Abstract
Coronavirus Disease 2019 (COVID-19) Pandemic is a big threat for all
countries’ health systems. In India, the first case of coronavirus
occurred in Kerala on 30 January 2020, followed by 2 more until 2
February 2020, but all three were cured, according to the World Health
Organization (WHO) India situation report-2. Since this, no single case
of coronavirus has been recorded across the country for a month.
Nevertheless, from 2 March onwards, the number of cases rose on a
regular basis. As of 21 June 2020, 410,461 confirmed cases and 13,254
total deaths, as stated in the World Health Organization (WHO) India
situation report-21. This research presents significant findings about
the early outbreak of COVID-19 in India. Due to the recent rapid rise in
new cases of COVID-19, the pre-evolution of pandemic coronavirus is a
pre-eminence in India. The susceptible-infectious-recovered (SIR) model
was developed to estimate the reproductive number R0 at
the early stage of the outbreak of COVID-19 and to evaluate this
outbreak with available data on confirmed, deaths and recovered cases in
India from 02 February 2020 to 26 June 2020.
Keywords: COVID-19; SIR model; Reproductive number; Prediction;
India
Severe acute respiratory syndrome coronavirus 2 (SARS-CoV-2) is a major
pathogen of the fast expanding coronavirus disease outbreak (COVID-19)
originating in the city of Wuhan, Hubei Province, China (Qun, 2020).
More than 9.8 million people worldwide have been affected and in India,
with 528,859 reported cases, 4th place, with a global percentage of
5.40, 8th place in 16,095 deaths, with a global percentage of 3.00 and
4th position in 321,723 cases recovered as of 28 June 2020 (WHO, 2020,
Coronavirus disease 2019 (COVID-19) situation reports - 160). Despite no
alternatives and an inevitable possibility of developing a vaccine, the
Government of India had to concentrate on inhibiting the 5-phase
national lockdown of virus control from 24 March to 30 June 2020
(Wikipedia, COVID-19 pandemic lockdown in India, 2020). The goal of the
lockdown is to reduce the progression of coronavirus. On the one side,
this lockdown could never continue forever due to its impact on people’s
everyday lives and health and on the economy of the country, and on the
other side, relaxing the lockdown would possibly lead to an increase in
the number of diseases. During this stage, the issue that seems very
important to both people and the government is the spread of the disease
as it reduces and the time will be right to drop the lockdown. In this
paper, we modeled the coronavirus spread to address this issue.
Predictions can be used to shape government actions to contain the virus
and to try to prepare for what’s going to happen sensitively and
economically. In the subsequent part of the paper, we initially
presented a model that is being used to examine the outbreak of
Coronavirus in India. We also briefly address and demonstrate how to
estimate an important epidemiological metric, the reproduction number.
We then use our model to examine the spread of disease if there is no
public health intervention.
There are various epidemiological models (Hethcote, 2009), but we have
used one of the most well-known ones, the
susceptible-infectious-recovered (SIR model). The underlying concept
behind the SIR model of transferable chronic diseases is that there are
three groups also called compartments denoted as S, I and R. S: those
who are safe but susceptible to disease (i.e., at risk of
contamination). In the beginning of the pandemic, S is that the
population as a whole is not infected by the virus; I: Infectious people
(and thus infected); R: individuals who have been contaminated but have
either recovered or died, they’re no longer contagious. These groups
grow over time as the virus evolves in the population as: S reduces once
people are infected and transfers to contagious group I; As people
recover or die, they move from the infected group I to the recovered
group R. In order to model the composition of the outbreak, we require
three differential equations to portray the rate of change in each
group, which are getting to know by: β, the rate of infection that
controls the transition between S and I; γ, the rate of removal or
recovery that governs the change between I and R. Expressly, this gives
the following:
\(\frac{\text{dS}}{\text{dt}}=\ -\ \frac{\text{β\ I\ S}}{N}\) (1)
\(\frac{\text{dI}}{\text{dt}}=\ \frac{\text{β\ I\ S}}{N}-\gamma I\)(2)
\(\frac{\text{dR}}{\text{dt}}=\ \gamma I\) (3)
The equation (Eq. 1) states that the amount of susceptible people (S)
decreases with the amount of actively infected people, where new
contaminated cases are the results of the infection rate (β) multiplied
by the amount of susceptible people (S) in contact with infectious
individuals (I). The equation (Eq. 2) expresses that the quantity of
infectious people (I) increments with the newly contaminated people (β I
S), minus the previously infected people who recovered (i.e., γ I which
is the evacuation rate γ multiplied by the infectious people I). The
equation (Eq. 3) expresses that the recovered group (R) increments with
the quantity of people who were infectious and who either recovered or
died (γ I).
