# Effective lockdown and role of hospital-based COVID-19 transmission in some
Indian states: An outbreak risk analysis

Tridip Sardar 111Email<EMAIL_ADDRESS>Sourav Rana 222Corresponding
author. Email<EMAIL_ADDRESS>Department of Mathematics,
Dinabandhu Andrews College, Kolkata, West Bengal, India  Department of
Statistics, Visva-Bharati University, Santiniketan, West Bengal, India

###### Abstract

There are several reports in India that indicate hospitals and quarantined
centers are COVID-19 hotspots. In the absence of efficient contact tracing
tools, Govt. and the policymakers may not be paying attention to the risk of
hospital-based transmission. To explore more on this important route and its
possible impact on lockdown effect, we developed a mechanistic model with
hospital-based transmission. Using daily notified COVID-19 cases from six
states (Maharashtra, Delhi, Madhya Pradesh, Rajasthan, Gujarat, and Uttar
Pradesh) and overall India, we estimated several important parameters of the
model. Moreover, we provided an estimation of the basic ($R_{0}$), the
community ($R_{C}$), and the hospital ($R_{H}$) reproduction numbers 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
hybrid statistical model. Using the ensemble model, we forecast COVID-19
notified cases (daily and cumulative) from May 3, 2020, till May 20, 2020,
under five different lockdown scenarios in the mentioned locations. Our
analysis of the mechanistic model suggests that most of the new COVID-19 cases
are currently undetected in the mentioned seven locations. Furthermore, a
global sensitivity analysis of four epidemiologically measurable &
controllable parameters on $R_{0}$ and as well on the lockdown effect,
indicate that if appropriate preventive measures are not taken immediately, a
much larger COVID-19 outbreak may trigger from hospitals and quarantined
centers. In most of the locations, our ensemble model forecast indicates a
substantial percentage of increase in the COVID-19 notified cases in the
coming weeks in India. Based on our results, we proposed a containment policy
that may reduce the threat of a larger COVID-19 outbreak in the coming days.

###### keywords:

COVID-19; Hospital-based transmission; Ensemble model forecast; Outbreak risk
analysis; Effective lockdown policy;

††journal: Elsevier

## Introduction

Coronavirus disease of 2019 (COVID-19) was first observed in Wuhan, China and
rapidly spread across the globe in a short duration Chen2020Lancet . World
Health Organization (WHO) declares COVID-19 as pandemic after assessing its
various characteristics WHO_Pandemic . As of April 29, 2020–over three million
cases and over two hundred thousand deaths due to COVID-19 is reported across
the globe worldcovid2020track . In India, the first confirmed case of COVID-19
was reported on $30/01/2020$ WHO2020_1 , a student from Kerala studying in a
university in Wuhan India_Today5 . As of April 29, 2020–$33065$ confirmed
cases and $1079$ deaths due to COVID-19 are reported in India
indiacovid2020track .

According to a daily monitoring report published by WHO, $22,073$ healthcare
workers across $52$ countries are being tested positive to COVID-19
WHOreportheathcare2020 . The report also noted that the number provided may be
an underestimation as there is no systematic reporting of infections among the
healthcare workers WHOreportheathcare2020 . In India, there are several
reports that indicate hospitals and quarantined centers are COVID-19 hotspots
TheHINDUsuperspreaders ; India_Today1 ; Economic_Times1 ; Print_In1 ; NDTV2 .
The doctors, nurses and other health workers are mainly vulnerable as they are
in close proximity with the COVID-19 patients India_Today1 ; Economic_Times1 ;
Print_In1 ; NDTV2 ; NDTVheathcareworkers . The close relatives of the notified
COVID-19 patients in the quarantine centers may also be at risk of getting
infection. In addition, those journalists who are continuously visiting
hospitals and quarantine centers to get updated reports on COVID-19, may also
be at risk of getting infection IndiaTVjournalistscovid19 ; Scroll_In1 .
Therefore, a significant percentage of susceptible population in the community
may be exposed to COVID-19 infection occurring from the contacts with patents
in hospitals and quarantine centers. Due to unavailability of efficient
contact tracing tools, Govt. and the policy makers may be ignoring this
important transmission route of COVID-19.

Currently there is no vaccine and effective medicine available for COVID-19.
Therefore, to break 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 nation may not be
a very feasible and effective solution. In addition, lockdown already have a
huge impact on the Indian economy specially on the short scale industries
Economic_Times2 ; India_Today4 . To partially overcome this economic crisis as
well reducing COVID-19 transmission, Govt. has proposed some amendments (known
as cluster containment strategy) on the lockdown rules from April 20, 2020
Economictimes2020a ; financialexpress20a . In these revised rules, Govt. has
provided some relaxation in current rules by dividing 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 Economictimes2020a ; financialexpress20a ; businesstoday2020a .
However, question remains whether this cluster containment strategy may be
successful in reducing COVID-19 transmission or not? If not then what may be
other alternative solutions to reduce COVID-19 transmission? These question
can only be answered by studying the dynamics and prediction of a mechanistic
mathematical model for COVID-19 transmission 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 effective control measures for reducing
COVID-19 transmission Moghadas2020PNAS ; Tang2020 ; sardar2020assessment . In
their study, Moghdas et al Moghadas2020PNAS  developed an age-structure model
on COVID-19 and projected required ICU beds for different outbreak scenarios
in USA. To improve the hospital capacity and possible containment of COVID-19,
they recommended self isolation and better hygiene practices in the community.
Using a mechanistic model, Tang et al Tang2020  estimated the control
reproduction numbers and studied the effect of various interventions on
COVID-19 transmission in China. Recently, Sardar et al sardar2020assessment
provided an effective lockdown strategy to control COVID-19 transmission in
India. They recommended cluster specific lockdown policy in different parts of
India for effective reduction in transmission of COVID-19. There are some
recent studies that uses different statistical modeling techniques to provide
reliable real-time forecast of COVID-19 cases Chakraborty2020Arxiv ;
Tandon2020Arxiv ; Hu2020Arxiv . Therefore, a combination of mathematical and
statistical models may be effective in determining the control measures as
well providing a robust real-time forecast of future 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. We assume that COVID-19 patients from the hospitals and
quarantine centers can only be in contact 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. In the
mechanistic model, we have incorporated the lockdown effect through home
quarantine of a certain percentage of susceptible population from the
community. Using daily notified COVID-19 cases from six states (Maharashtra,
Delhi, Madhya Pradesh, Rajasthan, Gujarat, and Uttar Pradesh) and overall
India, we estimated several important parameters of the mechanistic model.
Furthermore, we estimated the basic ($R_{0}$), the community ($R_{C}$), and
the hospital ($R_{H}$) reproduction numbers for the seven locations under
study. To obtain a reliable forecast of future COVID-19 notified cases in the
above mentioned locations, we used a hybrid statistical model that can
efficiently capture fluctuations in the daily time series data. BMA post-
processing technique based on DRAM algorithm is used to ensemble our
mechanistic mathematical model with the hybrid statistical model. Using the
ensemble model, we forecast COVID-19 notified cases (daily and cumulative)
from May 3, 2020 to May 20, 2020, under five different lockdown scenarios in
the seven locations. To determine an effective lockdown policy, we carried out
a global sensitivity analysis of four epidemiologically measurable &
controllable parameters on the lockdown effect (number of cases reduction) and
as well on $R_{0}$.

## Method

We extend a previous mechanistic model sardar2020assessment  by considering
hospital-based COVID-19 transmission (see Fig 1 and supplementary). We assume
that hospitalized & notified infected population can only be in contact with a
small fraction ($\rho$) of the susceptible population from the community (see
Table 1 and supplementary). We assume different transmission rates
($\beta_{1}$ and $\beta_{2}$, respectively) for community and hospital-based
infection. As it is very difficult to detect asymptomatic infected in the
community therefore, we assume that only a fraction of symptomatic infected
population being notified & hospitalized by COVID-19 testing at a rate, $\tau$
(see Fig 1 and Table 1). Following sardar2020assessment , the disease related
deaths are considered only for the notified & hospitalized population at a
rate $\delta$. We incorporated lockdown effect in our model (see Fig 1 and
supplementary) by home quarantined a fraction of susceptible population at a
rate $l$. We also assume that after the current lockdown period
$\left(\displaystyle\frac{1}{\omega}=40\hskip 2.84544ptdays\right)$ the home
quarantined individuals will returned to the general susceptible population
(see Fig 1 and supplementary). Moreover, we assume that the home quarantined
individuals do not mixed with the general population (see Fig 1) i.e. this
class of individuals do not contribute in the disease transmission. A flow
diagram and the information on our mechanistic ODE model parameters are
provided in Fig 1 and Table 1, respectively.

Mechanistic ODE model (see Fig 1 and supplementary) we are using for this
study may be efficient in capturing overall trend of the time-series data and
the transmission dynamics of COVID-19. However, as solution of the ODE model
is always smooth therefore, our mechanistic model may not be able to capture
fluctuations occurring in the daily time-series data. Several complex 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 using this daily time-series data. To resolve this
issue, we considered a Hybrid statistical model which is a combination of five
forecasting models namely, Auto-regressive Integrated Moving Average model
(ARIMA); Exponential smoothing state space model (ETS); Theta Method Model
(THETAM); Exponential smoothing state space model with Box-Cox transformation,
ARMA errors, Trend and Seasonal components (TBATS); and Neural Network Time
Series Forecasts (NNETAR). Finally, the Hybrid statistical model and the
mechanistic ODE model (see Fig 1 and supplementary) are combined together by a
post-processing Bayesian Model Averaging (BMA) technique which we discussed
later in the manuscript.

We used daily confirmed COVID-19 cases from Maharashtra (MH), Delhi (DL),
Madhya Pradesh (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), March 19, 2020 to April 16, 2020 (GJ),
March 14, 2020 to April 18, 2020 (UP), and March 2, 2020 to April 19, 2020
(IND) for our study. As of April 29, 2020, these referred six states
contributes 74% of the total COVID-19 notified cases in India
indiacovid2020track . Confirmed daily COVID-19 cases form these mentioned
seven locations are collected from indiacovid2020track . State-wise population
data are taken from aadhaar20 .

