# Forecasting the daily and cumulative number of cases for the COVID-19
pandemic in India

Subhas Khajanchi<EMAIL_ADDRESS>Department of Mathematics,
Presidency University, 86/1 College Street, Kolkata 700073, India.  Kankan
Sarkar<EMAIL_ADDRESS>Department of Mathematics, Malda College, Malda,
West Bengal 732101, India

###### Abstract

The ongoing novel coronavirus epidemic has been announced a pandemic by the
World Health Organization on March 11, 2020, and the Govt. of India has
declared a nationwide lockdown from March 25, 2020, to prevent community
transmission of COVID-19. Due to absence of specific antivirals or vaccine,
mathematical modeling play an important role to better understand the disease
dynamics and designing strategies to control rapidly spreading infectious
diseases. In our study, we developed a new compartmental model that explains
the transmission dynamics of COVID-19. We calibrated our proposed model with
daily COVID-19 data for the four Indian provinces, namely Jharkhand, Gujarat,
Andhra Pradesh, and Chandigarh. We study the qualitative properties of the
model including feasible equilibria and their stability with respect to the
basic reproduction number $\mathcal{R}_{0}$. The disease-free equilibrium
becomes stable and the endemic equilibrium becomes unstable when the recovery
rate of infected individuals increased but if the disease transmission rate
remains higher then the endemic equilibrium always remain stable. For the
estimated model parameters, $\mathcal{R}_{0}>1$ for all the four provinces,
which suggests the significant outbreak of COVID-19. Short-time prediction
shows the increasing trend of daily and cumulative cases of COVID-19 for the
four provinces of India.

###### pacs:

05.45.-a, 03.50.Kk, 03.65.-w

††preprint: AIP/123-QED

In India, 173,763 confirmed cases, 7,964 confirmed new cases and 4,971
confirmed deaths due to COVID-19 were reported as of May 30, 2020. As the
ongoing COVID-19 outbreak is quickly spreading throughout India and globe,
short-term modeling predictions give time-critical statistics for decisions on
containment and mitigation policies. A big problem for short-term prediction
is the evaluation of important parameters and how they alter when the first
interventions reveal an effect. In the absence of any therapeutics or licensed
vaccine, antivirals, isolation of populations diagnosed with COVID-19 and
quarantine of populations feared exposed to COVID-19 were used to control the
rapid spread of infection. During this alarming situation, forecasting is of
utmost priority for health care planning and to control the SARS-CoV-2 virus
with limited resources. We proposed a mathematical model that monitors the
dynamics of six compartments, namely susceptible (S), asymptomatic (A),
reported symptomatic (I), unreported symptomatic (U), quarantine (Q), and
recovered (R) individuals, collectively termed SAIUQR, that predicts the
course of the epidemic. Our SAIUQR model discriminates between reported and
unreported infected individuals, which is important as the former are
typically isolated and hence less likely to spread the infection. A detailed
theoretical analysis has been done for our SAIUQR model in terms of the basic
reproduction number $\mathcal{R}_{0}$. All the analytical findings are
verified numerically for the estimated model parameters. We have calibrated
our SAIUQR model with real observed data on the COVID-19 outbreak in the four
provinces of India. The basic reproduction number for all the provinces is
greater than unity, which resulted in a substantial outbreak of COVID-19.
Based on the simulation, our SAIUQR model predict that on June 13, 2020, the
daily new COVID-19 cases will be around 15, 454, 12, 96, and cumulative number
of COVID-19 cases will be around 661, 23955, 514, 4487, in Jharkhand, Gujarat,
Chandigarh and Andhra Pradesh, respectively.

##  I Introduction

After a novel strain of COVID-19, was detected in Wuhan, the city of Hubei
province, China, in December 2019 Wu20 , an exponentially increasing number of
patients in mainland China were identified with SARS-CoV-2, immediately the
Chinese Health authorities to initiate radical measures to control the
epidemic coronavirus. In spite of these radical measures, a SARS-CoV-2
coronavirus pandemic ensued in the subsequent months and China became the
epicenter. SARS-CoV-2 viruses are enveloped non-segmented positive-sense RNA
viruses that belong to the Coronaviridae family and the order Nidovirales, and
are extensively disseminated among humans as well as mammals Chen20 . The
COVID-19 is responsible for a range of symptoms together with fever, dry
cough, breathing difficulties, fatigue and lung infiltration in severe cases,
similar to those created by SARS-CoV (severe acute respiratory syndrome
coronavirus) and MERS-CoV (Middle East respiratory syndrome coronavirus)
infections Huang20 . SARS-CoV-2 has already crossed the earlier history of two
coronavirus epidemics SARS-CoV and MERS-CoV, posing the substantial threat to
the world people health problem as well as economical problem after the second
world war bbc1 . According to the World Health Organization report, dated 29
May 2020 reported 5,596,550 total cases and 353,373 deaths worldwide WHO129 .

Till date, there is no licensed vaccine, drugs and effective therapeutics
available for SARS-CoV-2 or COVID-19. Due to absence of pharmaceutical
interventions, Govt. of various countries are adopting different strategies to
control the outbreak and the most common one is the nation-wide lockdown. It
was started with the local Govt. of Wuhan by temporarily prevention of all
public traffics within the city on January 23, 2020 and soon followed by other
cities in Hubei province Lina20 . In the absence of drug or specific
antivirals for SARS-CoV-2 viruses, maintaining social distancing is the only
way to mitigate the human-to-human transmission for coronavirus diseases, and
thus the other countries also incorporated the strict lockdown, quarantines
and curfews.

In India, the first coronavirus case was reported in Kerala’s Thrissur
district on January 30, 2020 when a student returned back from Wuhan, the
sprawling capital of China’s Hubei province IndCov . The Govt. of India has
implemented a complete nationwide lockdown throughout the country on and from
March 25, 2020 for 21 days, following one day ‘Janata Curfew’ on March 22,
2020 to control the coronavirus or SARS-CoV-2 pandemic in India Pulla20 . Due
to massive spread of coronavirus diseases, the Govt. of India has extended the
lockdown and it is going on Phase 4; from May 18, 2020 to May 31, 2020.
Besides the implementation of nationwide lockdown, the Ministry of Health and
Family Welfare (MOHFW) of India, recommended different individual hygiene
measures, for example, frequent hand washing, social distancing, use of mask,
avoid gathering and touching eyes, mouth and nose etc. mohfw .

The Govt. also ceaselessly using different media and social network to aware
the public regarding coronavirus diseases and its precautions. Albeit, the
factors such as diverse and huge population, the unavailability of specific
therapeutics, drugs or licensed vaccines, inadequate evidences regarding the
mechanism of disease transmission make it strenuous to combat against the
coronavirus diseases throughout India. To control the transmission of
COVID-19, lockdown is a magnificent measure but testing is also an important
factor to identify the symptomatic and asymptomatic individuals. The
symptomatic individuals should be reported by the public health agencies to
separate them from the uninfected or asymptomatic individuals for their ICU
(Intensive Care Unit) treatment. Also, from an economic viewpoint, the strict
lockdown may be the cause of a substantial financial crisis in near future. In
particular, the lockdown in high dense countries can mitigate the disease
transmission rate, although entirely control not be obtainable. Thus, to
survive the economical status of a country, a strict lockdown for a long
period is not advisable at all in any situations. Hence there should be a
acceptable balance between the two different characteristics of governmental
strategies: strict lockdown and healthy economical situations. Albeit, few
questions remains to answer whether this cluster containment policy can be
effective in mitigating SARS-CoV-2 transmission or not ? If not then what can
be the possible solutions to mitigate the transmission of SARS-CoV-2 viruses ?
These questions can only be replied by investigating the dynamics and
forecasting of a mechanistic compartmental model for SARS-CoV-2 transmission
and comparing the outcomes with real scenarios.

A plenty of mathematical models has been investigated to study the
transmission dynamics and forecasting of COVID-19 outbreak Kucharski20 ;
Tang20 ; WuJT20 ; Fanelli20 ; Ribeiro20 ; Chakraborty20 ; KhCov20 ; Sarkar20 ;
He20 ; Anastassopoulou ; Giordano20 . Kucharski and colleagues Kucharski20
performed a model-based analysis for SARS-CoV-2 viruses and calculate the
reproduction number $\mathcal{R}_{0}=2.35$, where the authors have taken into
account all the positive cases of Wuhan, China, till March 05, 2020. Wu and
colleagues WuJT20  studied a susceptible-exposed-infectious-recovered (SEIR)
model to simulate the epidemic in Wuhan city and compute the basic
reproduction number $\mathcal{R}_{0}=2.68$ and predict their model based on
the data recorded from December 31, 2019 to January 28, 2020. Tang and
colleagues Tang20  developed a compartmental model to study the transmission
dynamics of COVID-19 and calculate the basic reproduction number
$\mathcal{R}_{0}=6.47$, which is very high for the infectious diseases.
Recently, Fanelli & Piazza Fanelli20  analyzed and predicted the
characteristics of SARS-CoV-2 viruses in the three mostly affected countries
till March 2020 with an aid of the mathematical modeling. Stochastic based
regression model also been studied by Ribeiro and Colleagues Ribeiro20  to
predict the scenarios of the most affected states of Brazil. Chakraborty &
Ghosh Chakraborty20  investigate a hybrid ARIMA-WBF model to predict the
various SARS-CoV-2 affected countries throughout the world. Khajanchi and
Colleagues KhCov20  developed a compartmental model to forecast and control of
the outbreak of COVID-19 in the four states of India and the overall India.
Sarkar & Khajanchi Sarkar20  developed a mathematical model to study the model
dynamics and forecast the SARS-CoV-2 viruses in seventeenth provinces of India
and the overall India. A discrete-time SIR model introducing dead compartment
system studied by Anastassopoulou et al. Anastassopoulou  to portray the
dynamics of SARS-CoV-2 outbreak. Giordano and colleagues Giordano20
established a new mathematical model for COVID-19 pandemic and predict that
restrictive social distancing can mitigate the widespread of COVID-19 among
the human. A couple of seminal papers has been investigated to study the
transmission dynamics of COVID-19 or SARS-CoV-2 viruses for different
countries, including Mexico city, Chicago and Wuhan, the sprawling capital of
Central China’s Hubei province Xiao-Lin20 ; Ndairou20 ; Mena20 ; Wong20 .
Short-term prediction is too important as it gives time-critical information
for decisions on containment and mitigation strategies KhCov20 ; Dehning20 . A
major problem for short-term predictions is the evaluation of important
epidemiological parameters and how they alter when first interventions reveal
an effect.

