Form this definition, however by defining

$\left\{\langle {Z}_{7}\cup I\rangle ,\u2018+\u2019\text{modulo}7\right\}=\left\{a+bI/a,b\left(b\in {Z}_{7}\right)\right\},$ then $N\left(G\right)$ can be made a group.

Graph:

We know that a graph G consists of two things:

1) A set V of elements called nodes (or points or vertices).

2) A set E of edges such that each edge in E is identified with a unique (unordered) pair $\left\{u,v\right\}$ of nodes in V, denoted by $e=\left\{u,v\right\}$ . So $G=\left(V,E\right)$ . So from the fundamental concepts on graph theory and the knowledge of neutrosophic group we can construct a neutrosophic graph which will serve as our viral transmission model.

From the review of literature we discovered that the first representation of groups by graphs is the Cayley graphs. These graphs were introduced by Arthor Cayley in 1878 and it shows pictorial representations of finite groups [11] .

Cayley graph:

Giving a group A and a set of generators for A. The Cayley digraph

$G=\left(V,E\right)$ denoted by $\langle A,S\rangle $ is constructed as follows:

1) The elements of the group A forms the vertices $V$ of the digraph $G$ .

2) The edge $\left(a,b\right)$ is in $E$ if and only if $ag=b$ for some generators g in S.

If $S=S\cup {S}^{\prime},$ then G is Cayley graph [12] [13] .

G graph:

For a group G with generating set $S=\left\{{s}_{1},{s}_{2},\cdots {s}_{k}\right\},$ the G-graph of G, denoted by $\text{\Gamma}\left(G,S\right)$ is the graph whose vertices are distinct cosets of $\langle {s}_{i}\rangle $ in G. Two distinct vertices are joined by an edge when the set intersection of the cosets is nonempty [14] .

Combining the above two definitions, Cayley graph and G-graph it is evidently clear that:

The generators of a cyclic group plays a vital role in constructing a graph from group and there is a relationship between a generator and its generating set which can be considered to be an arc, whether directed or undirected.

3. Methodology

1) Assumptions:

a) The total population is taken as a constant.

b) An initial infected class is introduced to the total population.

c) The dead or recovered people in the community belongs to the same class.

d) The population’s contact is heterogeneous.

2) Variables Declaration:

${S}^{*}$ = Not Susceptible, represents those members of the population who are negated from the disease or whose probability of being contracted with the disease is zero.

S = Susceptible, represents those members of the population who stand the risk of being infected.

L = Incubation, represents those members of the population who are infected and the body’s defence mechanisms are unable to destroy the virus.

E = Exposed, This represents the period when the symptoms start to shows up.

I = Infected, This represents a fraction of the population who are infected and remain in the infected class but potentially cannot transmit the disease. From Figure 2 this indicates that the curve increase in the infected class and then drops without touching the infective class. Epidemiologically this means it remains stagnant in the infected class, until being removed or return to susceptible.

T = Transmitter, This class of the population are infected and are capable of transmitting the infection.

R = Recovered, These are fraction of the population, that are immune, vaccinated, dead, isolated.

Figure 2. Neutrosophic viral transmission graph.

G = Represents the total population. In this case S, L, E, I, T, R, ${S}^{*}$ are subset of G.

Hence $\left(S,\text{}L,\text{}E,\text{}I,\text{}T,\text{}R,\text{}{S}^{*}\right)\subset G$ .

Let us use the integer modulo arithmetic ${Z}_{n}$ to represents the subsets of G, that is

${S}^{*}=0,\text{}S=1,\text{}L=2,\text{}E=3,\text{}I=4,\text{}T=5,\text{}R=6$ .

Observed that $G=\left\{0,1,2,3,4,5,6\right\}$ . Consider the elements of G to be integer modulo 7 hence $\left({Z}_{7}+\right)$ is a group.

Lets define $N\left(G\right)=\left\{\langle {Z}_{7}\cup I\rangle ,\u2018+\u2019\text{modulo7}\right\}$ is a neutrosophic group which is in fact a group. For $N\left(G\right)=\left\{a+bI/a,b\in \left({Z}_{7}\right)\right\}$ is a group under “+” modulo 7. Thus this neutrosophic group is also a group.

