On Transmission of COVID-19 in Terms of Semigraph

KAMAL BHATTARAI, ABDUR ROHMANa, SURAJIT KR. NATHb*

Department of Mathematical Sciences,

Bodoland University, Kokrajhar, 783370,

INDIA

a https://orcid.org/0009-0006-1353-6009

b https://orcid.org/0000-0002-2142-393X

*Corresponding Author

Abstract: - Semigraph Theory plays a significant role in most of the areas of science and technology. Every

situation can be understandably articulated in terms of suitable graphs by using various approaches of

Semigraph theory. Considering the recent advent of the pandemic in the world and the precautions taken for

prevention of the COVID-19, it is the most appropriate way to utilize the Semigraph models with practical as

well as theoretical aspects to prevent this epidemic. This work defines the two types of variable sets depending

on the time factor. In this project, the mechanism of infection of the virus has been described in a simple way.

The prevention method of the virus infection includes the partition of the semigraph i.e. isolating from other

non-infected persons. The whole world is using the same method while controlling the infection of viruses.

Key-Words: - COVID-19, Semigraph, Bipartite semigraph, Dendroid, Transmission, Social distancing.

Received: November 28, 2022. Revised: August 13, 2023. Accepted: September 20, 2023. Published: October 11, 2023.

1 Introduction

COVID-19 is a transferrable disease caused by the

coronavirus that recently started in Wuhan, China.

This highly transferrable virus and subsequently the

disease were completely shadowy to the world

before its outbreak. Considering the recent COVID-

19 virus and its transmission across the world, it is

important to understand and study the virus's spread

and impact. The disease caused by this virus has

become a pandemic and many countries have been

devastated badly. COVID-19 has devastated almost

all the countries in various dimensions. Using the

semigraph theory approach, this work helps users

to understand and visualize this disease, its impact,

and its spread. The different semigraph method

presented in this project shows the virus, and its

growth type is presented using semigraph theory.

Almost 50 million cases of COVID-19

(coronavirus) and more than 6 million deaths have

now been reported worldwide. The largest part of

the epidemic in the world comes into sight to be

stationary or declining. A good number of countries

have undergone worse conditions in the early stages

of their epidemics and few of them were affected

early in the pandemic and are now starting to see an

improvement in their cases. Hence, it is the most

important and inevitable to prevent the spread of

such types of epidemics. As we know, mathematical

modeling awards different astonishing inspirations

and tools to study different communal as well as

technical problems and interpret solutions. This will

show the way to find practical solutions to a variety

of problems and help to continue the harmony of

mankind.

The studies, [1], [2], [3], [4], [5], [6], and

especially, the authors in, [4], in the year 1994

generalized the definition of graph to semigraph.

Some definitions of semigraphs are given below:

Semigraph: A semigraph  is an ordered pair

󰇛 󰇜 where   󰇝    󰇞 is a nonempty set

whose elements are called vertices of , and  

󰇝    󰇞 is a set of r-tuples, called edges of .

The edges consist of distinct vertices, for various

  , satisfying the following conditions:

i. Any two edges have at most one vertex in

common.

ii. Two edges    and

󰇛  󰇜 are considered to be

equal if and only if

(a)    and

(b) either   for     , or 

 for     .

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Thus the edge 󰇛  󰇜 is the

same as the edge 󰇛      󰇜, where  and 

are said to be the end vertices, whereas

  are called the middle vertices of the

edge .

Subedges and Partial Edges: A subedge of an edge

E = (v1,v2 ,…………vm) is a k-tuple 

󰆷 =

( ………) where 1 ≤  <………< ≤ n.

A partial edge of E is a ( j−i+1)-tuple

E(vi,vj) = (vi, vi+1,…., vj ), where 1≤ i ≤ n. Thus a

subedge E′ of an edge E is a partial edge if, and only

if, any two consecutive vertices in E′ are also

consecutive vertices of E.

Removal of Vertex from an edge:

Bipartite Semigraph: Let G= (V, X) be a

semigraph. G is bipartite if its vertex set V can be

partitioned into sets {V1,V2} such that V1&V2 are

independent.

e-Bipartite Semigraph: G is e-bipartite if V can be

partitioned into sets {V1,V2} such that both V1 and

V2 are e-independent.

Strongly Bipartite Semigraph: G is Strongly

bipartite if V can be partitioned into sets {V1,V2}

such that V1&V2 are Strongly independent.

Dendroid: A dendroid is a connected Semigraph

without Strong cycles and a forest is a semigraph in

which every component is dendroid. Further, every

dendroid is e-bipartite and hence bipartite. Clearly,

every dendroid is also edge bipartite.

