# Assessment of Lockdown Effect in Some States and Overall India: A Predictive
Mathematical Study on COVID-19 Outbreak

Tridip Sardar 111Corresponding author. Email<EMAIL_ADDRESS>Sk Shahid
Nadim  Sourav Rana  Joydev Chattopadhyay  Department of Mathematics,
Dinabandhu Andrews College, Kolkata, India  Agricultural and Ecological
Research Unit, Indian Statistical Institute, Kolkata, India  Department of
Statistics, Visva-Bharati University, Santiniketan, West Bengal, India

###### Abstract

In the absence of neither an effective treatment or vaccine and with an
incomplete understanding of the epidemiological cycle, Govt. has implemented a
nationwide lockdown to reduce COVID-19 transmission in India. To study the
effect of social distancing measure, we considered a new mathematical model on
COVID-19 that incorporates lockdown effect. By validating our model to the
data on notified cases from five different states and overall India, we
estimated several epidemiologically important parameters as well as the basic
reproduction number ($R_{0}$). Combining the mechanistic mathematical model
with different statistical forecast models, we projected notified cases in the
six locations for the period May 17, 2020, till May 31, 2020. A global
sensitivity analysis is carried out to determine the correlation of two
epidemiologically measurable parameters on the lockdown effect and also on
$R_{0}$. Our result suggests that lockdown will be effective in those
locations where a higher percentage of symptomatic infection exists in the
population. Furthermore, a large scale COVID-19 mass testing is required to
reduce community infection. Ensemble model forecast suggested a high rise in
the COVID-19 notified cases in most of the locations in the coming days.
Furthermore, the trend of the effective reproduction number ($R_{t}$) during
the projection period indicates if the lockdown measures are completely
removed after May 17, 2020, a high spike in notified cases may be seen in
those locations. Finally, combining our results, we provided an effective
lockdown policy to reduce future COVID-19 transmission in India.

###### keywords:

COVID-19; Mathematical model; Lockdown effect; Outbreak containment policy

††journal: Elsevier

##  1 Introduction

As of May 15, 2020, 4593395 cases and 306376 deaths from 2019 novel
coronavirus disease (COVID-19), caused by severe acute respiratory syndrome
coronavirus 2 (SARS-CoV-2), were recorded worldwide csse2020 . Coronaviruses
are enveloped non-segmented positive-sense RNA viruses that belong to the
Coronaviridae family and the order Nidovirales, and are widely distributed
among humans and other mammals richman2016clinical . The novel coronavirus,
COVID-19 started in mainland China, with a geographical emphasis at Wuhan, the
capital city of Hubei province wang2020novel  and has widely spread all over
the world. Many of the initial cases were usually introduced to the wholesale
Huanan seafood market, which also traded live animals. Clinical trials of
hospitalized patients found that patients exhibit symptoms consistent with
viral pneumonia at the onset of COVID-19, most commonly fever, cough, sore
throat and fatigue cdcgov2020 . Some patients reported changes in their
ground-glass lungs; normal or lower than average white lymphocyte blood cell
counts and platelet counts; hypoxemia; and deranged liver and kidney function.
Most were said to be geographically related to the wholesale market of Huanan
seafood chinadaily2019 . Severe outbreaks occur in USA ($1457593$ cases),
Spain ($272646$ cases), Russia (262843 cases), UK (236711 cases), Italy
(223885) and so many countries and the disease continues to spread globally.
This has been declared a pandemic by the World Health Organization. It is the
third zoonotic human coronavirus that has arisen in the present century, after
the $2002$ severe acute respiratory syndrome coronavirus (SARS-CoV), which
spread to 37 countries and the 2012 Middle East respiratory syndrome
coronavirus (MERS-CoV), which spread to 27 countries.

The 2019 pandemic novel coronavirus was first confirmed in India on January
30, 2020, in the state of Kerala. A total of $82087$ confirmed cases and
$2648$ deaths in the country have been reported as of May 15, 2020
indiacovid2020track . The Indian government has introduced social distance as
a precaution to avoid the possibility of a large-scale population movement
that can accelerate the spread of the disease. India government implemented a
14-hour voluntary public curfew on 22 March 2020. Furthermore, the Prime
Minister of India also ordered a nationwide lockdown at midnight on March 24,
2020 to slow the spread of COVID-19. Despite no vaccine, social distancing has
identified as the most commonly used prevention and control strategy
ferguson2020report . The purpose of these initiatives is the restriction of
social interaction in workplaces, schools, and other public spheres, except
for essential public services such as fire, police, hospitals. No doubt the
spread of this virus outbreak has seriously disrupted the life, economy and
health of citizens. This is a great concern for everyone how long this
scenario will last and when the disease will be controlled.

Mathematical modeling based on system of differential equations may provide a
comprehensive mechanism for the dynamics of a disease transmission
sardar2020realistic . Several modeling studies have already been performed for
the COVID-19 outbreak tang2020updated ; quilty2020effectiveness ;
shen2020modelling ; tang2020estimation ; wu2020nowcasting . Based on data
collected from December 31, 2019 till January 28, 2020, Wu et al
wu2020nowcasting  developed a susceptible exposed infectious recovered model
(SEIR) to clarify the transmission dynamics and projected national and global
spread of disease. They also calculated around 2.68 is the basic reproductive
number for COVID-19. Tang et al tang2020estimation  proposed a compartmental
deterministic model that would combine the clinical development of the
disease, the epidemiological status of the patient and the measures for
intervention. Researchers found that the amount of control reproduction number
may be as high as 6.47, and that methods of intervention including intensive
touch tracing followed by quarantine and isolation would effectively minimize
COVID cases tang2020estimation . For the basic reproductive number, Read et
al. reported a value of 3.1 based on the data fitting of an SEIR model, using
an assumption of Poisson-distributed daily time increments read2020novel . A
report by Cambridge University has indicated that India’s countrywide three-
week lockdown would not be adequate to prevent a resurgence of the new
coronavirus epidemic that could bounce back in months and cause thousands of
infections singh2020age . They suggested that two or three lockdowns can
extend the slowdown longer with five-day breaks in between or a single 49-day
lockdown. Data-driven mathematical modeling plays a key role in disease
prevention, planning for future outbreaks and determining the effectiveness of
control. Several data-driven modeling experiments have been performed in
various regions tang2020estimation ; chen2020mathematical . Currently, there
are very limited works that studied the impact of lockdown on COVID-19
transmission dynamics in India.

In the present manuscript, we proposed a new mathematical model for COVID-19
that incorporates the lockdown effect. We also considered variability in
transmission between symptomatic and asymptomatic populations with former
being a fast spreader of the disease. Analyzing COVID-19 daily notified cases
from five affected states (Maharashtra, Delhi, Tamil Nadu, Gujarat and Punjab)
and from overall India, we studied the effect of social distancing measures
implemented by the Govt. on notified cases reduction in those regions. We also
estimates the basic reproduction numbers ($R_{0}$) for these six locations.
Using a post-processing BMA technique, we ensemble our COVID-19 mathematical
model with different statistical forecast model to obtain a projection of
notified cases in those six locations for the period May 17, 2020 till May 31,
2020. A global sensitivity analysis is carried out to determine the
correlation of two epidemiologically measurable parameters on lockdown effect
and also on $R_{0}$. Finally to determine the COVID-19 transmission trend
during the projection period (May 17, 2020 till May 31, 2020), we estimate the
effective reproduction number ($R_{t}$) for the mentioned six locations.

##  2 Method

Based on the development and epidemiological characteristics of COVID-19, a
SEIR type model is more appropriate to study the dynamics of this current
pandemic (kucharski2020early, ; peng2020epidemic, ; sardar2020realistic, ).
The model we developed in this paper is based on the interaction of seven
mutually exclusive sub-classes namely, Susceptible ($S$), Lockdown ($L$),
Exposed ($E$), Asymptomatic ($A$), Symptomatic ($I$), Hospitalized ($C$), and
Recovered ($R$).

Susceptible population ($S$) increased due to constant recruitment rate
$\Pi_{H}$ and those individuals coming back from lockdown compartment after
the lockdown period $\displaystyle\frac{1}{\omega}$. Population in the
susceptible class decreased due to new infection in contact with symptomatic
and asymptomatic infected population, natural death and also a fraction of the
susceptible individuals become home-quarantine due to lockdown at a rate $l$.
We also assumed variability in disease transmission in asymptomatic and
symptomatic population with later being a fast spreader of infection with a
variability factor $\left(0\leq\rho\leq 1\right)$ (gumel2004modelling, ;
mandal2020prudent, ).

Lockdown population ($L$) increased by those susceptible who are home-
quarantined during the lockdown period $\displaystyle\frac{1}{\omega}$, at a
rate $l$. Population under lockdown is decreased due to natural death and
those individuals who become susceptible again after the lockdown period
$\displaystyle\frac{1}{\omega}$. For simplicity, we assume ideal lockdown
scenario i.e. all population under lockdown maintained proper social
distancing and do not contribute to new infection.

Population in the exposed compartment ($E$) increased by new infection coming
from susceptible compartment. A fraction $\kappa$ of the exposed individuals
become symptomatic infected and remaining fraction ($1-\kappa$) become
asymptomatic infected after the disease incubation period $\frac{1}{\sigma}$.
Exposed population also decreased due to natural death at a rate $\mu$.

Asymptomatic infected compartment ($A$) increased due to a fraction
($1-\kappa$) of infection coming from exposed compartment. Since, asymptomatic
COVID-19 cases are hard to detect therefore, we assume that asymptomatic
infection are not notified. Population in this compartment is decreased due to
natural recovery and deaths at a rate $\gamma_{1}$ and $\mu$, respectively.