The main objective of this work is to develop a new mathematical model that
describes the transmission dynamics and forecasting of COVID-19 or SARS-CoV-2
pandemic in the four different provinces of India, namely Jharkahnd, Andhra
Pradesh, Chandigarh, and Gujarat. We estimated the model parameters of four
different states of India and fitted our compartmental model to the daily
confirmed cases and cumulative confirmed cases reported between March 15, 2020
to May 24, 2020. We compute the basic reproduction number $\mathcal{R}_{0}$
for the four different states based on the estimated parameter values. We also
perform the short-term predictions of the four different states of India from
May 25, 2020 to June 13, 2020, and it shows the increasing trends of COVID-19
pandemic in four different provinces of India.

The remaining part of this manuscript has been organized in the following way.
In the Section 2, we describe the formulation of the compartmental model for
COVID-19 and its basic assumptions. Section 3 describes the theoretical
analysis of the model, which incorporates the positivity and boundedness of
the system, computation of the basic reproduction number $\mathcal{R}_{0}$,
existence of the biologically feasible singular points and their local
stability analysis. In the same section, we perform the global stability
analysis for the infection free equilibrium point $E^{0}$ and the existence of
transcritical bifurcation at the threshold $\mathcal{R}_{0}=1.$ In the Section
4, we conduct some model simulations to validate our analytical findings by
using the estimated model parameters for Jharkhand, the state of India. The
parameters are estimated for the real world example on COVID-19 for four
different states of India and perform a short-term prediction based on the
estimated parameter values. A discussion in the Section 5 concludes the
manuscript.

##  II Mathematical model

A compartmental mathematical model has been developed to study the
transmission dynamics of COVID-19 outbreak in India and throughout the world.
We adopt a variant that focuses some important epidemiological properties of
COVID-19 or SARS-CoV-2 coronavirus diseases. Based on the health status, we
stratify the total human population into six compartments, namely susceptible
or uninfected $(S)$, asymptomatic or pauci-symptomatic infected $(A)$,
symptomatic reported infected $(I)$, unreported infected $(U)$, quarantine
$(Q)$, and recovered $(R)$ individuals, collectively termed SAIUQR. At any
instant of time, the total population is denoted by $N=S+A+I+U+Q+R.$ Depending
on the six state variables, we aim to develop an autonomous system using first
order nonlinear ordinary differential equations.

In the model formulation, quarantine refers to the separation of coronavirus
infected population from the general population when the individuals are
infected but clinical symptoms has not yet developed, whereas isolation refers
to the separation of coronavirus infected population when the population
already identified the clinical symptoms. Our mathematical model introduces
some demographic effects by assuming a proportional natural mortality rate
$\delta>0$ in each of the six compartments. In addition, our model
incorporates a constant recruitment of susceptible populations into the region
at the rate $\Lambda_{s}$ per unit time. This parameter represents new birth,
immigration and emigration. The parameter $\beta_{s}$ represents the
probability of disease transmission rate. However the disease transmission
from vulnerable to infected individuals (for our model, the class is A) depend
on various factors, namely safeguard precautions (use of mask, social
distancing, etc.) and hygienic safeguard (use of hand sanitizer) taken by the
susceptible individuals as well as infected population. In our model
formulation, we incorporate the asymptomatic or pauci-symptomatic infected
(undetected) individuals, which is important to better understand the
transmission dynamics of COVID-19, which also studied by Giordano et al.
Giordano20  and Xiao-Lin et al. Xiao-Lin20 .

In our model formulation, we assumed that the COVID-19 virus is spreading when
a vulnerable person come into contact with an asymptomatic infected
individuals. The uninfected individuals decreases after infection, obtained
through interplays among a susceptible population and an infected individuals
who may be asymptomatic, reported symptomatic and unreported symptomatic. For
these three compartments of infected population, the transmission coefficients
are $\beta_{s}\alpha_{a}$, $\beta_{s}\alpha_{i}$, and $\beta_{s}\alpha_{u}$
respectively. We consider $\beta_{s}$ as the disease transmission rate along
with the adjustment factors for asymptomatic $\alpha_{a}$, reported
symptomatic $\alpha_{i}$ and unreported symptomatic $\alpha_{u}$ individuals.
The interplays among infected populations (asymptomatic, reported symptomatic,
and unreported symptomatic) and susceptible individuals can be modeled in the
form of total individuals using standard mixing incidence Anderson91 ;
Diekmann00 ; Hethcote00 ; Gumel04 .

The quarantined population can either move to the susceptible or infected
compartment (reported and unreported), depending on whether they are infected
or not Keeling08 , with a portion $\rho_{s}$. Here, $\gamma_{q}$ is the rate
at which the quarantined uninfected contacts were released into the wider
community. Asymptomatic individuals were exposed to the virus but clinical
symptoms of SARS-CoV-2 viruses has not yet developed. The asymptomatic
individuals decreases due to contact with reported and unreported symptomatic
individuals at the rate $\gamma_{a}$ with a portion $\theta\in(0,1)$, and
become quarantine at the rate $\xi_{a}$. Also, the asymptomatic individuals
become recovered at the rate $\eta_{a}$ and has a natural mortality rate
$\delta$. A fraction of quarantine individuals become reported infected
individuals at the rate $\gamma_{q}$ with a portion $\rho_{s}$ (where
$\rho_{s}\in(0,1))$.

As we know that an individual is whether infected by coronavirus diseases or
not can be identified by RT-PCR screening test and a person with negative
results with the RT-PCR screening test may yet be coronavirus positive as it
may take around 7-21 days to express the coronavirus symptoms Lan20 . Thus, a
fraction of coronavirus positive class can be considered as reported
symptomatic individuals $(\theta)$ and unreported symptomatic individuals
$(1-\theta)$. The reported symptomatic individuals separated from the general
populations and move to the isolated class or hospitalized class for clinical
treatment.

Also, it can be noticed that once an individual recovered from the SARS-CoV-2
diseases has a very little chance to become infected again for the same
disease Lan20 . Therefore, we assume that none of the recovered individuals
move to the susceptible or uninfected class again. In our mathematical model
formulation, we assume that the reported infected individuals $(I)$ are unable
to spread or transmit the viruses as they are kept completely isolated from
the susceptible or uninfected individuals. As the reported infected
individuals are moved to the hospital or Intensive Care Unit (ICU) Liu20 . For
our modeling perspective, we are mainly interested in predictions over a
relatively short time window within which the temporary immunity is likely
still to be in placed, and the possibility of reinfection would negligibly
affect the total number of uninfected populations and so there would be no
considerable difference in the evolution of the epidemic curves we consider.
Social mixing patterns are introduced into our contagion parameters in an
average fashion over the entire individuals, irrespective of age. Based on
these biological assumptions, we develop the following mathematical model
using a system of nonlinear ordinary differential equations to study the
outbreak of COVID-19 or SARS-CoV-2 coronavirus diseases:

$\displaystyle\left\\{\begin{array}[]{ll}S^{\prime}(t)=\Lambda_{s}-\beta_{s}S\bigg{(}\alpha_{a}\frac{A}{N}+\alpha_{i}\frac{I}{N}+\alpha_{u}\frac{U}{N}\bigg{)}+\rho_{s}\gamma_{q}Q-\delta
S,\\\
A^{\prime}(t)=\beta_{s}S\bigg{(}\alpha_{a}\frac{A}{N}+\alpha_{i}\frac{I}{N}+\alpha_{u}\frac{U}{N}\bigg{)}-(\xi_{a}+\gamma_{a})A-\eta_{a}A-\delta
A,\\\ I^{\prime}(t)=\theta\gamma_{a}A+(1-\rho_{s})\gamma_{q}Q-\eta_{i}I-\delta
I,\\\ U^{\prime}(t)=(1-\theta)\gamma_{a}A-\eta_{u}U-\delta U,\\\
Q^{\prime}(t)=\xi_{a}A-\gamma_{q}Q-\delta Q,\\\
R^{\prime}(t)=\eta_{u}U+\eta_{i}I+\eta_{a}A-\delta R,\end{array}\right.$ (7)

the model is supplemented by the following initial values:

$\displaystyle S(t_{0})$ $\displaystyle=$ $\displaystyle S_{0}\geq
0,~{}A(t_{0})=A_{0}\geq 0,~{}Q(t_{0})=Q_{0}\geq 0,$ $\displaystyle I(t_{0})$
$\displaystyle=$ $\displaystyle I_{0}\geq 0,~{}U(t_{0})=U_{0}\geq
0,~{}R(t_{0})=R_{0}\geq 0.$ (8)

In our model $t\geq t_{0}$ is the time in days, $t_{0}$ represents the
starting date of the outbreak for our system (7). The transmission dynamics of
the COVID-19 is illustrated in the Figure 1. The description of the model
parameters are presented in the Table 1.

Figure 1:  A schematic representation of the mechanistic SAIUQR model for the
transmission dynamics of COVID-19 or SARS-CoV-2. The interaction among
different stages of individuals is shown in the graphical scheme: S,
susceptible or uninfected population; A, asymptomatic infected population; I,
COVID-19 reported symptomatic infected individuals; U, COVID-19 unreported
symptomatic infected individuals; Q, quarantine individuals; and R, COVID-19
recovered individuals. Biological interpretations of the model parameters are
given in the Table 1.

##  III SAIUQR model analysis

In this section, we provide the basic properties of the SAIUQR model (7),
including positivity and boundedness of the solutions, basic reproduction
number and the biologically feasible singular points and their stability
analysis, subject to the non-negative initial values
$(S_{0},A_{0},Q_{0},I_{0},U_{0},Q_{0},R_{0})\in\mathbf{R}^{6}.$

###### Theorem III.1

The solutions of the SAIUQR system (7) with the initial values (II) are
defined on $\mathbf{R}_{+}^{6}$ remain positive for all $t>0$.

###### Proof III.1

The proof of this theorem is given in the Appendix A.

###### Theorem III.2

The solutions of the SAIUQR system (7) with the initial conditions (II) are
uniformly bounded in the region $\Omega$.

###### Proof III.2

The proof of this theorem is given in the Appendix B.