We estimated several uninformative parameters (see Table 1) of our mechanistic
model (see Fig 1 and supplementary) by calibrating our mathematical model (see
Fig 1 and supplementary) 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 and supplementary) are also
unknown therefore, we prefer to estimate these uninformative initial
conditions 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) starting from March 25, 2020 to April 14, 2020
Economictimes2020  and Govt. then extend the lockdown period up to May 3, 2020
(lockdown 2.0) timesofindia20b ; financialexpress20 . Therefore, the daily
COVID-19 time series data contains the effect of with and without lockdown
scenario, therefore, we prefer to use a combination of two mathematical models
(without and with lockdown) for calibration. An elaboration on the combination
technique of the mathematical models without and with lockdown (see Fig 1 and
supplementary) is provided below:

$\bullet$

We first use the mechanistic model without lockdown (see Eq S1 in
supplementary) starting from the first date of the daily COVID-19 data up to
end of March 24, 2020 for the seven locations MH, DL, MP, RJ, GJ, UP, and IND,
respectively.

$\bullet$

Using values of the state variables of the model without lockdown (see Model
S1 in supplementary method) onset of March 24, 2020 as initial conditions, we
run the mechanistic model with lockdown (see Fig 1 and Model S2 in
supplementary) up to the end date of the daily COVID-19 data for the seven
locations MH, DL, MP, RJ, GJ, UP, and IND, respectively.

The nonlinear least square function ${}^{\prime}lsqnonlin^{\prime}$ in the
MATLAB based optimization toolbox is called to fit the simulated and observed
daily COVID-19 notified cases in those seven locations mentioned earlier.
Bayesian based ${}^{\prime}DRAM^{\prime}$ algorithm haario2006dram  is used to
sample the uninformative parameters and initial conditions (see Table S1 and
Table S2 in supplementary) of the mathematical models combination without and
with lockdown (see Fig 1 and supplementary). A details on mechanistic model
fitting is provided in sardar2017mathematical .

Calibration of the Hybrid statistical model for the mentioned seven states are
done using the $R$ package ${}^{\prime}forecastHybrid^{\prime}$. First, we
fitted the individual models ARIMA, ETS, THETAM, TBATS, and NNETAR by calling
the functions ${}^{\prime}auto.arima^{\prime}$, ${}^{\prime}ets^{\prime}$,
${}^{\prime}thetam^{\prime}$, ${}^{\prime}tbats^{\prime}$, and
${}^{\prime}nnetar^{\prime}$ respectively. The results generated from each of
the above models are combined with equal weights to determine the Hybrid
statistical model. Equal weight among the five individual models are taken as
it generates a robust result (see Table S4 in supplementary) for the Hybrid
statistical model lemke2010meta .

Post-processing BMA technique for combining the mechanistic model (see Fig 1
and supplementary) and the Hybrid statistical model is based on
${}^{\prime}DRAM^{\prime}$ algorithm haario2006dram . Let, $\displaystyle
Y^{ODE}=\\{y_{j}^{ODE}\\}_{j=1}^{n}$ and $\displaystyle
Y^{HBD}=\\{y_{j}^{HBD}\\}_{j=1}^{n}$ be $n$ simulated observations from our
mechanistic ODE model (see Fig 1 and supplementary) and the Hybrid statistical
model respectively and let, $\displaystyle\hat{Y}=\\{y_{j}^{obs}\\}_{j=1}^{n}$
be $n$ observation from the data. Then

$\displaystyle Y^{E}=w_{1}\hskip 5.69046ptY^{ODE}+w_{2}\hskip
5.69046ptY^{HBD},$ (0.1)

is our ensemble model, where, the weights $w_{1}$ and $w_{2}$ satisfies the
constraints

$\displaystyle\Delta=\left\\{w_{1},w_{2}\geq 0:w_{1}+w_{2}=1\right\\}.$

We assume $w_{1}$ and $w_{2}$ follows Gaussian proposal distribution. Then the
error sum of square function haario2006dram  is defined as:

$\displaystyle
SS(\tilde{\theta})=\sum_{i=1}^{n}\left(\hat{Y}-Y^{E}(\tilde{\theta})\right)^{2}.$

Posterior distribution of the weights
$\tilde{\theta}=\left(\tilde{w_{1}},\tilde{w_{2}}\right)$ for the ensemble
model (0.1) are generated using Bayesian based ${}^{\prime}DRAM^{\prime}$
algorithm haario2006dram  (see Table S3 and Fig S1 to S7 in supplementary).

To save the countries short-scale industries and the agricultural sectors,
Indian Govt. has proposed some amendments on current lockdown rules from April
20, 2020 Economictimes2020a ; financialexpress20a . In these revised rules,
Govt. has provided some relaxation in current rules by dividing different
districts of the various states into three red (hotspot), orange (limited
human movement), and green (Economic activity) zones depending on the number
of COVID-19 cases Economictimes2020a ; financialexpress20a ;
businesstoday2020a . Implementation of these new rules in our mechanistic
models combination (see Fig 1 and supplementary) are based on the following
assumptions:

$\bullet$

Lockdown rule will be relaxed from April 20, 2020 in those states where the
current estimate of the lockdown rate (see Table 2) is higher than a threshold
value. This relaxation in lockdown is based on the fact that locations where
lockdown are strictly implemented before April 20, 2020 are likely to have
more impact on the economic growth.

$\bullet$

Lockdown rule will be more intensive from April 20, 2020 in those states where
the current estimate of the lockdown rate (see Table 2) is below a threshold
value. This assumption is made because locations where lockdown are not
implemented properly before April 20, 2020 are likely to have more red
(hotspot) zones.

$\bullet$

50% lockdown success is taken as the threshold value for our study. Here, 50%
lockdown success in Delhi means that 50% of the susceptible population in this
state is successfully home-quarantined during the period March 25, 2020 till
April 20, 2020.

$\bullet$

Insensitivity and relaxation in lockdown are measured in a same scale namely
10%, 20% and 30% 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 forecast of 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 continuously rising in these seven
locations therefore, it is more likely that lockdown period will be extended
beyond May 3, 2020. Therefore, forecast using the ensemble model (0.1) during
the mentioned time duration in those seven locations are based on the
following scenarios:
(A1) Using our mechanistic models combination (see Fig 1 and supplementary)
and the current estimate of the lockdown rates (see Table 2), we forecast
notified COVID-19 cases up to May 20, 2020. This forecast is combined together
with the results obtain from the hybrid statistical model by using our
ensemble model (0.1).

(A2) Using our mechanistic models combination (see Fig 1 and supplementary)
and the current estimate of the lockdown rates (see Table 2), we forecast
notified COVID-19 cases up to April 20, 2020. From April 21, 2020 till May 20,
2020, forecast are made using 10% increment or decrement (depending on the
state) in the estimate of current lockdown rate. This forecast is combined
together with the results obtain from the hybrid statistical model by using
our ensemble model (0.1).

(A3) Using our mechanistic models combination (see Fig 1 and supplementary)
and the current estimate of the lockdown rates (see Table 2), we forecast
notified COVID-19 cases up to April 20, 2020. From April 21, 2020 till May 20,
2020, forecast are made using 20% increment or decrement (depending on the
state) in the estimate of current lockdown rate. This forecast is combined
together with the results obtain from the hybrid statistical model by using
our ensemble model (0.1).

(A4) Using our mechanistic models combination (see Fig 1 and supplementary)
and the current estimate of the lockdown rates (see Table 2), we forecast
notified COVID-19 cases up to April 20, 2020. From April 21, 2020 till May 20,
2020, forecast are made using 30% increment or decrement (depending on the
state) in the estimate of current lockdown rate. This forecast is combined
together with the results obtain from the hybrid statistical model by using
our ensemble model (0.1).

(A5) Using our mechanistic models combination (see Fig 1 and supplementary)
and the current estimate of the lockdown rates (see Table 2), we forecast
notified COVID-19 cases up to May 2, 2020. From May 3, 2020 till May 20, 2020,
forecast are made with no lockdown (see Eq. S1 in supplementary). This
forecast is combined together with the results obtain from the hybrid
statistical model by using our ensemble model (0.1).

As, we assumed that lockdown individuals do not mixed with the general
population therefore, the basic reproduction number ($R_{0}$) with and without
lockdown (see Fig 1 and supplementary) are equal van2002reproduction :

$\displaystyle
R_{0}=\frac{\beta_{1}\kappa\sigma}{(\mu+\sigma)(\gamma_{1}+\mu+\tau)}+\frac{\beta_{1}(1-\kappa)\sigma}{(\mu+\gamma_{2})(\mu+\sigma)}+\frac{\beta_{2}\kappa\rho\sigma\tau}{(\mu+\sigma)(\delta+\gamma_{3}+\mu)(\gamma_{1}+\mu+\tau)}.\\\
$

In the expression of $R_{0}$, sum of first two term indicate the community
infection occurring from symptomatic and asymptomatic infected population. The
third term in $R_{0}$ specify the hospital-based COVID-19 transmission. To
distinguish the community and hospital-based COVID-19 transmission, we defined
the community reproduction number ($R_{C}$), and the hospital reproduction
number ($R_{H}$) as follows:

$\displaystyle
R_{C}=\frac{\beta_{1}\kappa\sigma}{(\mu+\sigma)(\gamma_{1}+\mu+\tau)}+\frac{\beta_{1}(1-\kappa)\sigma}{(\mu+\gamma_{2})(\mu+\sigma)},\\\
$

and

$\displaystyle R_{H}=\frac{\beta_{2}\hskip 1.13791pt\kappa\hskip
1.13791pt\rho\hskip 1.13791pt\sigma\hskip
1.13791pt\tau}{(\mu+\sigma)(\delta+\gamma_{3}+\mu)(\gamma_{1}+\mu+\tau)}.\\\ $

Using estimated values of epidemiologically uninformative parameters (see
Table 2), we estimated $R_{0}$, $R_{C}$, 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 relation between effect of lockdown (number of COVID-19 case reduction)
with some important epidemiologically measurable & controllable parameters.
Our mechanistic ODE model (see Fig 1 and supplementary) has several important
parameters and among them measurable and controllable parameters are
$\beta_{2}$: average rate of transmission occurring from hospitalized &
notified based contacts (it can be controllable by following WHO guidelines);
$\rho$: fraction of susceptible population from the community that are exposed
to notified & hospitalized based contacts (it also can be minimized by
following proper guidelines from WHO); $\kappa$: fraction of infected that are
symptomatic (rapid COVID-19 testing can provide an accurate estimate); $\tau$:
notification & hospitalization rate of symptomatic infected population (it
also depend on number of COVID-19 testing). We perform a global sensitivity
analysis marino2008methodology  to determine the effect of these parameters on
the lockdown effect and on the basic reproduction number ($R_{0}$),
respectively. The effect of lockdown is measured in terms of differences in
the total number of COVID-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 the mentioned four parameters
(see Table 1) using Latin Hypercube Sampling (LHS) technique. Other
informative and uninformative parameters during simulation of the mechanistic
model are 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 four parameters on the lockdown
effect and the basic reproduction number ($R_{0}$), 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 mentioned seven locations, we
estimated the weights ($w_{1}$ and $w_{2}$) for our ensemble model (0.1) (see
Table S2 in supplementary). Our result suggest that the mechanistic ODE model
(see Fig 1 and supplementary) displayed a better performance in RJ, UP, and
IND compared to the hybrid statistical model (see Table S2 in supplementary).
For rest of the locations (MH, DL, MP, and GJ), the hybrid statistical model
has performed better than the mechanistic ODE model in terms 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 (see Fig 1 and supplementary) and the hybrid statistical model
respectively, has provided a robust result in all of these mentioned seven
locations in terms of capturing time-series data trend (see Fig. 2).