Population in the symptomatic infected compartment ($I$) increased due to a
fraction $\kappa$ of infection coming from exposed compartment after the
incubation period $\frac{1}{\sigma}$. This compartment decreased due to
natural recovery at a rate $\gamma_{2}$, natural death at a rate $\mu$ and
those infected population who are notified & hospitalized at a rate $\tau$.

Notified & hospitalized infected population ($C$) increased due to influx of
infection coming from symptomatic infected class at a rate $\tau$. This
population decreased due to natural death at a rate $\mu$, disease related
deaths at a rate $\delta$, and recovery from COVID-19 at a rate $\gamma_{3}$.
We assume that population of this compartment do not mix with the general
population in the community i.e. this compartment do not contribute in the
COVID-19 transmission.

Finally, recovered population ($R$) increased due to influx of individuals
coming from asymptomatic ($A$), symptomatic ($I$), and notified & hospitalized
individuals ($C$) at a rate $\gamma_{1}$, $\gamma_{2}$, and $\gamma_{3}$,
respectively. As we are analyzing this study in a shorter time frame
therefore, we assume definitive immunity i.e. recovered population do not
contribute to new COVID-19 infection. Thus recovered population decreased due
to natural death at a rate $\mu$.

Based on the above assumptions the system of equations that represent COVID-19
transmission with and without lockdown are provided below:

### Model without lock-down

$\displaystyle\displaystyle{\frac{dS}{dt}}$ $\displaystyle=$
$\displaystyle\Pi_{H}-\frac{\beta_{1}IS}{\left(N-C\right)}-\frac{\rho\beta_{1}AS}{\left(N-C\right)}-\mu
S$ $\displaystyle\displaystyle{\frac{dE}{dt}}$ $\displaystyle=$
$\displaystyle\frac{\beta_{1}IS}{\left(N-C\right)}+\frac{\rho\beta_{1}AS}{\left(N-C\right)}-(\mu+\sigma)E,$
$\displaystyle\displaystyle{\frac{dA}{dt}}$ $\displaystyle=$
$\displaystyle(1-\kappa)\sigma E-(\gamma_{1}+\mu)A,$
$\displaystyle\displaystyle{\frac{dI}{dt}}$ $\displaystyle=$
$\displaystyle\kappa\sigma E-(\gamma_{2}+\tau+\mu)I,$ (2.1)
$\displaystyle\displaystyle{\frac{dC}{dt}}$ $\displaystyle=$
$\displaystyle\tau I-(\delta+\gamma_{3}+\mu)C,$
$\displaystyle\displaystyle{\frac{dR}{dt}}$ $\displaystyle=$
$\displaystyle\gamma_{1}A+\gamma_{2}I+\gamma_{3}C-\mu R,$

### Model with lock-down

$\displaystyle\displaystyle{\frac{dS}{dt}}$ $\displaystyle=$
$\displaystyle\Pi_{H}+\omega
L-\frac{\beta_{1}IS}{\left(N-L-C\right)}-\frac{\rho\beta_{1}AS}{\left(N-L-C\right)}-\mu
S-lS$ $\displaystyle\displaystyle{\frac{dL}{dt}}$ $\displaystyle=$
$\displaystyle lS-(\mu+\omega)L,$ $\displaystyle\displaystyle{\frac{dE}{dt}}$
$\displaystyle=$
$\displaystyle\frac{\beta_{1}IS}{\left(N-L-C\right)}+\frac{\rho\beta_{1}AS}{\left(N-L-C\right)}-(\mu+\sigma)E,$
$\displaystyle\displaystyle{\frac{dA}{dt}}$ $\displaystyle=$
$\displaystyle(1-\kappa)\sigma E-(\gamma_{1}+\mu)A,$
$\displaystyle\displaystyle{\frac{dI}{dt}}$ $\displaystyle=$
$\displaystyle\kappa\sigma E-(\gamma_{2}+\tau+\mu)I,$ (2.2)
$\displaystyle\displaystyle{\frac{dC}{dt}}$ $\displaystyle=$
$\displaystyle\tau I-(\delta+\gamma_{3}+\mu)C,$
$\displaystyle\displaystyle{\frac{dR}{dt}}$ $\displaystyle=$
$\displaystyle\gamma_{1}A+\gamma_{2}I+\gamma_{3}C-\mu R.$

A diagram of our model is provided in Fig 1. Information of our model
parameters is provided in Table 1.

### Mathematical properties of the model

We studied the positivity and boundedness of solution of the model (2) (see
supplementary appendix). The system (2) demonstrates two equilibria, that is,
the disease-free equlibrium and an unique endemic equilibrium (see
supplementary appendix). The disease-free state is locally asymptotically
stable whenever the corresponding basic reproduction number ($R_{0}$) is less
than unity (see supplementary appendix). By using a nonlinear Lyapunov
function, it is also seen that the disease-free equilibrium is globally
asymptotically stable whenever $R_{0}<1$ (see supplementary appendix). In
addition, the model (2) has an unique endemic equilibrium if $R_{0}$ exceeds
unity. Furthermore, using the central manifold theory, the local stability of
the endemic equilibrium is established whenever $R_{0}>1$ (see supplementary
appendix).

### Data

Daily COVID-19 reported cases from Maharashtra (MH), Delhi (DL), Tamil Nadu
(TN), Gujarat (GJ), Punjab (PJ) and whole India (IND) for the time period
March 14, 2020 till May 3, 2020 are considered for our study. These five
states are deeply affected by current COVID-19 outbreak in India
(indiacovid2020track, ). Daily COVID-19 notified cases were collected from
(indiacovid2020track, ). Demographic data of the different locations are taken
from aadhaar20 ; Nitiayog2020 .

### Estimation procedure

Several important epidemiological parameters (see Table 1) of our mathematical
model (2) are estimated using COVID-19 daily reported cases from the mentioned
six locations. Total time duration of lockdown implemented by Govt. is 54 days
starting from March 25, 2020 till May 17, 2020. Time-series data of daily
COVID-19 cases in our study for the locations MH, DL, TN, GJ, PJ, and IND,
respectively contains both with and without lockdown effect. Therefore, a
combination of our mathematical models (2)& (2) (with and without lockdown)
are used for calibration. From our models (2)& (2), new COVID-19 notified
cases during the $i^{th}$ time interval $\left[t_{i},t_{i}+\Delta
t_{i}\right]$ is

$\displaystyle H_{i}(\hat{\theta})=\displaystyle\tau\int_{t_{i}}^{t_{i}+\Delta
t_{i}}I(\xi,\hat{\theta})\hskip 5.69046ptd\xi,\\\ $ (2.3)

where, $\Delta t_{i}$ is the time step length and $\hat{\theta}$ is the set of
unknown parameters of the models (2)& (2) that are estimated. Then $K$
observation from the data and from the models (2)& (2) are
$\left\\{D_{1},D_{2},...,D_{K}\right\\}$ and
$\\{H_{1}(\hat{\theta}),H_{2}(\hat{\theta}),....,H_{K}(\hat{\theta})\\}$,
respectively. Therefore, we constructed the sum of squares function
(sardar2013optimal, ) as:

$\displaystyle
SS(\hat{\theta})=\displaystyle\sum_{i=1}^{K}\left[D_{i}-H_{i}(\hat{\theta})\right]^{2},\\\
$ (2.4)

MATLAB based nonlinear least square solver $fmincon$ is used to fit simulated
and observed daily COVID-19 notified cases for the mentioned states and the
whole country. Delayed Rejection Adaptive Metropolis (haario2006dram, ) (DRAM)
algorithm is used to sample the 95% confidence region. An elaboration of this
model fitting technique is provided in (sardar2017mathematical, ).

### Statistical forecast models and the ensemble model

COVID-19 mathematical model we developed in this study may be efficient in
capturing the transmission dynamics. However, as solution of the mathematical
model is always smooth therefore, our model may not be able to replicate the
fluctuations occurring in daily time-series data. Moreover, forecast of future
COVID-19 cases based on a single mathematical model may not be very reliable
approach. For this purpose, we used two statistical forecast models namely,
Auto-regressive Integrated Moving Average (ARIMA); and ARMA errors, Trend and
Seasonal components (TBATS) respectively. A Hybrid statistical model (HYBRID)
based on the combination of ARIMA and TBATS is also used during forecast.
Calibration of three statistical forecast models (ARIMA, TBATS and HYBRID)
using COVID-19 daily notified cases from MH, DL, TN, GJ, PJ, and IND,
respectively during March 14, 2020 till May 3, 2020, are done using the $R$
package ’forecastHybrid’ Forecast2020 . Each individuals models (ARIMA and
TBATS) are first fitted to the aforesaid time-series data and then we combined
each models with weightage based on in sample error to obtain the HYBRID model
Forecast2020 . Prediction skill of the each three statistical forecast model
(ARIMA, TBATS and HYBRID) are tested on the daily COVID-19 notified cases
during May 4, 2020 till May 8, 2020 for each of the six locations (see
supplementary Table S2). Based on the prediction skill (see supplementary
Table S2), the best statistical forecast model is ensemble with our COVID-19
mathematical models (2) and (2). A post-processing BMA technique based on
’DRAM’ algorithm haario2006dram  is used to determine the weightage (see
supplementary Table S3 and Fig S6 to Fig S11) to combine the best statistical
model with the COVID-19 mathematical models (2) and (2).

