# Dynamics of the COVID-19 pandemic in India

Subhas Khajanchi<EMAIL_ADDRESS>Department of Mathematics,
Presidency University, 86/1 College Street, Kolkata 700073, India  Kankan
Sarkar  Department of Mathematics, Malda College, Malda, West Bengal 712101,
India  Jayanta Mondal  Department of Mathematics, Diamond Harbour Women s
University, Diamond Harbour Road, Sarisha 743368, India

###### Abstract

Understanding the dynamics of the COVID-19 pandemic is crucial for improved
control and social distancing strategies. To that effect, we have employed the
susceptible-exposed-infectious-recovered model, refined by contact tracing and
hospitalization data from Indian provinces Kerala, Delhi, Maharashtra, and
West Bengal, as well as from overall India. We have performed a sensitivity
analysis to identify the most crucial input parameters, and we have calibrated
the model to describe the data as best as possible. Short-term predictions
reveal an increasing and worrying trend of COVID-19 cases for all four
provinces and India as a whole, while long-term predictions also reveal the
possibility of oscillatory dynamics. Our research thus leaves the option open
that COVID-19 might become a seasonal occurrence. We also simulate and discuss
the impact of media on the dynamics of the COVID-19 pandemic.

##  I Introduction

According to the World Heath Organization (WHO), 1,878,489 confirmed cases
(119,044 confirmed deaths) of novel coronavirus disease 2019 (2019-nCoV) has
been reported throughout the world, as of April 15, 2020, including 11,439
confirmed cases (377 deaths) from India A-who . The outbreak was first
announced by the “Health Commission of Hubei province”, China, a cluster of
unexplained cases of pneumonia of unknown etiology (unknown cases) Zhu20 ,
which is lethal, was first identified in Wuhan, the capital city of Hubei and
seven largest city of China, on December 31, 2019 Chan20 ; Cohen20 ; Tang20 .
Later on, the novel coronavirus disease was called officially as COVID-19 by
WHO A-who . The epidemic was announced a major public concern worldwide on
January 30, 2020 by the World Health Organization Zhu20 . Coronavirus, an
enveloped virus described by a single-stranded, positive-sense RNA belonging
to the family of Coronaviridae and the Nidovirales order and extensively
disseminated among humans as well as mammals Chen20 . The ongoing coronavirus
epidemic has been announced a pandemic by the World Health Organization (WHO)
on January 30, 2020, and the Govt. of India has announced 21 days nationwide
lockdown from March 25, 2020 to April 14, 2020, to prevent stage-III spreading
of the virus or human-to-human transmission.

Coronavirus causes typically mild infections but sometimes lethal communicable
disorders leading to Severe Acute Respiratory Syndrome (SARS) and Middle East
Respiratory Syndrome (MERS) and in present COVID-19 pandemic, and in general
it can be thought as SARSr-CoV Gumel04 , which is mostly observed in bats-
could create a future disease outbreak Drosten03 ; Zhou20 . The most important
modes of transmission of coronavirus are respiratory droplets and contact
transmission (contaminated hands) and it has incubation period 2-14 days AA-
who ; Chan20 . From the confirmed cases of coronavirus, the symptoms range
from fever, sneeze, or even a runny nose, dry cough fatigue breathing problem
and lung infiltration to severely ill and dying AA-who . Shared surfaces are
also an important risky since the virus clings on to surfaces and propagates
as soon as another person touches the shared surface. The virus also can
remain active for about 3 hours after aerosolization (droplets carrying virus
mixed with air), on plastic for fifteen hours, and on metallic surfaces for
five-six hours.

The continuing coronavirus outbreak has been announced a widespread by the
World Health Organization, as of March 16, 2020, the United Nations’ Public
Health Agency has authenticated 1,67,511 confirmed cases, including 6,606
deaths across 151 countries throughout the world A-who ; Chan20 . As countries
start shutting down borders, cutting off international marketing and
quarantining their individuals, the key to safeguard our families, near ones
and communities lies in understanding the nature and behavior and why and how
this particular virus is developing. Some other factors like population
mobility by air travel, the documented human-to-human transmission, low air
temperature and low humidity highly affect the transmission of coronavirus
A-who ; Jingyuan20 .

Throughout the world, the air travel is one of the most important dynamic
network, and India is in 17th position worldwide the countries with highest
chance of importation of coronavirus through air traveler AA-trav ; Mandal20 .
The probability of an infected air traveler to reach to India at final
destination is 0.383% (as on April 07, 2020), with maximum relative import
risk in Delhi (0.124%, as on April 07, 2020), Mumbai (0.064%, as on April 07,
2020), Kolkata (0.021%, as on April 07, 2020), Madras, Kochi, Bangalore,
Hyderabad and so on AA-trav .

At the beginning, the Ministry of Health and Family Welfare (MoHFW), Govt. of
India suggested to discontinue to travel from China to India and send to
quarantine who are coming from China AA-CH-tra . Those who return from Wuhan,
the sprawling capital city of Hubei province in China, later on January 15,
2020 were to be examined for coronavirus. Also, those who are feeling sick
within one month of returning from China were suggested to admit the nearby
health center in addition to maintain self-isolation at home AA-CH-tra1 . At
the very beginning, thermal entry screening of travelers from Wuhan (China)
was set up around twenty one airports throughout the country with
international screening for flights from Tiwan, China, Singapore, South Korea,
Italy, Hong-Kong, Japan and so on. Symptomatic ill travelers are suggested to
volunteer for screening test. Similar screening test has also be taken into
account at the international harbor AA-CovIn .

Governments will not be competent to diminish both fatalities from coronavirus
epidemic and the economic impact of viral outbreak. Maintaining the fatality
rate as minimum as possible will be the utmost importance for the populations;
therefore the Governments must put in place measures to mitigate the
unavoidable economic downturn. In our viewpoint, coronavirus has turned into a
pandemic, with small chains of transmission in several countries and big
chains of transmission resulting in large-scale outbreak in most of the
countries, namely United States, Spain, Italy, South Korea, Germany, France
and so on A-who .

Till date, there is no specific licensed vaccine, antivirals or effective
therapeutics to treat coronavirus infections. Due to absence of coronavirus
therapeutics, we utilized the Non-Pharmacological interventions (NPIs) -
focused at minimizing transmission by reducing contact rates among
individuals. As for examples the measures adopted in this time incorporated
social distancing, closing schools, universities, offices, churches, bars,
avoid mass gatherings, other social places as well as contacts of cases
(quarantine, surveillance, contact tracing) Ferguson20 .

The route of an outbreak can be described by a series of important factors,
but some of which are extremely difficult to understand at present coronavirus
disease. The basic reproduction number $R_{0}$ is one of the most crucial
quantities in infectious diseases, as $R_{0}$ measures how contagious a
disease is. $R_{0}$ designates the number of secondary infections infected by
one infected individual in a whole susceptible class, or more specifically the
area under epidemic curve. For $R_{0}<1$, the disease is expected to stop
spreading, but for $R_{0}=1$ an infected individual can infect on an average 1
person, that is, the spread of the disease is stable. The disease can spread
and become epidemic if $R_{0}$ must be greater than 1. $R_{0}$ can help us to
understand the effectiveness of the disease, that is, under what condition the
disease can stop or spread ? The values for $R_{0}$ in China is around 2.5 in
the primary stage of the outbreak of coronavirus Anderson20 .

The spread of coronavirus outbreak depends on the infectivity of the virus and
the availability of susceptible individuals. Mathematical modeling play a
vital role to better understand the disease dynamics and designing policies to
manage quickly spreading infectious diseases in lack of effective vaccine or
specific antivirals Egger17 . Recently, many mathematical model have already
been studied to understand the complicated dynamics of novel coronavirus, and
some of these are cited in our references Chan20 ; Chen20a ; Imai20 ; Liu20 ;
Mandal20 ; Nadim20 ; Tang20 ; Tang20a ; Wu20 . A novel coronavirus model has
been established by Chen et al. Chen20a  and find the basic reproduction
number. Imai et al. Imai20  studied a computational model for novel
coronavirus disease 2019 in Wuhan, a city of China and their model mainly
focused on human-to-human transmission. Tang et al. Tang20  established a
compartmental model for coronavirus by incorporating symptomatic ill class to
obtain the patients’ epidemiological status. They obtained the basic
reproduction number 6.47, which is very high for the infectious diseases.
Nadim et al. Nadim20  studied a mathematical to investigate the coronavirus
disease, where they performed stability analysis and they validated their
model with the data from Hubei, the city of China. Their model is originally
established by Gumel et al. Gumel04  for Severe Acute Respiratory Syndrome
(SARS) outbreak. Wu et al. Wu20  established a coronavirus model by
considering four classes, namely susceptible, exposed, infected and recovered
to study the human-human transmission dynamics based on the data from December
31, 2019 to January 28, 2020. They computed the reproduction number
approximately 2.68 for coronavirus disease.