###  III.1 Basic reproduction number

In any infectious disease modeling, the basic reproduction number is the key
epidemiological parameter for describing the characteristics of the diseases.
The basic reproduction number symbolized by $\mathcal{R}_{0}$ and is defined
as “the number of secondary infected individuals caused by a single infected
individuals in the entire susceptible individuals” Diekmann90 . The
dimensionless basic reproduction number $\mathcal{R}_{0}$ quantifies the
expectation of the disease die out or the spreading of the diseases. For,
$\mathcal{R}_{0}<1$ describes on an average an infected population spreads
less than a new infective during the course of its infection period, and thus
the diseases can die out. For, $\mathcal{R}_{0}>1$ describes each infected
individuals spread on an average more than 1 new infection, and the disease
can spread throughout the population. Various techniques can be used to
compute the basic reproduction number $\mathcal{R}_{0}$ for an epidemic
outbreak. In our present study, we use the next generation matrix to evaluate
$\mathcal{R}_{0}$ Diekmann90 . In our compartmental model, the following
classes are explicitly related to the outbreak of the novel coronavirus
diseases: $A$, $I$, $U$, and $Q$. Thus, from the SAIUQR model system (7), we
get the matrices $\mathcal{F}$ for the new infection and $\mathcal{V}$ for the
transition terms are given by, respectively

$\displaystyle\mathcal{F}$ $\displaystyle=$
$\displaystyle\left[\begin{array}[]{c}\beta_{s}S\bigg{(}\alpha_{a}\frac{A}{N}+\alpha_{i}\frac{I}{N}+\alpha_{u}\frac{U}{N}\bigg{)}\\\
0\\\ 0\\\ 0\end{array}\right],$ $\displaystyle\mathcal{V}$ $\displaystyle=$
$\displaystyle\left[\begin{array}[]{c}(\xi_{a}+\gamma_{a}+\eta_{a}+\delta)A\\\
-\theta\gamma_{a}A-(1-\rho_{s})\gamma_{q}Q+(\eta_{i}+\delta)I\\\
-(1-\theta)\gamma_{a}A+(\eta_{u}+\delta)U\\\
-\xi_{a}A+(\gamma_{q}+\delta)Q\end{array}\right].$

The variational matrix for the SAIUQR system (7), can be evaluated at an
infection-free singular point
$E^{0}(S^{0},A^{0},Q^{0},I^{0},U^{0},R^{0})=(\frac{\Lambda_{s}}{\delta},0,0,0,0,0)$,
we have

$\displaystyle F$ $\displaystyle=$
$\displaystyle\left[\begin{array}[]{cccc}\beta_{s}\alpha_{a}\frac{S}{N}&\beta_{s}\alpha_{i}\frac{S}{N}&\beta_{s}\alpha_{u}\frac{S}{N}&0\\\
0&0&0&0\\\ 0&0&0&0\\\ 0&0&0&0\end{array}\right],$ $\displaystyle V$
$\displaystyle=$
$\displaystyle\left[\begin{array}[]{cccc}\xi_{a}+\gamma_{a}+\eta_{a}+\delta&0&0&0\\\
-\theta\gamma_{a}&(\eta_{i}+\delta)&0&-(1-\rho_{s})\gamma_{q}\\\
-(1-\theta)\gamma_{a}&0&(\eta_{u}+\delta)&0\\\
-\xi_{a}&0&0&\gamma_{q}+\delta\end{array}\right].$

The basic reproduction number $\mathcal{R}_{0}=\rho(FV^{-1})$, where
$\rho(FV^{-1})$ represents the spectral radius for a next generation matrix
$FV^{-1}$. Therefore, from the SAIUQR model (7), we get the basic reproduction
number $\mathcal{R}_{0}$ as

$\displaystyle\rho(FV^{-1})$ $\displaystyle=$
$\displaystyle\mathcal{R}_{0}~{}=~{}\frac{\beta_{s}\alpha_{a}}{\xi_{a}+\gamma_{a}+\eta_{a}+\delta}$
$\displaystyle+$
$\displaystyle~{}~{}\frac{(1-\theta)\beta_{s}\alpha_{u}\gamma_{a}}{(\eta_{u}+\delta)(\xi_{a}+\gamma_{a}+\eta_{u}+\delta)}$
$\displaystyle+$
$\displaystyle\frac{\beta_{s}\alpha_{i}(\theta\gamma_{a}(\gamma_{q}+\delta)+(1-\rho_{s})\xi_{a}\gamma_{q})}{(\xi_{a}+\gamma_{a}+\eta_{a}+\delta)(\gamma_{q}+\delta)(\eta_{i}+\delta)}.$

###  III.2 Equilibria

The SAIUQR model (7) has two biologically feasible equilibrium points, namely
(i) infection free steady state
$E^{0}(S^{0},A^{0},Q^{0},I^{0},U^{0},R^{0})=(\frac{\Lambda_{s}}{\delta},0,0,0,0,0)$,
and
(ii) the endemic equilibrium point
$E^{*}(S^{*},A^{*},Q^{*},I^{*},U^{*},R^{*})$, where
$S^{*}=\frac{\Lambda_{s}}{\delta}-\hat{S}A^{*},~{}I^{*}=\hat{I}A^{*},~{}U^{*}=\hat{U}A^{*},~{}Q^{*}=\hat{Q}A^{*}$
and $R^{*}=\hat{R}A^{*}$. The expression of $A^{*}$ is given by
$A^{*}=\frac{\Lambda_{s}(\mathcal{R}_{0}-1)}{\delta\left\\{\hat{S}(\mathcal{R}_{0}-1)+1+\hat{I}+\hat{U}+\hat{Q}+\hat{R}\right\\}}$,
where
$\hat{S}=\frac{1}{\delta}\left[\xi_{a}+\gamma_{a}+\eta_{a}+\delta-\frac{\xi_{a}\rho_{s}\gamma_{q}}{\gamma_{q}+\delta}\right]$,

$\hat{I}=\frac{\theta\gamma_{a}(\gamma_{q}+\delta)+(1-\rho_{s})\xi_{a}\gamma_{q}}{(\eta_{i}+\delta)(\gamma_{q}+\delta)}$,
$\hat{U}=\frac{(1-\theta)\gamma_{a}}{\eta_{u}+\delta}$,
$\hat{Q}=\frac{\xi_{a}}{\gamma_{q}+\delta}$ and
$\hat{R}=\frac{1}{\delta}\bigg{[}\frac{(1-\theta)\gamma_{a}\eta_{u}}{\eta_{u}+\delta}+\frac{\eta_{i}\left\\{\theta\gamma_{q}(\gamma_{q}+\delta)+(1-\rho_{s})\gamma_{q}\xi_{a}\right\\}}{(\gamma_{q}+\delta)(\eta_{i}+\delta)}+\eta_{a}\bigg{]}$.

It can be observed the the infection free singular point
$E^{0}(S^{0},A^{0},Q^{0},I^{0},U^{0},R^{0})$ is always feasible and the
endemic equilibrium point $E^{*}(S^{*},A^{*},Q^{*},I^{*},U^{*},R^{*})$ is
feasible if the following condition holds:

$\displaystyle(i)$ $\displaystyle\mathcal{R}_{0}~{}>~{}1,$ $\displaystyle(ii)$
$\displaystyle\frac{1}{\xi_{a}+\gamma_{a}+\eta_{a}+\delta}\bigg{[}\frac{\Lambda_{s}}{A^{*}}+\frac{\xi_{a}\rho_{s}\gamma_{q}}{\gamma_{q}+\delta}\bigg{]}~{}>~{}1.$

###  III.3 Stability analysis

In the present subsection, we investigate the linear stability analysis for
the SAIUQR model (7) for the two feasible steady states. By using the
techniques of linearization, we investigate the local dynamics of the
complicated system of the coronavirus compartmental model. Generally, we
linearize the SAIUQR model around each of the feasible steady states and
perturb the compartmental model by a very small amount, and observe whether
the compartmental model returns to that steady states or converges to any
other steady states or attractor. The local stability analysis aids in
understanding the qualitative behavior of the complex nonlinear dynamical
system. By using the following theorem, we prove the local stability of the
infection free singular point $E^{0}(S^{0},A^{0},Q^{0},I^{0},U^{0},R^{0})$:

###### Theorem III.3

The infection free steady state $E^{0}$ is locally asymptotic stable if
$\mathcal{R}_{0}<1$ and unstable if $\mathcal{R}_{0}>1$.

###### Proof III.3

The proof of this theorem is given in Appendix C.

###### Theorem III.4

The infection free steady state $E^{0}$ is globally asymptotic stable for
$\mathcal{R}_{0}<1$ in the bounded region $\Omega$.

###### Proof III.4

The proof of this theorem is given in Appendix D.

###### Theorem III.5

The SAIUQR model system (7) admits a locally asymptotic stable around the
endemic equilibrium point $E^{*}$ for $\mathcal{R}_{0}>1$. Also, the system
(7) experiences forward bifurcation at $\mathcal{R}_{0}=1$.

###### Proof III.5

The proof of this theorem is given in Appendix E.

Table 1: Table of the biologically relevant parameter values and their description for the SAIUQR model system (7). Symbol |  Biological interpretations | Values & Source
---|---|---
$\Lambda_{s}$ |  birth rate of the susceptible individuals | Table 3
$\beta_{s}$ |  probability of the disease transmission coefficient | Estimated
$\alpha_{a}$ |  modification factor for asymptomatic infected individuals | Estimated
$\alpha_{i}$ |  modification factor for symptomatic infected individuals | Estimated
$\alpha_{u}$ |  modification factor for unreported infected individuals | Estimated
$\rho_{s}$ |  fraction of quarantine individuals that become susceptible individuals | 0.5 (0, 1) Fixed
$\gamma_{q}$ |  rate at which the quarantined individuals becomes susceptible individuals | Estimated
$\delta$ |  natural death rate of entire individuals | 0.1945 $\times 10^{-4}$ KhCov20
$\xi_{a}$ |  rate at which the asymptomatic individuals become quarantined | 0.07151 Sarkar20
$\gamma_{a}$ |  rate of transition from the asymptomatic individuals to infected individuals | Estimated
$\eta_{a}$ |  average recovery rate of asymptomatic individuals |  $\frac{1}{7.48}$ Sarkar20
$\theta$ |  fraction of asymptomatic individuals that become reported infected individuals | 0.8 (0, 1) Fixed
$\eta_{i}$ |  average recovery rate of reported symptomatic infected individuals |  $\frac{1}{7}$ Liu20
$\eta_{u}$ |  average recovery rate of unreported symptomatic infected individuals |  $\frac{1}{7}$ Liu20
Figure 2: Model estimation based on the observed data. Model simulations
fitted with the daily new cases and the cumulative confirmed cases of COVID-19
for four provinces of India, namely Jharkhand, Gujarat, Chandigarh, and Andhra
Pradesh. Observed data points are shown in the red circle while the blue curve
represents the best fitting curve for the SAIUQR model. The first column
represents the daily new cases and the second column represents the cumulative
confirmed cases of COVID-19. The estimated parameter values are listed in the
Table 2 and other parameter values are listed in the Table 1. The initial
values used for this parameter values are presented in the Table 3. The RMSE
and the value of $R_{0}$ for each provinces are mentioned in the inset.

##  IV Numerical simulation

In this section, we conduct some numerical illustrations to validate our
analytical findings. Analytically, we perform the local stability analysis for
infection free steady state $E^{0}$ and a unique endemic equilibrium point
$E^{*}$. We also perform the transcritical bifurcation at the threshold
$\mathcal{R}_{0}=1$ and the global stability analysis for disease-free steady
state $E^{0}$. In order to validate the analytical calculations, we used the
estimated parameter values for Jharkhand, the state of India and the
techniques for parameter estimation are described in the subsection IV.1.

