . . − .
86) 1 . . − .
72) 93 . . − .
56) 0 . . − .
68) 6 . . − . UP . . − .
14) 1 . . − .
88) 84(39 . − .
96) 0 . . − .
47) 16(0 . − . IND . . − .
94) 2 . . − .
86) 91 . . − .
84) 0 . . − . . . − . MH , DL , MP , RJ , GJ , UP , and IND are same as Table 2. Regionswhere, current lockdown rate ( (cid:121) ) implies the ensemble model (0.1) projections for the Scenarios (A2) to(A4) with 10%, 20% and 30% decrement in the current estimate of lockdown rate (see Table 2) duringthe mentioned period. Whereas, current lockdown rate ( (cid:120) ) implies the ensemble model (0.1) projectionsfor the Scenarios (A2) to (A4) with 10%, 20% and 30% increment in the current estimate of lockdownrate (see Table 2) during the mentioned period. Scenario (A1) implies the ensemble model (0.1) forecastwith the current estimate of the lockdown rate (see Table 2) during May 3, 2020 till May 20, 2020.Finally, Scenario (A5) implies the ensemble model (0.1) forecast with no lockdown during May 3, 2020till May 20, 2020. All data are provided in the format
Estimate (95% CI) . Region Current lockdownrate Scenario A1 Scenario A2 Scenario A3 Scenario A4 Scenario A5MH (cid:121)(cid:121) − − − − − DL (cid:121)(cid:121) − − − − − MP (cid:120)(cid:120) − − − − − RJ (cid:120)(cid:120) − − − − − GJ (cid:120)(cid:120) − − − − − UP (cid:120)(cid:120) − − − − − IND (cid:121)(cid:121) − − − − − upplementary Method Model without lock-down
The basic structure of our model is an extension of our earlier model [23] by consideringhospital-based transmission. As hospitalized & notified infected ( H ) can only contact with avery small fraction of the susceptible class therefore, we assume the effective contact for a singlehospitalized & notified infected is β ( ρS ) N and ρ is the fraction of susceptible that are in contactwith the hospitalized & notified infected. We also assume that effective contact of a singlesymptomatic ( I S ) and a asymptomatic individual is same and at a rate β SN . Therefore, totalhuman population N ( t ) at time t is subdivided into six mutually exclusive sub-classes namely,susceptible ( S ( t )), exposed ( E ( t )), symptomatic ( I S ( t )), asymptomatic ( I A ( t )), hospitalized ¬ified ( H ( t )), and recovered ( R ( t )), respectively. Interaction between different sub-classes areprovided by the following system of differential equation: dSdt = Π − β SI S N − β SI A N − β ( ρS ) HN − µSdEdt = β SI S N + β SI A N + β ( ρS ) HN − ( µ + σ ) E,dI S dt = κσE − ( τ + µ + γ ) I S ,dI A dt = (1 − κ ) σE − ( µ + γ ) I A , (S-1) dHdt = τ I S − ( γ + δ + µ ) H,dRdt = γ I S + γ I A + γ H − µR. Model with lock-down
Lock-down effect in the model (S-1) can be included by home quarantine a fraction of thesusceptible population at a rate l . This home-quarantine populations are considered as a newsubclass namely, lock-down class L . We assume that population in the lock-down compartmentare home quarantined and they do not interact with other populations. Therefore, effectivecontact of a single symptomatic ( I S ), asymptomatic ( I A ), and hospitalized & notified infectedare β S ( N − L ) , β S ( N − L ) , and β ( ρS )( N − L ) , respectively. The interaction of all these sub-classes can be expressed as an lock-down ODE model as follows:25 Sdt = Π + ωL − β SI S ( N − L ) − β SI A ( N − L ) − β ( ρS ) H ( N − L ) − ( µ + l ) SdLdt = lS − ( µ + ω ) L,dEdt = β SI S ( N − L ) + β SI A ( N − L ) + β ( ρS ) H ( N − L ) − ( µ + σ ) E,dI s dt = κσE − ( τ + µ + γ ) I S ,dI A dt = (1 − κ ) σE − ( µ + γ ) I A , (S-2) dHdt = τ I S − ( δ + µ + γ ) H,dRdt = γ I S + γ I A + γ H − µR. A flow diagram of the model (S-1) and (S-2) are provided in
Fig 1 (see main text). Biologicalinterpretations of the model (S-1) and (S-2) parameters are provided in
Table 1 (see main text).
