Mohd Rahimie Bin Md Noor\[1\]<sup>\*</sup>, Noor Azreen Rosedee<sup>2</sup>, Nurulain Afiqah Abd Razak<sup>3</sup>, Wan Nur Aisyah Wan Roslan<sup>2</sup> , **Sharifah Athirah Syed Adnan<sup>3</sup>**  
**Mohd Zainuri Muhammad<sup>4</sup>**

<sup>1</sup>Faculty Mathematic and Computer Sciences Universiti Teknologi Mara Kelantan, 18500 Machang, Kelantan, Malaysia

*<sup>2</sup> Department of Mathematics, Faculty Mathematic and Computer Sciences Universiti Teknologi Mara Kelantan, 18500 Machang*

*<sup>3</sup>Medical Department, Klinik Kesihatan Padang Luas, 22000 Jertih, Terengganu, Malaysia*

*<sup>4</sup>Department of Business, Faculty Business Management Universiti Teknologi Mara Kelantan, 15050 Kota Bharu, Kelantan*

<table>
<tbody>
<tr class="odd">
<td>ARTICLE INFO</td>
<td></td>
<td>ABSTRACT</td>
</tr>
<tr class="even">
<td><p><em>Article history:</em></p>
<p>Received XX Month 2024</p>
<p>Revised XX Month 2024</p>
<p>Accepted XX Month 2024</p>
<p>Online first</p>
<p>Published 1 September 2024</p></td>
<td></td>
<td>The coronavirus (Covid-19) outbreak has triggered a global crisis. To curb the virus's spread, Standard Operating Procedures (SOPs) like movement control orders were enforced during the pandemic. This project focuses on examining Covid-19 transmission in India using the SIR model. The model employs differential equations derived from Covid-19 data starting in January 2020. It also analyzes the peak number of infectious individuals at varying contact ratios. The findings, obtained through MATLAB software, indicate that an increase in Covid-19’s reproduction number leads to a faster wave speed. To reduce the wave speed, controlling the contact ratio is crucial. Moreover, reducing the contact ratio effectively lowers the number of infectious individuals.</td>
</tr>
<tr class="odd">
<td><p><em>Keywords:</em></p>
<p>SIR Model</p>
<p>Reproduction number</p>
<p>Differential Equation</p>
<p>Matlab</p>
<p><em>DOI:</em></p>
<p>10.24191/jcrinn.v9i2</p></td>
<td></td>
<td></td>
</tr>
</tbody>
</table>

# Introduction

In December 2019, a surge in pneumonia cases caused by a novel coronavirus was reported in Wuhan city, China, triggering a global outbreak. This virus was later identified as SARS-CoV-2, leading to the emergence of Coronavirus disease (COVID-19). According to the World Health Organization (WHO), most individuals infected with Covid-19 experience mild to moderate respiratory symptoms. However, those in high-risk groups, such as the elderly and individuals with underlying health conditions like diabetes, cancer, heart disease, and chronic respiratory ailments, are more susceptible to severe illness.

Based on current understanding, the primary mode of transmission for the COVID-19 virus is through respiratory droplets released when an infected person coughs or sneezes, particularly in close contact situations (within 2 meters) with symptomatic individuals. While the virus has been detected in feces, urine, and blood, there have been no confirmed reports of fecal-oral transmission to date. Studies conducted by Public Health England (2020) have also indicated potential transmission in healthcare settings through aerosol-generating medical procedures, as well as increased transmission rates in poorly ventilated indoor spaces.

Research findings suggest that Covid-19 symptoms can vary widely in severity and presentation, with some individuals being asymptomatic. Common symptoms include fever, cough, shortness of breath, loss of appetite, anosmia (loss of smell), ageusia (loss of taste), and fatigue. Additional nonspecific symptoms may include sore throat, diarrhea, myalgia, nasal congestion, headache, vomiting, and nausea. Statistics indicate that among symptomatic individuals, 40% experience mild symptoms without pneumonia or hypoxia, 40% exhibit moderate symptoms with mild pneumonia, 15% develop severe disease such as acute respiratory distress syndrome (ARDS), and 5% progress to critical illness with life-threatening complications such as sepsis, cardiac issues, septic shock, and thromboembolic events. Long-term health complications have also been observed in individuals who have experienced mild or severe Covid-19 infections.