###  IV.1 Model calibration

We have calibrated our SAIUQR model (7) with the observed daily new COVID-19
cases. We have considered three states of India namely Jharkahnd, Gujarat,
Andhra Pradesh and one city of India namely Chandigarh. The daily new COVID-19
cases are collected from the first COVID-19 case reported and up to May 24,
2020. The daily reported COVID-19 data were obtained from COVID19 INDIA
(https://www.covid19india.org/) covid-19tr . We have estimated six model
parameters, namely $\beta_{s}$, $\alpha_{a}$, $\alpha_{i}$, $\alpha_{u}$,
$\gamma_{a}$ and $\gamma_{q}$, out of fourteen system parameters for the
system (7) by using least square method Banerjee15 . The values of these
parameters and the initial population size plays an important role in the
model simulation. The parameters are estimated by assuming the initial
population size. The initial population are presented in the Table 3. Three
days moving average filter has been applied to the daily COVID-19 cases to
smooth the data. The daily reported confirmed COVID-19 cases are fitted with
the model simulation by using the least square method. The estimated parameter
values are listed in the Table 2. Different set of parameter values locally
minimizes the Root Mean Square Error (RMSE) and we have considered the set of
parameter values, which gives the realistic value of the basic reproduction
number $\mathcal{R}_{0}$. RMSE is the measure of the accuracy of the fitting
data and the RMSE is defined as follows:

$\displaystyle\begin{aligned}
RMSE=\sqrt{\frac{\Sigma_{i=1}^{n}(O(i)-M(i))^{2}}{n}},\end{aligned}$

where $n$ represents the size of the observed data, $O(i)$ is the reported
daily confirmed COVID-19 cases and $M(i)$ represents the model simulation. The
Figure 2 shows the daily confirmed COVID-19 cases (first column), cumulative
confirmed COVID-19 cases (second column) and model simulations has been shown
in the blue curve for all the four provinces of India. The values of RMSE and
basic reproduction number $\mathcal{R}_{0}$ for all the four provinces are
presented in the inset of the figure. The SAIUQR model performs well for the
three provinces, namely Jharkhand, Chandigarh, and Andhra Pradesh. The RMSE
for Gujarat is higher than the other provinces as the number of daily
confirmed COVID-19 cases are higher than the other provinces. The value of the
basic reproduction number $\mathcal{R}_{0}$ for Jharkhand, Gujarat,
Chandigarh, and Andhra Pradesh are 1.6877, 1.8803, 1.4775, and 1.2435,
respectively and the trend of daily confirmed COVID-19 cases is increasing.
This increasing trend of the daily new COVID-19 cases for all the four
provinces of India are captured by our model simulation. In all the four
provinces $\mathcal{R}_{0}>1$, so the disease free equilibrium point $E^{0}$
is unstable. The basic reproduction number for the four states are greater
than unity, which indicates the substantial outbreak of the COVID-19 in the
states.

###  IV.2 Validation of analytical findings

In this section, we have validated our analytically findings by using
numerical simulations for the parameter values in the Table 1, and the
estimated parameter values in the Table 2 for our SAIUQR model for the
coronavirus diseases. The parameter values are estimated for the observed
COVID-19 data for the three states of India, namely Jharkhand, Gujarat and
Andhra Pradesh and for the city Chandigarh. Our analytical findings stated in
the Theorem III.3, shows that the disease free equilibrium point $E^{0}$ is
locally asymptotically stable with $\mathcal{R}_{0}<1$, and the Theorem III.5
stated that a unique endemic equilibrium point $E^{*}$ is locally
asymptotically stable for $\mathcal{R}_{0}>1$. The numerical simulations of
the SAIUQR model system (7) have been presented in the Figure 3 for all the
six individuals and for the different values of the disease transmission rate
$\beta_{s}$. The value of the parameters considered for numerical simulations
are $\alpha_{a}=0.264,$ $\alpha_{i}=0.76,$ $\alpha_{u}=0.96,$
$\gamma_{a}=0.0012,$ $\gamma_{q}=0.0015,$ $\delta=0.03,$ $\Lambda_{s}=1200$
and the other model parameter values are listed in the Table 1. Six initial
population sizes are considered for the model simulation, namely
$(39402,1500,2000,20,0,0)$, $(31402,1200,1500,16,0,0)$,
$(25402,900,1000,12,0,0)$, $(20402,600,500,8,0,0)$, $(15402,300,100,4,0,0)$
and $(15000,100,50,1,0,0)$. The time series simulation have been displayed for
$\beta_{s}=1.10$ (red curves in the Figure 3) and $\beta_{s}=0.55$ (blue
curves in the Figure 3). The values of $R_{0}$ are $1.2889$ and $0.7030$ for
$\beta_{s}=1.10$ and $\beta_{s}=0.55$, respectively. The blue curves in the
Figure 3 shows that the disease free equilibrium point
$E^{0}(40000,0,0,0,0,0)$ is locally asymptotically stable as well as globally
asymptotically stable with $R_{0}=0.7030<1$. Our SAIUQR model system (7)
converges to the endemic equilibrium point
$E^{*}(31035.0,1146.0,17.6,1.6,2601.5,5198.4)$ for $\beta_{s}=1.10$ and
$R_{0}=1.2889>1$ (red curves), which has been displayed in the Figure 3.
Hence, this numerical simulation verifies the analytical findings in Theorem
III.3 and Theorem III.5.

The Theorem III.5 states that the SAIUQR model system (7) undergoes a
transcritical bifurcation at the threshold $\mathcal{R}_{0}=1$. We have
plotted the COVID-19 reported symptomatic individuals $(I)$ in the $(R_{0},I)$
plane by gradually increasing the disease transmission rate $\beta_{s}$ (see
the Figure 4). The model parameter values are $\alpha_{a}=0.264,$
$\alpha_{i}=0.76,$ $\alpha_{u}=0.96,$ $\gamma_{a}=0.0012,$
$\gamma_{q}=0.0015,$ $\delta=0.03,$ $\Lambda_{s}=1200$ and the other parameter
values listed in the Table 1. We vary the disease transmission rate
$\beta_{s}$ from 0.67 to 1.10 and computed the basic reproduction number
$\mathcal{R}_{0}$ and the COVID-19 reported symptomatic individuals $(I)$. The
numerically computed values are presented in the Figure 4, which clearly shows
that the system (7) experiences transcritical bifurcation at the threshold
$\mathcal{R}_{0}=1$. The blue curve represents the stable endemic equilibrium
point $E^{*}$, black line represents the stable disease free equilibrium point
$E^{0}$ and the red line represents the unstable branch of the disease free
equilibrium point $E^{0}$. Hence, the Figure 4 shows that disease free
equilibrium point $E^{0}$ is stable for the reproduction number
$\mathcal{R}_{0}<1$ and an endemic equilibrium point $E^{*}$ is stable for the
reproduction number $\mathcal{R}_{0}>1$. From the biological point of view, it
can be described that the model system (7) will be free from COVID-19 for the
reproduction number $\mathcal{R}_{0}<1$ and the coronavirus diseases will
spread throughout the people for $\mathcal{R}_{0}>1.$

Figure 5(a) represents the reproduction number $\mathcal{R}_{0}$ decreases as
the recovery rate $\eta_{i}$ of reported infected individuals increases and
the reproduction number $\mathcal{R}_{0}$ becomes less than one for
$\beta_{s}=0.85$ and $\beta_{s}=0.76$. This indicates that the disease free
equilibrium point $E^{0}$ switches the stability of the model system (7) as
$\eta_{i}$ changes. But the reproduction number $\mathcal{R}_{0}$ remains
grater than one for the disease transmission rate $\beta_{s}=1.10$ and
$\beta_{s}=0.95$, that is, for large $\beta_{s}$ unique endemic equilibrium
point remains locally asymptotically stable even if $\eta_{i}$ changes. In
terms of COVID-19 diseases this interprets that if the rate of recovery for
infected individuals $(\eta_{i})$ be increased, which can be done by vaccinees
or specific therapeutics, the model system (7) changes its stability to
disease free equilibrium $E^{0}$ from endemic equilibrium $E^{*}$ but if the
transmission of the disease $(\beta_{s})$ is high enough then by vaccines or
specific therapeutics, the system (7) can not change its stability from
endemic equilibrium to disease free equilibrium.

Figure 5(b) represents the reproduction number $\mathcal{R}_{0}$ increases as
$\gamma_{a}$ (transition rate from asymptomatic individuals to symptomatic
individuals) increases but the reproduction number $\mathcal{R}_{0}$ remains
less than one for the disease transmission rate $\beta_{s}=0.55$ and
$\beta_{s}=0.45$. For $\beta_{s}=0.66$ and $\beta_{s}=0.76$, the basic
reproduction number $\mathcal{R}_{0}$ becomes grater than one and the SAIUQR
model system (7) loses the stability of disease free equilibrium point
$E^{0}$. Thus, to flatten the COVID-19 curve in any of the four provinces of
India, reduction of the transmission of the COVID-19 disease is utmost
priority even if the recovery rate increased by medication. Biologically it
means that to mitigate the COVID-19 diseases, the people must have to maintain
the social distancing, contact tracing by avoiding the mass gathering.

The predictive competency for the SAIUQR model system (7) requires valid
estimation of the system parameters $\gamma_{a}$ (rate of transition from
asymptomatic to symptomatic infected individuals), $\gamma_{q}$ (the rate that
the quarantine become susceptible), $\theta$ (fraction of asymptomatic
infectious that become reported symptomatic infectious), $\xi_{a}$ (rate at
which asymptomatic individuals become quarantined). In the Figure 6(a), we
plot the reproduction number $\mathcal{R}_{0}$ as a function of $\xi_{a}$ and
$\gamma_{q}$ for the parameter values in the Table 1 and estimated parameters
for the state Jharkhand, to encapsulate the significance of these values in
the evolution of COVID-19 outbreak. From the Figure 6(a), it can be observed
that the parameters have a little influence on the outbreak of the coronavirus
diseases as the parameters $\xi_{a}$ and $\gamma_{q}$ has a little
contribution for the reproduction number $\mathcal{R}_{0}$. In the Figure
6(b), we plot the reproduction number $\mathcal{R}_{0}$ as a function of
$\theta$ and $\gamma_{a}$ for the parameter values in the Table 1 and
estimated parameters for the state Jharkhand, to encapsulate the significance
of these values in the evolution of COVID-19 outbreak. The Figure 6(b) shows
that the parameters $\theta$ and $\gamma_{a}$ are more influential in
increasing the reproduction number $\mathcal{R}_{0}$. Thus, to control the
outbreak of COVID-19, we must control the parameters $\theta$ and
$\gamma_{a}$. The correctness of these values rely on the input of medical and
biological epidemiologists. Thus, the fraction $\theta$ of reported
symptomatic infected individuals may be substantially increased by public
health reporting measures, with greater efforts to recognize all the present
cases. Our model simulation reveals the effect of an increase in this fraction
$\theta$ in the value of the reproduction number $\mathcal{R}_{0}$, as evident
in the Figure 6(b), for the COVID-19 epidemic in the four sates of India.