The epidemic is developing as follows: S is equal to the whole
population before the outbreak of the disease starts, because no one has
immunity; at the start of the epidemic, as soon as the first person is
infected, S decreases by 1 and I increase by 1; this first infectious
human contaminant (before recovery or death) was found to be susceptible
to other individuals; the cycle persists, with newly infected
individuals who, in effect, harm other vulnerable people before they
recover, which is shown visually in Figure 1.
Figure 1. SIR model Source: Wikipedia
Next, in order to solve these differential equations and to find the
optimum values for the unknown parameters, the β and γ used R packages
on the Windows 10 operating system to fit the SIR model to the data of
India (Churches, 2020). Subsequently, the residual sum of squares (RSS)
was determined as indicated in Eq. 4 which minimizes the sum of the
squared differences between I(t), which is the amount of people in the
contagious compartment I at time t, and the associated number of cases
as expected by our model \(\hat{I}\) (t).
\(\text{RSS\ }\left(\beta,\gamma\right)=\ \sum_{t}{(I\left(t\right)-\hat{I}\ (t))}^{2}\)(4)
In order to fit the model of incidence data for India, we considered the
population of India to be 1,352,642,280 as of November 2019 according to
Wikipedia and the daily cumulative incidence extracted from the
{coronavirus} R package established by Rami Krispin from 2 February
2020 to 26 June 2020. Finally, the SIR model fitted with the available
data by finding the values for β and γ which minimize the residual sum
of squares between the observed cumulative incidence (observed in India)
and the predicted cumulative incidence (predicted by our model) as β is
0.5403282 and γ is 0.4596718. β controls the transition between S and I
(i.e., susceptible and infectious) and γ controls the transition between
I and R (i.e., infectious and recovered).
Figure 2. Show the number of cases observed that match the number of
confirmed cases expected by our model based on data of India from 02
February 2020 to 26 June 2020
Figure 2 shows that the number of observed cases fits the number of
confirmed cases predicted by our model. The fact that both phenomena
converge suggests that the pandemic is obviously at an exponential stage
in India. More data will be needed to see if this phenomenon is
sustained in the long term. Figure 3 displays the log-linear plot
converting the scale into log, which is easier to understand in terms of
the difference between the observed and the predicted number of
confirmed cases, and also displays how the number of confirmed cases
observed varies from the exponential trend. The plot shows that the
number of confirmed cases remained below what would have been expected
in the exponential phase at the beginning of the pandemic and until 4
March. In particular, the number of confirmed cases remained constant in
1 cases from February 16 to March 4. From 5 March to 26 June, the number
of confirmed cases continued to increase at a rate close to the
exponential rate.
Figure 3. Displays the difference between the number of reported cases
and the estimated number as a log-linear plot based on data of India
from 02 February 2020 to 26 June 2020
The basic reproduction number R0, as stated in Eq. 5,
also referred to as the basic reproduction ratio, is then determined
using the SIR model, which is closely related to β and γ,
which was estimated to be 1.175465.
\(R_{0}=\frac{\beta}{\gamma}\) (5)
The reproduction number is the average number of susceptible individuals
who are infected by each infectious person (Thompson, 2019; Yuan, 2020;
Zhang, 2020; Fu-Chang, 2020). In other words, the reproduction number
refers to the number of healthful people who are affected by the number
of vulnerable people. When R0 > 1 the
disease starts to spread across the whole population, but not when
R0 < 1. Typically, the higher the value of
R0, the more difficult it is to manage the disease and
the greater the risk of a pandemic. The R0 of 1.2 being
lower is due to the fact that the number of confirmed cases remained
stable and equal to 1 at the start of the pandemic. It indicates that,
on average, 1.2 people are infected for each infected person in India.
For dynamic systems, the proportion of the population needed to be
effectively immunized to prevent a sustained spread of the disease,
known as the ”herd immunity threshold,” must be greater than\(1-\ \frac{1}{R_{0}\ }\) (Fine, 2011). The reproduction number of 1.2
we have estimated indicates that, given formula 1- (1/1.2), 16.7% of
the population should be immunized to avoid the spread of the infection.
With a population of approximately 1.4 billion in India, this translates
into approximately 0.2 billion people. It is pertinent by using our
model, tailored to the available data from 02 February 2020 to 26 June
2020, 146 days on reported cases in India, to see what would happen if
the epidemic were to continue, without any public health intervention.
Figure 4. Displays cases of susceptible, recovered, observed and
infectious fitted to data from India between 02 February 2020 and 26
June 2020
Despite these forecasts, as seen in the figure 4, with the exact same
settings and no interference whatsoever to restrict the spread of the
pandemic, the peak in India is predicted to be reached by the beginning
of August. Approximately 15,881,175 people will have been diagnosed by
that time, which corresponds to about 3,176,235 serious injuries, about
952,870 people in need of intensive care and up to 714,652 deaths. We
understand at this point why these strict containment measures and
regulations are being enforced in India.