The estimates of uninformative parameters of the mechanistic ODE model (Fig 1
and supplementary) suggests that currently in the seven locations (MH, DL, MP,
RJ, GJ, UP, and IND) the community infection is mainly dominated by
contribution from the asymptomatic infected population (Table 2). Among the
seven locations, the lowest percentage of symptomatic infection in the
community is found in RJ (about 0.1%) and the highest percentage is found in
IND (about 35%) (Table 2). Our estimates suggest that currently in the seven
locations (MH, DL, MP, RJ, GJ, UP, and IND), the notification &
hospitalization rate of symptomatic infected population is about 0.2% to 23 %
(Table 2). Therefore, most of the COVID-19 infections in those mentioned seven
locations are currently undetected. Our result agrees with the recent report
by the Indian Council of Medical Research (ICMR) Indiatodayasymptomatic . Our
estimates of the lockdown rate for MH, DL, MP, RJ, GJ, UP, and IND,
respectively, suggest that lockdown is 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 61% to 77% of
the total susceptible population may be successfully home quarantined during
the present lockdown period (Table 2). However, for rest of the locations, our
results suggest that lockdown may not be successful as about 11% to 49% of the
total susceptible population may be 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 1% to 9% of the total susceptible
populations may be exposed to hospital (notified & hospitalized population)
related contacts (Table 2). Considering the fact that estimates of the average
hospital-based transmission 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 currently occurring due to
notified & hospitalized infected related contacts. These findings can be
further justified by analyzing the estimates of the basic ($R_{0}$), the
community ($R_{C}$) and the hospitalized ($R_{H}$) 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 1% to 16% of the total
COVID-19 transmission currently occurring from notified & hospitalized
infected related contacts (see Table 3). These figures can be increased up to
43% to 69% if proper 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-spreading incidents
occurring from hospital-based contacts. In RJ, low contribution of $R_{H}$ on
$R_{0}$ (see Table 3) may be due to existence of low percentage of the
symptomatic infected population in the community (Table 2) and that leads to
low percentage of notified & hospitalized COVID-19 cases. For further
investigation on super-spreading events, we carried a global uncertainty and
sensitivity analysis of some epidemiologically measurable and controllable
parameters from our mechanistic ODE model (Fig 1 and supplementary) namely,
$\boldsymbol{\beta_{2}}$: average rate of transmission occurring from notified
& hospitalized based contact (it can be controllable following the WHO
guidelines WHOhospitaltransmission ), $\boldsymbol{\rho}$: fraction of
susceptible population that are exposed to hospital-based contact (it can be
reduced by following proper guidelines from the WHO WHOhospitaltransmission ),
$\boldsymbol{\kappa}$: fraction of the newly infected that are symptomatic
(Rapid COVID-19 testing can provide an accurate estimate),
$\boldsymbol{\tau}$: hospitalization & notification rate of symptomatic
infected population (it also depend on number of COVID-19 testing) on the
basic reproduction number ($R_{0}$). Partial rank correlation coefficients
(PRCC) and its corresponding $p$-value suggested that all these four
parameters have significant positive correlation with $R_{0}$ (see Fig. 4 and
Fig. S7 to Fig.S12 in supplementary). Furthermore, high positive correlation
of $\rho$ on $R_{0}$ indicate that small increase in the percentage of
susceptible population from the community that are exposed hospital-based
transmission will leads to significant increase in COVID-19 force of
infection. Considering the fact that estimated value of $\beta_{2}$ (see Table
2) in MH, DL, MP, RJ, GJ, UP, and IND, respectively are very high (much higher
than community transmission rate), therefore, a small increase in $\rho$ may
leads to a larger COVID-19 outbreak in those seven locations. Therefore, until
and otherwise any preventive measures are taken in these locations, a larger
COVID-19 outbreak may trigger from hospitals and quarantine centers.

Using the ensemble model (0.1), we forecast daily as well as total COVID-19
cases under five different 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). Comparing the projected total COVID-19
cases during May 3, 2020 to May 20, 2020, (see Table 4) with the total
observed cases indiacovid2020track  during March 2, 2020 till April 29, 2020,
we found about two fold increase in the total cases in MH, MP, GJ, UP, and
IND, respectively. In summary, our forecast result suggest that in the coming
two weeks a significant increase in cases may be observed in most of these
locations.

To determine which epidemiologically measurable and controllable parameters
are most influencing the effect of lockdown, we carried out a global
uncertainty analysis of $\beta_{2}$, $\rho$, $\kappa$, and $\tau$ on the
lockdown effect. The lockdown effect is measured in terms of differences in
the total number 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 method section for details). For MH, PRCC and
its corresponding $p$-value suggested that all these four parameters have
significant influence on the lockdown effect (see Fig. S7 in supplementary).
Furthermore, significant negative correlation of $\beta_{2}$, and $\rho$ with
the lockdown effect (see Fig. S7 in supplementary) suggested that only home
quarantined the community may not be sufficient to reduce COVID-19
transmission in MH. Govt. and the policy makers may also have to focus on
reducing the transmission occurring from hospital premises based on the
guidelines from the WHO WHOhospitaltransmission . For DL, PRCC and its
corresponding $p$-value suggested that $\beta_{2}$, $\rho$, and $\kappa$ are
the main parameters that are influencing the lockdown effect (see Fig. S8 in
supplementary). Moreover, significant negative correlation of $\beta_{2}$ and
$\rho$ with the lockdown effect and as well as significant positive
correlation of $\kappa$ with the lockdown effect (see Fig. S8 in
supplementary) implies that an effective lockdown policy in DL may be a
combination of lockdown in the community, contact tracing of COVID-19 cases,
and with some effort in reducing hospital-based transmission following WHO
guidelines WHOhospitaltransmission . For MP, PRCC and its corresponding
$p$-value suggested that $\kappa$ and $\tau$ have high positive correlation
with the lockdown effect (see Fig. S9 in supplementary). Furthermore, $\rho$
have significant negative correlation with the lockdown effect (see Fig. S9 in
supplementary). Therefore an effective lockdown policy in MP may be a strict
implementation of lockdown in the red and orange zones, rapid COVID-19 testing
in the community and reducing hospital-based transmission by following
guidelines from WHO WHOhospitaltransmission . For RJ, PRCC and its
corresponding $p$-value suggested that only $\kappa$ have significant positive
correlation with the lockdown effect (see Fig. S10 in supplementary). No
significant correlation with hospital-based parameters may be due to existence
of low percentage of the symptomatic infected 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 on contact tracing
in the community with relaxation may be given in the Green and Orange zones to
increase the percentage of symptomatic infected in the community. For GJ, PRCC
and its corresponding $p$-value suggested that all these four parameters have
significant influence on the lockdown effect (see Fig. S11 in supplementary).
Furthermore, significant negative correlation of $\beta_{2}$, and $\rho$ with
the lockdown effect (see Fig. S11 in supplementary) indicate that only home
quarantined 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 occurring from hospital premises based on the
guidelines from the WHO WHOhospitaltransmission . For UP, PRCC and its
corresponding $p$-value suggested that $\beta_{2}$, $\rho$, and $\kappa$ are
the main parameters that are influencing the lockdown effect (see Fig. S12 in
supplementary). Moreover, significant negative correlation of $\beta_{2}$ and
$\rho$ with the lockdown effect and as well as significant positive
correlation of $\kappa$ with the lockdown effect (see Fig. S12 in
supplementary) implies that an effective lockdown policy in UP may be a
combination of lockdown (relaxation in Green zone), contact tracing in
community with effort in reducing hospital-based transmission following the
WHO guidelines WHOhospitaltransmission . Finally for IND, PRCC and its
corresponding $p$-value suggested that $\beta_{2}$ and $\rho$ are the main
parameters that are most influencing the lockdown effect (see Fig. 4).
Therefore, only home quarantined the community may not be sufficient to reduce
COVID-19 transmission in IND. Govt. of IND and the policy makers may also have
to focus on reducing the transmission occurring from hospital premises based
on the guidelines from the WHO WHOhospitaltransmission .

