Mathematical modeling and AI based decision making for COVID-19 suspects backed by novel distance and similarity measures on plithogenic hypersoft sets.

It goes without saying that coronavirus (COVID-19) is an infectious disease and many countries are coping with its different variants. Owing to the limited medical facilities, vaccine and medical experts, need of the hour is to intelligently tackle its spread by making artificial intelligence (AI) based smart decisions for COVID-19 suspects who develop different symptoms and they are kept under observation and monitored to see the severity of the symptoms. The target of this study is to analyze COVID-19 suspects data and detect whether a suspect is a COVID-19 patient or not, and if yes, then to what extent, so that a suitable decision can be made. The decision can be categorized such that an infected person can be isolated or quarantined at home or at a facilitation center or the person can be sent to the hospital for the treatment. This target is achieved by designing a mathematical model of COVID-19 suspects in the form of a multi-criteria decision making (MCDM) model and a novel AI based technique is devised and implemented with the help of newly developed plithogenic distance and similarity measures in fuzzy environment. All findings are depicted graphically for a clear understanding and to provide an insight of the necessity and effectiveness of the proposed method. The concept and results of the proposed technique make it suitable for implementation in machine learning, deep learning, pattern recognition etc.

Introduction

There is no gainsaying the fact that the 21st century has been a century of different challenges for the mankind. On the top of all is the emergence of COVID-19 in the city of Wuhan, China in December 2019. Owing to the uncertainty, limited information and resources, it became a challenging task to tackle the pandemic in terms of its spread, vaccine and treatment. A reasonable amount of literature has been written on the topic by the mathematicians and statisticians to develop emergency support systems for COVID-19 [1], [2], [3], [4], [5]. In this article, a very important question has been addressed about the COVID-19 suspects who develop mild, moderate or severe symptoms and a smart decision is required for their treatment. The target is achieved by developing distance and similarity measures on plithogenic hypersoft sets and implemented to the MCDM model of COVID-19 suspects in fuzzy environment.

In order to deal with uncertainty, fuzzy set was introduced in 1965 by Zadeh [6], as an extension of the notion of classical set in which an element is either a member of the set or not. There is no third possibility. In fuzzy set, a membership value is allocated to all the elements in the interval . An extension of the fuzzy set, called intuitionistic fuzzy set (IFS), was presented by Atanassov [7], [8], [9], in which a membership grade as well as a non-membership grade is assigned to each element of the set. IFSs have numerous applications in logic programming [10], pattern recognition [11], [12], [13], [14], [15], decision making and medical diagnosis [16], [17] etc. Fuzzy set and intuitionistic fuzzy set are unable to handle indeterminacy in the data. This gap was fulfilled by Smarandache who presented the idea of neutrosophic set [18]. A generalization of these concepts was proposed by Smarandache, and termed as plithogenic set [19], [20]. It was noted that, in all these concepts, there is an insufficiency of parametrization tool. In 1999, Molodtsov [21], defined soft set as a parameterized family of the subsets of the universe. Soft set has been widely studied and applied in different fields [22], [23], [24], [25]. In 2018, Smarandache [26] proposed the idea of hypersoft set, as a generalization of soft set. A lot of research work has been written on hypersoft set, its hybrids and applications [27], [28], [29], [30]. Smarandache introduced plithogenic hypersoft set and its mathematical definition was given by Rayees et al. [31].

Distance and similarity measures play a key role in medical diagnosis and decision making. Beg et al. [32] and Chen et al. [33] presented the similarity measures for fuzzy sets and their hybrids. A number of researchers proposed the distance and similarity measures on IFSs along with their applications [12], [13], [14], [34], [35], [36]. A ground breaking work was performed by Majumdar and Samanta [37], [38] who proposed the similarity measures on soft sets and fuzzy soft sets, based on the distance between them. Kharal [39] presented a modified version of the distance and similarity measures on soft sets given by Samanta et al. Similarity measures have extensive applications in psychology [40], decision making [41] etc.