Hence

$N\left(G\right)=\left\{\begin{array}{c}0,1,2,3,4,5,6,I,2I,3I,4I,5I,6I,1+I,1+2I,1+3I,1+4I,1+5I,\\ 1+6I,2+I,2+2I,2+3I,2+4I,2+5I,2+6I,3+I,3+2I,3+3I,\\ 3+4I,3+5I,3+6I,4+I,4+2I,4+3I,4+4I,4+5I,4+6I,5+I\\ 5+2I,5+3I,5+4I,5+5I,5+6I,6+I,6+2I,6+3I,6+4I,\\ 6+5I,6+6I.\end{array}\right\}$

Here “I” cannot stand for the group identity it is only an indeterminate and hence suitable for the nature of our disease propagation model.

Theorems:

Hence the following theorems were proposed based on the group $N\left(G\right)$ .

Theorem (1) Every stage of viral transmission is allocated a unique representation in $N\left(G\right)$ .

Proof: Suppose we have two representation for S i.e. $s,{s}^{\prime}\in S$ .

If $e\in N\left(G\right)$ is the identity element and be such that has no effect on the elements of $N\left(G\right),$ then $e*s=s*e=s$ since e is the identity element.

Also ${s}^{\prime}*e=e*{s}^{\prime}={s}^{\prime}$ since e is the identity element.

Hence $s={s}^{\prime}$ and the result follows.

Theorem (2) The operation $*$ on $N\left(G\right)$ , is well define.

Let $a,b\in N\left(G\right).$ By the definition of $*$ on $N\left(G\right)$ , we would have

$a*b=\stackrel{\xaf}{a+b}$ . Now suppose ${a}_{1}\in a$ and ${b}_{1}\in b$ , be chosen as class representative of $a\text{}$ and $b$ respectively by defining a suitable homomorphism on $N\left(G\right)$ , in this case we would have $a+b={a}_{1}+{b}_{1}=\stackrel{\xaf}{{a}_{1}+{b}_{1}}$ . Thus we can show that

$\stackrel{\xaf}{{a}_{1}+{b}_{1}}=\stackrel{\xaf}{a+b}$ .

Now ${a}_{1}\in a\Rightarrow {a}_{1}\equiv a\left(\mathrm{mod}n\right)\Rightarrow {a}_{1}=a+{k}_{n}$ for some $k\in Z$ , and

${b}_{1}\in b\Rightarrow {b}_{1}\equiv b\left(\mathrm{mod}n\right)\Rightarrow {b}_{1}=b+{h}_{n}$ for some $h\in Z$ , thus

${a}_{1}+{b}_{1}=a+b+\left(k+h\right)n\Rightarrow {a}_{1}+{b}_{1}\equiv a+b\left(\mathrm{mod}\right)n$ , this implies that $a*b=\stackrel{\xaf}{a+b}$ and the operation * on $N\left(G\right)$ , is well define.

Theorem (3) The order of the neutrosophic group $\left\{\langle {Z}_{n}\cup I\rangle ,\u2018+\u2019\text{modulo}n\right\}$ denoted by $\left|N\left(G\right)\right|$ is ${n}^{2}$ .

Proof: Since ${Z}_{n}$ has n elements, and the indeterminate “I” is such that:

$I+I+\cdots n$ times is $nI$ i.e. $\left|nI\right|$ is $n$ . Therefore $\langle {Z}_{n}\cup I\rangle $ is a combinations of the elements of ${Z}_{n}$ and $nI$ that is $n\times n={n}^{2}$ hence the proof.

3) Building graphs from neutrosophic group structure.

Kandasamy W and Smarandache F in 2015 defines neutosophic graphs as follows: “If the edge values are from the set $R\cup I$ or $Q\cup I$ or ${Z}_{n}\cup I$ or

$\text{Z}\cup I$ or $C\cup I$ they are term as neutrosophic graph” [4] . In our own case the edge are from $\left\{\langle {Z}_{7}\cup I\rangle ,\u2018+\u2019\text{modulo}7\right\}$ and the elements of $\left({Z}_{7}\right)$ where carefully allocated with a unique stages of the viral transmission. Now lets define a neutrosophic graph generated from neutrosophic group.