The conventional models governing the outbreak of infectious diseases mainly
depend on the interplays among susceptible, infected and exposed classes.
Moreover, the other aspects, such as vaccination, huge news coverage,
educational campaigns and rapid information flow can create great
psychological effects on the population, and thus remarkably change the
publics’ behavior and affect the implementation of individuals’ intervention
and control strategies Funk09 . It is mainly the awareness program due to
media which make the individuals enlighten regarding the disease to take
safeguards such as vaccination, wearing protective masks, social distancing
etc., to minimize their risks of being infected d'Onofrio07 ; d'Onofrio09 .
How long and how effectual media related impact persists is thus an important
issue for future outbreak, and quantifying this impact through a modeling
process falls within the scope of our investigation. Several mathematical
models have been incorporated with the assumption that the media related
awareness has an impact on reducing the contact rate of susceptible and
infected population Buonomo08 ; Funk09 ; Kumar17 ; Sun11 ; Das20 ; Xiao15 .
Influenced by those literatures we develop and analyze a mathematical model
for coronavirus to study the impact of media related awareness program.

To obtain a better insight into the important factors related with the control
of novel coronavirus in a community and throughout the world, we investigate a
dynamic model to study the epidemic of coronavirus and its control in four
mostly affected states, namely Maharashtra, Delhi, Kerala, West Bengal and the
overall India. We estimated most important parameters, namely $\beta_{s}$,
$\xi_{a}$, $q_{a}$, $\alpha_{h}$ $\xi_{h}$, $\xi_{i}$ and $\gamma_{i}$ by
using least square method. We calculate the basic reproduction for our model
for four different states as well as the India.

##  II Dynamic model without effective control measures

Here, we extend the classical deterministic susceptible-exposed-infectious-
removed (SEIR) compartmental model refined by introducing contact tracing-
hospitalization strategies to study the epidemiological properties of
coronavirus (COVID-19). We calibrate our mathematical model using data gained
form confirmed cases of coronavirus in India and estimated the basic
reproduction number for the disease transmission. In order to make our
mathematical model more realistic, we consider the following assumptions:

  * •

the model involves a net inflow rate of susceptible individuals $\Lambda_{s}$
per unit time,

  * •

the model has no zoonotic infections of coronavirus, considering only the
epidemic spread through human beings,

  * •

there is no effective control measures before April 10, 2020,

  * •

the model involves some demographic effects by accepting a proportional
natural mortality in each of the sub-populations.

In the underlying dynamic model, the total population $N(t)$ is classified
into six sub-populations (classes), namely susceptible $S(t)$, exposed $E(t)$,
asymptomatic $A(t)$, clinically ill or symptomatic $I(t)$, hospitalized $H(t)$
and recovered $R(t)$, respectively.

The coronavirus outbreaks in the population throughout the world and so the
news concerning coronavirus epidemics through different media including
television, social networking sites and newspaper as well as educational
programs from government, non-government organization and local bodies. The
source of information density of media is assumed to be proportional to the
number of symptomatic/infective individuals and will alter as the symptomatic
population alters. Herein, $M(t)$ denotes the cumulative density of media in
individual such that $M(t)$ = 0, for $I(t)$ = 0. This is mimic to the
mathematical model developed by d’Onofrio et al. d'Onofrio07 , in the case of
non-delayed model and mimics to the media variable as incorporated there. This
media creates the behavioral alteration in the susceptible populations to save
themselves from contracting infection. As for example, Government of Senegal
has taken an initiative through SMS campaign to spread the information
regarding Ebola awareness Govt . Albeit individuals are informed through
media, not everyone responses to it due to insufficiency of resources,
irresponsive nature and economic condition etc. Therefore a part of
susceptible individuals with news regarding coronavirus, responding to the
news for corona symptoms and altering their behavior to avoid infection, and
will move to the recovered individuals. The rate of behavioral response
through media has been considered to be a function of both the densities for
susceptible populations and media, that is, $f(S(t),M(t))$. Moreover, we
assume that the source of media is a function of $I(t)$, that is,
$f_{m}(I(t))$, as the source of media depends on the density of symptomatic
populations. A schematic representation of the above biological mechanism of
coronavirus in human upon which our model is based is shown in the Figure 1.
The dynamics of novel coronavirus model is governed by the following system of
nonlinear ordinary differential equations:

###  II.1 Dynamics of Susceptible individuals $S(t)$

The susceptible individual is recruited into the population with a constant
inflow rate $\Lambda_{s}$ and decreased by a natural mortality mortality rate
$\delta_{s}$. Here, $\beta_{s}\alpha_{a}$ represents the transmission
coefficient of the asymptomatic individuals to susceptible,
$\beta_{s}\alpha_{i}$ is the transmission coefficient of symptomatic infected
classes to the susceptible, and $\beta_{s}\alpha_{h}$ denotes transmission
coefficient of the hospitalized classes to susceptible population. We consider
$\beta_{s}$ is the disease transmission coefficient for both the
infectiousness of COVID-19 and contact rates, with adjustment factors
$\alpha_{a}$ (for asymptomatic classes), $\alpha_{i}$ (for symptomatic
classes) and $\alpha_{h}$ (for hospitalized individuals).

Albeit, coronavirus is supposed to be spread completely by symptomatic
populations, a very lower rate of transmission by asymptomatic populations
cannot even be ruled out. The adjustment parameter $\alpha_{a}\geq 0$ take
into accounts for varying levels of hygiene preventions through asymptomatic,
an analogous explanation can be drawn to the adjustment parameters
$\alpha_{i}$ and $\alpha_{h}$ during symptomatic and hospitalization,
respectively. As the asymptomatic, symptomatic and hospitalization programmes
and hygiene preventions during asymptomatic, symptomatic and hospitalization
were executed and enriched progressively after an epidemic of coronavirus, the
spreading coefficients $\alpha_{a}\beta_{s}$, $\alpha_{i}\beta_{s}$ and
$\alpha_{h}\beta_{s}$, asymptomatic, symptomatic and hospitalized rates could
be modeled as a time-dependent parameters in computations. Moreover, the
interplay among susceptible and infected population (asymptomatic, symptomatic
and hospitalized) is modeled utilizing standard homogeneous mixing incidence
Anderson91 ; Diekmann00 , in the form of total individuals. The rate of change
of susceptible individuals can be represented by the following ordinary
differential equation (ODE):

$\displaystyle\frac{dS}{dt}$ $\displaystyle=$
$\displaystyle\Lambda_{s}-\frac{\beta_{s}S(\alpha_{a}A+\alpha_{i}I+\alpha_{h}H)}{N}-\delta_{s}S-f(S,M).$
(1)

###  II.2 Dynamics of Exposed individuals $E(t)$

The population who are exposed individuals but not yet developed clinical
symptoms of coronavirus. The exposed individual is declined by asymptomatic
individuals at the rate $\gamma_{e}$ and natural death at the rate
$\delta_{e}$. The constant per-capita rate $\gamma_{e}$ modeled progression
from the exposed individuals $(E)$ to either the asymptomatic individuals
$(A)$ or symptomatic individuals $(I)$; the constant $q_{a}(0<q_{a}<1)$
represents budget for the rate of progress of population who move to either
the asymptomatic partially-infectious class $(A)$ or symptomatic class $(I)$
at the per-capita rate $\gamma_{e}$. The corresponding model dynamics for
exposed population can be presented by the following ODE:

$\displaystyle\frac{dE}{dt}$ $\displaystyle=$
$\displaystyle\frac{\beta_{s}S(\alpha_{a}A+\alpha_{i}I+\alpha_{h}H)}{N}-\delta_{e}E-\gamma_{e}E.$
(2)

###  II.3 Dynamics of Asymptomatic individuals $A(t)$

Asymptomatic populations were exposed to the coronavirus but not infectious
for the community. The exposed classes move to the asymptomatic classes at the
rate $\gamma_{e}$ by a constant portion $(1-q_{a})$. The asymptotic
individuals progress to the recovered class at the per-capita rate $\xi_{a}$
and natural mortality rate $\delta_{a}$. The rate of change of asymptomatic
individuals is govern by the following ODE:

$\displaystyle\frac{dA}{dt}$ $\displaystyle=$
$\displaystyle(1-q_{a})\gamma_{e}E-\delta_{a}A-\xi_{a}A.$ (3)

###  II.4 Dynamics of Symptomatic individuals $I(t)$

The symptomatic individuals are produced after the progression of clinical
symptoms of coronavirus (diagnosed by the clinicians) by the asymptomatic
individuals. The exposed individuals move to the symptomatic classes at the
per-capita rate $\gamma_{e}$ by a constant portion $q_{a}$. The symptomatic or
clinically ill individuals progress to the hospitalized or isolated class
$(H)$ at the per-capita rate $\gamma_{i}$ and natural mortality rate
$\delta_{i}$. It can be assumed that $\gamma_{i}$, the per-capita rate at
which symptomatic or clinically ill classes seek clinical attention and are
therefore put into the hospitalized class, is bigger than $\gamma_{e}$, the
per-capita rate at which the exposed individuals progress to the asymptomatic
classes. The clinically ill classes (class $I$) are moved to the recover
classes without being diagnosed at the per-capita rate $\xi_{i}$. The
transmission dynamics of symptomatic populations are governed by the following
ODE:

$\displaystyle\frac{dI}{dt}$ $\displaystyle=$ $\displaystyle
q_{a}\gamma_{e}E-\delta_{i}I-\xi_{i}I-\gamma_{i}I.$ (4)

###  II.5 Dynamics of Hospitalized individuals $H(t)$

These are the populations who have progressed clinical symptoms for
coronavirus and have been isolated at the hospital for medical treatment. The
hospitalized individuals are came from the clinically ill classes or
symptomatic community (class $I$) at the per-capita rate $\gamma_{i}$ and it
has natural mortality rate $\delta_{h}$. Hospitalized or isolated populations
recover at the per-capita rate $\xi_{h}$. We assumed that $\xi_{h}>\xi_{i}$,
as an hospitalized populations are likely to obtain a partially efficacious
during isolation or hospitalization. The dynamics of hospitalized individuals
are modeled by the following ODE:

$\displaystyle\frac{dH}{dt}$ $\displaystyle=$
$\displaystyle\gamma_{i}I-\delta_{h}H-\xi_{h}H.$ (5)

###  II.6 Dynamics of Recovered individuals $R(t)$

It can be assumed that the recovered population have enduring immunity against
coronavirus albeit to date this has neither been confirmed nor contradicted.
The asymptomatic, symptomatic and hospitalized individuals recover from the
coronavirus at the per-capita rates $\xi_{a}$, $\xi_{i}$ and $\xi_{h}$,
respectively. The recovered individuals die at the per-capita rate
$\delta_{r}$. The rate of change of recovered individuals can be written by
the ODE:

$\displaystyle\frac{dR}{dt}$ $\displaystyle=$
$\displaystyle\xi_{a}A+\xi_{i}I+\xi_{h}H-\delta_{r}R+f(S,M).$ (6)

###  II.7 Information variable or media awareness program $M(t)$

The function $f(S,M)$ indicates the behavioral response for the susceptible
population, induced by the effects of media related information of the
prevalence of coronavirus, by considering accessible precautionary measures to
become aware about the coronavirus disease. The behavioral response is not
completely efficacious due to the financial assistance, lack of the proper
resources and an irresponsive behavior among the populations, we consider the
media term as $f(S,M)=\eta_{m}d_{m}SM$. Here, $\eta_{m}$ represents the
corresponding response rate and $0\leq\eta_{m}\leq 1$ determines the response
intensity. The parameter $d_{m}$ indicates the media related information index
rate by which the population alter their behavior due to the prevalence of
coronavirus. The parameter $\delta_{m}$ represents the natural degradation of
media related information index, which can be occurred due to the fading of
memory about media related information with time in addition to the complacent
behavior of population.

We assume that the source of media related information index is proportional
to the number of symptomatic individuals. The function $f_{m}(I)$ represents
the growth of information index rate, which depend on the symptomatic
individuals. While the coronavirus outbreaks, the World Health Organization
(WHO), health agencies, social media, government and non-government
organization become dynamic to spread the information or media awareness in
concerning the prevalence of coronavirus.

As for the function $f_{m}(I)$, one can consider any function satisfying
$f_{m}(0)=0$ and $f^{\prime}_{m}(I)>0$ Buonomo08 . The most easiest example is
$f_{m}(I)=\alpha_{m}I$ with $\alpha_{m}\in(0,1)$, that is, $I$ is linear
function of the present prevalence of the coronavirus, which could represent
for instance current standardized incidence of serious cases of the
coronavirus d'Onofrio07 . One more example could be
$f_{m}(I)=\frac{\alpha_{m}I}{1+\beta_{m}I}$, where $I$ indicates the nonlinear
increasing function for the prevalence of the coronavirus, which could be
considered as a measure of the perceived risk for the infection rather than
the original one d'Onofrio09 ; Reluga06 . Here $\alpha_{m}$ represents the
source rate of the information index and $\beta_{m}$ is half saturation
constant. Thus, we consider the following ODE model for media awareness
program:

$\displaystyle\frac{dM}{dt}$ $\displaystyle=$ $\displaystyle
f_{m}(I)-\delta_{m}M.$ (7)

Based on the above biological assumptions and schematic representation of
coronavirus (see the Figure 1) together with specific forms, we formulate a
mathematical model of novel coronavirus with media awareness given by the
following seven-dimensional nonlinear system of ODEs:

$\displaystyle\left\\{\begin{array}[]{ll}\frac{dS}{dt}=\Lambda_{s}-\frac{\beta_{s}S(\alpha_{a}A+\alpha_{i}I+\alpha_{h}H)}{N}-\delta_{s}S-\eta_{m}d_{m}SM,\\\
\frac{dE}{dt}=\frac{\beta_{s}S(\alpha_{a}A+\alpha_{i}I+\alpha_{h}H)}{N}-\delta_{e}E-\gamma_{e}E,\\\
\frac{dA}{dt}=(1-q_{a})\gamma_{e}E-\delta_{a}A-\xi_{a}A,\\\
\frac{dI}{dt}=q_{a}\gamma_{e}E-\delta_{i}I-\xi_{i}I-\gamma_{i}I,\\\
\frac{dH}{dt}=\gamma_{i}I-\delta_{h}H-\xi_{h}H,\\\
\frac{dR}{dt}=\xi_{a}A+\xi_{i}I+\xi_{h}H-\delta_{r}R+\eta_{m}d_{m}SM,\\\
\frac{dM}{dt}=\alpha_{m}I-\delta_{m}M,\end{array}\right.$ (15)

the model is satisfied the following positive initial conditions:

$\displaystyle
S(0)=S_{0},~{}~{}~{}E(0)=E_{0},~{}~{}~{}A(0)=A_{0},~{}~{}~{}I(0)=I_{0},~{}~{}~{}H(0)=H_{0},~{}~{}~{}R(0)=R_{0},~{}~{}~{}M(0)=M_{0}.$
(16)

Since the model system (15) monitors the dynamics of human population, all
state variables are assumed to be positive.