## Conclusion

Our analysis of the mechanistic model with hospital-based COVID-19
transmission suggest that most of the new infections occurring in India as
well most of the states are currently undetected. Furthermore, a global
sensitivity analysis of two epidemiologically controllable parameters from the
hospital-based transmission on the basic reproduction nuber ($R_{0}$),
indicate that if appropriate preventive measures are not taken immediately, a
much larger COVID-19 outbreak may trigger due to the transmission occurring
from the hospitalized & notified based contacts. Moreover, our ensemble
forecast model (0.1) predicted a substantial percentage of increase in the
COVID-19 notified cases during May 3, 2020 till May 20, 2020, (see Table 4) in
most 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 low number of hospitalized and
reported cases in this state (see Table 2). However, cases may rise in
Rajasthan if relaxation in lockdown is applied. Furthermore, trend of the
forecast data in overall India (see Fig 3) during May 3, 2020 till May 20,
2020, indicating the fact that reaching the peak of the COVID-19 epidemic
curve may be a long way ahead for India. Finally, based on our results of
global sensitivity analysis of the four important epidemiologically measurable
& controllable parameters on the lockdown effect, we are suggesting the
following policy 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 the community. 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 in compare to
community transmission. To reduce the hospitalized & notified based contacts,
an efficient disaster management team is required. They will continuously
monitor the situations in different hospitals and quarantine centers across
India. This team must ensure that proper safety measures are being followed
based on the guidelines provided by ICMR and WHO WHOhospitaltransmission .

## Conflict of interests

The authors declare that they have no conflicts of interest.

## Acknowledgments

Dr. Tridip Sardar acknowledges the Science & Engineering Research Board (SERB)
major project 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.

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## Figures

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-19 symptomatic infected population, $I_{A}$: COVID-19
asymptomatic infected population, $H$: Notified & Hospital individuals
suffering from COVID-19 infection, $R$: COVID-19 recovered population, and
$L$: Home quarantined susceptible population during lockdown, respectively.
Epidemiological information of the parameters shown in this figure are
provided in Table 1. Figure 2: Combination of mechanistic ODE models (Fig 1
and supplementary), Hybrid statistical model, and Ensemble model (0.1) fitting
to the daily notified COVID-19 cases form six different states and overall
India. Respective subscripts are 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 No LD specifies the period before lockdown. Lockdown
effect is considered only for the mechanistic ODE model (Fig 1 and
supplementary) and consequently in the ensemble model (0.1). Shaded area
specifies the 95% confidence region. Figure 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. Figure 4: Effect of uncertainty
of four epidemiologically measurable and controllable parameters of the
mechanistic ODE model (see Fig 1, Table 1 and method section) on the effect of
lockdown and the basic reproduction number ($R_{0}$). Lockdown effect 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 of these 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 ranges provided in Table 1.

## Tables

Table 1: Parameters with their respective epidemiological information for the mechanistic ODE model (see Fig 1 and supplementary) for COVID-19. |  |  |
---|---|---|---
Parameters |  Biological Meaning |  Value/Ranges |  Reference
|  |  |
$\Pi$ $=$ $\mu$ $\times$ $N(0)$ |  Recruitment rate of human population |  Differs over states |  -
$\displaystyle\frac{1}{\mu}$ |  Average life expectancy at birth |  Differs over states |  nitiaayog20
$\displaystyle\beta_{1}$ |  Average transmission rate of a symptomatic and asymptomatic COVID-19 infected |  (0 - 500) $day^{-1}$ |  Estimated
$\displaystyle\beta_{2}$ |  Average transmission rate of a notified & hospitalized COVID-19 infected |  (0 - 500) $day^{-1}$ |  Estimated
$\displaystyle\rho$ |  Fraction of the susceptible population that are exposed to hospital-based transmission |  0 - 0.2 |  Estimated
$\displaystyle\frac{1}{\sigma}$ |  COVID-19 incubation period |  (1 - 14) days |  Estimated
$\displaystyle\kappa$ |  Fraction of the COVID-19 exposed population that become symptomatic infected |  0 - 1 |  Estimated
$\displaystyle\gamma_{1}$ |  Average recovery rate of symptomatic infection |  ($\displaystyle\gamma_{3}$ \- 1) $day^{-1}$ |  Estimated
$\displaystyle\gamma_{2}$ |  Average recovery rate of asymptomatic infection |  ($\displaystyle\gamma_{3}$ \- 1) $day^{-1}$ |  Estimated
$\displaystyle\tau$ |  Average hospitalization rate for the COVID-19 symptomatic individuals |  (0 - 1) $day^{-1}$ |  Estimated
$\displaystyle\delta$ |  Average death rate due to COVID-19 infection in hospital |  Differs over states |  indiacovid2020track
$\displaystyle\gamma_{3}$ |  Average recovery rate of the notified & hospitalized populations |  Differs over states |  indiacovid2020track
$\displaystyle l$ |  Average lockdown rate |  (0 - 0.9) $day^{-1}$ |  Estimated
$\displaystyle\frac{1}{\omega}$ |  Current lockdown period in India |  40 days |  timesofindia20b ; financialexpress20
|  |  |