The contributions made in this article are two fold. First of all, distance and similarity measures are defined on plithogenic hypersoft sets. The necessity of defining these measures arises due to the fact that plithogenic hypersoft set is a generalization of all the previously defined structures. Moreover, it takes into account all the necessary information as well the shortcomings in the process of mathematical modeling of an MCDM problem. Therefore, the problems involving decisions on COVID-19 suspects must be intelligently handled on a broad spectrum. Thus, a mathematical model of COVID-19 suspects is structured into an MCDM problem in the context of plithogenic hypersoft set. Secondly, the proposed plithogenic distance and similarity measures are implemented on the problem to make smart decisions on the infected ones depending on the severity of the infection. Last but not the least, the algorithm of the proposed AI based method not only detects the infection in the suspects [42], but also talks about its severity. It is capable of reading the input, processing it according to the proposed algorithm steps and concluding with useful output. It makes the proposed method applicable to a variety of medical diagnosis problems. The algorithm is tested on COVID-19 suspects data due to the ongoing situation in the world facing the pandemic.

Mathematically, the proposed plithogenic distance and similarity measures satisfy the basic properties of distance and similarity measures and obey the results of relevant theorems. For the validation of the proposed algorithm, the results are calculated with the corresponding Euclidean and Hamming distance and similarity measures and a comparative analysis is performed. The obtained results prove the validity of the proposed algorithm and its efficiency in producing AI based decision making. The scope of the study can be extended to pattern recognition, image processing, picture science, in different environments like pythagorean fuzzy environment [43], -rung orthopair fuzzy environment, spherical fuzzy environment, etc.

The remainder of the article is arranged such that, in Section 2, some basic concepts are given for a better understanding of the proposed work. In Section 3, plithogenic distance measure is defined, relevant theorems with proofs are given, types of plithogenic distance measures are given, and plithogenic similarity measure is defined. In Section 4, mathematical modeling and AI based decision making for COVID-19 suspects is given. In Section 5, the COVID-19 model is employed on COVID-19 suspects. Conclusion is given in Section 6 along with future directions of the study.

Preliminaries

Let  be an initial universe,  be its power set, and  be a finite set of alternatives under consideration. Let  be a finite set of parameters or attributes, whose attribute values are members of the sets  respectively, such that these sets are disjoint, i.e., , with  and . Let , and  be a collection of selected attributes, i.e., .

Soft set [21]

A soft set  is defined by the mapping . Mathematically, it can be written as

Hypersoft set [26]

Let  represents the cartesian product of  having elements of the form , where  for . Then, a hypersoft set  is defined by the mapping , and given by

Plithogenic set [19]

If all members of the set  are characterized by attributes such that the attributes may have been further divided into attribute values, along with a contradiction degree defined between each attribute value and its corresponding dominant attribute value, then the set  is called a plithogenic set.

Mathematically, it is represented by the notation , where  is a subset of ,  is an attribute,  is the set of attribute values of ,  is the degree of appurtenance function, and  is the contradiction degree function.

Plithogenic hypersoft set [31]

The set  is called a plithogenic hypersoft set (PHSS), if an appurtenance degree function  and a contradiction degree function  between two attribute values are defined as follows:  where  gives the fuzzy, intuitionistic fuzzy, and neutrosophic degree of appurtenance respectively, such that for the attribute values  and  of the same attribute satisfy the following constraints:  For -tuple , PHSS  can be mathematically represented as

Distance measure between fuzzy sets [32]

There are different measures in literature for distance which is calculated between the fuzzy sets under consideration. The most commonly used distances between fuzzy sets, were categorized as hamming distance measure , normalized hamming distance measure , Euclidean distance measure , and normalized Euclidean distance measure . For any two fuzzy sets  and , over a finite universe , with truth membership functions  and , respectively for  such that , the said distances are

Distance measure between soft sets [39]

Let  be finite universe,  be the attributes, such that , and  such that , then the commonly used distances between two soft sets  and , over , are given as

Plithogenic distance and similarity measures

In almost every field of science including pattern recognition, image processing, machine learning, artificial intelligence and related fields, whenever we need to distinguished two objects, the concept of distance and their similarity is required. There are different measures in literature for distance and similarity in which distance is calculated between the sets under consideration and the degree of similarity is calculated based on their associated distance. There are multiple applications in which the contradiction degree  need to be defined. So the effectiveness of our method can be seen there. The proposed distance formulas accommodated the contradiction degree  by maintaining the structure of original formula and also its conditions,