Positivity and boundedness of the solution for the Model (S-1)In this section, we provided a proof for the positivity and boundedness of solutions of thesystem (S-1) with initial conditions ( S (0) , E (0) , I S (0) , I A (0) , H (0) , R (0)) T ∈ R . We first statethe following lemma. Lemma S1. ([39]) Suppose Ω ⊂ R × C n is open, f i ∈ C (Ω , R ) , i = 1 , , , ..., n . If f i | x i ( t )=0 ,X t ∈ C n +0 ≥ , X t = ( x t , x t , ....., x n ) T , i = 1 , , , ...., n , then C n +0 { φ = ( φ , ....., φ n ) : φ ∈ C ([ − τ, , R n +0 ) } is the invariant domain of the following equations dx i ( t ) dt = f i ( t, X t ) , t ≥ σ, i = 1 , , , ..., n. where R n +0 = { ( x , ....x n ) : x i ≥ , i = 1 , ...., n } . Proposition S1.
The system (S-1) is invariant in R .Proof. By re-writing the system (S-1), we have dXdt = B ( X ( t )) , X (0) = X ≥ B ( X ( t )) = ( B ( X ) , B ( X ) , ..., B ( X )) T We note that dSdt | S =0 = Π ≥ , dEdt | E =0 = ( β I S + β I A + ρβ H ) SN ≥ ,dI S dt | I S =0 = κσE ≥ , dI A dt | I A =0 = (1 − κ ) σE ≥ ,dHdt | H =0 = τ I S ≥ , dRdt | R =0 = γ I S + γ I A + γ H ≥ . Then it follows from the Lemma S1 that R is an invariant set. Lemma S2.
The system (S-1) is bounded in the region
Ω = { ( S, E, I S , I A , H, R ∈ R | S + E + I S + I A + H + R ≤ Π µ } roof. We observed from the system that dNdt = Π − µN − δH ≤ Π − µN = ⇒ lim t →∞ supN ( t ) ≤ Π µ Hence the system (S-1) is bounded.
Local stability of disease-free equilibrium (DFE)
The DFE of the model (S-1) is given by ε = ( S , E , I S , I A , H , R )= (cid:16) Π µ , , , , , , (cid:17) The local stability of ε can be established on the system (S-1) by using the next generationoperator method. Using the notation in [34], the matrices F for the new infection and V forthe transition terms are given, respectively, by F = β β ρβ ,V = µ + σ − κσ τ + µ + γ − (1 − κ ) σ µ + γ − τ δ + µ + γ . It follows that the basic reproduction number [40], denoted by R = ρ ( F V − ), where ρ is thespectral radius, is given by R = β κσ ( µ + σ )( γ + µ + τ ) + β (1 − κ ) σ ( µ + γ )( µ + σ ) + β κρστ ( µ + σ )( δ + γ + µ )( γ + µ + τ )Using Theorem 2 in [34], the following result is established. Lemma S3.
The DFE, ε , of the model (S-1) is locally-asymptotically stable (LAS) if R < ,and unstable if R > .Global stability of DFE To show the global stability of ε , of the model (S-1), we rewrite the system (S-1) as follows dXdt = T ( X, I ) (S-4) dIdt = G ( X, I ) , G ( X,
0) = 0 , where, X = ( S, R ) ∈ R denotes (it components) the number of uninfected individuals and I = ( E, I S , I A , H ) ∈ R denotes (it components) the number of infected individuals including27atent, infectious, etc. ε = ( X ∗ ,