###  II.8 Basic reproduction number

The basic reproduction number, symbolized by $R_{0}$, is ‘the expected number
of secondary cases produced, in a completely susceptible population, by a
typical infective individual’ Diekmann90 ; PVDD02 . The dimensionless basic
reproduction number provides a threshold, which play a crucial role in
determining the disease persists or dies out from the individual. In a more
general way $R_{0}$ can be stated as the number of new infections created by a
typical infective population at an infection free equilibrium. $R_{0}<1$
determines on average an infected population creates less than one new
infected population during the course of its infective period, and the
infection can die out. In reverse way, $R_{0}>1$ determines each infected
population creates, on average, more than one new infection, and the disease
can spread over the population. The basic reproduction number $R_{0}$ can be
computed by using the concept of next generation matrix Diekmann90 ; PVDD02 .
In order to do this, we consider the nonnegative matrix $\mathcal{F}$ and the
non-singular $M-$matrix $\mathcal{V}$, expressing as the production of new-
infection and transition part respectively, for the system (15), are described
by

$\displaystyle\mathcal{F}$ $\displaystyle=$
$\displaystyle\left[\begin{array}[]{c}\frac{S}{N}\beta_{s}(\alpha_{a}A+\alpha_{i}I+\alpha_{h}H)\\\
0\\\ 0\\\
0\end{array}\right],~{}~{}~{}~{}~{}~{}~{}~{}~{}\mathcal{V}=\left[\begin{array}[]{c}(\delta_{e}+\gamma_{e})E\\\
-(1-q_{a})\gamma_{e}E+(\delta_{a}+\xi_{a})A\\\
-q_{a}\gamma_{e}E+(\delta_{i}+\xi_{i}+\gamma_{i})I\\\
-\gamma_{i}I+(\delta_{h}+\xi_{h})H\end{array}\right].$

The Jacobian matrix for the system (15), can be computed at an infection free
state $(E=A=I=H=0)$, we have

$\displaystyle F$ $\displaystyle=$
$\displaystyle\left[\begin{array}[]{cccc}0&\beta_{s}\alpha_{a}&\beta_{s}\alpha_{i}&\beta_{s}\alpha_{h}\\\
0&0&0&0\\\ 0&0&0&0\\\
0&0&0&0\end{array}\right]~{}~{}~{}~{}~{}~{}~{}~{}V~{}=~{}\left[\begin{array}[]{cccc}\delta_{e}+\gamma_{e}&0&0&0\\\
-(1-q_{a})\gamma_{e}&(\delta_{a}+\xi_{a})&0&0\\\
-q_{a}\gamma_{e}&0&(\delta_{i}+\xi_{i}+\gamma_{i})&0\\\
0&0&-\gamma_{i}&(\delta_{h}+\xi_{h})\end{array}\right]$

The basic reproduction number $R_{0}=\rho(FV^{-1})$, where $\rho(FV^{-1})$
represents the spectral radius for a next generation matrix $FV^{-1}$. Thus,
from the system (15), we get the basic reproduction number $R_{0}$ is

$\displaystyle\rho(FV^{-1})$ $\displaystyle=$ $\displaystyle
R_{0}=\frac{(1-q_{a})\gamma_{e}\beta_{s}\alpha_{a}}{(\delta_{e}+\gamma_{e})(\delta_{a}+\xi_{a})}+\frac{q_{a}\gamma_{e}\beta_{s}\alpha_{i}}{(\delta_{e}+\gamma_{e})(\delta_{i}+\xi_{i}+\gamma_{i})}+\frac{q_{a}\beta_{s}\alpha_{h}\gamma_{e}\gamma_{i}}{(\delta_{e}+\gamma_{e})(\delta_{i}+\xi_{i}+\gamma_{i})(\delta_{h}+\xi_{h})}.~{}~{}~{}~{}~{}~{}~{}~{}~{}$
(19)

##  III Model calibration and coronavirus data source

A detailed elucidation on the computer simulations for the coronavirus model
system (15) is investigated in this section. We start by building up on the
parameter estimations followed by some computer simulation results, which
provide insights on the eradication and development of the dynamics of novel
coronavirus (COVID-19) epidemic. Parameter values are taken from literature
and some of the parameters are estimated from the observed data. List of the
parameter values with description are given in the Table 1 and the values of
the remaining parameters estimated from the observed data, which are given in
the Table 2. Due to the lack of the data of coronavirus diseases related to
the effect of media related awareness, we have simulated our model without the
effect of media awareness. But, we have simulated our model with the effect of
media related awareness for four different states, namely Delhi, Maharashtra,
Kerala and West Bengal as well as the Republic of India. The initial
population sizes for different states, namely Kerala, Maharashtra, Delhi, West
Bengal and the Republic of India are stated in the Table 4. Also the
recruitment rate $\Lambda_{s}$ for four different states and overall India are
listed in the Table 4.

###  III.1 PRCC sensitivity analysis

To identify the most sensitive parameters with respect to symptomatic
population, we performed a sensitivity analysis by using Partial Rank
Correlation Coefficient (PRCC) technique for all the input parameters against
the variable $I$. The set of most sensitive parameters, which have the ability
to reduce the outbreak of coronavirus diseases, can be estimated from the
available data from different states (Kerala, Maharashtra, Delhi, West Bengal)
as well as the scenario of entire India. In our coronavirus system (15) has 17
parameters (without media) for which we varied all the system parameters
simultaneously. PRCC quantifies the relationship among a state equation of
interest and each of model parameters. As a result PRCC aid to obtain the
important parameters, which contribute most to the system variability. For
model simulation, we consider the PRCC values between -1.0 and +1.0.

Following the method established by Marino et al. Marino08 , we performed
Latin hypercube sampling and generated 2800 samples to compute the PRCC and
the p-values with reference to the symptomatic classes at the day 60. The
Figure 2 designates the PRCC results, which implies that the highest
positively correlated parameters are the disease transmission coefficient rate
$\beta_{s}$, proportion rate of exposed individuals $q_{a}$, the rate
$\xi_{h}$ by which the hospitalized class converted to recovered class, the
rate $\alpha_{h}$ by which susceptible individuals converted to exposed class
and the highly negatively correlated parameters are the rate $\xi_{i}$ by
which symptomatic individuals become recovered, the rate $\gamma_{i}$ by which
symptomatic individuals converted to hospitalized class, and the rate
$\xi_{a}$ by which the asymptomatic individuals moved to recovered
individuals, account for most uncertainty with respect to the symptomatic
population. Thus, our PRCC analysis yields these 7 parameters $\beta_{s}$,
$q_{a}$, $\xi_{i}$, $\gamma_{i}$, $\xi_{h}$, $\alpha_{h}$ and $\xi_{a}$ are
the most effective parameters out of 17 parameters. Therefore, we estimated
the 7 parameters by using least square method.

###  III.2 Parameter estimation

Now, we have calibrated our coronavirus disease model (15) with daily
confirmed new coronavirus cases for the entire India and four states namely,
Kerala, Delhi, West Bengal and Maharashtra. Data of daily confirmed new
coronavirus cases for whole India obtained from World Health Organization
(WHO) situation report for the period January 30, 2020 - March 30, 2020. Data
of daily confirmed new coronavirus cases for the state West Bengal obtained
from Health & Family Welfare Department, Government of West Bengal for the
period March 17, 2020 - March 31, 2020. Data of daily confirmed new
coronavirus cases for the state Kerala obtained from the Directorate of Health
Services, Government of Kerala for the period January 30, 2020 - March 31,
2020. The data of daily confirmed new coronavirus cases for the state Delhi
obtained from the Health & Family Welfare, Govt. of the National Capital
Territory of Delhi for the period March 4, 2020 - March 30, 2020. Also, the
data of total coronavirus cases for the state Maharashtra obtained from India
COVID-19 Tracker (https://www.covid19india.org/) for the period March 15, 2020
- March 31, 2020. We fit our coronavirus model system (15) (without media
related awareness program) with daily new coronavirus cases for the four
states (Delhi, Kerala, Maharashtra and West Bengal) and for India. Also, the
cumulative coronavirus cases has been fitted with our model system. The
observed data were fitted in the optimization toolbox of Matlab using
nonlinear least squares method and we have estimated the parameters
$\beta_{s}$, $\xi_{a}$, $q_{a}$, $\xi_{h}$, $\gamma_{i}$, $\xi_{i}$ and
$\alpha_{h}$ as these parameters found more sensitive in the PRCC analysis.
The biological interpretations of the model parameters are given in the Table
1. The estimated values of the model parameters for all the data sets are
given in the Table 2. The initial population sizes for all the data sets are
also estimated to match with the confirmed new coronavirus cases (see Table
4). The output of our model for daily new cases are shown in Figure 3. Also,
the observed daily new coronavirus cases are overlapped on the Fig. 3. It is
also important to observe the total number of coronavirus cases in a province
only. So, the observed daily cumulative confirmed coronavirus cases and the
model simulation has been plotted in the Fig. 4.