For this, let  be the finite universe,  be the set of attributes, and  be the set of corresponding attribute values sets of the attributes  such that  represents the number of chosen attributes and  represents the equal number of attribute values of each attributes. Let  be the cartesian product of attribute values  such that  be -tuple of the attribute values and  represents the fuzzy contradiction degree between the attribute value  and their corresponding dominant attribute value . Let  and  be two PHSSs such that  and , be their corresponding degree of appurtenance, respectively, where , then the following plithogenic distance and similarity measures are proposed as follows:

Plithogenic distance measures

The proposed plithogenic distance measures (PDM) are named as plithogenic hamming distance measure , normalized plithogenic hamming distance measure , plithogenic Euclidean distance measure , and normalized plithogenic Euclidean distance measure . For any two PHSSs  and , the said distances are proposed as follows:

Theorem 1

The distance measures between plithogenic hypersoft sets  and  satisfies the following inequalities.

It is obvious from the definitions described in the Eqs. (3.1),(3.2),(3.3), and (3.4), that the proposed plithogenic distance measures obey these laws.

Theorem 2

The distance functions  defined from  are metric.

Let  and  be three plithogenic hypersoft sets over the universe . then

Suppose ,

For any three plithogenic hypersoft sets  and , consider  This implies that

In similar way, we can prove the results for  and .

Categories of plithogenic distance measures

It is evident that PHSS is the generalization of crisp, fuzzy, intuitionistic fuzzy and neutrosophic set. On the basis of the degree of appurtenance, these distance measures are categorized as:

Plithogenic crisp distance measure

Plithogenic fuzzy distance measure

Plithogenic intuitionistic fuzzy distance measure

Plithogenic neutrosophic distance measure

Plithogenic crisp distance measure

For any two plithogenic crisp hypersoft sets  and  over a finite universe , the proposed distance measures are as follows:  where  and  represents the crisp value which are either  or .

Plithogenic fuzzy distance measure

For any two plithogenic fuzzy hypersoft sets  and  over a finite universe , the proposed distances are:  where  and  represents the truth membership values from the interval .

Plithogenic intuitionistic fuzzy distance measure

For any two plithogenic intuitionistic fuzzy hypersoft sets  and  over a finite universe , the proposed distances are:  where  and  represent the truth memberships and false memberships of the sets  and  respectively from the interval  with the defined constraints .

Plithogenic neutrosophic distance measure

For any two plithogenic neutrosophic hypersoft sets  and  over a finite universe , the proposed distances are:  where  represents the truth memberships,  represents the indeterminacy memberships and  represents the false memberships of the sets  and  from the interval  with the defined constraints .

Plithogenic similarity measure

It is a very emerging concept in every field of science to distinguished any two objects when they are closely similar to each other. A novel concept of PSM is proposed based on the newly developed PDM, given as  where the distance  represents the plithogenic distance, which may be plithogenic hamming distance , normalized plithogenic hamming distance , plithogenic Euclidean distance , or normalized plithogenic Euclidean distance . In addition, PSM is termed as plithogenic crisp similarity measure, plithogenic fuzzy similarity measure, plithogenic intuitionistic fuzzy similarity measure, or plithogenic neutrosophic similarity measure, if the degree of appurtenance between the sets under consideration with their associated distances is crisp, fuzzy, intuitionistic fuzzy, or neutrosophic respectively.

Mathematical modeling and AI based decision making for COVID-19 suspects

In this section, an algorithm is developed based on the newly developed plithogenic distance and similarity measures between PHSSs. Furthermore, the algorithm is implemented in a specific scenario to see the results.