###  IV.3 Short-term prediction

Mathematical modeling of infectious diseases can provide short-term and long-
term prediction of the pandemic Giordano20 ; KhCov20 ; Liu20 ; Sarkar20 . Due
to absence of any licensed vaccines or specific therapeutics, forecasting is
of utmost importance for strategies to control and prevention of the diseases
with limited resources. It should be noted that here we can predict the
epidemiological traits of SARS-CoV-2 or COVID-19 for short-term only as the
Governmental strategies can be altered time to time resulting in the
corresponding changes in the associated parameters of the proposed SAIUQR
model. Also, it is true that the scientists are working for drugs and/or
effective vaccines against COVID-19 and the presence of such pharmaceutical
interventions will substantially change the outcomes Corey20 . Thus, in this
study we performed a short-term prediction for our SAIUQR model system (7)
using the parameter values in the Table 1 and the estimated parameter values
in the Table 2. Using the observed data up to May 24, 2020, a short-term
prediction (for 20 days) has been done for daily new COVID-19 cases (first
column) and cumulative confirmed cases (second column) are presented in the
Figure 7. The black dot-dashed curve represents the short-term prediction of
our SAIUQR model from May 25, 2020 to June 13, 2020. Red shaded region is the
standard deviation band of our SAIUQR model simulated curve. The standard
deviations are computed from the model simulation based on the estimated data.
In each of four states, we plot the standard deviation bands at a standard
deviation level above and below the model simulation for different days.
Standard deviation band gives an estimation of the deviation of the actual
model data. The trend of the predicted daily COVID-19 cases is increasing for
all the four provinces of India. Prediction of the SAIUQR model should be
regarded as an estimation of the daily infected population and cumulative
confirmed cases of the four states of India. From the SAIUQR model simulation,
we can predict that the estimated daily new reported COVID-19 cases on June
13, 2020 will be approximately 15, 454, 12, and 96 in Jharkhand, Gujarat,
Chandigarh, and Andhra Pradesh, respectively (see the left column of the
Figure 7). Our SAIUQR model simulation predict that the confirmed cumulative
number of cases on June 13, 2020 will be approximately 661, 23955, 514, and
4487 in Jharkhand, Gujarat, Chandigarh, and Andhra Pradesh, respectively (see
the right column of the Figure 7).

Table 2: The SAIUQR model parameter values estimated from the observed daily new COVID-19 cases for four provinces of India, namely Jharkhand, Gujarat, Chandigarh and Andhra Pradesh. Six important parameters $\beta_{s}$, $\alpha_{a}$, $\alpha_{i}$, $\alpha_{u}$, $\gamma_{a}$ and $\gamma_{q}$ are estimated among fourteen system parameters. Provinces | $\beta_{s}$ | $\alpha_{a}$ | $\alpha_{i}$ | $\alpha_{u}$ | $\gamma_{a}$ | $\gamma_{q}$
---|---|---|---|---|---|---
Jharkhand | 0.760 | 0.264 | 0.760 | 0.9600 | 0.0012 | 0.0015
Gujarat | 1.006 | 0.342 | 0.168 | 0.1308 | 0.0004 | 0.0046
Chandigarh | 0.750 | 0.294 | 0.444 | 0.4600 | 0.0010 | 0.0011
Andhra Pradesh | 0.431 | 0.419 | 0.688 | 0.7100 | 0.0006 | 0.0280
Table 3: _Initial population size and the values of $\Lambda_{s}$ used in numerical simulations for four different provinces of India, namely Jharkhand, Gujarat, Chandigarh, and Andhra Pradesh._ Provinces | $S(0)$ | $A(0)$ | $Q(0)$ | $I(0)$ | $U(0)$ | $R(0)$ | $\Lambda_{s}$
---|---|---|---|---|---|---|---
Jharkhand | 39402 | 575 | 19 | 1 | 0 | 0 | 1200
Gujarat | 85402 | 1525 | 27 | 1 | 0 | 0 | 1300
Chandigarh | 20402 | 275 | 10 | 1 | 0 | 0 | 1200
Andhra Pradesh | 75401 | 355 | 12 | 1 | 0 | 0 | 970
Figure 3: Stability of the SAIUQR model system (7) around the disease free
equilibrium point $E^{0}$ and an unique endemic equilibrium point $E^{*}$. The
values of the estimated parameters are $\alpha_{a}=0.264,$ $\alpha_{i}=0.76,$
$\alpha_{u}=0.96,$ $\gamma_{a}=0.0012,$ $\gamma_{q}=0.0015,$ $\delta=0.03,$
$\Lambda_{s}=1200$ and other parameter values are listed in the Table 1.
Initial population sizes are $(39402,1500,2000,20,0,0)$,
$(31402,1200,1500,16,0,0)$, $(25402,900,1000,12,0,0)$,
$(20402,600,500,8,0,0)$, $(15402,300,100,4,0,0)$, $(15000,100,50,1,0,0)$. Time
series solution for $\beta_{s}=1.10$ (red curves) and $\beta_{s}=0.55$ (blue
curves). The values of $\mathcal{R}_{0}$ are $1.2889$ and $0.7030$ for
$\beta_{s}=1.10$ and $\beta_{s}=0.55$, respectively. Disease free equilibrium
point $E^{0}$ is locally asymptotically stable when $\mathcal{R}_{0}<1$ (blue
curves) and the endemic equilibrium point $E^{*}$ is locally asymptotically
stable when $\mathcal{R}_{0}>1$ (red curves). Figure 4:  The figure represents
the transcritical bifurcation diagram of the SAIUQR model system (7) with
respect to the basic reproduction number $\mathcal{R}_{0}$. The parameter
values are $\alpha_{a}=0.264,$ $\alpha_{i}=0.76,$ $\alpha_{u}=0.96,$
$\gamma_{a}=0.0012,$ $\gamma_{q}=0.0015,$ $\delta=0.03,$ $\Lambda_{s}=1200$
and other parameters as listed in the Table 1. Stability of the SAIUQR system
(7) exchange at the threshold $\mathcal{R}_{0}=1$. Figure 5: The figures
represents the basic reproduction number $\mathcal{R}_{0}$ in terms of (a)
$\eta_{i}$ (rate of recovery for infected individuals) and (b) $\gamma_{a}$
(rate at which asymptomatic individuals develops detected symptomatic infected
individuals). Green shaded region indicates $\mathcal{R}_{0}<1$ whereas the
pink shaded region indicates $\mathcal{R}_{0}>1$. The parameter values are
$\alpha_{a}=0.264,$ $\alpha_{i}=0.76,$ $\alpha_{u}=0.96,$ $\gamma_{a}=0.0012,$
$\gamma_{q}=0.0015,$ $\delta=0.03,$ $\Lambda_{s}=1200$ and the other parameter
values are listed in the Table 1.  Figure 6: The figures represents the
surface plot of the basic reproduction number $\mathcal{R}_{0}$ in (a)
($\gamma_{q}$, $\xi_{a}$)-plane and (b) ($\gamma_{a}$, $\theta$)-plane. Red
shading plane indicates $\mathcal{R}_{0}=1$. The parameter values for the sub-
figure (a): $\beta_{s}=0.76$, $\alpha_{a}=0.264,$ $\alpha_{i}=0.76,$
$\alpha_{u}=0.96,$ $\gamma_{a}=0.0012,$ $\delta=0.03,$ $\Lambda_{s}=1200$ and
for the sub-figure (b): $\beta_{s}=0.76$, $\alpha_{a}=0.264,$
$\alpha_{i}=0.76,$ $\alpha_{u}=0.96,$ $\gamma_{q}=0.0015,$ $\delta=0.03,$
$\Lambda_{s}=1200$ and the other parameter values are listed in the Table 1.
Figure 7: The figure represents the short-term prediction of the daily new
COVID-19 cases (first column) and the cumulative confirmed cases (second
column) for the three states of India namely Jharkhand, Gujarat, Andhra
Pradesh and one city of India namely Chandigarh. The black dot-dashed curve
represents the prediction from May 25, 2020 to June 13, 2020 (20 days). The
red shaded region is the standard deviation band of the SAIUQR model simulated
curve.

##  V Discussion and conclusion

The SARS-CoV-2 pandemic in India is a potential menace throughout the country
due to its exponential growth. Everyday the new cases are reported around 5-6
thousands and more than that from different states and territories of India,
which is an alarming situation as with the second most populated country
worldwide covid-19tr . Due to absence of any licensed vaccine, therapeutics or
treatment and with a peculiar epidemiological traits of SARS-CoV-2, one would
depend on the qualitative control of the diseases rather than complete
elimination. During this period of an epidemic when person-to-person
transmission is confirmed and the reported cases of SARS-CoV-2 viruses are
rising throughout the globe, prediction is of utmost priority for health care
planning and to manage the virus with limited resource. Furthermore,
mathematical modeling can be a powerful tool in designing strategies to manage
exponentially spreading coronavirus diseases in absence of any antivirals or
diagnostic test.

In this study, we proposed and analyzed a new compartmental mathematical model
for SARS-CoV-2 viruses to forecast and control the outbreak. In the model
formulation, we incorporate the transmission variability of asymptomatic and
unreported symptomatic individuals. We also incorporate the symptomatic
infected populations who are reported by the public health services. We assume
that reported infected individuals will no-longer associated into the
infection as they are isolated and move to the hospitals or Intensive Care
Unit (ICU). In our model, we incorporate the constant transmission rate in the
early exponential growth phase of the SARS-CoV-2 diseases as identified in
KhCov20 ; Gumel04 . We model the role of the Govt. imposed restrictions for
the public in India, beginning on March 25, 2020, as a time-dependent decaying
transmission rate after March 25, 2020. But, due to less stringent lockdown
exponentially increasing the disease transmission rate, we are able to fit
with increasing accuracy, our model simulations to the Indian reported cases
data up to May 24, 2020.