Definition: Henceforth we defined neutrosophic graph $\varnothing \left(V,E\right)$ generated from $N\left(G\right)$ as follows:

1. Given an element $a\in N\left(G\right)$ and generates a set B of elements in $N\left(G\right)$ then element a and set B connects or has a relation. Hence For any $b\in B,\left(a,b\right)$ is an edge.

2. The elements of $N\left(G\right)$ forms the vertices V of the neutrosophic graph.

As mentioned above and confirmed by Dinnen and Breto the generator(s) of a group G is very vital in the construction of a neutrosophic graph from group.

4) Finding set of generators

Since the neutrosophic group $N\left(G\right)$ , is generated by ${Z}_{7}\cup I$ , the generators are precisely the class of union of generators which partitioned $N\left(G\right)$ .

Hence in ${Z}_{7}$ the generators are $\left(1,2,3,4,5,6\right)$ . Moreover

$I$ is a generator of $2I,3I,4I,5I,6I$ ,

$1+I$ is a generator of $2+2I$ , $3+3I$ , $4+4I,5+5I$ , $6+6I$ ,

$1+2I$ is a generator of $2+4I$ , $3+6I$ , $4+I$ , $5+3I$ , $6+5I$ ,

$1+3I$ is a generator of $2+6I$ , $3+2I$ , $4+5I$ , $5+I$ , $6+4I$ ,

$1+4I$ is a generator of $2+I$ , $3+5I$ , $4+2I$ , $5+6I$ , $6+3I$ ,

$1+5I$ is a generator of $2+3I$ , $3+I$ , $4+6I$ , $5+4I$ , $6+2I$ ,

$1+6I$ is a generator of $2+5I$ , $3+4I$ , $4+3I$ . $5+2I$ , $6+I$ ,

$2+I$ is a generator of $4+2I$ , $6+3I$ , $1+4I$ , $3+5I$ , $5+6I$ ,

$2+2I$ is a generator of $4+4I$ , $6+6I$ , $1+I$ , $3+3I$ , $5+5I$ ,

$2+3I$ is a generator of $4+6I$ , $6+2I$ , $1+5I$ , $3+I5+4I$ ,

$2+4I$ is a generator of $4+I$ , $6+5I$ , $1+2I$ , $3+6I$ , $5+3I$ ,

$2+5I$ is a generator of $4+3I$ , $6+I$ , $1+6I$ , $3+4I$ , $5+2I$ ,

$2+6I$ is a generator of $4+5I$ , $6+4I$ , $1+3I$ , $3+2I$ , $5+I$ ,

$3+I$ is a generator of $6+2I2+3I$ , $5+4I$ , $1+5I$ , $4+6I$ ,

$3+2I$ is a generator of $6+4I$ , $2+6I$ , $5+I$ , $1+3I$ , $4+5I$ ,

$3+3I$ is a generator of $6+6I$ , $2+2I$ , $5+5I$ , $1+I$ , $4+4I$ ,

$3+4I$ is a generator of $6+I$ , $2+5I$ , $5+2I$ , $1+6I$ , $4+3I$ ,

$3+5I$ is a generator of $6+3I$ , $2+I$ , $5+6I$ , $1+4I$ , $4+2I$ ,

$3+6I$ is a generator of $6+5I$ , $2+5I$ , $5+3I$ , $1+2I$ , $4+I$ ,

$4+I$ is a generator of $1+2I$ , $5+3I$ , $2+4I$ , $6+5I$ , $3+6I,$

$4+2I$ is a generator of $1+4I$ , $5+6I$ , $2+I$ , $6+3I$ , $3+5I$ ,

$4+3I$ is a generator of $1+6I$ , $5+2I$ , $2+5I$ , $6+I$ , $3+4I$ ,

$4+4I$ is a generator of $1+I$ , $5+5I$ , $2+2I$ , $6+6I$ , $3+3I$ ,

$4+5I$ is a generator of $1+3I$ , $5+I$ , $2+6I$ , $6+4I$ , $3+2I$ ,

$4+6I$ is a generator of $1+5I$ , $5+4I$ , $2+3I$ , $5+2I$ , $3+I$ ,