0) denotes the disease-free equilibrium of the system (S-4).For the system (S-1), T ( X, I ) and G ( X, I ) are given as follows: T ( X, I ) = Π − β SI S N − β SI A N − β ( ρS ) HN − µSγ I S + γ I A + γ H − µR ,G ( X, I ) = β SI S N + β SI A N + β ( ρS ) HN − ( µ + σ ) EκσE − ( τ + µ + γ ) I S (1 − κ ) σE − ( µ + γ ) I A τ I S − ( γ + δ + µ ) H . It is clear from the expression of G ( X, I ) that G ( X,
0) = 0.To show global stability of ε = ( X ∗ ,
0) following two condition must hold: (H1)
For dXdt = T ( X, X ∗ is globally asymptotically stable. (H2) G ( X, I ) = AI − ˆ G ( X, I ), ˆ G ( X, I ) ≥ X, I ) ∈ Ω,where A = D I G ( X ∗ ,
0) is an M-matrix (the off diagonal elements are nonnegative) and Ωis the region where model (S-1) makes biological sense.Now, the system defined in (H1) can be written as dSdt = Π − µS, (S-5) dRdt = − µR. Solving analytically this system of equation we get, S ( t ) = Π µ + e − µt ( S (0) − Π µ ), R ( t ) = e − µt R (0).As t → ∞ , S ( t ) = Π µ , R ( t ) →
0. Therefore, X ∗ is globally asymptotically stable for dXdt = T ( X, (H1) holds for the system (S-1). Now matrices A and ˆ G ( X, I ) for the system (S-1) aregiven as follows: 28 = − ( µ + σ ) β β ρβ κσ − ( γ + τ + µ ) 0 0(1 − κ ) σ − ( γ + µ ) 00 τ − ( γ + δ + µ ) , ˆ G ( X, I ) = β I S (1 − SN ) + β I A (1 − SN ) + ρβ H (1 − SN )000 . Clearly, A is an M-matrix and as S ( t ) ≤ N ( t ) in Ω, therefore, ˆ G ( X, I ) ≥ X, I ) ∈ Ω.Following [41], the below result can be stated:
Theorem S1.
The DFE of the model (S-1) is globally asymptotically stable in Ω whenever R < .Existence of endemic equilibria In this section, the existence of the endemic equilibrium of the model (S-1) is established.Let us denote k = µ + σ, k = γ + µ + τ, k = µ + γ , k = δ + γ + µ. Let ε ∗ = ( S ∗ , L ∗ , E ∗ , I ∗ S , I ∗ A , H ∗ , R ∗ ) represents any arbitrary endemic equilibrium point (EEP)of the model (S-1). Further, define λ ∗ = β I ∗ S N ∗ + β I ∗ A N ∗ + ρβ H ∗ N ∗ (S-6)It follows, by solving the equations in (S-1) at steady-state, that S ∗ = Π λ ∗ + µ , E ∗ = λ ∗ S ∗ k , I ∗ S = κσλ ∗ S ∗ k k , I ∗ A = (1 − κ ) σλ ∗ S ∗ k k (S-7) H ∗ = τ κσλ ∗ S ∗ k k k , R ∗ = γ κσλ ∗ S ∗ µk k + γ (1 − κ ) σλ ∗ S ∗ µk k + γ τ κσλ ∗ S ∗ µk k k Substituting the expression in (S-7) into (S-6) shows that the non-zero equilibrium of the model(S-1) satisfy the following linear equation, in terms of λ ∗ : a λ ∗ + a = 0 (S-8)where a = µk k k + κσk k ( µ + γ ) + (1 − κ ) σk k ( µ + γ ) + τ κσk ( µ + γ ) a = µk k k k (1 − R ) 29ince a > µ > k > k > k > k >
0, it is clear that the model (S-1) has aunique endemic equilibrium point (EEP) whenever R > R <
1. This rules out the possibility of the existence of equilibrium other thanDFE whenever R <
1. Furthermore, it can be shown that, the DFE ε of the model (S-1) isglobally asymptotically stable (GAS) whenever R < Theorem S2.