### Disease forecasting under different lockdown scenario

Govt. have implemented lockdown all over India on March 25, 2020 and it will
continue till May 17, 2020. The short and medium scale industries are largely
affected by the lockdown Economic_Times2 ; India_Today4 . To partially recover
the economy, Govt. of India continuously relaxing the lockdown rules from
April 20, 2020 Economictimes2020a ; financialexpress20a ; TheHindu20a . To
forecast COVID-19 cases for the period May 17, 2020 till May 31, 2020, for the
six locations (MH, DL, TN, GJ, PJ and IND) based on the Govt. strategy, we
considered following scenarios:

Forecast based on current lockdown rate: We have estimated the average
lockdown rate for our COVID-19 mathematical model (see Table 1 and Table 2).
Using this lockdown rate and using other parameters (estimated and known) of
our mathematical models (2) & (2), we forecast COVID-19 notified cases during
May 17, 2020 till May 31, 2020 for the locations MH, DL, TN, GJ, PJ, and IND,
respectively. Finally, forecast based on our mathematical model is ensemble
with the result based on the best statistical forecast model for a location
mentioned earlier.

Forecast based on 15% reduction in current lockdown rate: We followed same
procedure as the previous scenario with 15% decrement in the estimate of
lockdown rate (see Table 1 and Table 2) to obtained the forecast during the
mentioned time period.

Forecast based on 20% reduction in current lockdown rate: we followed the same
procedure as previous two scenarios with 20% decrement in the estimate of
lockdown rate (see Table 1 and Table 2) to obtained the forecast during the
mentioned time period.

Forecast based on 30% reduction in current lockdown rate: we followed the same
procedure as previous three scenarios with 30% decrement in the estimate of
lockdown rate (see Table 1 and Table 2) to obtain the forecast during the
mentioned time period.

Forecast based on no lockdown: Continue as earlier and assuming lockdown is
lifted after May 17, 2020, we forecast COVID-19 notified cases during May 17,
2020 till May 31, 2020, for the six mentioned locations.

### Estimation of the basic and the effective reproduction number

Since we assumed that population under the lockdown do not contact with the
infection from the community therefore the basic reproduction number ($R_{0}$)
(van2002reproduction, ) for our mathematical model with and without lockdown
(see Fig 1 and supplementary method) are same and its expression is provided
below:

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

The effective reproductive number ($R_{t}$) is defined as the expected number
of secondary infection per infectious in a population made up of both
susceptible and non-susceptible hosts (rothman2008modern, ). If $R_{t}>1$, the
number of new cases will increase, for $R_{t}=1$, the disease become endemic,
and when $R_{t}<1$ there will be a decline in new cases.

Following (rothman2008modern, ), the expression of $R_{t}$ is given as
follows:

$\displaystyle\displaystyle R_{t}=R_{0}\times\hat{s},$

where, $\hat{s}$ is the fraction of the host population that is susceptible.

$R_{0}$ can easily be estimated by plugin the sample values of the unknown
parameters (see Table 2) of the model without lockdown (2) in the expressions
of $R_{0}$.

Following procedure is adapted to estimate $R_{t}$ during May 17, 2020 till
May 31, 2020 under two lockdown scenarios:

$\bullet$

Using current estimate of the lockdown rate and different parameters of our
mathematical model (see Table 1 and Table 2), we estimate $\hat{s}$ and
$R_{t}$ during May 17, 2020 till May 31, 2020, for the locations MH, DL, TN,
GJ, PJ and IND, respectively.

$\bullet$

Using different parameters (see Table 1 and Table 2) of our mathematical model
without lockdown (2), we estimated $\hat{s}$ and $R_{t}$ during May 17, 2020
till May 31, 2020 for the mentioned six locations.

### Sensitivity analysis and effective lockdown strategy

To determine an effective lockdown policy in those six locations (MH, DL, TN,
GJ, PJ and IND) will require some correlation between lockdown effect with
some epidemiologically measurable parameters of our mathematical model (see
Fig 1). There are several important parameters of our mathematical model (see
Table 1) and among them there are two parameters that are measurable namely,
$\kappa$: fraction of new infected that become symptomatic (COVID-19 testing
will provide an accurate estimate) and $\tau$: Average notification &
hospitalization rate of symptomatic COVID-19 infection (this parameter is
proportional to the number of COVID-19 testing). Lockdown effect is measured
as the difference between the total number of cases projected by our ensemble
model with and without lockdown. A global sensitivity analysis
marino2008methodology  is performed to determine the effect of the mentioned
two parameters on the lockdown effect and on the basic reproduction number
($R_{0}$). Using Latin Hyper cube sampling (LHS), we draw $1000$ samples for
$\kappa$ and $\tau$, respectively from their respective ranges (see Table 1).
Partial rank correlation and its corresponding $p$-value are examined to
determine the relation between two mentioned parameters with the lockdown
effect and $R_{0}$, respectively.

##  3 Results and discussion

Three models (mathematical, statistical forecast and ensemble) fitting to
daily COVID-19 notified cases during March 14, 2020 till May 3, 2020, for
Maharashtra (MH), Delhi (DL), Tamil Nadu (TN), Gujarat (GJ), Punjab (PJ), and
India (IND) is depicted in Fig 2. Among the three statistical forecast models
(ARIMA, TBATS and HYBRID), the ARIMA model performed better on the test
prediction data (May 4, 2020 till May 8, 2020) of DL and GJ (see supplementary
Table S2). Whereas, the TBATS model provide better result in compare to other
two models for the test prediction data (May 4, 2020 till May 8, 2020) of PJ
(see supplementary Table S2). For the remaining three locations (MH, TN and
IND), the HYBRID model provide the best result (see supplementary Table S2) on
the test prediction data (May 4, 2020 till May 8, 2020). The ensemble model,
which is a combination of our COVID-19 mathematical models (2) & (2) and the
best statistical forecast model (region specific), is performed well in
capturing COVID-19 daily time-series data trend in all the six locations.
Posterior distribution of the weights at which we combine our COVID-19
mathematical model with the best statistical forecast model in the six
different locations are provided in supplementary method (see Fig S6 to Fig
S11 and Table S3).

In MH, DL, and GJ, the estimate of the symptomatic influx fraction ($\kappa$)
suggest that low percentage (about 11% to 20%) of symptomatic infected in the
population (see Table 2). However, in TN and PJ, relatively higher percentage
(about 82% to 88%) of symptomatic infection is found (see Table 2). In overall
India, our estimate shows that currently about 62% of new infection are
symptomatic (see Table 2). Except for GJ, in other five locations, estimate of
the transmission rate ($\beta_{1}$) are found to be in same scale (see Table
2). Relatively higher value of $\beta_{1}$ is found in Gujrat (see Table 2).
Low value of the transmission variability factor ($\rho$) indicates most of
the community infection in GJ are due to contact with the symptomatic infected
population. As in GJ, the value of $\kappa$ is found to be small (about 11%)
therefore, relatively smaller symptomatic population producing most of the
infection in GJ. This indicate that there may be a possibility of existence of
super-spreaders among the symptomatic infected in GJ. This observation is
agree with a recent survey result in GJ NDTVsuperspreaders . Except for PJ, in
other five locations, the estimates of $\rho$ (below 50%) are found to be low
(see Table 2). This indicates small contribution of the asymptomatic infected
population towards the new infection produced in MH, DL, TN, GJ and IND,
respectively. Estimate of the lockdown rate in the five states (MH, DL, TN, GJ
and PJ) suggest that around 50% to 88% of the total susceptible population are
successfully home quarantined during the lockdown period (see Table 2). Thus,
lockdown is overall successful in those five states. However, this is not the
case for overall India, our estimate suggest that about 11% of the total
susceptible population in India maintained proper social distancing during the
lockdown period (see Table 2).

Our estimate of the basic reproduction number ($R_{0}$) (see Table 3), in the
six locations found to be in good agreement of the world-wide estimate
provided by WHO liu2020reproductive . We performed a global sensitivity
analysis of two epidemiologically measurable parameters of our mathematical
model (see Fig 1) namely $\kappa$: fraction of new infected that become
symptomatic and $\tau$: Average notification & hospitalization rate of
symptomatic COVID-19 infection, on $R_{0}$. Partial rank correlation and its
corresponding $p$-value (see Fig 3) suggest that $\tau$ has a negative
correlation on $R_{0}$. Thus, more testing will isolate more infection from
the community and therefore may reduce the COVID-19 community transmission.
Furthermore, high positive correlation of $\kappa$ with $R_{0}$ (see Fig 3)
indicates the possibility of high COVID-19 transmission in those areas where
population have higher percentage of symptomatic infection.

Ensemble model forecast of notified COVID-19 cases between May 17, 2020 till
May 31, 2020 (see Table 4, Fig. 4, and Fig S1 to Fig S10 in supplementary
appendix) indicate that in the coming few days, a high increment in the
COVID-19 notified cases may be observed in MH, DL, TN, GJ, PJ, and IND.
Furthermore, our ensemble model prediction during the mentioned period suggest
that around 117645 to 128379 cases may occurred in overall India (see Table
4). These numbers are much higher than the total cumulative cases between
March 2, 2020 till May 15, 2020, in whole India.

A global sensitivity analysis of $\kappa$ and $\tau$ on the lockdown effect
suggest that both of these parameters have high positive correlation with the
lockdown effect in all the six locations (see Fig. 5). Therefore, lockdown
will be effective in those region where higher percentage of symptomatic
infection is found in the population and also larger COVID-19 mass testing
will be required to isolate the cases.