Table 2: Estimated uninformative parameters of the mechanistic ODE model combinations (see Fig 1 and 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 Estimate (95% CI). |  |  |  |  |  |  |  |  |
---|---|---|---|---|---|---|---|---|---
Region | $\boldsymbol{\beta_{1}}$ | $\boldsymbol{\beta_{2}}$ |  $\boldsymbol{\rho}$ $\times$ $100$ | $\boldsymbol{\sigma}$ | $\boldsymbol{\kappa}$ | $\boldsymbol{\gamma_{1}}$ | $\boldsymbol{\tau\times 100}$ | $\boldsymbol{\gamma_{2}}$ | $\boldsymbol{l}\times 100$
|  |  |  |  |  |  |  |  |
MH | $\begin{subarray}{c}2.03\\\ \\\ (1.05-2.75)\end{subarray}$ | $\begin{subarray}{c}21.6\\\ \\\ (2.6-54.7)\end{subarray}$ | $\begin{subarray}{c}7.98\\\ \\\ (0.24-9.78)\end{subarray}$ | $\begin{subarray}{c}0.163\\\ \\\ (0.073-0.196)\end{subarray}$ | $\begin{subarray}{c}0.118\\\ \\\ (0.03-0.165)\end{subarray}$ | $\begin{subarray}{c}0.963\\\ \\\ (0.144-0.974)\end{subarray}$ | $\begin{subarray}{c}5.05\\\ \\\ (2.76-85.3)\end{subarray}$ | $\begin{subarray}{c}0.863\\\ \\\ (0.341-0.991)\end{subarray}$ | $\begin{subarray}{c}61.5\\\ \\\ (2.51-97.3)\end{subarray}$
|  |  |  |  |  |  |  |  |
DL | $\begin{subarray}{c}1.83\\\ \\\ (0.557-3.54)\end{subarray}$ | $\begin{subarray}{c}220.2\\\ \\\ (12.97-488.8)\end{subarray}$ | $\begin{subarray}{c}7.66\\\ \\\ (0.25-9.67)\end{subarray}$ | $\begin{subarray}{c}0.211\\\ \\\ (0.075-0.748)\end{subarray}$ | $\begin{subarray}{c}0.039\\\ \\\ (0.002-0.039)\end{subarray}$ | $\begin{subarray}{c}0.96\\\ \\\ (0.321-0.987)\end{subarray}$ | $\begin{subarray}{c}8.5\\\ \\\ (3.8-96.3)\end{subarray}$ | $\begin{subarray}{c}0.764\\\ \\\ (0.328-0.988)\end{subarray}$ | $\begin{subarray}{c}70.8\\\ \\\ (13.1-87.8)\end{subarray}$
|  |  |  |  |  |  |  |  |
MP | $\begin{subarray}{c}1.45\\\ \\\ (1.04-2.91)\end{subarray}$ | $\begin{subarray}{c}42.1\\\ \\\ (25.4-486)\end{subarray}$ | $\begin{subarray}{c}5.88\\\ \\\ (0.24-9.72)\end{subarray}$ | $\begin{subarray}{c}0.194\\\ \\\ (0.073-0.242)\end{subarray}$ | $\begin{subarray}{c}0.014\\\ \\\ (0.002-0.098)\end{subarray}$ | $\begin{subarray}{c}0.827\\\ \\\ (0.141-0.980)\end{subarray}$ | $\begin{subarray}{c}14\\\ \\\ (3.52-92.2)\end{subarray}$ | $\begin{subarray}{c}0.585\\\ \\\ (0.21-0.978)\end{subarray}$ | $\begin{subarray}{c}34.8\\\ \\\ (12.7-88.6)\end{subarray}$
|  |  |  |  |  |  |  |  |
RJ | $\begin{subarray}{c}1.16\\\ \\\ (0.557-2.74)\end{subarray}$ | $\begin{subarray}{c}325.6\\\ \\\ (3.71-484)\end{subarray}$ | $\begin{subarray}{c}1.33\\\ \\\ (0.07-9.56)\end{subarray}$ | $\begin{subarray}{c}0.937\\\ \\\ (0.148-0.949)\end{subarray}$ | $\begin{subarray}{c}0.0011\\\ \\\ (0.001-0.003)\end{subarray}$ | $\begin{subarray}{c}0.337\\\ \\\ (0.253-0.988)\end{subarray}$ | $\begin{subarray}{c}23.2\\\ \\\ (4.66-96.3)\end{subarray}$ | $\begin{subarray}{c}0.814\\\ \\\ (0.244-0.978)\end{subarray}$ | $\begin{subarray}{c}21\\\ \\\ (10.6-81.2)\end{subarray}$
|  |  |  |  |  |  |  |  |
GJ | $\begin{subarray}{c}0.90\\\ \\\ (0.53-2.10)\end{subarray}$ | $\begin{subarray}{c}186.3\\\ \\\ (9.64-490)\end{subarray}$ | $\begin{subarray}{c}2.32\\\ \\\ (0.21-9.20)\end{subarray}$ | $\begin{subarray}{c}0.54\\\ \\\ (0.101-0.94)\end{subarray}$ | $\begin{subarray}{c}0.028\\\ \\\ (0.01-0.092)\end{subarray}$ | $\begin{subarray}{c}0.84\\\ \\\ (0.15-0.98)\end{subarray}$ | $\begin{subarray}{c}14\\\ \\\ (4.02-76.2)\end{subarray}$ | $\begin{subarray}{c}0.52\\\ \\\ (0.202-0.98)\end{subarray}$ | $\begin{subarray}{c}11.4\\\ \\\ (2.31-32.2)\end{subarray}$
|  |  |  |  |  |  |  |  |
UP | $\begin{subarray}{c}0.53\\\ \\\ (0.52-1.92)\end{subarray}$ | $\begin{subarray}{c}242\\\ \\\ (8.23-353)\end{subarray}$ | $\begin{subarray}{c}4.29\\\ \\\ (0.1-9.51)\end{subarray}$ | $\begin{subarray}{c}0.66\\\ \\\ (0.078-0.71)\end{subarray}$ | $\begin{subarray}{c}0.12\\\ \\\ (0.077-0.95)\end{subarray}$ | $\begin{subarray}{c}0.78\\\ \\\ (0.19-0.988)\end{subarray}$ | $\begin{subarray}{c}2.38\\\ \\\ (0.2-5)\end{subarray}$ | $\begin{subarray}{c}0.408\\\ \\\ (0.218-0.982)\end{subarray}$ | $\begin{subarray}{c}48.9\\\ \\\ (12.3-87.6)\end{subarray}$
|  |  |  |  |  |  |  |  |
IND | $\begin{subarray}{c}2.25\\\ \\\ (1.29-3.24)\end{subarray}$ | $\begin{subarray}{c}499\\\ \\\ (494-499.8)\end{subarray}$ | $\begin{subarray}{c}8.99\\\ \\\ (0.32-9.70)\end{subarray}$ | $\begin{subarray}{c}0.098\\\ \\\ (0.072-0.15)\end{subarray}$ | $\begin{subarray}{c}0.35\\\ \\\ (0.07-0.96)\end{subarray}$ | $\begin{subarray}{c}0.72\\\ \\\ (0.38-0.99)\end{subarray}$ | $\begin{subarray}{c}0.23\\\ \\\ (0.03-0.53)\end{subarray}$ | $\begin{subarray}{c}0.995\\\ \\\ (0.25-0.9959)\end{subarray}$ | $\begin{subarray}{c}76.6\\\ \\\ (35.5-89.2)\end{subarray}$
|  |  |  |  |  |  |  |  |
Table 3: Estimates of the basic, the community and the hospital reproduction numbers. The informative and uninformative parameters of the mechanistic ODE model (see Fig 1 and supplementary) during the estimation of the different reproduction numbers are taken from Table 1 and Table 2, respectively. Respective subscripts MH, DL, MP, RJ, GJ, UP, and IND are same as Table 2. All data are given in the format Estimate (95% CI). |  |  |  |  |
---|---|---|---|---|---
Region |  $\boldsymbol{R_{0}}$ |  $\boldsymbol{R_{C}}$ |  % $\boldsymbol{R_{0}}$ |  $\boldsymbol{R_{H}}$ |  % $\boldsymbol{R_{0}}$
|  |  |  |  |
MH |  $\begin{subarray}{c}2.371\\\ \\\ (2.095-4.005)\end{subarray}$ |  $\begin{subarray}{c}2.309\\\ \\\ (1.542-3.894)\end{subarray}$ |  $\begin{subarray}{c}97.37\\\ \\\ (55.46-99.93)\end{subarray}$ |  $\begin{subarray}{c}0.0623\\\ \\\ (0.0019-1.254)\end{subarray}$ |  $\begin{subarray}{c}2.63\\\ \\\ (0.1-44.54)\end{subarray}$
|  |  |  |  |
DL |  $\begin{subarray}{c}2.54\\\ \\\ (1.37-5.52)\end{subarray}$ |  $\begin{subarray}{c}2.37\\\ \\\ (0.934-5.49)\end{subarray}$ |  $\begin{subarray}{c}93.46\\\ \\\ (56.91-99.95)\end{subarray}$ |  $\begin{subarray}{c}0.1658\\\ \\\ (0.0016-0.8236)\end{subarray}$ |  $\begin{subarray}{c}6.53\\\ \\\ (0.048-43.09)\end{subarray}$
|  |  |  |  |
MP |  $\begin{subarray}{c}2.497\\\ \\\ (2.21-6.11)\end{subarray}$ |  $\begin{subarray}{c}2.467\\\ \\\ (1.68-5.63)\end{subarray}$ |  $\begin{subarray}{c}98.82\\\ \\\ (49.22-99.81)\end{subarray}$ |  $\begin{subarray}{c}0.0296\\\ \\\ (0.0062-2.045)\end{subarray}$ |  $\begin{subarray}{c}1.18\\\ \\\ (0.19-50.78)\end{subarray}$
|  |  |  |  |
RJ |  $\begin{subarray}{c}1.43\\\ \\\ (1.42-3.78)\end{subarray}$ |  $\begin{subarray}{c}1.42\\\ \\\ (1.405-3.76)\end{subarray}$ |  $\begin{subarray}{c}99.46\\\ \\\ (94.40-100)\end{subarray}$ |  $\begin{subarray}{c}0.01\\\ \\\ (0-0.068)\end{subarray}$ |  $\begin{subarray}{c}0.54\\\ \\\ (0-3.66)\end{subarray}$
|  |  |  |  |
GJ |  $\begin{subarray}{c}1.835\\\ \\\ (1.51-4.86)\end{subarray}$ |  $\begin{subarray}{c}1.706\\\ \\\ (0.895-3.72)\end{subarray}$ |  $\begin{subarray}{c}93.02\\\ \\\ (31.4-99.56)\end{subarray}$ |  $\begin{subarray}{c}0.128\\\ \\\ (0.0083-2.68)\end{subarray}$ |  $\begin{subarray}{c}6.98\\\ \\\ (0.43-68.60)\end{subarray}$
|  |  |  |  |
UP |  $\begin{subarray}{c}1.46\\\ \\\ (1.35-3.14)\end{subarray}$ |  $\begin{subarray}{c}1.22\\\ \\\ (0.787-2.88)\end{subarray}$ |  $\begin{subarray}{c}84\\\ \\\ (39.21-99.96)\end{subarray}$ |  $\begin{subarray}{c}0.233\\\ \\\ (0.001-1.47)\end{subarray}$ |  $\begin{subarray}{c}16\\\ \\\ (0.039-60.79)\end{subarray}$
|  |  |  |  |
IND |  $\begin{subarray}{c}2.81\\\ \\\ (2.15-4.94)\end{subarray}$ |  $\begin{subarray}{c}2.56\\\ \\\ (1.92-4.86)\end{subarray}$ |  $\begin{subarray}{c}91.28\\\ \\\ (84.63-99.84)\end{subarray}$ |  $\begin{subarray}{c}0.245\\\ \\\ (0.0048-0.388)\end{subarray}$ |  $\begin{subarray}{c}8.72\\\ \\\ (0.16-15.37)\end{subarray}$
|  |  |  |  |

Table 4: Ensemble model (0.1) forecast of the total notified COVID-19 cases during May 3, 2020 till May 20, 2020. Respective subscripts MH, DL, MP, RJ, GJ, UP, and IND are same as Table 2. Regions where, current lockdown rate ($\boldsymbol{\big{\downarrow}}$) 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) during the mentioned period. Whereas, current lockdown rate ($\big{\uparrow}$) implies the ensemble model (0.1) projections for the Scenarios (A2) to (A4) with 10%, 20% and 30% increment in the current estimate of lockdown rate (see Table 2) during the mentioned period. Scenario (A1) implies the ensemble model (0.1) forecast with 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, 2020 till May 20, 2020. All data are provided in the format Estimate (95% CI). |  |  |  |  |  |
---|---|---|---|---|---|---
Region |  Current lockdown rate | Scenario A1 | Scenario A2 | Scenario A3 | Scenario A4 | Scenario A5
|  |  |  |  |  |
MH |  red$\big{\downarrow}$ | $\begin{subarray}{c}16790\\\ \\\ (16056-18727)\end{subarray}$ | $\begin{subarray}{c}17410\\\ \\\ (16499-19812)\end{subarray}$ | $\begin{subarray}{c}18147\\\ \\\ (17026-21101)\end{subarray}$ | $\begin{subarray}{c}19036\\\ \\\ (17662-22656)\end{subarray}$ | $\begin{subarray}{c}23000\\\ \\\ (20499-29596)\end{subarray}$
|  |  |  |  |  |
DL |  red$\big{\downarrow}$ | $\begin{subarray}{c}2642\\\ \\\ (1791-3294)\end{subarray}$ | $\begin{subarray}{c}1840\\\ \\\ (1337-3297)\end{subarray}$ | $\begin{subarray}{c}2693\\\ \\\ (1901-3300)\end{subarray}$ | $\begin{subarray}{c}2729\\\ \\\ (1979-3304)\end{subarray}$ | $\begin{subarray}{c}2886\\\ \\\ (2319-3320)\end{subarray}$
|  |  |  |  |  |
MP |  red$\big{\uparrow}$ | $\begin{subarray}{c}4036\\\ \\\ (3599-4578)\end{subarray}$ | $\begin{subarray}{c}3924\\\ \\\ (3531-4410)\end{subarray}$ | $\begin{subarray}{c}3825\\\ \\\ (3472-4263)\end{subarray}$ | $\begin{subarray}{c}3738\\\ \\\ (3419-4134)\end{subarray}$ | $\begin{subarray}{c}5058\\\ \\\ (4216-6101)\end{subarray}$
|  |  |  |  |  |
RJ |  red$\big{\uparrow}$ | $\begin{subarray}{c}1184\\\ \\\ (654-1624)\end{subarray}$ | $\begin{subarray}{c}1156\\\ \\\ (611-1608)\end{subarray}$ | $\begin{subarray}{c}1133\\\ \\\ (576-1595)\end{subarray}$ | $\begin{subarray}{c}1114\\\ \\\ (547-1585)\end{subarray}$ | $\begin{subarray}{c}1318\\\ \\\ (858-1699)\end{subarray}$
|  |  |  |  |  |
GJ |  red$\big{\uparrow}$ | $\begin{subarray}{c}7707\\\ \\\ (7689-12589)\end{subarray}$ | $\begin{subarray}{c}7444\\\ \\\ (7427-11991)\end{subarray}$ | $\begin{subarray}{c}7206\\\ \\\ (7191-11451)\end{subarray}$ | $\begin{subarray}{c}6990\\\ \\\ (6976-10961)\end{subarray}$ | $\begin{subarray}{c}9337\\\ \\\ (9312-16292)\end{subarray}$
|  |  |  |  |  |
UP |  red$\big{\uparrow}$ | $\begin{subarray}{c}5599\\\ \\\ (3656-6478)\end{subarray}$ | $\begin{subarray}{c}5383\\\ \\\ (3564-6206)\end{subarray}$ | $\begin{subarray}{c}5184\\\ \\\ (3479-5954)\end{subarray}$ | $\begin{subarray}{c}4999\\\ \\\ (3401-5722)\end{subarray}$ | $\begin{subarray}{c}7217\\\ \\\ (4343-8517)\end{subarray}$
|  |  |  |  |  |
IND |  red$\big{\downarrow}$ | $\begin{subarray}{c}38134\\\ \\\ (36550-45296)\end{subarray}$ | $\begin{subarray}{c}40278\\\ \\\ (38877-46612)\end{subarray}$ | $\begin{subarray}{c}42824\\\ \\\ (41640-48174)\end{subarray}$ | $\begin{subarray}{c}45896\\\ \\\ (44975-50059)\end{subarray}$ | $\begin{subarray}{c}57159\\\ \\\ (56971-57201)\end{subarray}$
|  |  |  |  |  |