The model (S-1) has a unique endemic (positive) equilibrium, given by ε ∗ , when-ever R > and has no endemic equilibrium for R ≤ . igures Figure S1: Ensemble model (see main text) forecast for the daily notified COVID-19 cases in Maharashtra during May 3,2020 till May 20, 2020 under five different lockdown scenarios. Under lockdown scenarios ( A2 ) to ( A4 ), projections aremade with 10%, 20% and 30% decrement in the current estimate of the lockdown rate, respectively (see Table 2 in maintext). In lockdown scenario ( A1 ), projections are made with current estimated lockdown rate (see Table 2 in main text).Finally, in the lockdown scenario ( A5 ), projections are made with no lockdown during May 3, 2020 till May 20, 2020. igure S2: Ensemble model (see main text) forecast for the daily notified COVID-19 cases in Delhi during May 3, 2020till May 20, 2020 under five different lockdown scenarios. Lockdown Scenarios ( A1 ) to ( A5 ) are same as Figure S1.Figure S3: Ensemble model (see main text) forecast for the daily notified COVID-19 cases in Madhya Pradesh duringMay 3, 2020 till May 20, 2020 under five different lockdown scenarios. Lockdown Scenarios ( A1 ) and ( A5 ) are same asFigure S1. However, under lockdown scenarios ( A2 ) to ( A4 ), projections are made with 10%, 20% and 30% increment in the current estimate of the lockdown rate, respectively (see Table 2 in main text). igure S4: Ensemble model (see main text) forecast for the daily notified COVID-19 cases in Rajasthan during May 3,2020 till May 20, 2020 under five different lockdown scenarios. Lockdown Scenarios ( A1 ) and ( A5 ) are same as Figure S1.However, under lockdown scenarios ( A2 ) to ( A4 ), projections are made with 10%, 20% and 30% increment in the currentestimate of the lockdown rate, respectively (see Table 2 in main text).Figure S5: Ensemble model (see main text) forecast for the daily notified COVID-19 cases in Gujarat during May 3,2020 till May 20, 2020 under five different lockdown scenarios. Lockdown Scenarios ( A1 ) and ( A5 ) are same as Figure S1.However, under lockdown scenarios ( A2 ) to ( A4 ), projections are made with 10%, 20% and 30% increment in the currentestimate of the lockdown rate, respectively (see Table 2 in main text). igure S6: Ensemble model (see main text) forecast for the daily notified COVID-19 cases in Uttar Pradesh during May 3,2020 till May 20, 2020 under five different lockdown scenarios. Lockdown Scenarios ( A1 ) and ( A5 ) are same as Figure S1.However, under lockdown scenarios ( A2 ) to ( A4 ), projections are made with 10%, 20% and 30% increment in the currentestimate of the lockdown rate, respectively (see Table 2 in main text). igure S7: Effect of uncertainty of four epidemiologically measurable and controllable parameters of the mechanistic ODEmodel (see Table 1 and Fig 1 in the main text) on the effect of lockdown and the basic reproduction number ( R ). Lockdowneffect is measured in terms of the differences in total number of COVID-19 cases occurred during May 3, 2020 till May 20,2020 in Maharashtra under the lockdown scenarios ( A1 ) and ( A5 ), respectively (see main text). Effect of Uncertainty ofthese four parameters on the two mentioned responses are measured using Partial Rank Correlation Coefficients (PRCC).500 samples for each parameters were drawn using Latin hypercube sampling techniques (LHS) from their respective rangesprovided in Table 1 (main text). igure S8: Effect of uncertainty of four epidemiologically measurable and controllable parameters of the mechanistic ODEmodel (see Table 1 and Fig 1 in the main text) on the effect of lockdown and the basic reproduction number ( R ). Lockdowneffect is measured in terms of the differences in total number of COVID-19 cases occurred during May 3, 2020 till May20, 2020 in Delhi under the lockdown scenarios ( A1 ) and ( A5 ), respectively (see main text). Effect of Uncertainty ofthese four parameters on the two mentioned responses are measured using Partial Rank Correlation Coefficients (PRCC).500 samples for each parameters were drawn using Latin hypercube sampling techniques (LHS) from their respective rangesprovided in Table 1 (main text). igure S9: Effect of uncertainty of four epidemiologically measurable and controllable parameters of the mechanistic ODE model (seeTable 1 and Fig 1 in the main text) on the effect of lockdown and the basic reproduction number ( R ). Lockdown effect is measuredin terms of the differences in total number of COVID-19 cases occurred during May 3, 2020 till May 20, 2020 in Madhya Pradesh under the lockdown scenarios ( A1 ) and ( A5 ), respectively (see main text). Effect of Uncertainty of these four parameters on the twomentioned responses are measured using Partial Rank Correlation Coefficients (PRCC). 