To measure the COVID-19 transmission trend during May 17, 2020 till May 31,
2020, we estimated the effective reproduction number ($R_{t}$) during the
mentioned period for MH, DL, TN, GJ, PJ and IND, respectively (see Fig. 6).
Our result suggest that, a decreasing trend in new notified COVID-19 cases
($R_{t}<1$) may be seen after May 31, 2020 if current lockdown measures (see
Table 2) are maintained in DL, TN and PJ, respectively. Furthermore, if social
distancing measures are removed after May 17, 2020, we may see a rise in the
daily COVID-19 cases in all of the six locations (see Fig. 6).

##  4 Conclusion

Up to May 15, 2020, total number of reported COVID-19 cases and deaths in
India are 81794 and 2649, respectively (indiacovid2020track, ). This tally
rises with few thousand new notified cases every day reported from different
locations in India (indiacovid2020track, ). Currently, there is no treatment
or vaccine available for COVID-19. Therefore, only measure to control the
outbreak may be home quarantined (lockdown) a larger percentage of susceptible
population. However, this disease control strategy may have some negative
impact on the economy. Therefore, it is utmost important to determine an
effective lockdown policy that may reduce COVID-19 transmission in the
community as well as save the Indian economy from drowning. This policy may be
found by studying the dynamics and prediction of a mechanistic mathematical
model for COVID-19 transmission and testing the results in real life
situation.

In this present study, we consider a new mathematical model on COVID-19
transmission that incorporates the lockdown effect (see Fig 1). In our models
(2) & (2), we also considered transmission variability between symptomatic and
asymptomatic population with former being a fast spreader of the disease.
Using daily time-series data of notified COVID-19 cases from five states
(Maharashtra, Delhi, Tamil Nadu, Gujarat and Punjab) and overall India, we
studied the effect of lockdown measures on the reduction of notified cases in
those regions. Our result suggest that lockdown will be effective in those
locations where higher percentage of symptomatic infection exist in the
population. Furthermore, a large scale COVID-19 mass testing is required to
reduce community infection in those locations. Using a post-processing BMA
technique, we ensemble the prediction of our mathematical model with the
results obtained from different statistical forecast model. Our ensemble model
forecast of COVID-19 daily notified cases during May 17, 2020 till May 31,
2020, suggested a very high rise in the COVID-19 notified cases in the
mentioned time duration in most of the locations. Furthermore, estimation of
the effective reproduction number ($R_{t}$) during the mentioned time duration
indicates if the lockdown measures are completely removed after May 17, 2020,
in those locations, a high spike in COVID-19 notified cases may be seen during
the mentioned forecasting period. We provide a suggestion for the Indian Govt.
and policy makers to acquire the following steps for effective containment of
COVID-19 transmission:

  1. 1.

Perform a survey to find the percentage of symptomatic infection in different
states and regions.

  2. 2.

Focus implementing extensive lockdown in those locations only where the
percentage of symptomatic infection is high.

  3. 3.

Provide relaxation in lockdown in other locations for some time. This process
will increase the percentage of symptomatic infection.

  4. 4.

Repeat step-2, when a region has a sufficient percentage of symptomatic
infection.

There are some drawback in our study and may be modified in future. We assume
that lockdown population ($L$) and notified & hospitalized infection ($C$) do
not mix with the general population in the community. However, there are
numerous evidences where disease transmitted from the hospital and from home
confined individuals NDTVhospital20 . We shall leave these challenges for our
future objectives.

## Conflict of interests

The authors declare that they have no conflicts of interest.

## Acknowledgments

The authors are grateful to editor-in-chief, handling editor and learned
reviewers for their comments and suggestions on the earlier version of this
manuscript. The comments immensely improve the standard of this article.

Dr. Tridip Sardar acknowledges the Science & Engineering Research Board (SERB)
major project grant (File No: EEQ/2019/000008 dt. 4/11/2019), Government of
India.

Sk Shahid Nadim receives funding as senior research fellowship from Council of
Scientific & Industrial Research (Grant No: 09/093(0172)/2016/EMR-I),
Government of India, New Delhi.

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: Flow diagram of the mathematical model with lockdown.
Epidemiological information of different parameters shown in the above figure
are provided in Table 1. Figure 2: Three models (mechanistic mathematical
model, statistical forecast model and ensemble model) fitting to daily
notified COVID-19 cases from five different states and overall India for the
period March 14, 2020 till May 3, 2020. Respective subscript are MH:
Maharashtra, DL: Delhi, TN: Tamil Nadu, GJ: Gujarat, PJ: Punjab, and IND:
India. Here LD denotes lockdown period and No LD indicate the period before
lockdown implementation. Statistical forecast model for different locations
are ARIMA (DL and GJ), TBATS (PJ) and HYBRID (MH, TN and IND), respectively.
Lockdown effect is only considered for the mechanistic mathematical model and
consequently in the ensemble model. Shaded area indicate the 95% confidence
region. Figure 3: Global sensitivity analysis of two epidemiologically
measurable parameters namely $\kappa$: fraction of new infected that become
symptomatic and $\tau$: average notification & hospitalization rate of
symptomatic COVID-19 infection, on $R_{0}$. The subscripts MH, DL, TN, GJ, PJ
and IND, respectively are same as Fig 2. Figure 4: Ensemble model forecast for
the daily notified COVID-19 cases in India during May 17, 2020 till May 31,
2020, under five different social distancing measure. Various legends are
Current Rate: daily notified case projection using the estimated value of the
lockdown rate (see Table 2), 15% Reduction: daily notified case projection
using 15% reduction in the estimated value of the lockdown rate (see Table 2),
20% Reduction: daily notified case projection using 20% reduction in the
estimated value of the lockdown rate (see Table 2), 30% Reduction: daily
notified case projection using 30% reduction in the estimated value of the
lockdown rate (see Table 2), and No lockdown: daily notified case projection
based on no lockdown scenario, respectively.  Figure 5: Global sensitivity
analysis of two epidemiologically measurable parameters namely $\kappa$:
fraction of new infected that become symptomatic and $\tau$: average
notification & hospitalization rate of symptomatic COVID-19 infection, on the
lockdown effect. The effect of lockdown is measured as the difference between
the total number of cases projected by our ensemble model with and without
lockdown scenario during May 17, 2020 till May 31, 2020. The subscripts MH,
DL, TN, GJ, PJ and IND, respectively are same as Fig 2. Figure 6: Effective
reproduction number ($R_{t}$) for the period May 17, 2020 till May 31, 2020,
in MH, DL, TN, GJ, PJ and IND, respectively. The subscripts MH, DL, TN, GJ, PJ
and IND, respectively are same as Fig 2.