Algorithm of the proposed method

Let  be a universe of COVID-19 suspects such that  be the set of alternatives or suspects under consideration, , be a finite set of attributes or symptoms, and  be the set of chosen symptoms, i.e., . Let  be the cartesian product of the sets  containing the attribute values of the attributes  respectively, such that these sets are disjoint, i.e., , with . Each alternative  is allocated a degree of appurtenance  w.r.t. each attribute value , and  can be fuzzy, intuitionistic fuzzy or neutrosophic degree of appurtenance. The aim of the specialist (doctor) is to examine the chosen suspects whether they are infected from the COVID-19 or not, and if they are infected, then what is severity of the disease in order to draw a suitable conclusion on each suspect. For this purpose, an efficient AI based method is proposed. The construction steps of the proposed algorithm are as follows:

Step 1: Specialist allocates a fuzzy degree of appurtenance to each suspect w.r.t. each symptom, represented by PHSS .

Step 2: Construct a set of established values of the symptoms of disease under study according to the sub-symptoms based on their severity, represented by PHSS .

Step 3: Measure the distance between PHSSs  and , using the proposed novel distance measures, i.e., plithogenic hamming distance and plithogenic Euclidean distance measures given as follows:  where  represents the number of chosen attributes and  represents the number of attribute values of each attribute.

Step 4: To check how much they are similar, calculate the plithogenic similarity based on plithogenic distance measures between the specialist allocated values and their corresponding established values.  where  may be any plithogenic distance from the proposed PDM. The PSM between any two PHSSs  and  are plithogenic-similar if and only if , where 0.5 is threshold value, keeping in mind that this value may be different in normalized plithogenic distances.

Step 5: If , it implies that the suspect is not infected.

Step 6: If , the suspect is infected and the decision will be made based on the severity of the infection by following the cases mentioned below:

If , the suspect needs to be quarantined at home with proper isolation.

If , the suspect needs to be quarantined at a facilitation center.

If , the suspect must be sent to the hospital for proper treatment.

Implementation of the proposed method

Let  be the universe of discourse, such that  be a set of COVID-19 suspects. In order to monitor the suspects, the attributes or symptoms are as follows:

Each symptom is further categorized to low , medium  and high , which have mild symptoms, moderate symptoms, and severe symptoms, respectively, to analyze the severity of the disease in terms of their attribute values correspondingly for better diagnoses. That is,  where . The suspects under observations recorded by the symptoms lies in the specific region where these suspects come from are as . Each suspect under observation is allocated a hypothetical fuzzy degree of appurtenance correspondingly to each chosen attribute w.r.t some constraints designed by the specialist to check the validity of the method, is given in Table 1.

Table 1

Fuzzy degree of appurtenance of each alternative w.r.t. each attribute value.

Symptoms	Severity			Suspects
		s_1	s_2	s_3	s_4	s_5
	L	0.58	0.52	0.20	0.50	0.15
α_1	M	0.75	0.68	0.29	0.55	0.32
	H	0.93	0.86	0.19	0.72	0.22
	L	0.60	0.58	0.10	0.42	0.20
α_2	M	0.69	0.62	0.25	0.51	0.29
	H	0.83	0.76	0.30	0.64	0.36
	L	0.64	0.58	0.19	0.46	0.12
α_3	M	0.79	0.60	0.29	0.57	0.31
	H	0.77	0.72	0.20	0.65	0.20
	L	0.71	0.65	0.27	0.50	0.29
α_4	M	0.88	0.80	0.35	0.68	0.33
	H	0.97	0.90	0.40	0.65	0.50
	L	0.70	0.59	0.17	0.52	0.26
α_5	M	0.82	0.71	0.24	0.64	0.30
	H	0.96	0.85	0.30	0.79	0.42
	L	0.70	0.61	0.21	0.59	0.29
α_6	M	0.95	0.75	0.40	0.80	0.20
	H	0.97	0.92	0.28	0.70	0.17
	L	0.20	0.11	0.13	0.10	0.20
α_7	M	0.31	0.14	0.26	0.28	0.30
	H	0.60	0.24	0.35	0.39	0.40
	L	0.25	0.30	0.22	0.21	0.22
α_8	M	0.29	0.40	0.35	0.44	0.35
	H	0.32	0.51	0.44	0.30	0.40
	L	0.45	0.46	0.50	0.62	0.10
α_9	M	0.49	0.45	0.10	0.53	0.19
	H	0.60	0.59	0.12	0.60	0.18
	L	0.11	0.41	0.20	0.22	0.33
α_10	M	0.19	0.15	0.34	0.40	0.27
	H	0.32	0.38	0.51	0.35	0.43
	L	0.70	0.61	0.13	0.33	0.16
α_11	M	0.60	0.50	0.10	0.55	0.14
	H	0.34	0.62	0.18	0.60	0.10
	L	0.19	0.41	0.20	0.50	0.20
α_12	M	0.26	0.30	0.36	0.40	0.45
	H	0.35	0.45	0.60	0.46	0.52
	L	0.10	0.22	0.40	0.38	0.40
α_13	M	0.32	0.31	0.21	0.50	0.49
	H	0.40	0.60	0.43	0.60	0.58