We fit our SAIUQR model for the daily confirmed cases and cumulative confirmed
cases of the four different provinces of India, namely Jharkhand, Andhra
Pradesh, Chandigarh, and Gujarat with data up to May 24, 2020. The estimated
model parameters for different states of India are given in the Table 2 and
the corresponding initial population size are listed in the Table 3. It can be
observed that the basic reproduction number for four different provinces of
India, namely Jharkhand, Andhra Pradesh, Chandigarh, and Gujarat are 1.6877,
1.2435, 1.4775, and 1.8830, respectively, which demonstrates the disease
transmission rate is quite high that basically indicates the substantial
outbreak of the COVID-19 diseases. This higher value of reproduction number
$\mathcal{R}_{0}$ captures the outbreak of COVID-19 phenomena in India. Based
on the estimated parameter values our model simulation suggest that the rate
of disease transmission need to be controlled, otherwise India will enter in
stage-3 of SARS-CoV-2 disease transmission within in a short period of time.

Based on the estimated model parameters we have validated our detailed
analytical findings. Our proposed SAIUQR model has two biologically feasible
singular points, namely infection free steady state $E^{0}$ and a unique
endemic steady state $E^{*}$ and they become locally asymptotically stable for
$\mathcal{R}_{0}<1$ and $\mathcal{R}_{0}>1$, respectively. Analytically, we
have shown that the infection free steady state $E^{0}$ of the SAIUQR model
(7) is globally asymptotically stable for $\mathcal{R}_{0}<1.$ We also showed
that the SAIUQR model (7) experiences transcritical bifurcation at the
threshold parameter $\mathcal{R}_{0}=1$, which has been shown in the Figure 4.
The Figure 3 (blue curves) and the Figure 3 (red curves) represents the local
asymptotic stability as well as global asymptotic stability of the infection
free steady state $E^{0}$ for $\mathcal{R}_{0}<1$ and endemic steady state
$E^{*}$ for $\mathcal{R}_{0}>1$, respectively.

The calibrated model then utilized for short-term predictions in the four
different states of India. Our SAIUQR model performs well in case of all the
four provinces of India, namely Jharkahnd, Chandigarh, Gujarat, and Andhra
Pradesh for daily confirmed cases and cumulative confirmed cases. Albeit, the
increasing (or exponential) pattern of daily new cases and cumulative
confirmed cases of SARS-CoV-2 is well captured by our proposed model for all
the four states of India, which has been shown in the Figure 2. Our model
simulation showed a short-term prediction for 20 days (from May 25, 2020 to
June 13, 2020) for daily confirmed cases and cumulative confirmed cases of the
four provinces of India. The short-term prediction for the four provinces of
India will demonstrates the increasing pattern of the daily and cumulative
cases in the near future (see the Figure 7). From the simulation, our model
predict that on June 13, 2020 the daily confirmed cases of COVID-19 of the
four provinces of India, namely Jharkhand, Gujarat, Chandigarh, and Andhra
Pradesh will be 15, 454, 12, and 96, respectively (see the left column of the
Figure 7). Similarly, from the simulation, our model predict that on June 13,
2020 the cumulative confirmed cases of COVID-19 of the four provinces of
India, namely Jharkhand, Gujarat, Chandigarh, and Andhra Pradesh will be 661,
23955, 514, and 4487, respectively (see the right column of the Figure 7).

It is worthy to mention that the scientists or clinicians are working for
effective vaccine or therapeutics to eradicate and/or control the outbreak of
SARS-CoV-2 diseases and the existence of such pharmaceutical interventions
will substantially change the outcomes Corey20 ; Lurie20 . Thus, in this
study, we are mainly focusing on short-term predictions for the COVID-19
pandemic and subsequently there would be a very little chance to alter in
corresponding parametric space. But the framework of our present compartmental
model provides some significant insights into the dynamics and forecasting of
the spread and control of COVID-19. Moreover, our model simulation suggest
that the quarantine, reported and unreported symptomatic individuals as well
as government intervention polices like media effect, lockdown and social
distancing can play a key role in mitigating the transmission of COVID-19.

## Acknowledgments

This study is supported by Science and Engineering Research Board (SERB) (File
No. ECR/ 2017/000234), Department of Science & Technology, Government of
India. The authors are thankful to the anonymous reviewers for their careful
reading and constructive suggestions/comments which helped in better
exposition of the manuscript.

## Data availability

All the data used in this work has been obtained from official sources
covid-19tr . All data supporting the findings of this study are in the paper
and available from the corresponding author on request.

## Author contributions statement

Subhas Khajanchi and Kankan Sarkar designed and performed the research as well
as wrote the paper.

## Competing interests

The authors declare that they have no conflict of interest.

##  Appendix A Proof of Theorem III.1

Proof. To prove the positivity of the system (7), we show that any solution
initiating from the non-negative octant $\mathbf{R}_{+}^{6}$ remains positive
for all $t>0$. In order to do this, we have to prove that on each hyperplane
bounding the non-negative octant the vector field points into
$\mathbf{R}_{+}^{6}$. For our system (7), we observe that

$\displaystyle\frac{dS}{dt}\bigg{|}_{S=0}$ $\displaystyle=$
$\displaystyle\Lambda_{s}+\rho_{s}\gamma_{q}Q~{}\geq~{}0,$
$\displaystyle\frac{dA}{dt}\bigg{|}_{A=0}$ $\displaystyle=$
$\displaystyle\beta_{s}S\bigg{(}\alpha_{i}\frac{I}{N}+\alpha_{u}\frac{U}{N}\bigg{)}~{}\geq~{}0,$
$\displaystyle\frac{dI}{dt}\bigg{|}_{I=0}$ $\displaystyle=$
$\displaystyle\theta\gamma_{a}A+(1-\rho_{s})\gamma_{q}Q~{}\geq~{}0,$
$\displaystyle\frac{dU}{dt}\bigg{|}_{U=0}$ $\displaystyle=$
$\displaystyle(1-\theta)\gamma_{a}A~{}\geq~{}0,$
$\displaystyle\frac{dQ}{dt}\bigg{|}_{Q=0}$ $\displaystyle=$
$\displaystyle\xi_{a}A~{}\geq~{}0,$ $\displaystyle\frac{dR}{dt}\bigg{|}_{R=0}$
$\displaystyle=$ $\displaystyle\eta_{u}U+\eta_{i}I+\eta_{a}A~{}\geq~{}0.$

Therefore, the positivity of the solutions starting in the interior of
$\mathbf{R}_{+}^{6}$ is assured. $\mathbf{R}_{+}^{6}$ is positively invariant
set of the SAIUQR model system (7).

##  Appendix B Proof of Theorem III.2

Proof. To prove the boundedness of the SAIUQR system (7), we add all the model
equations, which gives $N=S+A+I+U+Q+R$. Taking the differentiation gives

$\displaystyle\frac{dN}{dt}$ $\displaystyle=$ $\displaystyle\Lambda_{s}-\delta
N,$

which gives

$\displaystyle\limsup_{t\rightarrow\infty}N(t)$ $\displaystyle\leq$
$\displaystyle\frac{\Lambda_{s}}{\delta}.$

Without any loss of generality, we can assume that
$\limsup_{t\rightarrow\infty}S(t)\leq\frac{\Lambda_{s}}{\delta},$
$\limsup_{t\rightarrow\infty}A(t)\leq\frac{\Lambda_{s}}{\delta},$
$\limsup_{t\rightarrow\infty}I(t)\leq\frac{\Lambda_{s}}{\delta},$
$\limsup_{t\rightarrow\infty}U(t)\leq\frac{\Lambda_{s}}{\delta},$
$\limsup_{t\rightarrow\infty}Q(t)\leq\frac{\Lambda_{s}}{\delta}$, and
$\limsup_{t\rightarrow\infty}R(t)\leq\frac{\Lambda_{s}}{\delta}.$ Thus, we
have a bounded set

$\displaystyle\Omega$ $\displaystyle=$
$\displaystyle\bigg{\\{}(S,A,I,U,Q,R)\in\mathbf{R}_{+}^{6}:0\leq
S,A,I,U,Q,R\leq\frac{\Lambda_{s}}{\delta}\bigg{\\}},$

which is also a positively invariant set with respect to the SAIUQR model (7).
Thus, any solution trajectory starting from an interior point of
$\mathbf{R}_{+}^{6}$ ultimately enter into the region $\Omega$ and remains
there for all finite time. This results indicates that none of the individuals
grow unboundedly or exponentially for a finite time window.

##  Appendix C Proof of Theorem III.3

Proof. The variational matrix around the infection free steady state $E^{0}$
for the SAIUQR model system (7) is given by

$\displaystyle J_{E^{0}}$ $\displaystyle=$
$\displaystyle\left(\begin{array}[]{cccccc}-\delta&-\beta_{s}\alpha_{a}&-\beta_{s}\alpha_{i}&-\beta_{s}\alpha_{u}&\rho_{s}\gamma_{q}&0\\\
0&\beta_{s}\alpha_{a}-(\xi_{a}+\gamma_{a}+\eta_{a}+\delta)&\beta_{s}\alpha_{i}&\beta_{s}\alpha_{u}&0&0\\\
0&\theta\gamma_{a}&-(\eta_{i}+\delta)&0&(1-\rho_{s})\gamma_{q}&0\\\
0&(1-\theta)\gamma_{a}&0&-(\eta_{u}+\delta)&0&0\\\
0&\xi_{a}&0&0&-(\gamma_{q}+\delta)&0\\\
0&\eta_{a}&\eta_{i}&\eta_{u}&0&-\delta\end{array}\right).$

The above Jacobian matrix $J_{E^{0}}$ has two repeated eigenvalues which are
-$\delta$, while the other four eigenvalues are the roots of the following
characteristics equation $|J_{E^{0}}-\lambda I|=0$,

$\displaystyle(\xi_{a}$ $\displaystyle+$
$\displaystyle\gamma_{a}+\eta_{a}+\delta+\lambda)(\eta_{i}+\delta+\lambda)(\eta_{u}+\delta+\lambda)(\gamma_{q}+\delta+\lambda)$
$\displaystyle-$
$\displaystyle\beta_{s}\alpha_{a}(\eta_{i}+\delta+\lambda)(\eta_{u}+\delta+\lambda)(\gamma_{q}+\delta+\lambda)$
$\displaystyle-$
$\displaystyle\theta\gamma_{a}\beta_{s}\alpha_{i}(\eta_{u}+\delta+\lambda)(\gamma_{q}+\delta+\lambda)$
$\displaystyle-$
$\displaystyle(1-\theta)\gamma_{a}\beta_{s}\alpha_{u}(\eta_{i}+\delta+\lambda)(\gamma_{q}+\delta+\lambda)$
$\displaystyle-$
$\displaystyle(1-\rho_{s})\gamma_{q}\xi_{a}\beta_{s}\alpha_{i}(\eta_{u}+\delta+\lambda)~{}=~{}0,$

which can be rewritten in the following form:

$\displaystyle\frac{\beta_{s}\alpha_{a}}{\xi_{a}+\gamma_{a}+\eta_{a}+\delta+\lambda}+\frac{\theta\gamma_{a}\beta_{s}\alpha_{i}}{(\xi_{a}+\gamma_{a}+\eta_{a}+\delta+\lambda)(\eta_{i}+\delta+\lambda)}$
$\displaystyle+$
$\displaystyle\frac{(1-\theta)\gamma_{a}\beta_{s}\alpha_{u}}{(\xi_{a}+\gamma_{a}+\eta_{a}+\delta+\lambda)(\eta_{u}+\delta+\lambda)}$
$\displaystyle+$
$\displaystyle\frac{(1-\rho_{s})\gamma_{q}\xi_{a}\beta_{s}\alpha_{i}}{(\xi_{a}+\gamma_{a}+\eta_{a}+\delta+\lambda)(\eta_{i}+\delta+\lambda)(\gamma_{q}+\delta+\lambda)}~{}=~{}1.$

Denote the above expression as the following:

$\displaystyle m_{1}(\lambda)$ $\displaystyle=$
$\displaystyle\frac{\beta_{s}\alpha_{a}}{\xi_{a}+\gamma_{a}+\eta_{a}+\delta+\lambda}$
$\displaystyle+$
$\displaystyle\frac{\theta\gamma_{a}\beta_{s}\alpha_{i}}{(\xi_{a}+\gamma_{a}+\eta_{a}+\delta+\lambda)(\eta_{i}+\delta+\lambda)}$
$\displaystyle+$
$\displaystyle\frac{(1-\theta)\gamma_{a}\beta_{s}\alpha_{u}}{(\xi_{a}+\gamma_{a}+\eta_{a}+\delta+\lambda)(\eta_{u}+\delta+\lambda)}$
$\displaystyle+$
$\displaystyle\frac{(1-\rho_{s})\gamma_{q}\xi_{a}\beta_{s}\alpha_{i}}{(\xi_{a}+\gamma_{a}+\eta_{a}+\delta+\lambda)(\eta_{i}+\delta+\lambda)(\gamma_{q}+\delta+\lambda)},$
$\displaystyle=$ $\displaystyle
m_{11}(\lambda)+m_{22}(\lambda)+m_{33}(\lambda)+m_{44}(\lambda)~{}~{}~{}\textrm{(say)}.$

Substitute $\lambda=x+iy$, and we know that $Re(\lambda)\geq 0$, then the
above expression leads to

$\displaystyle|m_{11}(\lambda)|$ $\displaystyle\leq$
$\displaystyle\frac{\beta_{s}\alpha_{a}}{|\xi_{a}+\gamma_{a}+\eta_{a}+\delta+\lambda|}~{}\leq~{}m_{11}(x)$
$\displaystyle\leq$ $\displaystyle~{}m_{11}(0),$
$\displaystyle|m_{22}(\lambda)|$ $\displaystyle\leq$
$\displaystyle\frac{\theta\gamma_{a}\beta_{s}\alpha_{i}}{|\xi_{a}+\gamma_{a}+\eta_{a}+\delta+\lambda||\eta_{i}+\delta+\lambda|}~{}\leq~{}m_{22}(x)$
$\displaystyle\leq$ $\displaystyle~{}m_{22}(0),$
$\displaystyle|m_{33}(\lambda)|$ $\displaystyle\leq$
$\displaystyle\frac{(1-\theta)\gamma_{a}\beta_{s}\alpha_{u}}{|\xi_{a}+\gamma_{a}+\eta_{a}+\delta+\lambda||\eta_{u}+\delta+\lambda|}$
$\displaystyle\leq$ $\displaystyle~{}m_{33}(x)~{}\leq~{}m_{33}(0),$
$\displaystyle|m_{44}(\lambda)|$ $\displaystyle\leq$
$\displaystyle\frac{(1-\rho_{s})\gamma_{q}\xi_{a}\beta_{s}\alpha_{i}}{|\xi_{a}+\gamma_{a}+\eta_{a}+\delta+\lambda||\eta_{i}+\delta+\lambda||\gamma_{q}+\delta+\lambda|}$
$\displaystyle\leq$ $\displaystyle~{}m_{44}(x)~{}\leq~{}m_{44}(0).$

Thus,
$m_{11}(0)+m_{22}(0)+m_{33}(0)+m_{44}(0)~{}=~{}m_{1}(0)~{}=~{}\mathcal{R}_{0}<1,$
which gives that $m_{1}(\lambda)\leq 1.$ Therefore for $\mathcal{R}_{0}<1$,
all the eigenvalues of the characteristics equation $m_{1}(\lambda)=1$ are
real or have negative real parts. Thus for $\mathcal{R}_{0}<1$, all the
eigenvalues are negative and hence the infection free steady state $E^{0}$ is
locally asymptotically stable.

Now, if we consider $\mathcal{R}_{0}>1$, that is, $m_{1}(0)>1$, then

$\displaystyle\lim_{\lambda\rightarrow\infty}m_{1}(\lambda)$ $\displaystyle=$
$\displaystyle 0,$

then there exists $\lambda_{1}^{*}>0$ in such a way that
$m_{1}(\lambda_{1}^{*})=1.$ This indicates that there exists non-negative
eigenvalue $\lambda_{1}^{*}>0$ for the variational matrix $J_{E^{0}}$. Hence,
the infection free steady state $E^{0}$ is unstable for $\mathcal{R}_{0}>1$.

##  Appendix D Proof of Theorem III.4

Proof. To prove the global asymptotic stability of the infection free steady
state $E^{0}(S^{0},A^{0},Q^{0},I^{0},U^{0},R^{0})$, we can rewrite the SAIUQR
model (7) in the following form:

$\displaystyle\frac{dX}{dt}$ $\displaystyle=$ $\displaystyle F(X,V),$
$\displaystyle\frac{dV}{dt}$ $\displaystyle=$ $\displaystyle
G(X,V),~{}~{}~{}G(X,0)=0,$

where $X=(S,R)\in\mathbf{R}^{2}$ represents (its components) the number of
susceptible or uninfected individuals and $V=(A,I,U,Q)\in\mathbf{R}^{4}$
represents (its components) the number of infected individuals incorporating
asymptomatic, quarantine, and infectious etc. $E^{0}=(X^{*},0)$ designates the
infection free steady state for the SAIUQR model system (7). For the
compartmental model (7), $F(X,V)$ and $G(X,V)$ are defined as follows:

$\displaystyle F(X,V)$ $\displaystyle=$
$\displaystyle\left(\begin{array}[]{c}\Lambda_{s}-\beta_{s}\frac{S}{N}(\alpha_{a}A+\alpha_{i}I+\alpha_{u}U)+\rho_{s}\gamma_{q}Q-\delta
S\\\ \eta_{u}U+\eta_{i}I+\eta_{a}A-\delta R\\\ \end{array}\right),$ and
$\displaystyle G(X,V)$ $\displaystyle=$
$\displaystyle\left(\begin{array}[]{c}\beta_{s}\frac{S}{N}(\alpha_{a}A+\alpha_{i}I+\alpha_{u}U)-(\xi_{a}+\gamma_{a})A-\eta_{a}A-\delta
A\\\ \theta\gamma_{a}A+(1-\rho_{s})\gamma_{q}Q-\eta_{i}I-\delta I\\\
(1-\theta)\gamma_{a}A-\eta_{u}U-\delta U\\\ \xi_{a}A-\gamma_{q}Q-\delta Q\\\
\end{array}\right).$

From the above expression of $G(X,V)$, it is clear that $G(X,0)=0.$

The following two conditions $(C1)$ and $(C2)$ must be met to assure the
global asymptotic stability:
(C1) For $\frac{dX}{dt}=F(X,0),$ $X^{*}$ is globally asymptotically stable,
(C2) $G(X,V)=BV-\widehat{G}(X,V),$ $\widehat{G}(X,V)\geq 0$ for
$(X,V)\in\Omega$,
where $B=D_{I}G(X^{*},0)$ is an M-matrix (the non-diagonal components are non-
negative) and in the region $\Omega$, the SAIUQR model system (7) is
biologically feasible. The compartmental model (7) stated in the condition
$(C1)$ can be expressed as

$\displaystyle\frac{d}{dt}\left(\begin{array}[]{c}S\\\ R\\\
\end{array}\right)~{}=~{}\left(\begin{array}[]{c}\Lambda_{s}-\delta S\\\
-\delta R\\\ \end{array}\right).$

Analytically solve the above system of equations, we obtain that
$S(t)=\frac{\Lambda_{s}}{\delta}+\exp(-\delta
t)(S(0)-\frac{\Lambda_{s}}{\delta}),$ and $R(t)=\exp(-\delta t)R(0).$
Considering $t\rightarrow\infty$, $S(t)=\frac{\Lambda_{s}}{\delta}$ and
$R(t)\rightarrow 0.$ Thus, $X^{*}$ is globally asymptotically stable for
$\frac{dX}{dt}=F(X,0).$ Thus, the first condition $(C1)$ holds for the system
(7).

Now the matrices $B$ and $\widehat{G}(X,V)$ for the SAIUQR model system (7)
can be expressed as

$\displaystyle B$ $\displaystyle=$
$\displaystyle\left(\begin{array}[]{cccc}-(\xi_{a}+\gamma_{a}+\eta_{a}+\delta)+\alpha_{a}\beta_{s}&\alpha_{i}\beta_{s}&\alpha_{u}\beta_{s}&0\\\
\theta\gamma_{a}&-(\eta_{i}+\delta)&0&(1-\rho_{s})\gamma_{q}\\\
(1-\theta)\gamma_{a}&0&-(\eta_{u}+\delta)&0\\\
\xi_{a}&0&0&-(\gamma_{q}+\delta)\\\ \end{array}\right),$
$\displaystyle\widehat{G}(X,V)$ $\displaystyle=$
$\displaystyle\left(\begin{array}[]{c}\alpha_{a}\beta_{s}A\bigg{(}1-\frac{S}{N}\bigg{)}+\alpha_{i}\beta_{s}I\bigg{(}1-\frac{S}{N}\bigg{)}+\alpha_{u}\beta_{s}U\bigg{(}1-\frac{S}{N}\bigg{)}\\\
0\\\ 0\\\ 0\\\ \end{array}\right).$

It is clear that $B$ is a M-matrix as all its non-diagonal components are non-
negative. Also, $\widehat{G}(X,V)\geq 0$ in the region $\Omega$ as $S(t)\leq
N(t).$ Also, we showed that $X^{*}$ is a globally asymptotically stable steady
state of the system $\frac{dX}{dt}=F(X,0)$. Therefore, the infection free
steady state $E^{0}$ of the SAIURQ model (7) is globally asymptotically stable
in the region $\Omega$ for $\mathcal{R}_{0}<1.$