To assure the performance of our coronavirus system without media (15), we
have computed Mean Absolute Error $(E_{MAE})$ and Root Mean Square Error
$(E_{RMSE})$. Here, $E_{MAE}$ and $E_{RMSE}$ are given by

$\displaystyle E_{MAE}$ $\displaystyle=$
$\displaystyle\frac{\Sigma_{i=1}^{n}|O(i)-Q(i)|}{n},$ $\displaystyle E_{RMSE}$
$\displaystyle=$
$\displaystyle\sqrt{\frac{\Sigma_{i=1}^{n}(O(i)-Q(i))^{2}}{n}},$

where $O(i)$ represents the observed value, $Q(i)$ is the model output and $n$
is the sample size of the observed data. The values of $E_{MAE}$ and
$E_{RMSE}$ for the four states (Delhi, Kerala, Maharashtra and West Bengal)
and the Republic of India are presented in Table 3.

The fitting data enables us to quantify the basic reproduction number $R_{0}$
from the expression of (19). The basic reproduction number for coronavirus in
different states, for Kerala, $R_{0}=4.78435$; Delhi, $R_{0}=3.28271$; West
Bengal, $R_{0}=2.73854$; Maharashtra, $R_{0}=3.03971$; and for India,
$R_{0}=2.58792$. Let us observe that the basic reproduction is relatively high
in Kerala, relative to the other three states and the overall India considered
here. From the basic reproduction number $R_{0}$ in overall India, it can be
conclude that the coronavirus outbreaks is pandemic and the Govt. should take
a necessary action to keep social distancing and necessary precautions. The
basic reproduction number $R_{0}$ for four provinces and overall India are
presented the Table 5.

From the data fitting, it can be observed that our model performs relatively
good in all the states namely, Kerala, Delhi, West Bengal, Maharashtra. This
is showing a good agreement with the observed data and curve fitting with the
coronavirus disease model (15) under consideration. However, the model
performance less well in the case of whole India as the number of confirmed
coronavirus cases is relatively high. Though, the increasing trend of the
daily confirmed coronavirus cases have been nicely captured by the model for
all the data sets. We also compute the numerical errors for computing the
numerical simulations for our model and the corresponding lowest value of
$E_{MAE}$ and $E_{RMSE}$ have been obtained for the state West Bengal, since
the size of the data is lower than the other data sets.

###  III.3 Short-term prediction

Due to absence of any effective vaccine or treatment strategies for novel
coronavirus, predictive mathematical models can aid in understanding the
complicated dynamics of coronavirus transmission and its control as well as
its intricate epidemiological cycle. In order to study the short term
prediction for our model system (15), we have simulated the cumulative
symptomatic classes of coronavirus cases for the time period April 01, 2020 to
April 10, 2020 using the estimated parameters values in the Table 2 with
initial population sizes in the Table 4. The other baseline parameter values
are specified in the Table 1. The simulated results for the period April 01,
2020 to April 10, 2020 for the three provinces, namely Kerala, West Bengal,
Maharashtra and the union territory Delhi and the Republic of India have shown
in the Figure 5. The black solid curves are the model simulation and the solid
black circles are the observed values, as plotted in the Figure 5. Our model
simulation represents that the model performs well and successfully able to
capture the increasing trends of all the four cases (Delhi, Maharashtra, West
Bengal and the overall India) whereas the slope of the simulated curve is
relatively high for the state Kerala.

###  III.4 Sensitivity analysis for $R_{0}$

To describe how best to diminish human impermanency and morbidity due to novel
coronavirus, it is very essential to see the relative significance of various
factors responsible for its transmission. The initial disease transmission is
completely associated to the reproduction number $R_{0}$, and we calculate the
sensitivity indices with regard to $R_{0}$, to the parameters for the system
(15). The sensitivity indices describe us whether or not the infectious
diseases will develop throughout the individulas. Sensitivity analysis is
mainly utilized to describe the robustness of the reproduction number $R_{0}$
to fluctuation in the system parameters. It also give us to identify the
relative change in a state variable when a system parameter alters. The
normalized forward sensitivity index for $R_{0}$ with respect to the parameter
$p$ is defined as follows:

$\displaystyle\Gamma^{R_{0}}_{p}$ $\displaystyle=$
$\displaystyle\frac{\partial R_{0}}{\partial p}\times\frac{p}{R_{0}}.$

As we have the explicit expression for $R_{0}$ in (19), we obtain the
analytical formulae for sensitivity indices of $R_{0}$, $\Gamma^{R_{0}}_{p}$,
to each of fourteen parameters stated in the Table 6.

As for example, the sensitivity index of $R_{0}$ with regards to $\beta_{s}$
is

$\displaystyle\Gamma^{R_{0}}_{\beta_{s}}$ $\displaystyle=$
$\displaystyle\frac{\partial
R_{0}}{\partial\beta_{s}}\times\frac{\beta_{s}}{R_{0}}~{}=~{}1,$

independent of any parameters. This represents that $R_{0}$ is the increasing
function with respect to $\beta_{s}$. It implies that the probability of
disease transmission has a high impact on coronavirus control and management.
Other parameters have their sensitivity indices are specified in the Table 6.

From the Table 6, it can be noticed that some of the indices are positive and
some of the indices are negative, which indicate that the indices having ‘+’
signs increase the value of $R_{0}$ as one increase them and those having ‘-’
signs decrease the value of $R_{0}$, as they are increased. The most sensitive
parameter is the disease transmission coefficient $\beta_{s}$, where
sensitivity index $\Gamma^{R_{0}}_{\beta_{s}}~{}=~{}1$ indicates that the
increasing $\beta_{s}$ by 10% will increase $R_{0}$ by 10%. Also, the
sensitivity index of $\xi_{a}$, that is,
$\Gamma^{R_{0}}_{\beta_{s}}~{}=~{}-0.844286$ represents that increasing
$\xi_{a}$ by 10% will decrease $R_{0}$ by 8.44286%. Similarly, we can
interpret the other parameters too.

We plot the sensitivity indices of $R_{0}$, for each of the model parameters
using in the Table 6 has been shown in the Figure 6. In order to control the
coronavirus outbreaks, we must target the most sensitive parameters who has
the great impact in controlling the system dynamics. For instance, the disease
transmission coefficient $\beta_{s}$ is the most effective parameter in
controlling the coronavirus diseases, which can easily be identified from the
Figure 6 as well as in the Table 6.

To control the coronavirus outbreaks, the draw the contour plots for the
$R_{0}$ with respect to the positively correlated parameters, namely disease
transmission coefficient $\beta_{s}$ and the clinical outbreak rate $q_{a}$
for all the infected classes in the Figure 7. The contour plot represents the
dependence of $R_{0}$ for the four states (Kerala, Delhi, West Bengal,
Maharashtra) and the Republic of India. Contour plot indicates that for the
higher values of $\beta_{s}$ and $q_{a}$ the reproduction number $R_{0}$
become high, which means the coronavirus may enter the stage-3 and it spread
through the community. Therefore, to control $R_{0}$ must be restrict the
values of the transmission coefficient $\beta_{s}$ and the portion $q_{a}$ by
which the exposed class become symptomatically ill. To keep the mortality rate
as low as possible, we must be take the values of $\beta_{s}$ and $q_{a}$ as
small as possible. The Figure 6 and Table 6 represents that the adjust factors
$\alpha_{a}$, $\alpha_{i}$ and $\alpha_{h}$ are positively correlated with the
basic reproduction number $R_{0}$. Thus, we must have to reduce the adjust
factors too. Thus, we may conclude that to reduce the coronavirus disease
outbreak we need to maintain social distancing, limit or stop taking fairs and
theaters performances etc.