Each symptom has a dominant attribute value  from the chosen symptom, and the fuzzy contradiction degree  of each chosen attribute between the attribute value and the dominant attribute value is given in Table 2.

Table 2

Contradiction degrees with corresponding dominant value.

Symptoms	Dominant	L	M	H
α_1	H	0.95	0.70	0.00
α_2	H	1.00	0.60	0.00
α_3	M	0.45	0.00	0.60
α_4	H	1.00	0.75	0.00
α_5	H	0.97	0.80	0.00
α_6	M	0.40	0.00	0.58
α_9	L	0.00	0.50	0.90
α_11	L	0.00	0.58	0.95

A PHSS  is constructed and represented the data in tabular form with the opinion of the specialist field expert doctor, given in Table 3.

Table 3

Fuzzy degree of appurtenance allocated by the specialist to the suspects.

Symptoms	Severity			Suspects
		s_1	s_2	s_3	s_4	s_5
	L	0.58	0.52	0.20	0.50	0.15
α_1	M	0.75	0.68	0.29	0.55	0.32
	H	0.93	0.86	0.19	0.72	0.22
	L	0.60	0.58	0.10	0.42	0.20
α_2	M	0.69	0.62	0.25	0.51	0.29
	H	0.83	0.76	0.30	0.64	0.36
	L	0.64	0.58	0.19	0.46	0.12
α_3	M	0.79	0.60	0.29	0.57	0.31
	H	0.77	0.72	0.20	0.65	0.20
	L	0.71	0.65	0.27	0.50	0.29
α_4	M	0.88	0.80	0.35	0.68	0.33
	H	0.97	0.90	0.40	0.65	0.50
	L	0.70	0.59	0.17	0.52	0.26
α_5	M	0.82	0.71	0.24	0.64	0.30
	H	0.96	0.85	0.30	0.79	0.42
	L	0.70	0.61	0.21	0.59	0.29
α_6	M	0.95	0.75	0.40	0.80	0.20
	H	0.97	0.92	0.28	0.70	0.17
	L	0.45	0.46	0.50	0.62	0.10
α_9	M	0.49	0.45	0.10	0.53	0.19
	H	0.60	0.59	0.12	0.60	0.18
	L	0.70	0.61	0.13	0.33	0.16
α_11	M	0.60	0.50	0.10	0.55	0.14
	H	0.34	0.62	0.18	0.60	0.10

The established values of chosen symptoms by the experts are mentioned in Table 4.

Table 4

Established values of chosen symptoms.

Attributes	L	M	H
α_1	0.59	0.75	0.95
α_2	0.62	0.70	0.85
α_3	0.68	0.77	0.82
α_4	0.71	0.90	1.00
α_5	0.70	0.85	1.00
α_6	0.76	0.90	1.00
α_9	0.40	0.59	0.70
α_11	0.43	0.65	0.74

To measure the distance between the specialist recorded values and established values, plithogenic hamming and Euclidean distances are measured between them using the Eqs. (4.1), (4.2), and the results are given in Table 5.

Table 5

Distance measures.

Suspects	dHP(P_1,P_2)	dEP(P_1,P_2)
s_1	0.0710	0.0905
s_2	0.2219	0.2557
s_3	1.3272	1.4428
s_4	0.4397	0.5105
s_5	1.3055	1.4131

Now, plithogenic similarity based on plithogenic distance is evaluated to examine the severity of the infectedness of the suspects under observation to measure plithogenic-similarity is mentioned in Table 6.