## Tables

Table 1: Parameters with their respective epidemiological explanation for the mechanistic mathematical model (see Fig 1) for COVID-19. Parameters |  Epidemiological Meaning |  Ranges |  Reference
---|---|---|---
$N$ |  Total population |  Varies over different region |  aadhaar20
$\Pi_{H}$ $=$ $\displaystyle\mu\times N$ |  Average recruitment rate |  Varies over different region |  –
$\displaystyle\frac{1}{\mu}$ |  Average life expectancy at birth |  Varies over states |  Nitiayog2020
$\beta_{1}$ |  Transmission rate of symptomatic infected |  (0 - 200) $day^{-1}$ |  Estimated
$\rho$ |  Reduction in COVID-19 transmission for Asymptomatic infected |  0 - 1 |  Estimated
$\displaystyle\frac{1}{\sigma}$ |  COVID-19 incubation period |  (2 - 14) $days$ |  Estimated
$\displaystyle\kappa$ |  Fraction of exposed population that become symptomatic infected |  0 - 1 |  Estimated
$\gamma_{1}$ |  Recovery rate for asymptomatic infected |  ($\gamma_{3}$ \- 1) $day^{-1}$ |  Estimated
$\gamma_{2}$ |  Recovery rate for symptomatic infected |  ($\gamma_{3}$ \- 1) $day^{-1}$ |  Estimated
$\tau$ |  Average notification & hospitalization rate of symptomatic infection |  (0 - 1) $day^{-1}$ |  Estimated
$\delta$ |  Average case fatality rate |  Varies over states |  indiacovid2020track
$\gamma_{3}$ |  Average recovery rate for COVID-19 hospitalized & notified infection |  Varies over states |  indiacovid2020track
$l$ |  Average lockdown rate |  (0 - 1) $day^{-1}$ |  Estimated
$\frac{1}{\omega}$ |  lockdown period |  54 days |  TimesofIndia54days
Table 2: Estimated parameter values of the mechanistic mathematical model (see Fig 1). Respective row subscripts are same as Fig. 2. All data are given in the format Estimate (95% CI). Location | $\boldsymbol{\beta_{1}}$ | $\boldsymbol{\rho}$ | $\boldsymbol{\sigma}$ | $\boldsymbol{\kappa}$ | $\boldsymbol{\gamma_{1}}$ | $\boldsymbol{\gamma_{2}}$ | $\boldsymbol{\tau\times 100}$ | $\boldsymbol{l\times 100}$
---|---|---|---|---|---|---|---|---
MH | $\begin{subarray}{c}5.58\\\ \\\ (2.62-9.79)\end{subarray}$ | $\begin{subarray}{c}0.015\\\ \\\ (0.01-0.15)\end{subarray}$ | $\begin{subarray}{c}0.39\\\ \\\ (0.07-0.45)\end{subarray}$ | $\begin{subarray}{c}0.20\\\ \\\ (0.11-0.53)\end{subarray}$ | $\begin{subarray}{c}0.63\\\ \\\ (0.27-0.99)\end{subarray}$ | $\begin{subarray}{c}0.69\\\ \\\ (0.22-0.98)\end{subarray}$ | $\begin{subarray}{c}24.36\\\ \\\ (11.34-93.53)\end{subarray}$ | $\begin{subarray}{c}57.84\\\ \\\ (11.77-86.19)\end{subarray}$
DL | $\begin{subarray}{c}5.93\\\ \\\ (0.78-6.30)\end{subarray}$ | $\begin{subarray}{c}0.13\\\ \\\ (0.02-0.82)\end{subarray}$ | $\begin{subarray}{c}0.11\\\ \\\ (0.07-0.47)\end{subarray}$ | $\begin{subarray}{c}0.16\\\ \\\ (0.11-0.93)\end{subarray}$ | $\begin{subarray}{c}0.99\\\ \\\ (0.31-0.99)\end{subarray}$ | $\begin{subarray}{c}0.47\\\ \\\ (0.33-0.98)\end{subarray}$ | $\begin{subarray}{c}37.86\\\ \\\ (12.48-98.27)\end{subarray}$ | $\begin{subarray}{c}87.03\\\ \\\ (11.25-87.11)\end{subarray}$
TN | $\begin{subarray}{c}1.47\\\ \\\ (1.23-3.63)\end{subarray}$ | $\begin{subarray}{c}0.37\\\ \\\ (0.18-0.91)\end{subarray}$ | $\begin{subarray}{c}0.30\\\ \\\ (0.15-0.48)\end{subarray}$ | $\begin{subarray}{c}0.88\\\ \\\ (0.03-0.91)\end{subarray}$ | $\begin{subarray}{c}0.84\\\ \\\ (0.29-0.98)\end{subarray}$ | $\begin{subarray}{c}0.60\\\ \\\ (0.30-0.98)\end{subarray}$ | $\begin{subarray}{c}0.56\\\ \\\ (0.22-9.65)\end{subarray}$ | $\begin{subarray}{c}52.16\\\ \\\ (12.38-88.92)\end{subarray}$
GJ | $\begin{subarray}{c}10.73\\\ \\\ (5.96-20.53)\end{subarray}$ | $\begin{subarray}{c}0.03\\\ \\\ (0.002-0.15)\end{subarray}$ | $\begin{subarray}{c}0.10\\\ \\\ (0.07-0.20)\end{subarray}$ | $\begin{subarray}{c}0.11\\\ \\\ (0.10-0.15)\end{subarray}$ | $\begin{subarray}{c}0.24\\\ \\\ (0.21-0.98)\end{subarray}$ | $\begin{subarray}{c}0.91\\\ \\\ (0.66-0.99)\end{subarray}$ | $\begin{subarray}{c}16.79\\\ \\\ (10.10-23.56)\end{subarray}$ | $\begin{subarray}{c}77.68\\\ \\\ (37.15-89.36)\end{subarray}$
PJ | $\begin{subarray}{c}1.63\\\ \\\ (1.31-5.35)\end{subarray}$ | $\begin{subarray}{c}0.68\\\ \\\ (0.016-0.95)\end{subarray}$ | $\begin{subarray}{c}0.26\\\ \\\ (0.076-0.45)\end{subarray}$ | $\begin{subarray}{c}0.82\\\ \\\ (0.11-0.90)\end{subarray}$ | $\begin{subarray}{c}0.43\\\ \\\ (0.12-0.97)\end{subarray}$ | $\begin{subarray}{c}0.98\\\ \\\ (0.24-0.99)\end{subarray}$ | $\begin{subarray}{c}2.11\\\ \\\ (0.12-2.24)\end{subarray}$ | $\begin{subarray}{c}48.18\\\ \\\ (12.50-87.76)\end{subarray}$
IND | $\begin{subarray}{c}2.60\\\ \\\ (1.04-2.92)\end{subarray}$ | $\begin{subarray}{c}0.02\\\ \\\ (0.02-0.77)\end{subarray}$ | $\begin{subarray}{c}0.08\\\ \\\ (0.07-0.14)\end{subarray}$ | $\begin{subarray}{c}0.62\\\ \\\ (0.30-0.98)\end{subarray}$ | $\begin{subarray}{c}0.73\\\ \\\ (0.31-0.98)\end{subarray}$ | $\begin{subarray}{c}0.79\\\ \\\ (0.33-0.98)\end{subarray}$ | $\begin{subarray}{c}0.6\\\ \\\ (0.2-0.9)\end{subarray}$ | $\begin{subarray}{c}10.41\\\ \\\ (10.37-42.56)\end{subarray}$
Table 3: Estimated values of the basic reproduction number ($R_{0}$) in MH, DL, TN, GJ, PJ and IND. Respective row subscripts are same as Fig 2. All data are given in the format Estimate (95% CI). Location |  Basic Reproduction Number ($\boldsymbol{R_{0}}$)
---|---
MH |  $\begin{subarray}{c}1.3320\\\ \\\ (1.2369-2.3053)\end{subarray}$
DL |  $\begin{subarray}{c}1.7569\\\ \\\ (1.2409-2.2392)\end{subarray}$
TN |  $\begin{subarray}{c}2.2169\\\ \\\ (1.78-5.0374)\end{subarray}$
GJ |  $\begin{subarray}{c}2.3911\\\ \\\ (1.6223-2.7436)\end{subarray}$
PJ |  $\begin{subarray}{c}1.8077\\\ \\\ (1.7342-6.3960)\end{subarray}$
IND |  $\begin{subarray}{c}2.0337\\\ \\\ (1.6645-2.3318)\end{subarray}$
Table 4: Ensemble model forecast of the total COVID-19 cases during May 17, 2020 till May 31, 2020 in five different states and overall India under five different lockdown scenario. Respective row subscripts are same as Fig. 2. Different lockdown scenarios are Current Rate: cumulative case projection using the estimated value of the lockdown rate (see Table 2), 15% Reduction: cumulative case projection using 15% reduction in the estimated value of the lockdown rate (see Table 2), 20% Reduction: cumulative case projection using 20% reduction in the estimated value of the lockdown rate (see Table 2), 30% Reduction: cumulative case projection using 30% reduction in the estimated value of the lockdown rate (see Table 2), and No lockdown: cumulative case projection based on no lockdown scenario, respectively. All data are provided in the format Estimate (95% CI). Location |  Current Rate |  15% Reduction |  20% Reduction |  30% Reduction |  No lockdown
---|---|---|---|---|---
MH |  $\begin{subarray}{c}33138\\\ \\\ (31910-34046)\end{subarray}$ |  $\begin{subarray}{c}34602\\\ \\\ (34430-34730)\end{subarray}$ |  $\begin{subarray}{c}35144\\\ \\\ (34983-35361)\end{subarray}$ |  $\begin{subarray}{c}36318\\\ \\\ (35532-37383)\end{subarray}$ |  $\begin{subarray}{c}48635\\\ \\\ (41285-58578)\end{subarray}$
DL |  $\begin{subarray}{c}6247\\\ \\\ (5857-6811)\end{subarray}$ |  $\begin{subarray}{c}6504\\\ \\\ (6253-6866)\end{subarray}$ |  $\begin{subarray}{c}6601\\\ \\\ (6403-6887)\end{subarray}$ |  $\begin{subarray}{c}6816\\\ \\\ (6734-6933)\end{subarray}$ |  $\begin{subarray}{c}9623\\\ \\\ (7535-11068)\end{subarray}$
TN |  $\begin{subarray}{c}19344\\\ \\\ (19319-21503)\end{subarray}$ |  $\begin{subarray}{c}20958\\\ \\\ (20946-21975)\end{subarray}$ |  $\begin{subarray}{c}21598\\\ \\\ (21591-22162)\end{subarray}$ |  $\begin{subarray}{c}23076\\\ \\\ (22562-23107)\end{subarray}$ |  $\begin{subarray}{c}60390\\\ \\\ (31726-62172)\end{subarray}$
GJ |  $\begin{subarray}{c}8227\\\ \\\ (8128-9039)\end{subarray}$ |  $\begin{subarray}{c}8414\\\ \\\ (8273-9572)\end{subarray}$ |  $\begin{subarray}{c}8485\\\ \\\ (8328-9776)\end{subarray}$ |  $\begin{subarray}{c}8646\\\ \\\ (8452-10233)\end{subarray}$ |  $\begin{subarray}{c}11208\\\ \\\ (10433-17549)\end{subarray}$
PJ |  $\begin{subarray}{c}7427\\\ \\\ (7128-7517)\end{subarray}$ |  $\begin{subarray}{c}8188\\\ \\\ (7702-8333)\end{subarray}$ |  $\begin{subarray}{c}8480\\\ \\\ (7924-8647)\end{subarray}$ |  $\begin{subarray}{c}9139\\\ \\\ (8421-9354)\end{subarray}$ |  $\begin{subarray}{c}19715\\\ \\\ (16406-20705)\end{subarray}$
IND |  $\begin{subarray}{c}117645\\\ \\\ (110999-122572)\end{subarray}$ |  $\begin{subarray}{c}119204\\\ \\\ (111569-124863)\end{subarray}$ |  $\begin{subarray}{c}119740\\\ \\\ (111765-125650)\end{subarray}$ |  $\begin{subarray}{c}120831\\\ \\\ (112164-127255)\end{subarray}$ |  $\begin{subarray}{c}128379\\\ \\\ (114923-138352)\end{subarray}$

##  5 Supplementary Appendix

### Positivity and boundedness of the solution for the Model (2)

This subsection is provided to prove the positivity and boundedness of
solutions of the system (2) with initial conditions
$(S(0),L(0),E(0),A(0),I(0),C(0),R(0))^{T}\in\mathbb{R}_{+0}^{7}$. We first
state the following lemma.