500 samples for each parameters were drawnusing Latin hypercube sampling techniques (LHS) from their respective ranges provided in Table 1 (main text). igure S10: Effect of uncertainty of four epidemiologically measurable and controllable parameters of the mechanistic ODE model (seeTable 1 and Fig 1 in the main text) on the effect of lockdown and the basic reproduction number ( R ). Lockdown effect is measuredin terms of the differences in total number of COVID-19 cases occurred during May 3, 2020 till May 20, 2020 in Rajasthan under thelockdown scenarios ( A1 ) and ( A5 ), respectively (see main text). Effect of Uncertainty of these four parameters on the two mentionedresponses are measured using Partial Rank Correlation Coefficients (PRCC). 500 samples for each parameters were drawn using Latinhypercube sampling techniques (LHS) from their respective ranges provided in Table 1 (main text). igure S11: Effect of uncertainty of four epidemiologically measurable and controllable parameters of the mechanistic ODE model (seeTable 1 and Fig 1 in the main text) on the effect of lockdown and the basic reproduction number ( R ). Lockdown effect is measuredin terms of the differences in total number of COVID-19 cases occurred during May 3, 2020 till May 20, 2020 in Gujarat under thelockdown scenarios ( A1 ) and ( A5 ), respectively (see main text). Effect of Uncertainty of these four parameters on the two mentionedresponses are measured using Partial Rank Correlation Coefficients (PRCC). 500 samples for each parameters were drawn using Latinhypercube sampling techniques (LHS) from their respective ranges provided in Table 1 (main text). igure S12: Effect of uncertainty of four epidemiologically measurable and controllable parameters of the mechanistic ODE model (seeTable 1 and Fig 1 in the main text) on the effect of lockdown and the basic reproduction number ( R ). Lockdown effect is measuredin terms of the differences in total number of COVID-19 cases occurred during May 3, 2020 till May 20, 2020 in Uttar Pradesh under the lockdown scenarios ( A1 ) and ( A5 ), respectively (see main text). Effect of Uncertainty of these four parameters on the twomentioned responses are measured using Partial Rank Correlation Coefficients (PRCC). 500 samples for each parameters were drawnusing Latin hypercube sampling techniques (LHS) from their respective ranges provided in Table 1 (main text).Figure S13: Posterior distribution of the weights for the mechanistic ODE model combinations (S-1) & (S-2)and the Hybrid statistical model (see main text), respectively for Maharashtra. igure S14: Posterior distribution of the weights for the mechanistic ODE model combinations (S-1) & (S-2)and the Hybrid statistical model (see main text), respectively for Delhi.Figure S15: Posterior distribution of the weights for the mechanistic ODE model combinations (S-1) & (S-2)and the Hybrid statistical model (see main text), respectively for Madhya Pradesh. igure S16: Posterior distribution of the weights for the mechanistic ODE model combinations (S-1) & (S-2)and the Hybrid statistical model (see main text), respectively for Rajasthan.Figure S17: Posterior distribution of the weights for the mechanistic ODE model combinations (S-1) & (S-2)and the Hybrid statistical model (see main text), respectively for Gujarat. igure S18: Posterior distribution of the weights for the mechanistic ODE model combinations (S-1) & (S-2)and the Hybrid statistical model (see main text), respectively for Uttar Pradesh.Figure S19: Posterior distribution of the weights for the mechanistic ODE model combinations (S-1) & (S-2)and the Hybrid statistical model (see main text), respectively for India. ables Table S1: Estimated uninformative initial conditions for the mechanistic ODE model (S-1). Respectivesubscripts are MH : Maharashtra, DL : Delhi, MP : Madhya Pradesh, RJ : Rajasthan, GJ : Gujarat, UP : Uttar Pradesh, and IND : India. All data are provided in the format
Estimate (95% CI) . Region S (0) E (0) I (0) A (0)MH − . − . . . − .
93) 2316 . . − . DL − − . − .
29) 245(144 . − MP − . − . . − .
10) 4095(93 . − RJ − . . − . − . − GJ − . − . . − . . − . UP − . . − .
34) 3 . . − .
35) 4764(379 − IND − − − − MH , DL , MP , RJ , GJ , UP , and IND are same as Table S1. All data are provided in the format
Estimate (95% CI) . Weights MH DL MP RJ GJ UP IND w . . − . . . − . . . − . . . − . . . − .
38) 0 . . − . . . − . w . . − . . . − . . . − . . . − . . . − . . . − . . . − . Table S3: Goodness of fit (RMSE and MAE) of the Hybrid statistical model (see main text) for the testdata from MH , DL , MP , RJ , GJ , UP , and IND , respectively. Respective subscripts MH , DL , MP , RJ , GJ , UP , and IND are same as Table S1.
Goodness of fit MH DL MP RJ GJ UP IND
RM SE . .
855 28 . . . . . M AE . . . . . . .61564