Table 6

Similarity measures.

Suspects	SHP(P_1,P_2)	SEP(P_1,P_2)
s_1	0.9338	0.9170
s_2	0.8184	0.7963
s_3	0.4297	0.4094
s_4	0.6946	0.6620
s_5	0.4337	0.4144

From Table 6, the values of the hamming and euclidean similarity measures are obtained efficiently. From these values, a decision for the suspects are taken based on the obtained values which is represented in Table 7.

Table 7

Suspects decision.

Similarity index	Decision	Suspects
0≤S^P<0.5	Safe zone	s_3,s_5
0.5≤S^P<0.7	Home Isolation	s_4
0.7≤S^P<0.9	Quarantine center	s_2
0.9≤S^P≤1	Hospital treatment	s_1

The graphical representation of these suspects is given in Fig. 4.2.

Fig. 4.2

Plithogenic-similarity of suspects.

It is obvious from the graph, there is a danger zone above the line 0.5, while below the line, there is a safe zone. Hence, the vertical bars of the suspects which are below the danger zone line, are safe, while those who are above this line, are in danger (see Fig. 4.1).

Fig. 4.1

Proposed COVID-19 model flowchart.

Comparative analysis

To authenticate the proposed plithogenic distances, we compare our results with the usual hamming and Euclidean distance measures given in the Eqs. (5.1), (5.2) given below:  where  represents the number of chosen attributes and  represents the equal number of attribute values of each attributes.

Now, plithogenic similarity based on plithogenic distance is compared with the usual similarity based on usual distances to examine the severity of the infection in the suspects. The comparison analysis of usual similarity with plithogenic similarity is shown in Table 8.

Table 8

Comparison analysis of COVID-19 suspects using hamming similarity measure (HSM) and plithogenic hamming similarity measure (PHSM)

Suspect	HSM	Decision	PHSM	Decision
s_1	0.9259	Hospital treatment	0.9338	Hospital treatment
s_2	0.7897	Quarantine center	0.8184	Quarantine center
s_3	0.3941	Safe zone	0.4297	Safe zone
s_4	0.6633	Home Isolation	0.6946	Home Isolation
s_5	0.3949	Safe zone	0.4337	Safe zone

The comparison analysis of usual similarity with plithogenic similarity is shown in Table 9.

Table 9

Comparison analysis of COVID-19 suspects using Euclidean similarity measure (ESM) and plithogenic Euclidean similarity measure (PESM)

Suspect	ESM	Decision	PESM	Decision
s_1	0.9132	Hospital treatment	0.9170	Hospital treatment
s_2	0.7782	Quarantine center	0.7963	Quarantine center
s_3	0.3910	Safe zone	0.4094	Safe zone
s_4	0.6465	Home Isolation	0.6620	Home Isolation
s_5	0.3924	Safe zone	0.4144	Safe zone

It can been seen that the proposed plithogenic distance and similarity measures are reliable and efficient in producing results.

Conclusion

In this article, plithogenic distance and similarity measures on plithogenic hypersoft sets are defined for making intelligent decisions on COVID-19 suspects. For this purpose, mathematical modeling of COVID-19 suspects is structured into an MCDM problem by using the concept of plithogenic hypersoft set in a fuzzy environment and a novel algorithm is implemented on the problem, based on newly developed plithogenic distance and similarity measures. The algorithm of the proposed method not only detects the infection in the suspects, but also explain its severity. Mathematically, the proposed plithogenic distance and similarity measures satisfy the basic properties of distance and similarity measures and obey the results of relevant theorems. For the validation of the proposed algorithm, the results are calculated with the corresponding Euclidean and Hamming distance and similarity measures and a comparative analysis is performed. The obtained results prove the validity of the proposed algorithm and its efficiency in producing AI based decision making. The scope of the study can be extended to pattern recognition, image processing, picture science, in different environments like pythagorean fuzzy environment, -rung orthopair fuzzy environment, spherical fuzzy environment, etc.

Declaration of Competing Interest

The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.