######  Lemma 5.1.

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 $\mathbb{R}_{+0}^{n}=\\{(x_{1},....x_{n}):x_{i}\geq 0,i=1,....,n\\}$
yang1996permanence .

######  Proposition 5.1.

The system (2) is invariant in $\mathbb{R}_{+0}^{7}$.

###### Proof.

By re-writing the system (2) we have:

$\displaystyle\frac{dX}{dt}$ $\displaystyle=B(X(t)),X(0)=X_{0}\geq 0$ (S-1)

$B(X(t))=(B_{1}(X),B_{1}(X),...,B_{7}(X))^{T}$
We note that

$\displaystyle\frac{dS}{dt}|_{S=0}$ $\displaystyle=\Pi_{H}+\omega
L>0,\frac{dL}{dt}|_{L=0}=lS\geq 0,\frac{dE}{dt}|_{E=0}=\frac{\beta_{1}S(I+\rho
A)}{N-C}\geq 0,$ $\displaystyle\frac{dA}{dt}|_{A=0}$
$\displaystyle=(1-\kappa)\sigma E\geq 0,\frac{dI}{dt}|_{I=0}=\kappa\sigma
E\geq 0,\frac{dC}{dt}|_{C=0}=\tau I\geq 0,$
$\displaystyle\frac{dR}{dt}|_{R=0}$
$\displaystyle=\gamma_{1}A+\gamma_{2}I+\gamma_{3}C\geq 0.$

Then it follows from the Lemma 5.1 that $\mathbb{R}_{+0}^{7}$ is an invariant
set for the COVID-19 system (2) with lockdown. ∎

######  Corollary 5.1.

The system (2) is invariant in $\mathbb{R}_{+0}^{6}$.

###### Proof.

Proceeding as proposition 5.1, we can easily show that $\mathbb{R}_{+0}^{6}$
is an invariant set for the COVID-19 system (2) without lockdown. ∎

######  Lemma 5.2.

The system (2) is bounded in the region
$\Omega=\\{(S,L,E,A,I,C,R)\in\mathbb{R}_{+0}^{7}|S+L+E+A+I+C+R\leq\frac{\Pi_{H}}{\mu}\\}$

###### Proof.

We have from the system (2):

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

Hence the system (2) is bounded. ∎

######  Corollary 5.2.

The system (2) is bounded in the region
$\Omega^{*}=\\{(S,E,A,I,C,R)\in\mathbb{R}_{+0}^{6}|S+E+A+I+C+R\leq\frac{\Pi_{H}}{\mu}\\}$

###### Proof.

proceeding same as lemma 5.2, we can easily show that the system (2) is
bounded in $\Omega^{*}$. ∎

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

The DFE of the model (2) is provided as follows:

$\displaystyle\varepsilon_{0}$
$\displaystyle=(S^{0},L^{0},E^{0},A^{0},I^{0},C^{0},R^{0})$
$\displaystyle=\Big{(}\frac{\Pi_{H}(\mu+\omega)}{\mu\left(\mu+\omega+l\right)},\frac{\Pi_{H}l}{\mu\left(\mu+\omega+l\right)},0,0,0,0,0\Big{)}$

The local stability of $\varepsilon_{0}$ can be established for the COVID-19
system (2) 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

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

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

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

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

######  Lemma 5.3.

The DFE, $\varepsilon_{0}$, of the model (2) is locally-asymptotically stable
(LAS) if $R_{0}<1$, and unstable if $R_{0}>1$.

The threshold quantity, $R_{0}$ is the basic reproduction number of the
disease hethcote2000mathematics ; anderson1979population ;
anderson1992infectious . This represent the average number of secondary cases
generated by a infected person in a fully susceptible population. The
epidemiological significance of 5.3 is that when $R_{0}$ is less than unity, a
low influx of infected individuals into the population will not cause major
outbreaks, and the disease would die out in time.

### Global stability of DFE

######  Theorem 5.1.

The DFE of the model (2) is globally asymptotically stable in $\Omega$
whenever $R_{0}\leq 1$.

###### Proof.

Consider the following Lyapunov function

$\displaystyle\mathcal{L}=\Big{(}\frac{\sigma(\kappa
k_{2}+\rho(1-\kappa)k_{3})}{k_{1}k_{2}}\Big{)}E+\Big{(}\frac{\rho
k_{3}}{k_{2}}\Big{)}A+I$

where $k_{1}=\mu+\sigma$, $k_{2}=\gamma_{1}+\mu$ and
$k_{3}=\gamma_{2}+\tau+\mu$.
We take the Lyapunov derivative with respect to $t$,

$\displaystyle\dot{\mathcal{L}}$ $\displaystyle=\Big{(}\frac{\sigma(\kappa
k_{2}+\rho(1-\kappa)k_{3})}{k_{1}k_{2}}\Big{)}\dot{E}+\Big{(}\frac{\rho
k_{3}}{k_{2}}\Big{)}\dot{A}+\dot{I}$ $\displaystyle=\frac{\sigma(\kappa
k_{2}+\rho(1-\kappa)k_{3})}{k_{1}k_{2}}\Big{[}\frac{\beta_{1}S(I+\rho
A)}{N-L-C}-k_{1}E\Big{]}+\frac{\rho k_{3}}{k_{2}}[(1-\kappa)\sigma
E-k_{2}A]+(\kappa\sigma E-k_{3}I)$
$\displaystyle\leq\frac{\beta_{1}\sigma(\kappa
k_{2}+\rho(1-\kappa)k_{3})}{k_{1}k_{2}}(I+\rho A)-\frac{\sigma(\kappa
k_{2}+\rho(1-\kappa)k_{3})}{k_{2}}E+\frac{\rho(1-\kappa)k_{3}\sigma}{k_{2}}E$
$\displaystyle-\rho k_{3}A+\kappa\sigma E-k_{3}I\text{ (Since $S\leq N-L-C$ in
$\Omega$)}$ $\displaystyle=\frac{\beta_{1}\sigma(\kappa
k_{2}+\rho(1-\kappa)k_{3})}{k_{1}k_{2}}(I+\rho A)-\rho k_{3}A-k_{3}I$
$\displaystyle=\frac{\beta_{1}\sigma(\kappa
k_{2}+\rho(1-\kappa)k_{3})}{k_{1}k_{2}k_{3}}k_{3}(I+\rho A)-\rho
k_{3}A-k_{3}I$ $\displaystyle\leq k_{3}(R_{0}-1)(I+\rho A)\leq 0,\text{
whenever $R_{0}\leq 1$.}$

Since all the variables and parameters of the model (2) are non-negative, it
follows that $\dot{\mathcal{L}}\leq 0$ for $R_{0}\leq 1$ with
$\dot{\mathcal{L}}=0$ at diseases free equilibrium. Hence, $\mathcal{L}$ is a
Lyapunov function on $\Omega$. Therefore, followed by LaSalle’s Invariance
Principle lasalle1976stability , that

$(E(t),A(t),I(t))\rightarrow(0,0,0)\text{ as }t\rightarrow\infty$ (S-2)

Since $\lim\limits_{t\rightarrow\infty}supI(t)=0$ (from S-2), it follows that,
for sufficiently small $\epsilon>0$, there exist constants $B_{1}>0$ such that
$\lim\limits_{t\rightarrow\infty}supI(t)\leq\epsilon$ for all $t>B_{1}$.
Hence, it follows from the sixth equation of the model (2) that, for
$t>B_{1}$,

$\displaystyle\frac{dC}{dt}\leq\tau\epsilon-k_{4}C$

Therefore using comparison theorem smith1995theory

$\displaystyle
C^{\infty}=\lim\limits_{t\rightarrow\infty}supC(t)\leq\frac{\tau\epsilon}{k_{4}}$

So as $\epsilon\rightarrow 0$,
$C^{\infty}=\lim\limits_{t\rightarrow\infty}supC(t)\leq 0$
Similarly by using $\lim\limits_{t\rightarrow\infty}infI(t)=0$, it can be
shown that

$\displaystyle C_{\infty}=\lim\limits_{t\rightarrow\infty}infC(t)\geq 0$

Thus, it follows from above two relations

$\displaystyle C_{\infty}\geq 0\geq C^{\infty}$

Hence $\lim\limits_{t\rightarrow\infty}C(t)=0$
Similarly, it can be shown that

$\displaystyle\lim\limits_{t\rightarrow\infty}R(t)=0,\lim\limits_{t\rightarrow\infty}S(t)=\frac{\Pi_{H}\left(\mu+\omega\right)}{\mu\left(\mu+\omega+l\right)},\text{and}\lim\limits_{t\rightarrow\infty}L(t)=\frac{\Pi_{H}l}{\mu\left(\mu+\omega+l\right)}.$

Therefore by combining all above equations, it follows that each solution of
the model equations (2), with initial conditions $\in\Omega$ , approaches
$\varepsilon_{0}$ as $t\rightarrow\infty$ for $R_{0}\leq 1$. ∎

### Existence and stability of endemic equilibria

In this section, the existence of the endemic equilibrium of the model (2) is
established. Let us denote

$\displaystyle M_{1}$
$\displaystyle=\frac{\mu+\omega}{\mu+\sigma},M_{2}=\frac{\left(1-\kappa\right)\sigma\left(\mu+\omega\right)}{\left(\mu+\gamma_{1}\right)\left(\mu+\sigma\right)},M_{3}=\frac{\kappa\sigma\left(\mu+\omega\right)}{\left(\mu+\gamma_{2}+\tau\right)\left(\mu+\sigma\right)},$
$\displaystyle M_{4}$
$\displaystyle=\frac{\kappa\tau\sigma\left(\mu+\omega\right)}{\left(\mu+\gamma_{2}+\tau\right)\left(\mu+\gamma_{3}+\delta\right)\left(\mu+\sigma\right)}.$