$5+I$ is a generator of $3+2I$ , $1+3I$ , $6+4I$ , $4+5I$ , $2+6I,$

$5+2I$ is a generator of $3+4I$ , $1+6I$ , $6+I$ , $4+3I$ , $2+5I$ ,

$5+3I$ is a generator of $3+6I$ , $1+2I$ , $6+5I$ , $4+I$ , $2+4I$ ,

$5+4I$ is a generator of $3+I$ , $1+5I$ , $6+2I$ , $4+6I$ , $2+3I,$

$5+5I$ is a generator of $3+3I$ , $1+I$ , $6+6I$ , $4+4I$ , $2+2I$ ,

$5+6I$ is a generator of $3+5I$ , $1+4I$ , $6+3I$ , $4+2I$ , $2+I$ ,

$6+I$ is a generator of $5+2I$ , $4+3I$ , $3+4I$ , $2+5I$ , $1+6I$ ,

$6+2I$ is a generator of $5+4I$ , $4+6I$ , $3+I$ , $2+3I$ , $1+5I$ ,

$6+3I$ is a generator of $5+6I$ , $4+2I$ , $3+5I$ , $2+I$ , $1+4I$ ,

$6+4I$ is a generator of $5+I$ , $4+5I$ , $3+2I$ , $2+6I$ , $1+3I$ ,

$6+5I$ is a generator of $5+3I$ , $4+I$ , $3+6I$ , $2+4I$ , $1+2I$ ,

$6+6I$ is a generator of $5+5I$ , $4+4I$ , $3+3I$ , $2+2I$ , $1+I$ ,

4. Discussion

From the result so far generated above we can see that every element $a+bI$ in $N\left(G\right)$ partitioned $N\left(G\right)$ into subset which are themselves groups each when the identity $0+0I$ is embedded in it. Therefore a neutrosophic group $N\left(G\right)$ is cyclic if for any $a+bI\in \left({Z}_{n}\cup \text{I}\right)$ , there exist an integer $r\in {Z}_{n}$ such that

$r\left(a+bI\right)=c+dI$ for some $c+dI\in N\left(G\right)$ . Then $a+bI$ is a generator of

$c+dI$ .

Observed that the principal generators denoted by $\left\{\langle {p}_{g}\rangle \right\}$ are those point in $N\left(G\right)$ whose generating cycles are distinct. Also the sub generators are those who are either subset of $\left\{\langle {p}_{g}\rangle \right\}$ or parts of its generating cycles have been generated by $\left\{\langle {p}_{g}\rangle \right\}$ .

Corollary If $\left\{\langle {Z}_{n}\cup I\rangle ,\u2018+\u2019\text{modulo}n\right\}$ is a cyclic neutrosophic group and $\langle a\rangle $ is a generator, then it generates only $n-2$ elements.

Proof: The generalization of $N\left(G\right)$ as a group is false, but $N\left(G\right)$ always contained a group.

Let ${Z}_{n}$ be the cyclic group in $N\left(G\right)$ by the theorem “let $G=\langle {Z}_{n}\rangle $ be a finite cyclic group of order n. The generators of $G$ are the elements $ra$ where the $\mathrm{gcd}\left(r,n\right)=1$ coprime or relatively prime”.

Now $r$ is taking from $\langle {Z}_{7}\rangle $ where $r=\left(0,1,2,3,4,5,6\right)$ , $\mathrm{gcd}\left(0,7\right)=1$ is impossible, $\mathrm{gcd}\left(1,7\right)=1$ Obvious.

Hence each generator, generate only $n-2$ elements in $N\left(G\right)$ .