## Research Background

According to the World Health Organization (2020), the global tally of reported Covid-19 cases continues to surge, raising concerns about the strain it may place on healthcare systems worldwide. The escalating trend underscores potential challenges stemming from limited treatment facilities, shortages in respiratory aids, insufficient manpower, and other critical resources. To better understand and anticipate the trajectory of the pandemic, researchers have turned to mathematical models like the Susceptible-Infectious-Recovered (SIR) model.

Malavika et al (2021) , researchers employed a deterministic approach using the SIR model to forecast the maximum number of infections and the rate of disease transmission. Focusing on Covid-19 cases in India, they examined the model's differential equation, assuming no migration cases, and calculated disease spread rates across various reproduction number values, including 1.55, 1.36, and 1.13. Through MATLAB simulations, they analyzed the impact of different initial conditions and parameters on the SIR model, ultimately deriving equations to estimate the peak number of infectious individuals.

This project seeks to leverage SIR model equations to assess Covid-19 dynamics in India comprehensively. By exploring disease spread patterns, peak infectious levels, extent of transmission, and migration scenarios, researchers aim to provide valuable insights into epidemic control strategies. The model will account for both migrating and non-migrating populations, with key parameters such as contact ratio and reproduction number undergoing variation. Mitigation efforts will focus on controlling the contact ratio, with interventions like social distancing and adherence to standard operating procedures aimed at reducing infectious individuals. Strategies will be tested with contact ratio values set at 50% and 20%, aiming to curb the spread of Covid-19 cases effectively.

##  Literature Review

According to the World Health Organization (2020), COVID-19 is caused by a novel coronavirus named SARS-CoV-2. The WHO first became aware of this new virus on December 31, 2019, after receiving a report about a cluster of cases of 'viral pneumonia' in Wuhan, China. As the COVID-19 pandemic spread across the country, hospitals faced severe shortages of staff and beds, struggling to provide adequate care for patients. Abelson (2020) noted in a New York Times article on the coronavirus outbreak that COVID-19 patients filled hospital beds across the United States, with hospital staff working around the clock to treat them and struggling to find beds for incoming patients.

In an article titled “COVID-19: We Are Flattening the Curve, but Still in an Uphill Battle,” Aqilah (2020) discusses the situation in Malaysia. Dr. Noor Hisham, the Director General of Health in Malaysia, emphasized that the early period of the pandemic is critical for determining the success of efforts to combat COVID-19. This success is not solely dependent on the number of experts and health facilities in the country but also on public compliance with the Movement Control Order (MCO). This compliance is a crucial factor for the mathematical model aimed at flattening the curve to succeed. Black et al. (2020) state that the growth rate of COVID-19 cases can be slowed by reducing the average number of cases that each infected person generates. While this approach may prolong the outbreak and spread the number of severe cases over time, it reduces the risk of overwhelming the healthcare system. Without a vaccine, the primary methods to reduce transmission are good hygiene and social distancing measures.

Kennedy (2020) illustrates how efforts to slow the spread of COVID-19 and the characteristics of the global pandemic can be understood through a simple logistic model for infection spread. This model assumes society consists of two types of individuals: those infected and those uninfected. If these individuals encounter each other, the infection will spread. Kennedy explains that the probability of a new infection is highest when exactly half of the population is infected. The model posits that the rate of new infections is proportional to both the fraction of the infected and uninfected populations.

Zou et al. (2020) also employed the logistic growth model in their study, using the built-in “SSlogis” function in R to fit logistic growth curves across 20 Chinese provinces. The results indicated that the model accurately described the cumulative number of cases in all provinces except Shandong, with an R² value greater than 0.99. The study suggests that suppression measures can eliminate active virus cases, but the virus might still be present in the environment or reintroduced from unknown sources. Therefore, suppression must be supported by containment strategies, isolation of infected individuals, and widespread testing. The SIR model, a classical yet highly relevant model for analyzing infectious diseases, assumes a constant total population and focuses on the spread of infection over time ,Murray (2002). The model divides the population into three classes: Susceptible (S), Infective (I), and Removed (R), which includes those who recover or die from the disease. Other models, such as SI and SEIR, may include additional classes depending on the disease's characteristics.