## Supplementary Method

### Model without lock-down

The basic structure of our model is an extension of our earlier model
sardar2020assessment  by considering hospital-based transmission. As
hospitalized & notified infected ($H$) can only contact with a very small
fraction of the susceptible class therefore, we assume the effective contact
for a single hospitalized & notified infected is
$\displaystyle\beta_{2}\frac{(\rho S)}{N}$ and $\rho$ is the fraction of
susceptible that are in contact with the hospitalized & notified infected. We
also assume that effective contact of a single symptomatic ($I_{S}$) and a
asymptomatic individual is same and at a rate
$\displaystyle\beta_{1}\frac{S}{N}$. Therefore, total human 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 & notified ($H(t)$), and recovered ($R(t)$),
respectively. Interaction between different sub-classes are provided by the
following system of differential equation:

$\displaystyle\displaystyle{\frac{dS}{dt}}$ $\displaystyle=$
$\displaystyle\Pi-\frac{\beta_{1}SI_{S}}{N}-\frac{\beta_{1}SI_{A}}{N}-\frac{\beta_{2}(\rho
S)H}{N}-\mu S$ $\displaystyle\displaystyle{\frac{dE}{dt}}$ $\displaystyle=$
$\displaystyle\frac{\beta_{1}SI_{S}}{N}+\frac{\beta_{1}SI_{A}}{N}+\frac{\beta_{2}(\rho
S)H}{N}-(\mu+\sigma)E,$ $\displaystyle\displaystyle{\frac{dI_{S}}{dt}}$
$\displaystyle=$ $\displaystyle\kappa\sigma E-(\tau+\mu+\gamma_{1})I_{S},$
$\displaystyle\displaystyle{\frac{dI_{A}}{dt}}$ $\displaystyle=$
$\displaystyle(1-\kappa)\sigma E-(\mu+\gamma_{2})I_{A},$ (S-1)
$\displaystyle\displaystyle{\frac{dH}{dt}}$ $\displaystyle=$
$\displaystyle\tau I_{S}-(\gamma_{3}+\delta+\mu)H,$
$\displaystyle\displaystyle{\frac{dR}{dt}}$ $\displaystyle=$
$\displaystyle\gamma_{1}I_{S}+\gamma_{2}I_{A}+\gamma_{3}H-\mu R.$

### Model with lock-down

Lock-down effect in the model (Model without lock-down) can be included by
home quarantine a fraction of the susceptible population at a rate $l$. This
home-quarantine populations are considered as a new subclass namely, lock-down
class $L$. We assume that population in the lock-down compartment are home
quarantined and they do not interact with other populations. Therefore,
effective contact of a single symptomatic ($I_{S}$), asymptomatic ($I_{A}$),
and hospitalized & notified infected are
$\displaystyle\beta_{1}\frac{S}{(N-L)}$,
$\displaystyle\beta_{1}\frac{S}{(N-L)}$, and
$\displaystyle\beta_{1}\frac{(\rho S)}{(N-L)}$, respectively. The interaction
of all these sub-classes can be expressed as an lock-down ODE model as
follows:

$\displaystyle\displaystyle{\frac{dS}{dt}}$ $\displaystyle=$
$\displaystyle\Pi+\omega
L-\frac{\beta_{1}SI_{S}}{(N-L)}-\frac{\beta_{1}SI_{A}}{(N-L)}-\frac{\beta_{2}(\rho
S)H}{(N-L)}-(\mu+l)S$ $\displaystyle\displaystyle{\frac{dL}{dt}}$
$\displaystyle=$ $\displaystyle lS-(\mu+\omega)L,$
$\displaystyle\displaystyle{\frac{dE}{dt}}$ $\displaystyle=$
$\displaystyle\frac{\beta_{1}SI_{S}}{(N-L)}+\frac{\beta_{1}SI_{A}}{(N-L)}+\frac{\beta_{2}(\rho
S)H}{(N-L)}-(\mu+\sigma)E,$ $\displaystyle\displaystyle{\frac{dI_{s}}{dt}}$
$\displaystyle=$ $\displaystyle\kappa\sigma E-(\tau+\mu+\gamma_{1})I_{S},$
$\displaystyle\displaystyle{\frac{dI_{A}}{dt}}$ $\displaystyle=$
$\displaystyle(1-\kappa)\sigma E-(\mu+\gamma_{2})I_{A},$ (S-2)
$\displaystyle\displaystyle{\frac{dH}{dt}}$ $\displaystyle=$
$\displaystyle\tau I_{S}-(\delta+\mu+\gamma_{3})H,$
$\displaystyle\displaystyle{\frac{dR}{dt}}$ $\displaystyle=$
$\displaystyle\gamma_{1}I_{S}+\gamma_{2}I_{A}+\gamma_{3}H-\mu R.$

A flow diagram of the model (Model without lock-down) and (Model with lock-
down) are provided in Fig 1 (see main text). Biological interpretations of the
model (Model without lock-down) and (Model with lock-down) parameters are
provided in Table 1 (see main text).

### Positivity and boundedness of the solution for the Model (Model without
lock-down)

In this section, we provided a proof for the positivity and boundedness of
solutions of the system (Model without lock-down) with initial conditions
$(S(0),E(0),I_{S}(0),I_{A}(0),H(0),R(0))^{T}\in\mathbb{R}_{+}^{6}$. We first
state the following lemma.

######  Lemma S1.

(yang1996permanence ) Suppose $\Omega\subset\mathbb{R}\times\mathbb{C}^{n}$ is
open, $f_{i}\in C(\Omega,\mathbb{R}),i=1,2,3,...,n$. If
$f_{i}|_{x_{i}(t)=0,X_{t}\in\mathbb{C}_{+0}^{n}}\geq 0$,
$X_{t}=(x_{1t},x_{2t},.....,x_{1n})^{T},i=1,2,3,....,n$, then
$\mathbb{C}_{+0}^{n}\\{\phi=(\phi_{1},.....,\phi_{n}):\phi\in\mathbb{C}([-\tau,0],\mathbb{R}_{+0}^{n})\\}$
is the invariant domain of the following equations

$\displaystyle\frac{dx_{i}(t)}{dt}=f_{i}(t,X_{t}),t\geq\sigma,i=1,2,3,...,n.$

where $\mathbb{R}_{+0}^{n}=\\{(x_{1},....x_{n}):x_{i}\geq 0,i=1,....,n\\}$.

######  Proposition S1.

The system (Model without lock-down) is invariant in $\mathbb{R}_{+}^{6}$.

###### Proof.

By re-writing the system (Model without lock-down), we have

$\displaystyle\frac{dX}{dt}$ $\displaystyle=B(X(t)),X(0)=X_{0}\geq 0$ (S-3)

$B(X(t))=(B_{1}(X),B_{1}(X),...,B_{6}(X))^{T}$
We note that

$\displaystyle\frac{dS}{dt}|_{S=0}=\Pi\geq
0,\frac{dE}{dt}|_{E=0}=\frac{(\beta_{1}I_{S}+\beta_{1}I_{A}+\rho\beta_{2}H)S}{N}\geq
0,$ $\displaystyle\frac{dI_{S}}{dt}|_{I_{S}=0}=\kappa\sigma E\geq
0,\frac{dI_{A}}{dt}|_{I_{A}=0}=(1-\kappa)\sigma E\geq 0,$
$\displaystyle\frac{dH}{dt}|_{H=0}=\tau I_{S}\geq
0,\frac{dR}{dt}|_{R=0}=\gamma_{1}I_{S}+\gamma_{2}I_{A}+\gamma_{3}H\geq 0.$

Then it follows from the Lemma S1 that $\mathbb{R}_{+}^{6}$ is an invariant
set. ∎

######  Lemma S2.

The system (Model without lock-down) is bounded in the region
$\Omega=\\{(S,E,I_{S},I_{A},H,R\in\mathbb{R}_{+}^{6}|S+E+I_{S}+I_{A}+H+R\leq\frac{\Pi}{\mu}\\}$

###### Proof.

We observed from the system that

$\displaystyle\frac{dN}{dt}=\Pi-\mu N-\delta H\leq\Pi-\mu N$
$\displaystyle\Longrightarrow\lim\limits_{t\rightarrow\infty}supN(t)\leq\frac{\Pi}{\mu}$

Hence the system (Model without lock-down) is bounded. ∎

### Local stability of disease-free equilibrium (DFE)

The DFE of the model (Model without lock-down) is given by

$\displaystyle\varepsilon_{0}$
$\displaystyle=(S^{0},E^{0},I_{S}^{0},I_{A}^{0},H^{0},R^{0})$
$\displaystyle=\Big{(}\frac{\Pi}{\mu},0,0,0,0,0,0\Big{)}$

The local stability of $\varepsilon_{0}$ can be established on the system
(Model without lock-down) by using the next generation operator method. Using
the notation in van2002reproduction , the matrices $F$ for the new infection
and $V$ for the transition terms are given, respectively, by

$\displaystyle F$
$\displaystyle=\begin{bmatrix}0&\beta_{1}&\beta_{1}&\rho\beta_{2}\\\
0&0&0&0\\\ 0&0&0&0\\\ 0&0&0&0\\\ \end{bmatrix},$ $\displaystyle V$
$\displaystyle=\begin{bmatrix}\mu+\sigma&0&0&0\\\
-\kappa\sigma&\tau+\mu+\gamma_{1}&0&0\\\
-(1-\kappa)\sigma&0&\mu+\gamma_{2}&0\\\ 0&-\tau&0&\delta+\mu+\gamma_{3}\\\
\end{bmatrix}.$

It follows that the basic reproduction number hethcote2000mathematics ,
denoted by $R_{0}=\rho(FV^{-1})$, where $\rho$ is the spectral radius, is
given by

$\displaystyle
R_{0}=\frac{\beta_{1}\kappa\sigma}{(\mu+\sigma)(\gamma_{1}+\mu+\tau)}+\frac{\beta_{1}(1-\kappa)\sigma}{(\mu+\gamma_{2})(\mu+\sigma)}+\frac{\beta_{2}\kappa\rho\sigma\tau}{(\mu+\sigma)(\delta+\gamma_{3}+\mu)(\gamma_{1}+\mu+\tau)}$

Using Theorem 2 in van2002reproduction , the following result is established.