###  III.5 Effect of media related awareness

To study the impact of media related awareness for the coronavirus disease we
simulate our system (15) for four provinces, namely Kerala, Maharashtra,
Delhi, West Bengal and the Republic of India, with respect to the parameter
values specified in the Table 1 and rest of the parameter values estimated and
are given in the Table 2 with initial population sizes are given in the Table
4. The Figure 8 represents the simulation of our model with media awareness
for daily confirmed new cases of coronavirus of infected patients. The first
sub-figure of the Figure 8 represents the corresponding profiles of
symptomatic individuals for the overall India in absence and presence of media
awareness programme with the available time period 60 days. Observe that the
blue curve has the highest peak or the proliferation of symptomatic population
is high and sharp in compare to the other curves, which indicates that the
sizes of symptomatic individuals increases and reaches to its maximum peak in
absence of media awareness as the source of information index $(\alpha_{m})$
and the rate of media response $(\eta_{m})$ both are zero. To investigate how
often and/or how long the media awareness effect remains dynamic, we simulate
our coronavirus disease model (15). But in presence of active media awareness,
both $\alpha_{m}$ and $\eta_{m}$ are nonzero, and for fixed $\alpha_{m}=0.01$
when we varied the rate of media response $\eta_{m}=0.1$ and $\eta_{m}=0.5$,
the thick red curve and dotted red curve decreases but the peak of both the
red curves are below the thick blue curve. For fixed media response
$\eta_{m}=0.15$ when we varied the source rate of information index
$\alpha_{m}=0.015$ and $\alpha_{m}=0.025$, it can be noted that the thick
green curve and dotted green curve decreases but the peak of both the green
curves are below the thick blue cure. The model simulation (in case of overall
India) showed in the first figure of Figure 8, it can be noted that in
presence of media effect the count of symptomatic individuals decreases. It is
worthy to mention that the media has an impact on the prevalence of the
outbreak of coronavirus diseases. It is fascinating to observe that the media
effect remains dynamic almost until the highest peak of the disease outbreak,
has been shown in the Figure 8. Similar interpretations can be made for the
four provinces of the Republic of India, namely Delhi, Maharashtra, Kerala and
West Bengal, in presence of media related awareness programme for daily
confirmed new cases of coronavirus diseases of symptomatic individuals. From
the model simulation, we can conclude that the media awareness has an impact
on the reduction of coronavirus diseases for the symptomatic individuals, as
shown in the Figure 8.

We also investigate the media related awareness for the novel coronavirus
diseases for the Republic of India and four provinces, namely Kerala,
Maharashtra, West Bengal and Delhi, with respect to parameters listed in the
Table 1 and rest of the estimated parameters listed in the Table 2 with
initial individuals are listed in the Table 4. The impact of media awareness
effect for the daily cumulative confirmed cases of coronavirus diseases for
symptomatic classes are shown in the Figure 9. In the first sub-figure of the
Figure 9 indicates the model simulation of (15) corresponding to the profiles
of symptomatic individuals for overall Indian scenarios with the available
time period 60 days. In this case the thick blue curve showing the maximum
peak or the growth of cumulative symptomatic individuals in compare to other
curves, which represents that the sizes of cumulative symptomatic or
clinically ill population reached their maximum peak in absence of media
information as the source of information index $(\alpha_{m})$ and the rate of
media response $(\eta_{m})$ both are considered as zero. In presence of
effective media awareness, that is, both $\alpha_{m}$ and $\eta_{m}$ are
nonzero, and for fixed $\alpha_{m}=0.01$ when varied the media response rate
$\eta_{m}=0.1$ and $\eta_{m}=0.5$ the thick red curve and dotted red curve
decreases significantly and both the peak remains below the thick blue curve.
The same scenarios has been observed when we vary information index rate
$\alpha_{m}=0.015$ and $\alpha_{m}=0.025$ with fixed value for media response
rate $\eta_{m}=0.15$ the thick green curve and dotted blue curve reduces
noteworthy and remains below the thick blue curve. Thus, the media related
awareness reduces the peak of symptomatic individuals in case of cumulative
confirmed coronavirus diseases, which shows that the media awareness play a
key role on the prevalence of the outbreak of coronavirus diseases, as shown
in the Figure 9. It is very interesting to note that the media effect remains
effectual almost until the maximum peak of the disease epidemic. Similar
conclusions can be drawn for four provinces of India, namely Kerala,
Maharashtra, Delhi and West Bengal in presence of effective media awareness
for daily cumulative confirmed cases of symptomatic populations. Thus, the
media awareness has a great impact to control the coronavirus diseases, as
shown in the Figure 9.

###  III.6 Long-term dynamics

To study the future outbreaks of coronavirus diseases, we simulate our system
(15) for a long days of the infected individuals, namely asymptomatic
individuals and symptomatic or clinically ill individuals for the parameters
are listed in the Table 1 and we have considered the estimated parameters for
the Table 2 in case of Indian data only with initial population sizes are
given in the Table 4. The model simulation represented that the people from
overall India can experience future epidemics if the control policies are not
implemented more effectively, as shown in the Figure 10. The number of
infected individuals (asymptomatic and symptomatic) are oscillate around 130
days and around 320 days. This can be interpreted as the reproduction number
$R_{0}$ is always greater than unity in India (see the Table 5 for India).
Otherwise, we can say that the impact of media related awareness is a dynamic
process and the media has a great impact in reducing the transmission of
coronavirus outbreaks at the initial stage of an epidemic. The factors like
low air temperature and as well low humidity has a great impact on the
transmission of novel coronavirus Jingyuan20 . Moreover, to control the future
outbreaks of novel coronavirus the media related awareness programme and the
control strategy must be implemented properly.

##  IV Discussion

According to the INDIA COVID-19 TRACKER report, 7,598 confirmed cases and 246
deaths due to novel coronavirus outbreaks in India Indiacov19 , as of April
10, 2020. The total number of new cases and new deaths are occurred everyday
reported from different cities across the India Indiacov19 . This is a
terrifying circumstances as India has approximately 139 crores population and
may enter stage-3 of coronavirus disease transmission. Till date, there is no
specific vaccine, antivirals or effective therapeutics to treat coronavirus
diseases. During this period, forecasting is the utmost priority for the
planning of heath care and control the novel coronavirus diseases. Thus,
mathematical modeling can aid us in designing to control the outbreak of
coronavirus diseases in absence of any treatment or any specific diagnostic
test.

In this manuscript, we propose a compartmental epidemic model on novel
coronavirus diseases by incorporating the effect of media awareness to predict
and control the outbreak. In our model we noted that media awareness play a
key role in generating public consciousness and encouraging disease measures
Funk10 . In our proposed model, we consider two infectious individuals, namely
asymptomatic and symptomatic individuals with former being a fast spreader of
the coronavirus diseases. Our model formulation is based on the work by Gumel
et al. Gumel04 , where we estimate the system parameters based on the PRCC
sensitivity analysis at an early stage of the outbreak, in which the number of
cumulative classes grow exponentially.

We calibrated the proposed model to fit with the data from four different
states, namely Maharashtra, Kerala, Delhi, West Bengal and the Republic of
India. The most important part is to perform the sensitivity analysis of 17
model parameters and its estimation. To identify the key input parameters
which contributed to the symptomatic outcome, strongly assisted the
implementation of an integrated policy for various measures, involving the
epidemic during different phases of the diseases, we use the sensitivity
analysis by using PRCC techniques. PRCC analysis yields the most sensitive
parameters are $\beta_{s}$, $q_{a}$, $\gamma_{i}$, $\alpha_{h}$, $\xi_{a}$ and
$\xi_{h}$ out of 17 input parameters without incorporating media impact. Then
we estimated these 7 parameters based on the data from four different states
and from the entire India. By fitting data on novel coronavirus cases during
January 30, 2020 to March 31, 2020, in the four different states of India and
the overall India to our proposed model, we were able to get estimates of the
unknown parameters for our system.

By fitting our proposed model (simply the classical model without considering
media impact) system (15) with real data obtained from India COVID-19 Tracker
(https://www.covid19india.org/), we were able to get acceptable estimations
for the system parameters and curve fitting, as shown in the Figures 3 and 4.
To ensure the performance of our system without media, we compute the
numerical errors, namely $E_{MAE}$ and $E_{RMSE}$. Within the reasonable set
of parameters, the model simulations without incorporating media awareness,
obtained relatively good fits to novel coronavirus data, as plotted in the
Figures 3 and 4.