Results

Hence given that ${y}_{0}=\langle 1+I\rangle ,$ is a generator with its generating sets as

${y}_{1}=2+2I$ , ${y}_{2}=3+3I$ , ${y}_{3}=4+4I,\text{}{y}_{4}=5+5I$ , ${y}_{5}=6+6I$ , then the nodes are $V=\left({y}_{0},{y}_{1},{y}_{2}{y}_{3},{y}_{4},{y}_{5}\right)$ . The dotted lines shows neutrosophic edges, indicating that the connections between those distinct nodes are indeterminate as shown in Figure 2.

Since ${y}_{0}$ is the generator, then connections to any of its generating sets is guarantee and the tendency of contracting is high, meaning it has a direct communication to those neighbors. But ${y}_{1}$ to ${y}_{3}$ , ${y}_{2}$ to ${y}_{5}$ , ${y}_{4}$ to ${y}_{2}$ all has indeterminate connections, this indicates that the tendencies of contraction is vague.

Figure 3. Cyclic neutrosophic viral graph.

The figure above is a cyclic representation of the neutrosophic group $\left(G\right)$ . It depicts the connections between the nodes in order of their generation. Thus eight (8) cycles were formed with six (6) elements each together with the terminating cycle for the null neutrosophic or identity as shown in Figure 3.

This is indicating that circulation of viral infection within the same class is faster than its interconnectivity. And from the large circle we can see that, connection from node $6+I$ to node $6+2I$ indicates the reversal since the graph is undirected, which shows involution. Moreover the distance between these three nodes $6+6I$ , $6+I$ , $6+2I$ remains equal when considering the shortest possible routes.

5. Conclusions

In this paper we represented an entire epidemic population with integer modulo 7, i.e., $\left({Z}_{7}\right)$ and realize that $\left({Z}_{7}+\right)$ is a group. Then we transform this group $\left({Z}_{7}+\right)$ into a neutrosophic group $N\left(G\right),$ by defining

$N\left(G\right)=\left\{\langle {Z}_{7}\cup I\rangle ,\u2018+\u2019\text{modulo7}\right\}=\left\{a+bI/a,b\left({Z}_{7}\right)\right\}$ . We found that this is a special type of neutrosophic group which is also a group and in particular a cyclic group and having a set of generators each, which partitioned $N\left(G\right)$ into classes of subset which are themselves groups when the identity $0+0I$ is embedded to each subset.

The results obtained is being converted into neutrosophic graph $\varnothing \left(V,E\right)$ with elements of $N\left(G\right)$ forming the vertices V and E is an arc determined by defining a suitable homeomorphisms between the groups $\left({Z}_{7}+\right)$ and $N\left(G\right)$ as shown above.

Cite this paper

Zubairu, A. and Ibrahim, A. (2017) The Spread of Infectious Disease on Network Using Neutrosophic Algebraic Structure.*Open Journal of Discrete Mathematics*, **7**, 77-86. doi: 10.4236/ojdm.2017.72009.

Zubairu, A. and Ibrahim, A. (2017) The Spread of Infectious Disease on Network Using Neutrosophic Algebraic Structure.

References

[1] Kermack, W.O. and McKendrick, A.G. (1927) A Contribution to the Mathematical Theory of Epidemics. Proceedings of the Royal Society Lond. A, 115, 700-721.

https://doi.org/10.1098/rspa.1927.0118

[2] Brauer, A., Haug, G.H., Dulski, P., Sigman, D.M. and Negendank, J.F.W. (2008) An Abrupt Wind Shift in Western Europe at the Onset of the Younger Dryas Cold Period. Nature Geoscience, 1, 520-523.

https://doi.org/10.1038/ngeo263

[3] Smarandache, F. (2014) Neutrosophic logic and set.

http://fs.gallup.unme.edu/neutrosophy

[4] Vasantha Kandasamy, W.B and Smarandache, F. (2015) Neutrosophic Graphs, a New Dimension to Graph Theory. EuropaNova, USA.

[5] Feinleib, M. (2001) A Dictionary of Epidemiology. In: John, M. and Last, R.A. Eds., 4th Edition, Chicago, USA, 93-101.

[6] Kramer, et al. (2010) Principles of Infectious Disease Epidemiology. Modern Infectious Disease Epidemiology, Springer, New York.

https://doi.org/10.1007/978-0-387-93835-6

[7] Claude, B., Perrin, D. and Ruskin, H.J. (2009) Considerations for a Social and Geographical Framework for Agent-Based Epidemics. International Conference Computational Aspects of Social Networks CASON, 9, 149-154.