######  Lemma S3.

The DFE, $\varepsilon_{0}$, of the model (Model without lock-down) is locally-
asymptotically stable (LAS) if $R_{0}<1$, and unstable if $R_{0}>1$.

### Global stability of DFE

To show the global stability of $\varepsilon_{0}$, of the model (Model without
lock-down), we rewrite the system (Model without lock-down) as follows

$\displaystyle\displaystyle{\frac{dX}{dt}}$ $\displaystyle=$ $\displaystyle
T(X,I)$ (S-4) $\displaystyle\displaystyle{\frac{dI}{dt}}$ $\displaystyle=$
$\displaystyle G(X,I),\hskip 8.5359ptG(X,0)=0,$

where, $X=(S,R)\in\mathbb{R}_{+}^{2}$ denotes (it components) the number of
uninfected individuals and $I=(E,I_{S},I_{A},H)\in\mathbb{R}_{+}^{4}$ denotes
(it components) the number of infected individuals including latent,
infectious, etc. $\varepsilon_{0}=(X^{*},0)$ denotes the disease-free
equilibrium of the system (S-4). For the system (Model without lock-down),
$T(X,I)$ and $G(X,I)$ are given as follows:

$\displaystyle T(X,I)$
$\displaystyle=\begin{bmatrix}\Pi-\frac{\beta_{1}SI_{S}}{N}-\frac{\beta_{1}SI_{A}}{N}-\frac{\beta_{2}(\rho
S)H}{N}-\mu S\\\ \\\ \gamma_{1}I_{S}+\gamma_{2}I_{A}+\gamma_{3}H-\mu R\\\
\end{bmatrix},$ $\displaystyle G(X,I)$
$\displaystyle=\begin{bmatrix}\frac{\beta_{1}SI_{S}}{N}+\frac{\beta_{1}SI_{A}}{N}+\frac{\beta_{2}(\rho
S)H}{N}-(\mu+\sigma)E\\\ \\\ \kappa\sigma E-(\tau+\mu+\gamma_{1})I_{S}\\\ \\\
(1-\kappa)\sigma E-(\mu+\gamma_{2})I_{A}\\\ \\\ \tau
I_{S}-(\gamma_{3}+\delta+\mu)H\\\ \end{bmatrix}.$

It is clear from the expression of $G(X,I)$ that $G(X,0)=0$.

To show global stability of $\varepsilon_{0}=(X^{*},0)$ following two
condition must hold:

(H1) For $\displaystyle{\frac{dX}{dt}}=T(X,0)$, $X^{*}$ is globally
asymptotically stable.

(H2) $G(X,I)=AI-\hat{G}(X,I)$, $\hat{G}(X,I)\geq 0$ for $(X,I)\in\Omega$,

where $A=D_{I}G(X^{*},0)$ is an M-matrix (the off diagonal elements are
nonnegative) and $\Omega$ is the region where model (Model without lock-down)
makes biological sense.

Now, the system defined in (H1) can be written as

$\displaystyle\frac{dS}{dt}$ $\displaystyle=\Pi-\mu S,$ (S-5)
$\displaystyle\frac{dR}{dt}$ $\displaystyle=-\mu R.$

Solving analytically this system of equation we get,
$S(t)=\frac{\Pi}{\mu}+e^{-\mu t}(S(0)-\frac{\Pi}{\mu})$, $R(t)=e^{-\mu
t}R(0)$. As $t\rightarrow\infty$, $S(t)=\frac{\Pi}{\mu},\leavevmode\nobreak\
R(t)\rightarrow 0$. Therefore, $X^{*}$ is globally asymptotically stable for
$\displaystyle{\frac{dX}{dt}}=T(X,0)$.

So (H1) holds for the system (Model without lock-down). Now matrices $A$ and
$\hat{G}(X,I)$ for the system (Model without lock-down) are given as follows:

$\displaystyle A$
$\displaystyle=\begin{bmatrix}-(\mu+\sigma)&\beta_{1}&\beta_{1}&\rho\beta_{1}\\\
\\\ \kappa\sigma&-(\gamma_{1}+\tau+\mu)&0&0\\\ \\\
(1-\kappa)\sigma&0&-(\gamma_{2}+\mu)&0\\\ \\\
0&\tau&0&-(\gamma_{3}+\delta+\mu)\\\ \end{bmatrix},$
$\displaystyle\hat{G}(X,I)$
$\displaystyle=\begin{bmatrix}\beta_{1}I_{S}(1-\frac{S}{N})+\beta_{1}I_{A}(1-\frac{S}{N})+\rho\beta_{2}H(1-\frac{S}{N})\\\
\\\ 0\\\ \\\ 0\\\ \\\ 0\\\ \end{bmatrix}.$

Clearly, $A$ is an M-matrix and as $S(t)\leq N(t)$ in $\Omega$, therefore,
$\hat{G}(X,I)\geq 0$ for $(X,I)\in\Omega$. Following castillo2002computation ,
the below result can be stated:

######  Theorem S1.

The DFE of the model (Model without lock-down) is globally asymptotically
stable in $\Omega$ whenever $R_{0}<1$.

### Existence of endemic equilibria

In this section, the existence of the endemic equilibrium of the model (Model
without lock-down) is established. Let us denote

$\displaystyle k_{1}$
$\displaystyle=\mu+\sigma,k_{2}=\gamma_{1}+\mu+\tau,k_{3}=\mu+\gamma_{2},k_{4}=\delta+\gamma_{3}+\mu.$

Let $\varepsilon^{*}=(S^{*},L^{*},E^{*},I_{S}^{*},I_{A}^{*},H^{*},R^{*})$
represents any arbitrary endemic equilibrium point (EEP) of the model (Model
without lock-down). Further, define

$\displaystyle\lambda^{*}=\frac{\beta_{1}I_{S}^{*}}{N^{*}}+\frac{\beta_{1}I_{A}^{*}}{N^{*}}+\frac{\rho\beta_{2}H^{*}}{N^{*}}$
(S-6)

It follows, by solving the equations in (Model without lock-down) at steady-
state, that

$\displaystyle S^{*}$
$\displaystyle=\frac{\Pi}{\lambda^{*}+\mu},E^{*}=\frac{\lambda^{*}S^{*}}{k_{1}},I_{S}^{*}=\frac{\kappa\sigma\lambda^{*}S^{*}}{k_{1}k_{2}},I_{A}^{*}=\frac{(1-\kappa)\sigma\lambda^{*}S^{*}}{k_{1}k_{3}}$
(S-7) $\displaystyle H^{*}$
$\displaystyle=\frac{\tau\kappa\sigma\lambda^{*}S^{*}}{k_{1}k_{2}k_{4}},R^{*}=\frac{\gamma_{1}\kappa\sigma\lambda^{*}S^{*}}{\mu
k_{1}k_{2}}+\frac{\gamma_{2}(1-\kappa)\sigma\lambda^{*}S^{*}}{\mu
k_{1}k_{3}}+\frac{\gamma_{3}\tau\kappa\sigma\lambda^{*}S^{*}}{\mu
k_{1}k_{2}k_{4}}$

Substituting the expression in (S-7) into (S-6) shows that the non-zero
equilibrium of the model (Model without lock-down) satisfy the following
linear equation, in terms of $\lambda^{*}$:

$\displaystyle a_{0}\lambda^{*}+a_{1}=0$ (S-8)

where

$\displaystyle a_{0}$ $\displaystyle=\mu k_{2}k_{3}k_{4}+\kappa\sigma
k_{3}k_{4}(\mu+\gamma_{1})+(1-\kappa)\sigma
k_{2}k_{4}(\mu+\gamma_{2})+\tau\kappa\sigma k_{3}(\mu+\gamma_{3})$
$\displaystyle a_{1}$ $\displaystyle=\mu k_{1}k_{2}k_{3}k_{4}(1-R_{0})$

Since $a_{0}>0$, $\mu>0$, $k_{1}>0$, $k_{2}>0$, $k_{3}>0$ and $k_{4}>0$, it is
clear that the model (Model without lock-down) has a unique endemic
equilibrium point (EEP) whenever $R_{0}>1$ and no positive endemic equilibrium
point whenever $R_{0}<1$. This rules out the possibility of the existence of
equilibrium other than DFE whenever $R_{0}<1$. Furthermore, it can be shown
that, the DFE $\varepsilon_{0}$ of the model (Model without lock-down) is
globally asymptotically stable (GAS) whenever $R_{0}<1$.

From the above discussion we have concluded that

######  Theorem S2.

The model (Model without lock-down) has a unique endemic (positive)
equilibrium, given by $\varepsilon^{*}$, whenever $R_{0}>1$ and has no endemic
equilibrium for $R_{0}\leq 1$.