Jo et al. (2020) applied the SIR model to analyze COVID-19 in South Korea, incorporating deep learning and time-dependent parameters. The Susceptible-Infectious-Recovery (SIR) model is controlled by transition rates between parameters β and γ. Some studies consider these parameters as constant, while others view them as time-varying. Jo et al. argue that past research using constant parameters is inadequate as it fails to accurately reflect actual data. Ahmetolan et al. suggest that the SEIR model, which accounts for an incubation period, is more suitable for COVID-19. However, they note that the SEIR model's parameters cannot be accurately determined from the normalized curve of removed individuals without clinical data.

Another study by Malavika et al. (2021) used the logistic growth curve model to predict new COVID-19 cases, forecast the maximum number of active cases for India, and evaluate the impact of lockdowns in China, Italy, and South Korea. The model helps prepare the healthcare system and determine measures to reduce transmission. For India, with R₀ fixed at 2.5 and D at 7 days, the values of β and γ were found to be 0.14 and 0.36, respectively. The study found that active cases increased after three weeks of lockdown, suggesting that the lockdown did not significantly impact the number of positive cases, possibly due to increased testing.

According to Alanazi et al. (2020), the SIR model divides the total population into Susceptible, Infectious, and Recovered categories. The transition from Susceptible to Infectious is influenced by the contact rate, which determines the disease's spread in the population. Alanazi et al.(2020) argue that this transition is stochastic rather than deterministic. In conclusion, while numerous methods exist for solving infectious disease problems, the SIR model remains the simplest and most relevant for analyzing diseases like COVID-19.

1.  > **Method**
    
    1.  **Construction Of SIR Model**

The SIR model, credited to Kermack and McKendrick in 1927 (Brauer, 2005), serves as a fundamental framework widely used for analyzing disease transmission dynamics. The model considers three key elements: Susceptible (S), Infectious (I), and Recovered (R). In this model, individuals who have not been infected are classified as susceptible. Given the absence of a vaccine for COVID-19, the entire population is susceptible to infection. When a susceptible individual comes into close contact with a Covid-19 patient, they transition to the infectious stage. Due to the contagious nature of the disease, the number of infected individuals increases over time, thereby elevating the risk of infection for those in the susceptible stage, leading to their progression to the infected stage. Recovered individuals, who are no longer infectious, represent the third category. The SIR model is governed by equations derived from studies such as the one cited in the journal "Systematic Approach for COVID-19 Predictions and Parameter Estimation" by Srivasta et al. \[7\].

(1)

(2)

(3)

(4)

\= Rate of contact between susceptible and infectious

\= Rate of recovery

With the initial conditions given as :

, ,

This SIR model is designed for scenarios without migration, wherein the following assumptions are applied :

1\. Constant Population.

2\. Rate of Infection Proportional to Contacts.

3\. Constant Rate of Recovery or Mortality.

2.  **Deriving the Equation for the Speed of Disease Spread**

The SIR model undergoes further analysis to incorporate migration conditions. As individuals within the population travel and migrate, the incidence of infection can surge, potentially leading to an epidemic. As outlined in reference Srivastava et. al (2020), the following assumptions are considered in the modified SIR model :

1.  The susceptible population remains stationary in space.

2.  The infectious population migrates randomly at a constant rate.

3.  The recovered population remains stationary in space.

Introducing migration into the population necessitates a partial derivative analysis, as it introduces both time and space dependencies into the model. The modified SIR equation under these conditions will be :

(5)

(6)

Where,

D = constant rate of diffusion

x = space in which the individuals are migrating

(7)

Then, a new variables y is created as this model is a function of time and space

(8)

Where,

\= travelling speed

Differentiating the equation (8) with respect to time :

(9)

Dividing the equation (5) and (6) by (9) and rearrange space x to y using the non-dimensionalize analysis, another equation is obtained :

(10)

(11)

With **r**, rate of susceptible and **a**, rate of recovery, the equation (11) can be modified to :

and

(12)

1.  > **Analysis of COVID-19 using SIR model**

**2.3.1 The spread of the disease**

According to Srivastava et al. (2020), the reproductive ratio is

(13)

Where, number of secondary infections in population caused by initial primary infection. In which the contact ratio is shown by

(14)

Therefore, is the contact ratio. If , thus the disease will spread. The equality of initial susceptible was gained

from the equation below by substituting the given condition in (3),

(15)

Where the given conditions are , and must be positive and in order for to be negative.