From the basic reproduction number $R_{0}$ in Table 5, it can be compared that
the province Kerala is highly effective by coronavirus diseases compare to
other three provinces and the Republic of India. Nevertheless, our model-based
analysis also demonstrates that the basic reproduction number in India and
four different provinces remains greater than unity. The estimation of basic
reproduction number $R_{0}$ indicates that the strengthen control involvements
are mandatory to reduce and/or eradicate the future coronavirus diseases.
Thus, in this scenario the public must have to give utmost priority to stay
home quarantine and successful the lock-down to combat against coronavirus
pandemic. In this situation, media related awareness is mandatory to aware
each and every people such that they can maintain social distancing, wear mask
to protect themselves and avoid cultural programme.

From the sensitivity indices (see Table 6 and the Figure 6) for the basic
reproduction number $R_{0}$, we observed that the disease transmission
coefficient $\beta_{s}$ and the portion $q_{a}$ clinical outbreak rates in all
infected classes are positively correlated with $R_{0}$. This indicates that
to reduce coronavirus diseases the social distancing is the most important
factor and we reduce the values for $\beta_{s}$ and $q_{a}$ to control the
diseases. As we know, for $R_{0}<1$ the diseases can be died out. Thus, to
keep control the value of $R_{0}$ we must reduce $\beta_{s}$ and $q_{a}$.
While investigating the contour plot, we observe that for the lower values for
$\beta_{s}$ and $q_{a}$, the reproduction number can be controlled and reduced
to less than unity, as shown in the Figure 7. Thus, if we are able to control
the social distancing, then we are able to control the basic reproduction
number $R_{0}$ and hence we are able to eradicate the coronavirus diseases.
Since $R_{0}$ quantifies the initial epidemic transmission, its sensitivity
indices enable us to establish the relative significance of various parameters
in coronavirus transmission. From the sensitivity of basic reproduction number
$R_{0}$, we can conclude that the most effective control measures are the
isolation of all close contacts and strengthening the self-protection ability
for susceptible individuals.

We have checked our calibrated epidemic model for the short term prediction in
the four provinces and the republic of India. The simulation of our calibrated
model successfully able to capture the increasing growth patterns for three
different provinces, namely Delhi, Maharashtra, West Bengal and the Republic
of India, whereas in case of the province Kerala, the model fitting is not
good compare to other states and overall India, as plotted in the Figure 5. To
get an answer regarding the question why the model prediction is perfectly
matched with Delhi, Maharashtra, West Bengal and the overall India but not
effective for Kerala, we have closely look into the estimated parameters
(Table 2) and come to the conclusion that in Delhi, West Bengal, Maharashtra
and overall India there is a large number of symptomatic individuals whereas
in Kerala the symptomatic population are decreasing due to effective
precautions for coronavirus diseases. Thus, our calibrated model predicted
that the increasing pattern of coronavirus diseases will be continued if the
people not aware about the diseases. Thus, the public must have to take the
precautions and the health care agencies should concentrate on successful
implementation of control mechanisms to reduce and/or eradicate the disease.

In order to study the long-term dynamics of our model, we observed the most
interesting switching phenomena, as plotted in the Figure 10. Maybe this is
happen due to the impact of media awareness as it is a dynamic process on
disease epidemic. Also, the media awareness has a good impact in mitigating
the disease transmission at an initial stage of an epidemic. It is worthy to
mention that the media impact is not always remain efficacious for mitigating
disease transmission throughout the epidemic Collinson14 . A detailed
understanding of media awareness during an outbreak can help in the
progression of an implementable public health strategy. A particular interest
to the designers of such strategies are the effects of media awareness on some
significant epidemic features such as the size of the peak, its timing and the
whole number of diseases. Our proposed model and its simulation, like those in
earlier investigations Buonomo08 ; Collinson14 ; d'Onofrio07 ; d'Onofrio09 ;
Funk09 ; Funk10 ; Kumar17 ; Reluga06 ; Sun11 ; Das20 ; Xiao15 , allow that
quantifying these media effects gives additional insights.

Our model simulation and prediction suggests that the novel coronavirus
diseases has a potentiality to exhibit oscillatory dynamics in the future but
can be controllable by maintaining social distances and effectiveness of
isolation or hospitalization. Our model forecasts that isolation or
hospitalization of populations with coronavirus symptoms, under stringent
hygiene safeguards and social distancing, be able to effectual control in a
community and may even eliminate the diseases. Our study also suggests that
the size and duration of an epidemic can be considerably affected by timely
implementation of the hospitalization or isolation programme. Based on the
model simulations and prediction to control the coronavirus diseases, we
provide some particular inferences to face with emerging diseases:

i.

avoidance of mass gatherings, rallies, social distancing and implement
extensive lock-down;

ii.

provide essential personal safeguard to the staff who are involved in
emergency services;

iii.

timely give the statistics of the coronavirus diseases to the community,
including the number of asymptomatic cases, symptomatic cases, hospitalized
cases and so on;

iv.

increase the media awareness to aware the population, organize the targeted
health care education and self-protection and remove the public mental
disorder.

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Data availability
All data supporting the findings of this study are in the paper and available
from the corresponding author on reasonable request.

Author contributions statement
Subhas Khajanchi, Kankan Sarkar, and Jayanta Mondal designed and performed the
research as well as wrote the paper.

Competing interests
The authors declare that they have no conflict of interest.

Figure 1:  The schematic flow diagram represents the biological mechanism of
novel coronavirus (COVID-19) infection in India, which influences the
formulation of the mathematical model (15). The mathematical model consists of
seven sub-populations including media related information index: susceptible
$S(t)$, exposed $E(t)$, asymptomatic $A(t)$, symptomatic or clinically ill
$I(t)$, hospitalized or isolated $H(t)$, recovered $R(t)$ and media awareness
$M(t)$ individuals in a total population of $N(t)$ = $S(t)$ \+ $E(t)$ \+
$A(t)$ \+ $I(t)$ \+ $H(t)$ \+ $R(t)$ individuals.

Figure 2:  Partial rank correlation coefficients illustrating the dependence
of symptomatic individuals $I$ on each of the system parameters at the day 60
with $p<0.02$.

Figure 3: Daily new confirmed positive coronavirus cases. Observed data are
shown in red circles, whereas the blue curve is the best fitting curve of the
model system (15), without media awareness.

Figure 4: Daily cumulative confirmed coronavirus cases. Observed data are
shown in red circles, whereas the blue curve is the best fitting curve of the
model system (15), without media awareness.

Figure 5: Model simulations shows the short term predictions for the four
provinces, namely Kerala, Delhi, West Bengal, Maharashtra and the Republic of
India. The solid black curve represents the model predictions without
incorporating media impact for the new infected cumulative coronavirus cases,
whereas the solid black circles are the real data. The baseline parameter
values are listed in the Table 1 and rest of estimated parameters are listed
in the Table 2. Our model able to capture the increasing tends for three
states, namely Delhi, West Bengal, Maharashtra and the Republic of India, of
newly infected cumulative coronavirus diseases. In case of Kerala, model
performance is slightly difference in compare to other states and the overall
India.

Figure 6:  Result shows the normalized forward sensitivity indices of basic
reproduction number $R_{0}$ with respect to each of the baseline parameter
values using in the Table 6.

Figure 7:  Contour plots of basic reproduction number $R_{0}$ for the four
different provinces and the Republic of India. Plot contours of $R_{0}$ versus
the disease transmission coefficient $\beta_{s}$ and the portion $q_{a}$ of
exposed class after being clinically ill due to novel coronavirus. For (a) the
Republic of India, (b) the province Kerala, (c) Delhi, (d) West Bengal and (e)
Maharashtra. The contour plots exhibits that the higher disease transmission
probability of coronavirus disease will remarkably increase the basic
reproduction number. The baseline parameter values are obtained from the Table
1 and rest of the parameter values are estimated and obtained from the Table
2.

Figure 8: Illustrations of the media related effect for the daily confirmed
new cases of coronavirus infected patients for four different states (Kerala,
Delhi, Maharashtra, West Bengal) and the Republic of India. The model baseline
parameters are taken from the Table 1 and rest of parameters are estimated and
taken from the Table 2 for different states and overall India. The initial
values are taken from the Table 4.