## Figures

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 are made with 10%, 20% and 30% decrement in the current estimate
of the lockdown rate, respectively (see Table 2 in main text). 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. Figure S2:
Ensemble model (see main text) forecast for the daily notified COVID-19 cases
in Delhi during May 3, 2020 till 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 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 current estimate of the lockdown rate,
respectively (see Table 2 in main text). Figure 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 current estimate 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 current estimate of the lockdown
rate, respectively (see Table 2 in main text). Figure 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 current estimate of the lockdown rate, respectively (see
Table 2 in main text). Figure S7: Effect of uncertainty of four
epidemiologically measurable and controllable parameters of the mechanistic
ODE model (see Table 1 and Fig 1 in the main text) on the effect of lockdown
and the basic reproduction number ($R_{0}$). Lockdown effect 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 of these 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 ranges
provided in Table 1 (main text). Figure S8: Effect of uncertainty of four
epidemiologically measurable and controllable parameters of the mechanistic
ODE model (see Table 1 and Fig 1 in the main text) on the effect of lockdown
and the basic reproduction number ($R_{0}$). Lockdown effect is measured in
terms of the differences in total number of COVID-19 cases occurred during May
3, 2020 till May 20, 2020 in Delhi under the lockdown scenarios (A1) and (A5),
respectively (see main text). Effect of Uncertainty of these 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 ranges provided in
Table 1 (main text). Figure S9: Effect of uncertainty of four
epidemiologically measurable and controllable parameters of the mechanistic
ODE model (see Table 1 and Fig 1 in the main text) on the effect of lockdown
and the basic reproduction number ($R_{0}$). Lockdown effect is measured in
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 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 ranges
provided in Table 1 (main text). Figure S10: Effect of uncertainty of four
epidemiologically measurable and controllable parameters of the mechanistic
ODE model (see Table 1 and Fig 1 in the main text) on the effect of lockdown
and the basic reproduction number ($R_{0}$). Lockdown effect is measured in
terms of the differences in total number of COVID-19 cases occurred during May
3, 2020 till May 20, 2020 in Rajasthan under the lockdown scenarios (A1) and
(A5), respectively (see main text). Effect of Uncertainty of these 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 ranges
provided in Table 1 (main text). Figure S11: Effect of uncertainty of four
epidemiologically measurable and controllable parameters of the mechanistic
ODE model (see Table 1 and Fig 1 in the main text) on the effect of lockdown
and the basic reproduction number ($R_{0}$). Lockdown effect is measured in
terms of the differences in total number of COVID-19 cases occurred during May
3, 2020 till May 20, 2020 in Gujarat under the lockdown scenarios (A1) and
(A5), respectively (see main text). Effect of Uncertainty of these 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 ranges
provided in Table 1 (main text). Figure S12: Effect of uncertainty of four
epidemiologically measurable and controllable parameters of the mechanistic
ODE model (see Table 1 and Fig 1 in the main text) on the effect of lockdown
and the basic reproduction number ($R_{0}$). Lockdown effect is measured in
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 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 ranges
provided in Table 1 (main text). Figure S13: Posterior distribution of the
weights for the mechanistic ODE model combinations (Model without lock-down) &
(Model with lock-down) and the Hybrid statistical model (see main text),
respectively for Maharashtra. Figure S14: Posterior distribution of the
weights for the mechanistic ODE model combinations (Model without lock-down) &
(Model with lock-down) and the Hybrid statistical model (see main text),
respectively for Delhi. Figure S15: Posterior distribution of the weights for
the mechanistic ODE model combinations (Model without lock-down) & (Model with
lock-down) and the Hybrid statistical model (see main text), respectively for
Madhya Pradesh. Figure S16: Posterior distribution of the weights for the
mechanistic ODE model combinations (Model without lock-down) & (Model with
lock-down) and the Hybrid statistical model (see main text), respectively for
Rajasthan. Figure S17: Posterior distribution of the weights for the
mechanistic ODE model combinations (Model without lock-down) & (Model with
lock-down) and the Hybrid statistical model (see main text), respectively for
Gujarat. Figure S18: Posterior distribution of the weights for the mechanistic
ODE model combinations (Model without lock-down) & (Model with lock-down) 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 (Model without lock-down) & (Model with lock-down) and the
Hybrid statistical model (see main text), respectively for India.

## Tables

Table S1: Estimated uninformative initial conditions for the mechanistic ODE model (Model without lock-down). Respective subscripts 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 |  $\boldsymbol{S(0)}$ |  $\boldsymbol{E(0)}$ |  $\boldsymbol{I(0)}$ |  $\boldsymbol{A(0)}$
|  |  |  |
MH |  $\begin{subarray}{c}120428468\\\ \\\ (101502357-124478508)\end{subarray}$ |  $\begin{subarray}{c}0.001\\\ \\\ (0-0.003)\end{subarray}$ |  $\begin{subarray}{c}332.43\\\ \\\ (6.14-633.93)\end{subarray}$ |  $\begin{subarray}{c}2316.93\\\ \\\ (230.8-7103.05)\end{subarray}$
|  |  |  |
DL |  $\begin{subarray}{c}12173507\\\ \\\ (10289760-19705021)\end{subarray}$ |  $\begin{subarray}{c}3743\\\ \\\ (179-9704)\end{subarray}$ |  $\begin{subarray}{c}0.14\\\ \\\ (0-0.29)\end{subarray}$ |  $\begin{subarray}{c}245\\\ \\\ (144.5-9409)\end{subarray}$
|  |  |  |
MP |  $\begin{subarray}{c}84632517\\\ \\\ (70665560-89661041)\end{subarray}$ |  $\begin{subarray}{c}1541\\\ \\\ (65.96-4956)\end{subarray}$ |  $\begin{subarray}{c}0.2914\\\ \\\ (0.01-1.10)\end{subarray}$ |  $\begin{subarray}{c}4095\\\ \\\ (93.46-7852)\end{subarray}$
|  |  |  |
RJ |  $\begin{subarray}{c}74502407\\\ \\\ (70099616-79723261)\end{subarray}$ |  $\begin{subarray}{c}195.9\\\ \\\ (43.74-9770)\end{subarray}$ |  $\begin{subarray}{c}129\\\ \\\ (0.7408-132)\end{subarray}$ |  $\begin{subarray}{c}6209\\\ \\\ (40.86-9404)\end{subarray}$
|  |  |  |
GJ |  $\begin{subarray}{c}62744520\\\ \\\ (60628093-69846136)\end{subarray}$ |  $\begin{subarray}{c}1280\\\ \\\ (58.78-2238)\end{subarray}$ |  $\begin{subarray}{c}79.29\\\ \\\ (3.70-237)\end{subarray}$ |  $\begin{subarray}{c}1.17\\\ \\\ (0.39-2.60)\end{subarray}$
|  |  |  |
UP |  $\begin{subarray}{c}224225631\\\ \\\ (200822802-229090013)\end{subarray}$ |  $\begin{subarray}{c}0.49\\\ \\\ (0.01-1.34)\end{subarray}$ |  $\begin{subarray}{c}3.07\\\ \\\ (0.17-7.35)\end{subarray}$ |  $\begin{subarray}{c}4764\\\ \\\ (379-8031)\end{subarray}$
|  |  |  |
IND |  $\begin{subarray}{c}1219512468\\\ \\\ (1219512465-1219512469)\end{subarray}$ |  $\begin{subarray}{c}49998\\\ \\\ (49995-49999)\end{subarray}$ |  $\begin{subarray}{c}9997\\\ \\\ (9994-9999)\end{subarray}$ |  $\begin{subarray}{c}9995\\\ \\\ (9991-9999)\end{subarray}$
|  |  |  |
Table S2: Weight estimates for the mechanistic ODE model combinations (Model without lock-down) & (Model with lock-down) and the Hybrid statistical model (see main text), respectively. Respective subscripts 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
|  |  |  |  |  |  |
$\boldsymbol{w_{1}}$ | $\begin{subarray}{c}0.3019\\\ \\\ (0.2160-0.5284)\end{subarray}$ | $\begin{subarray}{c}0.3226\\\ \\\ (0.1149-0.6594)\end{subarray}$ | $\begin{subarray}{c}0.1599\\\ \\\ (0.0966-0.2384)\end{subarray}$ | $\begin{subarray}{c}0.5490\\\ \\\ (0.3091-0.8384)\end{subarray}$ | $\begin{subarray}{c}0.1672\\\ \\\ (0.1665-0.38)\end{subarray}$ | $\begin{subarray}{c}0.6612\\\ \\\ (0.2807-0.8333)\end{subarray}$ | $\begin{subarray}{c}0.8176\\\ \\\ (0.5017-0.8874)\end{subarray}$
|  |  |  |  |  |  |
$\boldsymbol{w_{2}}$ | $\begin{subarray}{c}0.6981\\\ \\\ (0.4716-0.7840)\end{subarray}$ | $\begin{subarray}{c}0.6774\\\ \\\ (0.3405-0.8851)\end{subarray}$ | $\begin{subarray}{c}0.8401\\\ \\\ (0.7616-0.9034)\end{subarray}$ | $\begin{subarray}{c}0.4510\\\ \\\ (0.1616-0.6909)\end{subarray}$ | $\begin{subarray}{c}0.8328\\\ \\\ (0.62-0.8335)\end{subarray}$ | $\begin{subarray}{c}0.3388\\\ \\\ (0.1667-0.7193)\end{subarray}$ | $\begin{subarray}{c}0.1824\\\ \\\ (0.1126-0.4983)\end{subarray}$
|  |  |  |  |  |  |
Table S3: Goodness of fit (RMSE and MAE) of the Hybrid statistical model (see main text) for the test data 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
|  |  |  |  |  |  |
$\boldsymbol{RMSE}$ | $\begin{subarray}{c}35.15368\end{subarray}$ | $\begin{subarray}{c}57.855\end{subarray}$ | $\begin{subarray}{c}28.59711\end{subarray}$ | $\begin{subarray}{c}18.4018\end{subarray}$ | $\begin{subarray}{c}21.77785\end{subarray}$ | $\begin{subarray}{c}17.60281\end{subarray}$ | $\begin{subarray}{c}114.3764\end{subarray}$
|  |  |  |  |  |  |
$\boldsymbol{MAE}$ | $\begin{subarray}{c}22.2375\end{subarray}$ | $\begin{subarray}{c}34.4592\end{subarray}$ | $\begin{subarray}{c}20.01179\end{subarray}$ | $\begin{subarray}{c}13.22069\end{subarray}$ | $\begin{subarray}{c}12.91013\end{subarray}$ | $\begin{subarray}{c}10.16784\end{subarray}$ | $\begin{subarray}{c}65.61564\end{subarray}$
|  |  |  |  |  |  |