**2.3.2 Maximum number of infectious individuals and size of the disease spread**

The number of infected individuals is important to be determine in order to plan and distribute health resources. The equation of maximum number of infectious is

(16)

The equation (19) was gained by dividing (3) and (2) :

(17)

(18)

(19)

After integrating (19) and substitute the initial values,

(20)

From (20), (21) and (22) were achieved,

(21)

(22)

By putting the above values into (20), then the maximum number of infectious as,

Next, Substituting , and equation (23) and (24) was achieved,

(23)

(24)

Where,

(25)

From [(25),](#_bookmark31) is subtracted from the entire population as is a logarithmic function. When the symptoms might not be observed as the contact ratio, is very high, will be very low and maximum number of infectious can be written as shown in [(17).](#_bookmark24) The equation for the final value of is

(26)

When the number of infected individual become zero, it means that the spread of the disease is stopped. By substituting into (20), then

(27)

(28)

Where and are the final value of and respectively.

2.  > **Analysis**

**3.1 Initial Conditions for SIR Model**

Based on the first case COVID-19 reported in India on January 30, 2020 there was only a single case on that day and a total population in India was 1.38 billion. Then, according to Srivastava et al.(2020),

(29)

(30)

(31)

In the beginning of the COVID-19 cases in India, the and are as follow

(32)

(33)

The value of can be determine using the formula (13) and rewrite it as

(34)

**3.1 Derive Equation of the Speed of Disease Spread**

Next, the spatiotemporal spread of the disease will be determined. From equation (5) and (6) the non- dimentionalization method has been used. By rearranging equation (4), and (5), equations below were obtained:

On the left hand side, taking out *r* and *S*<sub>0</sub> as they are constant,

Dividing both sides with *rS*<sub>0</sub> and then by dropping the asterisks for notational simplicity, the equation below was obtained

(35)

Next, substituting the assumptions into (6), and simplify

To make the equation dimensionless, all the terms were divided with the coefficient of the highest order derivative. In this case, the highest order derivative is second derivative with the coefficient of then, dropping the asterisks notation for simplicity

Now, the parameter and from the dimensional model have been reduced to one dimensionless grouping, . According to Murray (2003), can be referred as the number of secondary infections caused by the primary infective in a susceptible population, or can be describe as basic reproduction number, . By changing the terms into ,

(36)

Then, to determine the speed of the disease spread in the population, a new variable has been introduced as (8). Deriving (8) with respect to time while assuming as a constant,

Dividing this equation with the equation of . To divide to equation, the partial derivatives have been converted into the full derivatives in terms of single variables y as it is now dimensionless.

(37)

And,

(38)

To analyse these equations, consider some initial values of from (8), which are (past) and (future), substituting the value into (8),

For ,

(39)

As approaches negative infinity, which is going back to the past, it means the disease does not exist yet so I → 0 and S → 1 which is the full proportion of susceptible.

For ,

(40)

As is approaching infinity, it means the future value, the infection will eventually going to be zero as the disease gone by time.

To further analyse the equation, linearization method is used with the aid of the value of from the past which is 1, to make an approximation, thus equation as below is obtained

(41)

Where is a constant with a small value. Differentiate the equation with respect to

(42)

Substitute the value from (41) and (42) into (37) and (38), then equations below are obtained

(43)

(44)

Using the phase plane analysis on (43) and (44), then, for the travelling waves solution to exist (Crawford, 2020),

(45)

Which refers to the minimum possible wave speed for a travelling waves solution to exist for the entire problem. Then, the wave speed (speed of the travelling waves/speed of the spread of the disease as it is

transmitted through the population) is

(46)

Then, using three values of taken from Srivastava et al. \[7\], the values of were calculated, which are presented in Table 3.1 below.