Figure 9: Illustrations of the media related effect for the daily cumulative
confirmed cases of coronavirus infected patients for four different states
(Kerala, Delhi, Maharashtra, West Bengal) and the Republic of India. The model
baseline parameters are taken from the Table 1 and rest of parameters are
estimated and taken from the Table 2 for different states and overall India.
The initial values are taken from the Table 4.

Figure 10: Long term dynamics for the coronavirus disease system (15) for the
infectious individuals (asymptomatic and symptomatic populations). The value
of the parameters used for the numerical simulation is given in Table 1. The
other parameters values are $\beta_{s}=1.8326$, $\gamma_{i}=0.1423$,
$q_{a}=0.4110$,$\xi_{a}=0.1738$, $\xi_{h}=0.2100$ and $\Lambda_{s}=1600$ with
the initial individuals $S(0)=50000$, $E(0)=5000$, $A(0)=10$, $I(0)=1$,
$H(0)=1$, $R(0)=10$.

Table 1: Table of biologically relevant parameter values and their description for the coronavirus model system (15). Parameter | Biological meaning | Values (Unit) | Source
---|---|---|---
$\Lambda_{s}$ | Inflow rate of susceptible individual | - | -
$\beta_{s}$ | Disease transmission coefficient | - | Estimated
$\alpha_{a}$ | Adjustment factor for asymptomatic classes | 0.20 $day^{-1}$ | Gumel04
$\alpha_{i}$ | Adjustment factor for symptomatic classes | 0.45 $day^{-1}$ | Nadim20
$\alpha_{h}$ | Adjustment factor for hospitalized individuals | - | Estimated
$\delta_{s}$ | Natural death rate of susceptible classes | 0.1945 $\times 10^{-4}$ $day^{-1}$ | data-gov
$\delta_{e}$ | Mortality rate of exposed individuals | 0.1945 $\times 10^{-4}$ $day^{-1}$ | data-gov
$\delta_{a}$ | Mortality rate of asymptomatic individuals | 0.1945 $\times 10^{-4}$ $day^{-1}$ | data-gov
$\delta_{i}$ | Mortality rate of symptomatic individuals | 0.03 $day^{-1}$ | Who-46
$\delta_{h}$ | Mortality rate of hospitalized individuals | 0.1945 $\times 10^{-4}$ $day^{-1}$ | data-gov
$\delta_{r}$ | Mortality rate of recovered individuals | 0.1945 $\times 10^{-4}$ $day^{-1}$ | data-gov
$\gamma_{e}$ | Conversion rate from exposed to asymptomatic individuals | 1/7 $day^{-1}$ | A-who
$\gamma_{i}$ | Rate at which symptomatic individuals become hospitalized | - | Estimated
$q_{a}$ | Proportion of exposed individuals | - | Estimated
$\xi_{a}$ | Rate of recovery from asymptomatic individuals | - | Estimated
$\xi_{i}$ | Rate of recovery from symptomatic individuals | - | Estimated
$\xi_{h}$ | Rate of recovery from hospitalized individuals | - | Estimated
$\delta_{m}$ | Degradation of media related information | 0.06 | Kumar17
$\eta_{m}$ | Rate of media response | (0, 1) | Xiao15
$d_{m}$ | Media related information index rate | 0.017 $day^{-1}$ | Kumar17
$\alpha_{m}$ | Source rate of the information index | (0, 1) | Buonomo08 ; d'Onofrio07

Table 2: _Estimated parameter values from the observed data._ Parameters |  India |  Kerala |  Delhi |  West Bengal | Maharashtra
---|---|---|---|---|---
$\beta_{s}$ |  1.8326 |  2.6822 |  1.7880 |  1.5013 | 1.7781
$\xi_{a}$ |  0.1738 |  0.1121 |  0.1105 |  0.1101 | 0.1203
$q_{a}$ |  0.4110 |  0.0028 |  0.0430 |  0.0125 | 0.0448
$\xi_{h}$ |  0.2100 |  0.2137 |  0.1358 |  0.1332 | 0.1124
$\gamma_{i}$ |  0.1423 |  0.0348 |  0.8900 |  0.8953 | 0.5213
$\xi_{i}$ |  0.2100 |  0.2105 |  0.2091 |  0.2097 | 0.2109
$\alpha_{h}$ |  0.3500 |  0.3491 |  0.3506 |  0.3489 | 0.3501

Table 3: _Accuracy of the model ( 15) for the four different states and the Republic of India._ Performance Metrics |  India |  Kerala |  Delhi |  West Bengal | Maharashtra
---|---|---|---|---|---
$E_{MAE}$ |  10.0600 |  2.4286 |  3.3326 |  1.0294 | 8.7160
$E_{RMSE}$ |  17.6314 |  4.9185 |  5.8537 |  1.4416 | 15.4680

Table 4: _Estimated values values of the initial population size and the constant inflow rate_ Parameters |  India |  Kerala |  Delhi |  West Bengal |  Maharashtra | Source
---|---|---|---|---|---|---
$S(0)$ |  50000 |  40000 |  2500 |  12000 |  15000 | Estimated
$E(0)$ |  0 |  8 |  300 |  1000 |  1700 | Estimated
$A(0)$ |  0 |  1 |  13 |  100 |  250 | Estimated
$I(0)$ |  1 |  1 |  1 |  1 |  1 | Data
$H(0)$ |  1 |  1 |  1 |  1 |  1 | Estimated
$R(0)$ |  0 |  0 |  0 |  0 |  0 | Data
$M(0)$ |  0 |  0 |  0 |  0 |  0 | Estimated
$\Lambda_{s}$ |  1600 |  500 |  500 |  1100 |  1100 | Estimated

Table 5: _Basic reproduction number of the four different provinces and the Republic of India._ Basic reproduction number |  India |  Kerala |  Delhi |  West Bengal | Maharashtra
---|---|---|---|---|---
$R_{0}$ |  2.58792 |  4.78435 |  3.28271 |  2.73854 | 3.03971

Table 6: _Table of sensitivity indices of basic reproduction number $R_{0}$ for the coronavirus model system (15) for four different states and the Republic of India. The baseline parameter values taken from the Table 1 and other parameters are estimated and taken from the Table 2._ Parameters |  |  | Sensitivity | Indices |
---|---|---|---|---|---
| India | Kerala | Delhi | West Bengal | Maharashtra
$\beta_{s}$ | 1.0000 | 1.0000 | 1.0000 | 1.0000 | 1.0000
$\alpha_{a}$ | 0.478462 | 0.99711 | 0.943025 | 0.983684 | 0.928715
$\alpha_{i}$ | 0.341547 | 0.00256553 | 0.00933187 | 0.0021075 | 0.0154709
$\alpha_{h}$ | 0.179991 | 0.000324077 | 0.0476427 | 0.0142081 | 0.0558138
$q_{a}$ | 0.187671 | 0.0000898542 | 0.0146025 | 0.0038639 | 0.0277269
$\gamma_{e}$ | 0.000136131 | 0.000136131 | 0.000136131 | 0.000136131 | 0.000136131
$\gamma_{i}$ | -0.0141361 | -0.0000411902 | 0.00273315 | 0.00133818 | 0.00705925
$\delta_{e}$ | -0.000136131 | -0.000136131 | -0.000136131 | -0.000136131 | -0.000136131
$\delta_{a}$ | -0.0000535388 | -0.000172974 | -0.00016596 | -0.000173745 | -0.00015013
$\xi_{a}$ | -0.478408 | -0.996937 | -0.942859 | -0.983511 | -0.928565
$\delta_{i}$ | -0.0409264 | -0.000314886 | -0.0015138 | -0.000431249 | -0.00280575
$\xi_{i}$ | -0.286484 | -0.00220945 | -0.0105512 | -0.00301443 | -0.0197244
$\delta_{h}$ | -0.0000166691 | -2.94934$\times 10^{-8}$ | -6.82266$\times 10^{-6}$ | -2.07434$\times 10^{-6}$ | -9.6565$\times 10^{-6}$
$\xi_{h}$ | -0.179975 | -0.000324048 | -0.0476359 | -0.014206 | -0.0558041