Let $\varepsilon^{*}=(S^{*},E^{*},A^{*},I^{*},C^{*},R^{*})$ represents any
arbitrary endemic equilibrium point (EEP) of the model (2). Further, define

$\displaystyle\lambda^{*}=\frac{\beta_{1}(I^{*}+\rho
A^{*})}{N^{*}-L^{*}-C^{*}}$ (S-3)

It follows, by solving the equations in (2) at steady-state, that

$\displaystyle S^{*}$
$\displaystyle=\frac{\left(\mu+\omega\right)L^{*}}{l},L^{*}=\frac{\Pi_{H}l}{\lambda^{*}\left(\mu+\omega\right)+\mu\left(\mu+\omega+l\right)},E^{*}=\frac{M_{1}L^{*}\lambda^{*}}{l},$
(S-4) $\displaystyle A^{*}$
$\displaystyle=\frac{M_{2}L^{*}\lambda^{*}}{l},I^{*}=\frac{M_{3}L^{*}\lambda^{*}}{l},C^{*}=\frac{M_{4}L^{*}\lambda^{*}}{l}$
$\displaystyle R^{*}$
$\displaystyle=\frac{\left(\gamma_{1}M_{2}+\gamma_{2}M_{3}+\gamma_{3}M_{4}\right)L^{*}\lambda^{*}}{l\mu}$

Substituting the expression in (S-4) into (S-3) shows that the non-zero
equilibrium of the model (2) satisfy the following linear equation, in terms
of $\lambda^{*}$:

$\displaystyle A\lambda^{*}=B$ (S-5)

where

$\displaystyle A$ $\displaystyle=\mu
M_{1}+M_{2}\left(\mu+\gamma_{1}\right)+M_{3}\left(\mu+\gamma_{2}\right)+\gamma_{3}M_{4}$
$\displaystyle B$ $\displaystyle=\mu\left(\mu+\omega\right)(R_{0}-1)$

Since, $M_{1}>0$, $M_{2}>0$, $M_{3}>0$, and $M_{4}>0$ $\implies$ $A>0$, it is
clear that the model (2) has an 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 $R_{0}<1$. Therefore, we have the following result:

######  Theorem 5.2.

The model (2) has a unique endemic (positive) equilibrium, given by
$\varepsilon^{*}$, whenever $R_{0}>1$ and has no endemic equilibrium for
$R_{0}\leq 1$.

Now we will prove the local stability of endemic equilibrium.

######  Theorem 5.3.

The endemic equilibrium $\varepsilon^{*}$ of the COVID-19 system (2) with
lockdown is locally asymptotically stable if $R_{0}>1$.

###### Proof.

The Jacobian matrix of the system (2) $J_{\varepsilon_{0}}$ at DFE is given by

$\displaystyle
J_{\varepsilon_{0}}={\begin{pmatrix}-\left(\mu+l\right)&\omega&0&-\rho\beta_{1}&-\beta_{1}&0&0\\\
l&-\left(\mu+\omega\right)&0&0&0&0&0\\\
0&0&-(\mu+\sigma)&\rho\beta_{1}&\beta_{1}&0&0\\\
0&0&(1-\kappa)\sigma&-(\mu+\gamma_{1})&0&0&0\\\
0&0&\kappa\sigma&0&-(\mu+\gamma_{2}+\tau)&0&0\\\
0&0&0&0&\tau&-(\mu+\gamma_{3}+\delta)&0\\\
0&0&0&\gamma_{1}&\gamma_{2}&\gamma_{3}&-\mu\\\ \end{pmatrix}},$

Here, by taking $\beta_{1}$ as a bifurcation parameter, we use the central
manifold theory method to determine the local stability of the endemic
equilibrium castillo2004dynamical . Taking $\beta_{1}$ as the bifurcation
parameter and gives critical value of $\beta_{1}$ at $R_{0}=1$ is given as

$\beta_{1}^{*}=\frac{(\mu+\sigma)(\gamma_{1}+\mu)(\gamma_{2}+\tau+\mu)}{[\kappa\sigma(\gamma_{1}+\mu)+(1-\kappa)\rho\sigma(\gamma_{2}+\tau+\mu)]}$

The Jacobian of (2) at $\beta=\beta_{1}^{*}$, denoted by
$J_{\varepsilon_{0}}|_{\beta=\beta_{1}^{*}}$ has a right eigenvector
(corresponding to the zero eigenvalue) given by
$w=(w_{1},w_{2},w_{3},w_{4},w_{5},w_{6},w_{7})^{T}$ , where

$\displaystyle w_{1}$
$\displaystyle=-\frac{\left(\mu+\sigma\right)\left(\mu+\omega\right)}{\mu\left(\mu+\omega+l\right)},w_{2}=-\frac{\left(\mu+\sigma\right)l}{\mu\left(\mu+\omega+l\right)},w_{3}=1,w_{4}=\frac{\left(1-\kappa\right)\sigma}{\mu+\gamma_{1}},$
$\displaystyle w_{5}$
$\displaystyle=\frac{\kappa\sigma}{\mu+\gamma_{2}+\tau},w_{6}=\frac{\kappa\sigma\tau}{\left(\mu+\gamma_{2}+\tau\right)\left(\mu+\gamma_{3}+\delta\right)}$
$\displaystyle w_{7}$
$\displaystyle=\frac{\gamma_{1}(1-\kappa)\sigma}{\mu(\gamma_{1}+\mu)}+\frac{\gamma_{2}\kappa\sigma}{\mu(\gamma_{2}+\tau+\mu)}+\frac{\gamma_{3}\tau\kappa\sigma}{\mu(\gamma_{2}+\tau+\mu)(\delta+\gamma_{3}+\mu)}$

Similarly, from $J_{\varepsilon_{0}}|_{\beta=\beta_{1}^{*}}$, we obtain a left
eigenvector $v=(v_{1},v_{2},v_{3},v_{4},v_{5},v_{6},v_{7})$ (corresponding to
the zero eigenvalue), where

$\displaystyle
v_{1}=0,v_{2}=0,v_{3}=1,v_{4}=\frac{\rho\beta_{1}^{*}}{\gamma_{1}+\mu},v_{5}=\frac{\beta_{1}^{*}}{\gamma_{2}+\tau+\mu},v_{6}=0,v_{7}=0.$

Selecting the notations $S=x_{1}$, $L=x_{2}$, $E=x_{3}$, $A=x_{4}$, $I=x_{5}$,
$C=x_{6}$, $R=x_{7}$ and $\frac{dx_{i}}{dt}=f_{i}$. Now we calculate the
following second-order partial derivatives of $f_{i}$ at the disease-free
equilibrium $\varepsilon_{0}$ and obtain

$\displaystyle\frac{\partial^{2}f_{3}}{\partial x_{4}\partial
x_{3}}=-\frac{\rho\beta_{1}^{*}\mu\left(\mu+\omega+l\right)}{\Pi_{H}\left(\mu+\omega\right)}=\frac{\partial^{2}f_{2}}{\partial
x_{3}\partial x_{4}},$ $\displaystyle\frac{\partial^{2}f_{3}}{\partial
x_{5}\partial
x_{3}}=-\frac{\beta_{1}^{*}\mu\left(\mu+\omega+l\right)}{\Pi_{H}\left(\mu+\omega\right)}=\frac{\partial^{2}f_{3}}{\partial
x_{3}\partial x_{5}},$ $\displaystyle\frac{\partial^{2}f_{3}}{\partial
x_{4}^{2}}=-\frac{2\rho\beta_{1}^{*}\mu\left(\mu+\omega+l\right)}{\Pi_{H}\left(\mu+\omega\right)},$
$\displaystyle\frac{\partial^{2}f_{3}}{\partial
x_{5}^{2}}=-\frac{2\beta_{1}^{*}\mu\left(\mu+\omega+l\right)}{\Pi_{H}\left(\mu+\omega\right)},$
$\displaystyle\frac{\partial^{2}f_{3}}{\partial x_{4}\partial
x_{5}}=-\frac{\left(1+\rho\right)\beta_{1}^{*}\mu\left(\mu+\omega+l\right)}{\Pi_{H}\left(\mu+\omega\right)}=\frac{\partial^{2}f_{3}}{\partial
x_{5}\partial x_{4}},$ $\displaystyle\frac{\partial^{2}f_{3}}{\partial
x_{7}\partial
x_{4}}=-\frac{\rho\beta_{1}^{*}\mu\left(\mu+\omega+l\right)}{\Pi_{H}\left(\mu+\omega\right)}=\frac{\partial^{2}f_{3}}{\partial
x_{4}\partial x_{7}},$ $\displaystyle\frac{\partial^{2}f_{3}}{\partial
x_{7}\partial
x_{5}}=-\frac{\beta_{1}^{*}\mu\left(\mu+\omega+l\right)}{\Pi_{H}\left(\mu+\omega\right)}=\frac{\partial^{2}f_{3}}{\partial
x_{5}\partial x_{7}}.$

Now we calculate the coefficients $a$ and $b$ defined in Theorem 4.1
castillo2004dynamical  of Castillo-Chavez and Song as follow