Table 3.1: Speed of Disease Spread

<table>
<thead>
<tr class="header">
<th><blockquote>
<p><strong>R<sub>0</sub></strong></p>
</blockquote></th>
<th><strong>C</strong></th>
</tr>
</thead>
<tbody>
<tr class="odd">
<td><blockquote>
<p>1.55</p>
</blockquote></td>
<td><blockquote>
<p>1.191366794</p>
</blockquote></td>
</tr>
<tr class="even">
<td><blockquote>
<p>1.36</p>
</blockquote></td>
<td><blockquote>
<p>1.028991511</p>
</blockquote></td>
</tr>
<tr class="odd">
<td><blockquote>
<p>1.13</p>
</blockquote></td>
<td><blockquote>
<p>0.6783634654</p>
</blockquote></td>
</tr>
</tbody>
</table>

**3.3 Maximum Number of Infectious Individuals**

**3.3.1 The spread of the disease**

By integrating equation (19) directly,

Hence, by substituting initial values of and , new equation achieved as

(47)

**3.3.2 Maximum number of infectious individuals and size of the disease spread**

> According to journal [Srivastava et al.](#_bookmark68) \[7\][,](#_bookmark68) parameter value given by Table 3.2 .
> 
> Table 3.2: List of Parameter Value

<table>
<thead>
<tr class="header">
<th><strong>Parameter</strong></th>
<th><blockquote>
<p><strong>Description</strong></p>
</blockquote></th>
<th><blockquote>
<p><strong>Constant Value</strong></p>
</blockquote></th>
</tr>
</thead>
<tbody>
<tr class="odd">
<td><blockquote>
<p>a</p>
</blockquote></td>
<td><blockquote>
<p>Rate of recovery</p>
</blockquote></td>
<td><blockquote>
<p><span class="underline">1</span></p>
<p>7</p>
</blockquote></td>
</tr>
<tr class="even">
<td><blockquote>
<p>r</p>
</blockquote></td>
<td><blockquote>
<p>Rate of contact between susceptible and</p>
<p>infectious</p>
</blockquote></td>
<td>2<em>.</em>3602<em>e</em><sup>−10</sup></td>
</tr>
<tr class="odd">
<td><blockquote>
<p>q</p>
</blockquote></td>
<td><blockquote>
<p>Contact ratio</p>
</blockquote></td>
<td>1<em>.</em>65214<em>e</em><sup>−9</sup></td>
</tr>
</tbody>
</table>

Basic reproduction number,

From (28) and (26) :

\= Number of susceptible left in pandemic

\= Number of people catch the disease

The values stated for and are taken from Srivastava et al. \[7\] after calcu lated, the number of each ***,*** and are given in the following table:

Table 3.3: Number of Maximum Infectious Individuals

<table>
<thead>
<tr class="header">
<th><blockquote>
<p><em><strong>q</strong></em></p>
</blockquote></th>
<th><blockquote>
<p><strong><em>R</em><sub>0</sub></strong></p>
</blockquote></th>
<th><blockquote>
<p><em><strong>Imax</strong></em></p>
</blockquote></th>
<th><blockquote>
<p><em><strong>Send</strong></em></p>
</blockquote></th>
<th><blockquote>
<p><em><strong>Rend</strong></em></p>
</blockquote></th>
</tr>
</thead>
<tbody>
<tr class="odd">
<td><blockquote>
<p>20%</p>
</blockquote></td>
<td><blockquote>
<p>1.55</p>
</blockquote></td>
<td><blockquote>
<p>1379999993</p>
</blockquote></td>
<td><blockquote>
<p>1379999895</p>
</blockquote></td>
<td><blockquote>
<p>2760000023</p>
</blockquote></td>
</tr>
<tr class="even">
<td><blockquote>
<p>50%</p>
</blockquote></td>
<td><blockquote>
<p>1.55</p>
</blockquote></td>
<td><blockquote>
<p>1379999997</p>
</blockquote></td>
<td><blockquote>
<p>1379999958</p>
</blockquote></td>
<td><blockquote>
<p>2760000007</p>
</blockquote></td>
</tr>
<tr class="odd">
<td><blockquote>
<p>20%</p>
</blockquote></td>
<td><blockquote>
<p>1.36</p>
</blockquote></td>
<td><blockquote>
<p>1379999993</p>
</blockquote></td>
<td><blockquote>
<p>1379999895</p>
</blockquote></td>
<td><blockquote>
<p>2760000023</p>
</blockquote></td>
</tr>
<tr class="even">
<td><blockquote>
<p>50%</p>
</blockquote></td>
<td><blockquote>
<p>1.36</p>
</blockquote></td>
<td><blockquote>
<p>1379999997</p>
</blockquote></td>
<td><blockquote>
<p>1379999958</p>
</blockquote></td>
<td><blockquote>
<p>2760000007</p>
</blockquote></td>
</tr>
<tr class="odd">
<td><blockquote>
<p>20%</p>
</blockquote></td>
<td><blockquote>
<p>1.13</p>
</blockquote></td>
<td><blockquote>
<p>1379999994</p>
</blockquote></td>
<td><blockquote>
<p>1379999895</p>
</blockquote></td>
<td><blockquote>
<p>2760000023</p>
</blockquote></td>
</tr>
<tr class="even">
<td><blockquote>
<p>50%</p>
</blockquote></td>
<td><blockquote>
<p>1.13</p>
</blockquote></td>
<td><blockquote>
<p>1379999998</p>
</blockquote></td>
<td><blockquote>
<p>1379999958</p>
</blockquote></td>
<td><blockquote>
<p>2760000007</p>
</blockquote></td>
</tr>
</tbody>
</table>