Variable set: Let S be the set of vertices of a

Semigraph G. If the strength of S changes with time

t then it is said Variable set, denoted by

Strictly Increasing Set: Let (V) be the set of

vertices at time t1, if |(V)| < |(V)|, then the set

is defined as Strictly increasingly Set. But, if

|(V)| > |(V)|, then the set can be defined as

Strictly Decreasing Set. Pictorially, as shown in

Figure 1.

Fig. 1: The strictly increasing set with respect to

time

),( 21 tt

In, [1], [2], [3], [4], [5], [6], and especially, the

authors in, [2], they construct the model of

transmission of Covid-19 which describes below:

2 Analysis of COVID-19 infection

defining some new term

Let us try to coin some new terms such that our

study becomes easy and simple. Since the infection

rate of the virus is rapid, many factors can be stated

as preventive methods such as SOCIAL

DISTANCING, VACCINATION, WEARING

MASKS, and so on.

Suppose

denotes the infected person at time

where

1,,...,2,1,0  nni

and

)( i

denotes

the rate of spread of the virus infection at time

by the carrier

. It is worth mentioning that in this

paper, we will be concerned with the middle

vertices only as the approach is being done through

semigraph.

Here

and

1n

are the person who was

initially infected by the virus but now are free of the

virus or died, i.e.

0)()( 10  n

xRxR

. Hence all

the responsibility of spreading the virus is borne by

.,...,2,1 ni 

We define some terms below:

(i)

be the term denoting Social

distancing, i.e.

mDsd 2|| 

, where m =

meter.

(ii)

denotes the people who are fully

vaccinated.

(iii)



denotes the people who are not fully

vaccinated yet.

Mathematically, Let

Vi EY 

then the rate of

being infected or spreading the virus gradually roll

down, i.e. for

ki ,...,2,1

%100)( 

, since

the efficiency of vaccines is not 100%.

Let



vi EZ

then the rate of being infected

by the virus is high and chances of spreading the

virus are also higher, i,e.

 

 %50)( i

, for

.,...,2,1 ki 

i.e. They have more than a 50% chance of

spreading the virus infection.

We describe the feasible infection of the COVID-19

virus diagrammatically below.

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Let

,...}2,1|{  ixS ii

. Gradually with the

increase of time, the cardinality of

increase i.e.

|||| 1ii SS 



at different time

,...2,1i

Specifically, the explosion of COVID-19 is

presented in Figure 2 and Figure 3.

Fig. 2: The explosion of COVID-19 in the type of

Chain Reaction

Now we use our semigraphical approach to the

spread of the virus:

Fig. 3: The explosion of COVID-19

3 Models Describing the Transmission

of COVID-19

In, [1], [2], [3], [4], [5], [6], and especially, the

authors in, [2], construct a semigraphical model of

transmission of COVID-19 virus.

Semigraphical Model -I :

In this model, we discuss the transmission of the

COVID-19 virus simply.

Let N = the set of non-infected persons and X the set

of infected persons. (Carrier of virus).

Let

Ny 

be any element. If y tries to make a bond

i.e. come close contact with the element of X, then N

becomes

}{yN 

and X becomes

}{yX 

Semigraphically, we illustrate with the help of

an example (Figure 4). Let

},,,{ xwvuN 

it is

written as in the set of edges

)},(),,,{( xvwvuEN

and

},{ baX 

then it is

written in the set of edge

)},{( baEX

. Here we

describe the mechanism of infection in

Semigraphical Model-I

Reduce Non-infected persons Here the vertex (person) x comes

with the contact with

vertices a and b, got infected. Now x gains huge momentum

to infect i.e. building edges with new vertex (Non-infected)

Fig. 4: The Semigraphical Model -I

Suppose the new carrier x has been exposed at

the initial stage, then its rate of spreading infection

will be zero i.e.

%0)( xR

Now if we isolate the person at a separate place

so that the person does not come in contact with

anyone. That means the vertex cannot form any

edge with the help of other new vertex/vertices,

which is expressed in Figure 5 below:

Fig. 5: Isolation procedure of the person infected by

the virus x.

This acts as a single edge

),,( bxa

where the

capacity of building the edges by the vertices x has

been broken down after it has been isolated.

Hence, for any number of elements from the set

N coming in close contact with any element of X

being exposed at the very initial stage, they create

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only a single edge with the capacity (rate of

spreading) of

)(xR

zero.

4 Semigraphical Model-II

In this section two Models using Semigraph of

transmission of Covid-19. Apart from this, the

growth rate of infected persons is analyzed.

