The Assessment of Medication Effects in Omicron Patients through MADM Approach Based on Distance Measures of Interval-Valued Fuzzy Hypersoft Set

Omicron, so-called COVID-2, is an emerging variant of COVID-19 which is proved to be the most fatal amongst the other variants such as alpha, beta and gamma variants (α, β, γ variants) due to its stern and perilous nature. It has caused hazardous effects globally in a very short span of time. The diagnosis and medication of Omicron patients are both challenging undertakings for researchers (medical experts) due to the involvement of various uncertainties and the vagueness of its altering behavior. In this study, an algebraic approach, interval-valued fuzzy hypersoft set (iv-FHSS), is employed to assess the conditions of patients after the application of suitable medication. Firstly, the distance measures between two iv-FHSSs are formulated with a brief description some of its properties, then a multi-attribute decision-making framework is designed through the proposal of an algorithm. This framework consists of three phases of medication. In the first phase, the Omicron-diagnosed patients are shortlisted and an iv-FHSS is constructed for such patients and then they are medicated. Another iv-FHSS is constructed after their first medication. Similarly, the relevant iv-FHSSs are constructed after second and third medications in other phases. The distance measures of these post-medication-based iv-FHSSs are computed with pre-medication-based iv-FHSS and the monotone pattern of distance measures are analyzed. It is observed that a decreasing pattern of computed distance measures assures that the medication is working well and the patients are recovering. In case of an increasing pattern, the medication is changed and the same procedure is repeated for the assessment of its effects. This approach is reliable due to the consideration of parameters (symptoms) and sub parameters (sub symptoms) jointly as multi-argument approximations.


Introduction
According to World Health Organization (WHO) [1], there are approximately 0.5 billion confirmed cases of COVID-19 and more than 6 million deaths reported as of 12 am coordinated universal time (UTC) 15 April 2022. This disease has been spreading rapidly for more than two and a half years. COVID-2 was recognized at the end of 2019 and a variety of other variations arose. On the basis of their origin and exposure, WHO has classified these variants into three main groups for screening and exploration purposes, which are named as variants under monitoring (VM), variants of interest (VI) and variants of concern (VC). Variants α, β, γ, δ are placed in the category of VC [2]. These variants are a major cause of deaths across the globe. In the last week of November 2021, Omicron arose as the fifth variant of VC, as declared by WHO, and has rapidly increased. 1.
An algebraic structure, interval-valued fuzzy hypersoft set (iv-FHSS), is employed to assess the conditions of Omicron patients after applying appropriate medication.

2.
A multi-attribute decision-making framework is designed through the proposal of an algorithm based on the distance measures between two iv-FHSSs. 3.
The proposed framework consists of three phases of medication. The Omicrondiagnosed patients are shortlisted and an iv-FHSS is constructed for such patients and then they are medicated in the first phase. Another iv-FHSS is constructed after their first medication. Similarly, the relevant iv-FHSSs are constructed after second and third medications in other phases. The distance measures of these post-medicationbased iv-FHSSs are computed with pre-medication-based iv-FHSS and the monotone pattern of distance measures are analyzed. 4.
It is observed that a decreasing pattern of computed distance measures assures that the medication is working well and the patients are recovering. In case of an increasing pattern, the medication is changed and the same procedure is repeated for the assessment of its effects. This approach is reliable due to the consideration of parameters (symptoms) and sub-parameters (sub-symptoms) jointly as multi-argument approximations.
The rest of the paper is organized as follows: Section 2 includes the basic notions of f-set, s-set, hs-set and ivf-set, etc. Section 3 presents some new operations of iv-FHSS. Section 4 investigates the distance measures of iv-FHSS. Section 5 presents the decision system of iv-FHSS along with application in the treatment of omicron patients. In the last section, the article is summarized with future directions.

Preliminaries
Let Z, P (Z ), C(Z ) represent the universe of discourse, collection of all subsets of Z and collection of all fuzzy sets of Z, respectively, and E be the set of parameters.

Definition 1 ([3]).
A fuzzy set F over Z is characterized by a membership function f F : Z → [0, 1] and is given by f F (z) = {(z, f F (z))|z ∈ Z } which assigns a real value within [0, 1] to each z ∈ Z and f F (z) is the membership-grade of z ∈ Z.

Definition 3 ([5]).
A fuzzy soft set ( f s-set) F S S over Z is defined as ∆ F S S : E → C(Z ) and is given by

Definition 4 ([7]
). An interval-valued fuzzy set (iv f -set) M iv f over Z is given by a function , and υ and ν denote lower and upper membership-grades of an element, respectively. For convenience, the set of all iv f -sets over Z is denoted by Γ(Z ).