3.  > **RESULT**

**4.1 Predicted Spread of COVID-19 in India Before Lockdown**

![](66bda56228a16_media/media/image128.png)

> Figure 4.1: SIR Model Prediction for Spread of COVID-19 in India

Figure 4.1 above shows the spread of COVID-19 virus starting from the first case reported in India which is on January 30, 2020. The initial reproduction value, is 2.28. Initially, the population of susceptible is increasing and on the 70th day, it starts to decrease drastically. After that, it only decreases slightly until the value approaching 0. For recovered population, as days goes by, the population keep increasing until it approaching the total population in India. Infectious population is highest on 92nd day. From this, it clearly shows that the number of infectious group could be very high if there is no action taken to limit the disease spread. It could infect other people in a big scale such as through community transmission that will lead to very serious problem.

**4.2 Discussion on the Speed of the Disease Spread**

From the results obtained in Figure 4.1, it can be concluded that as the is increasing then the will also increase. Besides, based on (12) and (13), it can be said that where and are both initial susceptible and contact ratio respectively. The value of is clearly a fixed number while the value of is changed by time. Hence, it can be concluded that to minimize the wave speed, the contact ratio must be controlled**.**

**4.3 Predicted Number of Infectious Individuals**

Table 4.1: Number of Maximum Infectious Individuals

<table>
<thead>
<tr class="header">
<th></th>
<th></th>
<th><blockquote>
<p><em><strong>Imax</strong></em></p>
</blockquote></th>
<th><blockquote>
<p><em><strong>Send</strong></em></p>
</blockquote></th>
<th><blockquote>
<p><em><strong>Rend</strong></em></p>
</blockquote></th>
</tr>
</thead>
<tbody>
<tr class="odd">
<td><blockquote>
<p>1<em>.</em>65214<em>e</em><sup>10</sup></p>
</blockquote></td>
<td><blockquote>
<p>2.28</p>
</blockquote></td>
<td><blockquote>
<p>275871147.1</p>
</blockquote></td>
<td><blockquote>
<p>−1<em>.</em>1358<em>e</em><sup>10</sup></p>
</blockquote></td>
<td><blockquote>
<p>16843464770</p>
</blockquote></td>
</tr>
<tr class="even">
<td><blockquote>
<p>20%</p>
</blockquote></td>
<td><blockquote>
<p>1.55</p>
</blockquote></td>
<td><blockquote>
<p>1379999993</p>
</blockquote></td>
<td><blockquote>
<p>1379999895</p>
</blockquote></td>
<td><blockquote>
<p>2760000023</p>
</blockquote></td>
</tr>
<tr class="odd">
<td><blockquote>
<p>50%</p>
</blockquote></td>
<td><blockquote>
<p>1.55</p>
</blockquote></td>
<td><blockquote>
<p>1379999997</p>
</blockquote></td>
<td><blockquote>
<p>1379999958</p>
</blockquote></td>
<td><blockquote>
<p>2760000007</p>
</blockquote></td>
</tr>
<tr class="even">
<td><blockquote>
<p>20%</p>
</blockquote></td>
<td><blockquote>
<p>1.36</p>
</blockquote></td>
<td><blockquote>
<p>1379999993</p>
</blockquote></td>
<td><blockquote>
<p>1379999895</p>
</blockquote></td>
<td><blockquote>
<p>2760000023</p>
</blockquote></td>
</tr>
<tr class="odd">
<td><blockquote>
<p>50%</p>
</blockquote></td>
<td><blockquote>
<p>1.36</p>
</blockquote></td>
<td><blockquote>
<p>1379999997</p>
</blockquote></td>
<td><blockquote>
<p>1379999958</p>
</blockquote></td>
<td><blockquote>
<p>2760000007</p>
</blockquote></td>
</tr>
<tr class="even">
<td><blockquote>
<p>20%</p>
</blockquote></td>
<td><blockquote>
<p>1.13</p>
</blockquote></td>
<td><blockquote>
<p>1379999994</p>
</blockquote></td>
<td><blockquote>
<p>1379999895</p>