##  Appendix E Proof of Theorem III.5

Proof. Now, we use the theory of center manifold to investigate the local
asymptotic stability of the interior equilibrium point
$E^{*}(S^{*},A^{*},Q^{*},I^{*},U^{*},R^{*})$ by considering the disease
transmission rate $\beta_{s}$ as a bifurcation parameter,
$\beta_{s}=\beta_{s}^{c}$ corresponding to $\mathcal{R}_{0}=1,$ is

$\displaystyle\beta_{s}^{c}$ $\displaystyle=$
$\displaystyle\frac{(\eta_{u}+\delta)(\gamma_{q}+\delta)(\eta_{i}+\delta)(\xi_{a}+\gamma_{a}+\eta_{a}+\delta)}{\alpha_{a}(\eta_{u}+\delta)(\gamma_{q}+\delta)(\eta_{i}+\delta)+(1-\theta)\alpha_{u}\gamma_{a}(\gamma_{q}+\delta)(\eta_{i}+\delta)+\alpha_{i}\left\\{\theta\gamma_{a}(\gamma_{q}+\delta)+(1-\rho_{s})\xi_{a}\gamma_{a}\right\\}(\eta_{u}+\delta)}.$

The variational matrix of the SAIUQR model (7) at $\beta_{s}=\beta_{s}^{c}$,
denoted by $J_{E^{0}}$ has the right eigenvector associated to zero is
eigenvalue given by
$\omega=[\omega_{1},~{}\omega_{2},~{}\omega_{3},~{}\omega_{4},~{}\omega_{5},~{}\omega_{6}]^{T}$,
where

$\displaystyle\omega_{1}$ $\displaystyle=$
$\displaystyle\frac{\omega_{2}}{\delta}\bigg{[}-\frac{(1-\theta)\gamma_{a}\beta_{s}\alpha_{u}}{\eta_{u}+\delta}+\frac{\rho_{s}\gamma_{q}\xi_{a}}{\eta_{q}+\delta}-(\xi_{a}+\gamma_{a}+\eta_{a}+\delta)$
$\displaystyle+$
$\displaystyle\frac{\beta_{s}\alpha_{u}(1-\theta)\gamma_{a}}{\eta_{u}+\delta}\bigg{]},$
$\displaystyle\omega_{2}$ $\displaystyle=$
$\displaystyle\omega_{2}~{}>~{}0,~{}~{}~{}\omega_{4}~{}=~{}\frac{(1-\theta)\gamma_{a}\omega_{2}}{\eta_{u}+\delta},~{}~{}~{}\omega_{5}~{}=~{}\frac{\xi_{a}\omega_{2}}{\eta_{q}+\delta},$
$\displaystyle\omega_{3}$ $\displaystyle=$
$\displaystyle\frac{\omega_{2}}{\beta_{s}\alpha_{i}}\bigg{[}(\xi_{a}+\gamma_{a}+\eta_{a}+\delta)-\beta_{s}\alpha_{a}-\frac{\beta_{s}\alpha_{u}(1-\theta)\gamma_{a}}{\eta_{u}+\delta}\bigg{]},$
$\displaystyle\omega_{6}$ $\displaystyle=$
$\displaystyle\frac{\omega_{2}}{\delta}\bigg{[}\eta_{a}+\frac{\eta_{u}(1-\theta)\gamma_{a}}{\eta_{u}+\delta}+\frac{\eta_{i}((\xi_{a}+\gamma_{a}+\eta_{a}+\delta)-\beta_{s}\alpha_{a})}{\beta_{s}\alpha_{i}}$
$\displaystyle-$
$\displaystyle\frac{\eta_{i}\beta_{s}\alpha_{u}(1-\theta)\gamma_{a}}{\beta_{s}\alpha_{i}(\eta_{u}+\delta)}\bigg{]}.$

Similarly, at the threshold $\beta_{s}=\beta_{s}^{c}$, the variational matrix
$J_{E^{0}}$ has the left eigenvector associated to zero eigenvalue is given by
$\upsilon=[\upsilon_{1},~{}\upsilon_{2},~{}\upsilon_{3},~{}\upsilon_{4},~{}\upsilon_{5},~{}\upsilon_{6}]$,
where

$\displaystyle\upsilon_{1}$ $\displaystyle=$ $\displaystyle
0,~{}~{}~{}\upsilon_{6}=0,~{}~{}~{}\upsilon_{2}=\upsilon_{2}~{}>~{}0,$
$\displaystyle~{}~{}~{}\upsilon_{3}$ $\displaystyle=$
$\displaystyle\frac{\beta_{s}\alpha_{i}\upsilon_{2}}{\eta_{i}+\delta},~{}~{}~{}\upsilon_{4}=\frac{\beta_{s}\alpha_{u}\upsilon_{2}}{\eta_{u}+\delta},$
$\displaystyle\upsilon_{5}$ $\displaystyle=$
$\displaystyle\frac{\upsilon_{2}}{\xi_{a}}\bigg{[}(\xi_{a}+\gamma_{a}+\eta_{a}+\delta)-\beta_{s}\alpha_{a}-\frac{\theta\gamma_{a}\beta_{s}\alpha_{i}}{\eta_{i}+\delta}$
$\displaystyle-$
$\displaystyle\frac{\beta_{s}\alpha_{u}(1-\theta)\gamma_{a}}{\eta_{u}+\delta}\bigg{]}.$

Let us introduce the notations for the SAIUQR model system (7): $S=x_{1}$,
$A=x_{2}$, $I=x_{3}$, $U=x_{4}$, $Q=x_{5}$, $R=x_{6}$, and
$\frac{dx_{i}}{dt}=f_{i}$, where $i=1,2,...,6.$ Now, we compute the following
nonzero second order partial derivatives of $f_{i}$ at the infection free
steady state $E^{0}$ and obtain

$\displaystyle\frac{\partial^{2}f_{2}}{\partial x_{2}\partial x_{3}}$
$\displaystyle=$
$\displaystyle-\beta_{s}(\alpha_{a}+\alpha_{i})\frac{\Lambda_{s}}{\delta},~{}\frac{\partial^{2}f_{2}}{\partial
x_{2}\partial
x_{4}}=-\beta_{s}(\alpha_{a}+\alpha_{u})\frac{\Lambda_{s}}{\delta},$
$\displaystyle\frac{\partial^{2}f_{2}}{\partial x_{2}\partial x_{5}}$
$\displaystyle=$
$\displaystyle-\beta_{s}\alpha_{a}\frac{\Lambda_{s}}{\delta},~{}~{}~{}\frac{\partial^{2}f_{2}}{\partial
x_{2}\partial x_{6}}~{}=~{}-\beta_{s}\alpha_{a}\frac{\Lambda_{s}}{\delta},$
$\displaystyle\frac{\partial^{2}f_{2}}{\partial x_{3}\partial x_{4}}$
$\displaystyle=$
$\displaystyle-\beta_{s}(\alpha_{i}+\alpha_{u})\frac{\Lambda_{s}}{\delta},~{}~{}~{}\frac{\partial^{2}f_{2}}{\partial
x_{3}\partial x_{5}}~{}=~{}-\beta_{s}\alpha_{i}\frac{\Lambda_{s}}{\delta},$
$\displaystyle\frac{\partial^{2}f_{2}}{\partial x_{3}\partial x_{6}}$
$\displaystyle=$
$\displaystyle-\beta_{s}\alpha_{i}\frac{\Lambda_{s}}{\delta},~{}~{}~{}\frac{\partial^{2}f_{2}}{\partial
x_{4}\partial x_{6}}~{}=~{}-\beta_{s}\alpha_{u}\frac{\Lambda_{s}}{\delta},$
$\displaystyle\frac{\partial^{2}f_{2}}{\partial x_{4}\partial x_{5}}$
$\displaystyle=$
$\displaystyle-\beta_{s}\alpha_{u}\frac{\Lambda_{s}}{\delta},~{}~{}\frac{\partial^{2}f_{2}}{\partial
x_{2}\partial x_{2}}~{}=~{}-2\beta_{s}\alpha_{a}\frac{\Lambda_{s}}{\delta},$
$\displaystyle\frac{\partial^{2}f_{2}}{\partial x_{3}\partial x_{3}}$
$\displaystyle=$
$\displaystyle-2\beta_{s}\alpha_{i}\frac{\Lambda_{s}}{\delta},~{}~{}~{}\frac{\partial^{2}f_{2}}{\partial
x_{4}\partial x_{4}}~{}=~{}-2\beta_{s}\alpha_{u}\frac{\Lambda_{s}}{\delta}.$

The rest of the partial derivatives at the infection free steady state $E^{0}$
remains zero. Now, we compute the coefficients $a$ and $b$ due to the well-
known Theorem 4.1 by Castillo-Chavez & Song Castillo04  as follows:

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

and

$\displaystyle b$ $\displaystyle=$
$\displaystyle\sum_{i,k=1}^{6}\upsilon_{k}\omega_{i}\frac{\partial^{2}f_{k}(0,0)}{\partial
x_{i}\partial\beta_{s}}.$

By substituting the values of all the nonzero second-order partial derivatives
and the left and right eigenvectors from above analysis at the threshold
$\beta_{s}=\beta_{s}^{c}$, we have

$\displaystyle a$ $\displaystyle=$
$\displaystyle-\frac{\beta_{s}\upsilon_{2}\Lambda_{s}}{\delta}(\omega_{2}\omega_{3}(\alpha_{a}+\alpha_{i})+\omega_{2}\omega_{4}(\alpha_{a}+\alpha_{u})$
$\displaystyle+$
$\displaystyle\omega_{2}\omega_{5}\alpha_{a}+\omega_{2}\omega_{6}\alpha_{a}+\omega_{3}\omega_{4}(\alpha_{i}+\alpha_{u})+\omega_{3}\omega_{5}\alpha_{i}$
$\displaystyle+$
$\displaystyle\omega_{3}\omega_{6}\alpha_{i}+\omega_{4}\omega_{6}\alpha_{u}+\omega_{4}\omega_{5}\alpha_{u}+2\omega_{2}^{2}\alpha_{a}$
$\displaystyle+$ $\displaystyle
2\omega_{3}^{2}\alpha_{i}+2\omega_{4}^{2}\alpha_{u}),$

and

$\displaystyle b$ $\displaystyle=$
$\displaystyle\frac{\upsilon_{2}\Lambda_{s}}{\delta}(\omega_{2}\alpha_{a}+\omega_{3}\alpha_{i}+\omega_{4}\alpha_{u}).$

From the above expressions, it can be observed that $a<0$ and $b>0$, therefore
due to the Remark 1 of the well-known Theorem 4.1 by Castillo-Chavez & Song
Castillo04 , and by Khajanchi et al. Khajanchi18  a transcritical bifurcation
occurs at the basic reproduction number $\mathcal{R}_{0}=1$ and the interior
equilibrium point $E^{*}$ is locally asymptotically stable for
$\mathcal{R}_{0}>1$.

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