4.1 Semigraphical Model-II

This model is one of the important models that

clearly show the transmission of the virus in several

stages. It is worth mentioning here that the person

(vertex) who has come in close contact with the

carrier has not been exposed initially. As described

above (Figure 5) if the person/vertex x is not

isolated at an early stage and makes it free then from

it the virus gets transmitted at an unknowing rate.

This fact is illustrated below (Figure 6):

Fig. 6: Transmission growing at an unknown rate.

In this stage, the infection rate increases rapidly.

Each infected person with the virus may have more

chance to spread the indefinitely.

4.2 Growth Rate

The Growth Rate of virus can be divided into two

types:

(i) Simple Growth Rate/ Linear Growth Rate

(ii) Multiple Growth Rate

(i) Simple growth rate: Let us explain this growth

with a suitable example. In this growth, the number

of infected people by the virus (carrier) remains

constant.

Let

},{ baX 

be the carrier of the virus infection.

i.e.

2|| X

and a person x comes close contact

with any element of X then the set X will be

}{

1xXX 

and

3|| 1X

. Suppose the infected

person (vertex) comes into contact with three other

non-infected persons (vertices)

and

. Then

the set

becomes enlarge to

},,{ 32112 xxxXX 

where

6|| 2X

. A figure

(Figure 7) given below shows this phenomenon

nicely.

Fig. 7: The growth of the virus linearly in

Semigraph (Model-I).

From the figure,

),,,( 321 xxxxe 

is the chain of infected persons

(vertices) infected by x.

),,,( 13121111 xxxxe 

created by

),,,( 23222122 xxxxe 

created by

),,,( 33323131 xxxxe 

created by

,...)(1111 xe 

created by

,...)(1212 xe 

created by

And so on.

The number of distinct new COVID-19 cases

(edges) is mentioned in Table 1 corresponding to the

number of infected persons (vertices).

Table 1. The number of distinct new COVID-19

cases

NUMBER OF

INFECTED PERSONS

(VERTICES)

DISTINCT NUMBER OF

COVID-19 CASES (EDGES)

331.3 

393.3 

3279.3 

38127.3 

k3

33.3.3 

 nn

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Hence we can conclude that each edge of a

chain of infection created by any carrier yields r no.

of COVID-19 Cases (new except itself).

Then K edges yields rk+1 no COVID-19 Cases.

Where K=rk no of edges.

Diagrammatically, Figure 8 presents the simple

growth in Semigraph (Model-II).

Fig. 8: The simple growth in Semigraph (Model-II)

5 Conclusion

This paper concludes that there is an infinite scope

of mathematics for the research as well as resolving

social problems like COVID-19 and technical

problems.

Semigraph Theory plays a significant role in

most of the areas of science and technology. Every

situation can be understandably articulated in terms

of suitable graphs by using various approaches of

Semigraph theory. Considering the recent advent of

the pandemic in the world and the precautions taken

for prevention of the COVID-19, it is the most

appropriate way to utilize the Semigraph models

with practical as well as theoretical aspects to

prevent this epidemic.

6 Compliance with Ethical Standards

Conflict of Interest: The authors declare that they

have no conflict of interest or other ethical conflicts

concerning this research article.

References:

[1] E. Sampathkumar, Semigraphs and their

Applications. Academy of Discrete

Mathematics and Application, India, 2019.

[2] K. Bhattarai, A study of COVID-19 by using

Semigraphs, Dissertation, Bododland

University,India 2022.

[3] M. Monod et al., Age Groups that sustain

resurging COVID-19 epidemics in the United

States, Science 371, eabe8372,

DOI:10.1126/Science.abe8372, 2021

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Volume 18, 2023

[4] N. Deo, Graph Theory with Applications to

Engineering and Computer Science, Prentice

– Hall of India.

[5] M. Varkey, T.K., S.R. Joseph, On Product of

Disemigraphs, Global Journal of Pure and

Applied Mathematics, Vol. 13, Number 9,

pp.4505-4514, 2017

[6] D. Crnkovic, A. Svob, Application of

Tolerance Graphs to Combat COVID-19

Pandemic, SN Computer Science(2021)2:83

Contribution of Individual Authors to the

Creation of a Scientific Article (Ghostwriting

Policy)

The author contributed in the present research, at all

stages from the formulation of the problem to the

final findings and solution.

Sources of Funding for Research Presented in a

Scientific Article or Scientific Article Itself

No funding was received for conducting this study.

Conflict of Interest

The author has no conflict of interest to declare.

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(Attribution 4.0 International, CC BY 4.0)

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