[8] Nelson, K.E. (2006) Infectious Disease Epidemiology. Theory and Practice, Jones & Bartlett Publishers.

[9] Vasantha Kandasamy, W.B., Smarandache, F. and Ilanthenral, K. (2005) Introduction to Linear Bialgebra, Hexis, Phoenix.

[10] Vasantha Kandasamy, W.B., and Florentin S. (2005) Basic Neutrosophic Algebraic Structures and Their Applications to Fuzzy and Neutrosophic Models, Hexis, Church Rock.

[11] Cayley, A. (1952) Collected Mathematical Papers. Cambridge University press.

https://doi.org/10.1017/CBO9780511703768

[12] Michael, J.D. and Michael, R.F. (1991) Algebraic Constructions of Efficient Broadcast Netwoks. Los Alamos National Laboratorym New Mexico.

[13] Michael, J.D. (1991) Algebraic Methods for Efficient Network Constructions. University of Victoria, Idaho, 19-25.

[14] Bretto, A. and Gillibert, L. (2004) Graphical and Computational Representation of Groups. In: Proceedings of ICCS, In: LNCS, Springer-Verlag, Berlin, 343-350.

https://doi.org/10.1007/978-3-540-25944-2_44

[1] Kermack, W.O. and McKendrick, A.G. (1927) A Contribution to the Mathematical Theory of Epidemics. Proceedings of the Royal Society Lond. A, 115, 700-721.

https://doi.org/10.1098/rspa.1927.0118

[2] Brauer, A., Haug, G.H., Dulski, P., Sigman, D.M. and Negendank, J.F.W. (2008) An Abrupt Wind Shift in Western Europe at the Onset of the Younger Dryas Cold Period. Nature Geoscience, 1, 520-523.

https://doi.org/10.1038/ngeo263

[3] Smarandache, F. (2014) Neutrosophic logic and set.

http://fs.gallup.unme.edu/neutrosophy

[4] Vasantha Kandasamy, W.B and Smarandache, F. (2015) Neutrosophic Graphs, a New Dimension to Graph Theory. EuropaNova, USA.

[5] Feinleib, M. (2001) A Dictionary of Epidemiology. In: John, M. and Last, R.A. Eds., 4th Edition, Chicago, USA, 93-101.

[6] Kramer, et al. (2010) Principles of Infectious Disease Epidemiology. Modern Infectious Disease Epidemiology, Springer, New York.

https://doi.org/10.1007/978-0-387-93835-6

[7] Claude, B., Perrin, D. and Ruskin, H.J. (2009) Considerations for a Social and Geographical Framework for Agent-Based Epidemics. International Conference Computational Aspects of Social Networks CASON, 9, 149-154.

[8] Nelson, K.E. (2006) Infectious Disease Epidemiology. Theory and Practice, Jones & Bartlett Publishers.

[9] Vasantha Kandasamy, W.B., Smarandache, F. and Ilanthenral, K. (2005) Introduction to Linear Bialgebra, Hexis, Phoenix.

[10] Vasantha Kandasamy, W.B., and Florentin S. (2005) Basic Neutrosophic Algebraic Structures and Their Applications to Fuzzy and Neutrosophic Models, Hexis, Church Rock.

[11] Cayley, A. (1952) Collected Mathematical Papers. Cambridge University press.

https://doi.org/10.1017/CBO9780511703768

[12] Michael, J.D. and Michael, R.F. (1991) Algebraic Constructions of Efficient Broadcast Netwoks. Los Alamos National Laboratorym New Mexico.

[13] Michael, J.D. (1991) Algebraic Methods for Efficient Network Constructions. University of Victoria, Idaho, 19-25.

[14] Bretto, A. and Gillibert, L. (2004) Graphical and Computational Representation of Groups. In: Proceedings of ICCS, In: LNCS, Springer-Verlag, Berlin, 343-350.

https://doi.org/10.1007/978-3-540-25944-2_44