$\displaystyle
a=\sum_{k,i,j=1}^{6}v_{k}w_{i}w_{j}\frac{\partial^{2}f_{k}(0,0)}{\partial
x_{i}\partial x_{j}}$

and

$\displaystyle
b=\sum_{k,i=1}^{6}v_{k}w_{i}\frac{\partial^{2}f_{k}(0,0)}{\partial
x_{i}\partial\beta}$

Replacing the values of all the second-order derivatives measured at DFE and
$\beta_{1}=\beta_{1}^{*}$, we get

$\displaystyle a$
$\displaystyle=-2\left(w_{3}+w_{4}+w_{5}+w_{7}\right)\left[\frac{\rho\beta_{1}^{*}\mu\left(\mu+\omega+l\right)}{\Pi_{H}\left(\mu+\omega\right)}w_{4}+\frac{\beta_{1}^{*}\mu\left(\mu+\omega+l\right)}{\Pi_{H}\left(\mu+\omega\right)}w_{5}\right]<0$

and

$\displaystyle b$ $\displaystyle=\rho w_{4}+w_{5}$
$\displaystyle=\frac{\left(1-\kappa\right)\sigma\rho}{\left(\mu+\gamma_{1}\right)}+\frac{\kappa\sigma}{\left(\mu+\gamma_{2}+\tau\right)}>0.$

Since $a<0$ and $b>0$ at $\beta=\beta_{1}^{*}$, therefore using the Remark 1
of the Theorem 4.1 stated in castillo2004dynamical , a transcritical
bifurcation occurs at $R_{0}=1$ and the unique endemic equilibrium of the
COVID-19 system (2) with lockdown is locally asymptotically stable for
$R_{0}>1$. ∎

## Figures

Figure S1: Ensemble model forecast for the daily notified COVID-19 cases in
Maharashtra during May 17, 2020 till May 31, 2020, under five different social
distancing measure. Various legends are Current Rate: daily notified case
projection using the estimated value of the lockdown rate (see Table 4 main
text), 15% Reduction: daily notified case projection using 15% reduction in
the estimated value of the lockdown rate (see Table 4 main text), 20%
Reduction: daily notified case projection using 20% reduction in the estimated
value of the lockdown rate (see Table 4 main text), 30% Reduction: daily
notified case projection using 30% reduction in the estimated value of the
lockdown rate (see Table 4 main text), and No lockdown: daily notified case
projection based on no lockdown scenario, respectively.  Figure S2: Ensemble
model forecast for the daily notified COVID-19 cases in Delhi during May 17,
2020 till May 31, 2020, under five different social distancing measure.
Various legends are same as Fig S1. Figure S3: Ensemble model forecast for the
daily notified COVID-19 cases in Tamil Nadu during May 17, 2020 till May 31,
2020, under five different social distancing measure. Various legends are same
as Fig S1. Figure S4: Ensemble model forecast for the daily notified COVID-19
cases in Gujarat during May 17, 2020 till May 31, 2020, under five different
social distancing measure. Various legends are same as Fig S1. Figure S5:
Ensemble model forecast for the daily notified COVID-19 cases in Punjab during
May 17, 2020 till May 31, 2020, under five different social distancing
measure. Various legends are same as Fig S1. Figure S6: Posterior density of
the weights for the mechanistic mathematical model (2 & 2) and the best
statistical forecast model (HYBRID), respectively for Maharashtra. Figure S7:
Posterior density of the weights for the mechanistic mathematical model (2 &
2) and the best statistical forecast model (ARIMA), respectively for Delhi
Figure S8: Posterior density of the weights for the mechanistic mathematical
model (2 & 2) and the best statistical forecast model (HYBRID), respectively
for Tamil Nadu. Figure S9: Posterior density of the weights for the
mechanistic mathematical model (2 & 2) and the best statistical forecast model
(ARIMA), respectively for Gujarat. Figure S10: Posterior density of the
weights for the mechanistic mathematical model (2 & 2) and the best
statistical forecast model (TBATS), respectively for Punjab. Figure S11:
Posterior density of the weights for the mechanistic mathematical model (2 &
2) and the best statistical forecast model (HYBRID), respectively for India.

## Tables

Table S1: Estimated initial state variables of the mathematical model (2). All data are given in the format Estimate (95% CI). Location |  $\boldsymbol{S(0)}$ |  $\boldsymbol{E(0)}$ |  $\boldsymbol{A(0)}$ |  $\boldsymbol{I(0)}$
---|---|---|---|---
Maharashtra |  $\begin{subarray}{c}123243118\\\ \\\ (114977907-124511349)\end{subarray}$ |  $\begin{subarray}{c}27.27\\\ \\\ (0.51-35.06)\end{subarray}$ |  $\begin{subarray}{c}9531\\\ \\\ (9483-9989)\end{subarray}$ |  $\begin{subarray}{c}24.76\\\ \\\ (14.17-30.26)\end{subarray}$
Delhi |  $\begin{subarray}{c}16070717\\\ \\\ (10346593-19737264)\end{subarray}$ |  $\begin{subarray}{c}14.74\\\ \\\ (0.05-14.97)\end{subarray}$ |  $\begin{subarray}{c}437\\\ \\\ (92.38-4017)\end{subarray}$ |  $\begin{subarray}{c}282\\\ \\\ (7.82-325)\end{subarray}$
Tamil Nadu |  $\begin{subarray}{c}75367306\\\ \\\ (70335585-79738740)\end{subarray}$ |  $\begin{subarray}{c}4.80\\\ \\\ (0.05-4.73)\end{subarray}$ |  $\begin{subarray}{c}0.40\\\ \\\ (0.03-2.52)\end{subarray}$ |  $\begin{subarray}{c}7.61\\\ \\\ (1.96-10.88)\end{subarray}$
Gujarat |  $\begin{subarray}{c}66848292\\\ \\\ (60218205-69744037)\end{subarray}$ |  $\begin{subarray}{c}7630\\\ \\\ (2622-9775)\end{subarray}$ |  $\begin{subarray}{c}26.10\\\ \\\ (1.26-28.70)\end{subarray}$ |  $\begin{subarray}{c}14.01\\\ \\\ (0.02-20.54)\end{subarray}$
Punjab |  $\begin{subarray}{c}26433551\\\ \\\ (26433543-26433556)\end{subarray}$ |  $\begin{subarray}{c}7.83\\\ \\\ (0.20-13.81)\end{subarray}$ |  $\begin{subarray}{c}19.26\\\ \\\ (0.29-29.54)\end{subarray}$ |  $\begin{subarray}{c}8\\\ \\\ (1.08-15.96)\end{subarray}$
India |  $\begin{subarray}{c}1226841787\\\ \\\ (1219848614-1297276832)\end{subarray}$ |  $\begin{subarray}{c}221104\\\ \\\ (21136-279521)\end{subarray}$ |  $\begin{subarray}{c}462997\\\ \\\ (27541-642484)\end{subarray}$ |  $\begin{subarray}{c}36043\\\ \\\ (3601-84059)\end{subarray}$
Table S2: Goodness of fit (RMSE) for the three statistical forecast model (ARIMA, TBATS and HYBRID), respectively. RMSE for different locations are calculated only for the test period data (May 4, 2020 till May 8, 2020). RMSE values of the best performed statistical forecast model in different locations are shown in red. Location |  ARIMA |  TBATS |  HYBRID
---|---|---|---
Maharashtra |  179.84368 |  110.57237 |  90.04999
Delhi |  92.17782 |  96.11452 |  94.14085
Tamil Nadu |  182.5897 |  244.8254 |  119.9740
Gujarat |  28.16381 |  37.36576 |  30.92572
Punjab |  292.5078 |  188.9780 |  238.9491
India |  457.7622 |  438.1160 |  368.0725
Table S3: Weight estimates for the mechanistic mathematical model (2 & 2) and the best statistical forecast model, respectively. Respective subscript are MH: Maharashtra, DL: Delhi, TN: Tamil Nadu, GJ: Gujarat, PJ: Punjab, and IND: India. $\boldsymbol{w_{1}}$ and $\boldsymbol{w_{2}}$ denote the weights of the COVID-19 mathematical model (2 & 2) and the best statistical forecast model, respectively for a region. All data are provided in the format Estimate (95% CI). Weights | MH | DL | TN | GJ | PJ | IND
---|---|---|---|---|---|---
$\boldsymbol{w_{1}}$ | $\begin{subarray}{c}0.48\\\ \\\ (0.2243-0.8262)\end{subarray}$ | $\begin{subarray}{c}0.5186\\\ \\\ (0.1112-0.8004)\end{subarray}$ | $\begin{subarray}{c}0.6158\\\ \\\ (0.1801-0.5848)\end{subarray}$ | $\begin{subarray}{c}0.2009\\\ \\\ (0.1554-0.5737)\end{subarray}$ | $\begin{subarray}{c}0.8331\\\ \\\ (0.6291-0.8942)\end{subarray}$ | $\begin{subarray}{c}0.3131\\\ \\\ (0.1695-0.6149)\end{subarray}$
$\boldsymbol{w_{2}}$ | $\begin{subarray}{c}0.52\\\ \\\ (0.1738-0.7757)\end{subarray}$ | $\begin{subarray}{c}0.4814\\\ \\\ (0.1996-0.8888)\end{subarray}$ | $\begin{subarray}{c}0.3842\\\ \\\ (0.4152-0.8199)\end{subarray}$ | $\begin{subarray}{c}0.7991\\\ \\\ (0.4263-0.8446)\end{subarray}$ | $\begin{subarray}{c}0.1669\\\ \\\ (0.1058-0.3709)\end{subarray}$ | $\begin{subarray}{c}0.6869\\\ \\\ (0.3851-0.8305)\end{subarray}$