</blockquote></td>
<td><blockquote>
<p>2760000023</p>
</blockquote></td>
</tr>
<tr class="odd">
<td><blockquote>
<p>50%</p>
</blockquote></td>
<td><blockquote>
<p>1.13</p>
</blockquote></td>
<td><blockquote>
<p>1379999998</p>
</blockquote></td>
<td><blockquote>
<p>1379999958</p>
</blockquote></td>
<td><blockquote>
<p>2760000007</p>
</blockquote></td>
</tr>
</tbody>
</table>

From the table 4.1 above, generally, in the current outbreak of COVID-19, the initial number of contact ratio produced was very high because the disease is easily transmitted and lots of people coming into contact with the infected people. Hence, the initial values of contact ratio, and reproduction number, were ***1.65214e<sup>10</sup>*** and ***2.28*** respectively were high. Then, and that has been calculated by the given values has been produced such a big number but value of was too small. In order to lower the number of infectious individuals, the contact ratio, need to be minimized. To minimize the contact ratio, strict social distancing has been implemented to the society. From the table, it can be seen clearly that 20% of contact ratio will produce less number of infectious individuals while 50% of contact ratio will produce high number of infectious individuals. Thus, it can be concluded that to reduce the number of infectious individuals, the value of contact ratio, need to be reduce too.

4.  > **conclusion**

In this paper, a mathematical model for COVID-19 pandemic is constructed with two assumptions which are no migration and with migration. The SIR model yields three outcomes, which are predicted spread of COVID-19 in India before lockdown, speed of disease for coronavirus and prediction for number of infectious individuals by using the different value of and . These analysis can lead to an improved utilization of health-care resources. From the result, it can be concluded that is very important to determine the spreading of the disease. The lower the value of , the lower the spread of virus will be. Furthermore, from the analysis of disease spread, we may conclude that when increases, so the potential of virus spread high and by controlling the contact ratio will reduce wave speed virus. Next, contact ratio also affecting the maximum number of infectious individuals. If we reduce the contact ratio, the maximum number of infectious individuals will drop as well. So, to limit this COVID-19 from spreading widely, the government need to use a lockdown policy to reduce the contact ratio beside enforcing the strict social distancing rule. With the developed of COVID-19 vaccine in 2021, it is also helping us to lower the number of infected people and the death cases due to the virus. As a consequence, the healthcare system still can cope with the COVID-19’s situation even though it may take some time to reach the endemic equilibria, where the spread of the disease will finally stop and the infected population reduced to zero.

5.  > **Acknowledgements/Funding**

There is no providing the facilities and financial support on this research.

6.  > **CONFLICT OF INTEREST STATEMENT**

I agree that this research was conducted in the absence of any self-benefits, commercial or financial conflicts and declare the absence of conflicting interests with the funders.

**8. AUTHORS’ CONTRIBUTIONS**

9.  > **References**

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1.  <sup>\*</sup> Corresponding author. *E-mail address*: <donottypehere@email.com> (Add the e-mail in the final camera-ready submission)