Set Theoretic Operations of Interval-Valued Fuzzy Hypersoft Sets
This section of the paper aims to characterize some new operations of iv-FHSS. Consider two iv-FHSSs F iv f 1 HS , Λ 1 and F iv f 2 HS , Λ 2 . 1.
The addition of these two iv-FHSSs F iv f 1 HS , Λ 1 and F iv f 2 HS , Λ 2 is defined as follows: 2.
The multiplication of two iv-FHSSs F iv f 1 HS , Λ 1 and F iv f 2 HS , Λ 2 is defined as follow: 3.
The union of two iv-FHSSs F iv f 1 HS , Λ 1 and F iv f 2 HS , Λ 2 is defined as follows:

4.
The intersection of two iv-FHSSs F iv f 1 HS , Λ 1 and F iv f 2 HS , Λ 2 is defined as fol-

5.
Partial membership of iv-FHSS F iv f 1 HS , Λ 1 is defined as follows: 6. Partial non-membership of iv-FHSS F iv f 1 HS , Λ 1 is defined as follows: Example 5. An application of a similarity measure on fuzzy soft sets in medical diagnosis is presented in [35]. In this section, an application on iv-FHSS is discussed. Let a businessman want to hire a contractor for the construction of a building. There are five contractors Z = {z 1 , z 2 , . . . , z 5 } under consideration. There are two committees of experts. One of the commit-tee considers a set of attributes The iv-FHSSF iv f 1 The other committee constructed iv-FHSS F iv f 2  The iv-FHSS (F iv f 1 HS , Ω) can also be written as (F iv f 1 Similarly, iv-FHSS (F iv f 2 HS , Ω) can also be written as (F iv f 2

Example 6.
Consider two iv-FHSS F iv f 1 HS , Ω and F iv f 2 HS , Ω as in Example 5. The addition of F iv f 1 HS , Ω and F iv f 2 HS , Ω is calculated as follows: Example 7. Consider two iv-FHSS F iv f 1 HS , Ω and F iv f 2 HS , Ω as in Example 5. The multiplication of iv-FHSSs F iv f 1 HS , Ω and F iv f 2 HS , Ω is calculated as follows: Example 8. Consider two iv-FHSS F iv f 1 HS , Ω and F iv f 2 HS , Ω as in Example 5. The union of iv-FHSSs F iv f 1 HS , Ω and F iv f 2 HS , Ω is calculated as follows:

Example 9.
Consider two iv-FHSS F iv f 1 HS , Ω and F iv f 2 HS , Ω as in Example 5. The intersection of iv-FHSSs F iv f 1 HS , Ω and F iv f 2 HS , Ω is calculated as follows: , ω 2 , , ω 8 , Example 10. Consider iv-FHSS F iv f 1 HS , Ω as in Example 5. Partial membership of iv-FHSS F iv f 1 HS , Ω is calculated as follows: Example 11. Consider iv-FHSS F iv f 1 HS , Ω as in Example 5. Partial non-membership of iv-FHSS F iv f 1 HS , Ω is as follows:

Distance Measures between Interval-Valued Fuzzy Hypersoft Sets
In this section, the distance measures of iv-FHSSs are discussed. Let (F iv f HS , E ) be an iv-FHSS over Z; then, for each η ∈ E approximate setF iv f HS (η) is given byF iv f HS (z) = (z, κFivf HS (η) (z)) where κFivf HS (η) (z) = κ lF iv f HS (η) (z), κ uF iv f HS (η) (z) . Euclidean distance measure d Euc , Hamming distance measure d Ham and Hausdorff distance measure d Hau for two iv-FHSSs are given as follows:

Definition 9. Let Euclidean distance between iv-FHSSs
F iv f 1 HS , Λ 1 and F iv f 2 HS , Λ 2 be denoted by Definition 10. Let Hamming distance between iv-FHSSs F iv f 1 HS , Λ 1 and F iv f 2 HS , Λ 2 be denoted by Definition 11. Let Hausdorff distance between iv-FHSSs F iv f 1 HS , Λ 1 and F iv f 2 HS , Λ 2 be denoted by

Proposed Algorithm and Implementation
The following algorithm i.e., Algorithm 1 is developed by using iv-FHSS for decision making.

Application
A brief introduction to the outbreak of the Omicron variant is discussed in this section along with the optimised effect of medication on the treatment of Omicron patients.

Outbreak of Omicron Variant
The first confirmed case of Omicron [36] was reported in the second week of November last year as per data collected by WHO. Since that time, the distinct proof of the Omicron variant appeared to quickly spread. A new genomic-arrangement investigation on 77 infection tests gathered in South Africa in mid-November indicated an alarming situation, as all the investigated variations were of Omicron [37]. The average number of cases grew from roughly 300 to 900 daily, and reached the milestone of 2000 in the last week of November 2021 [38]. Positive cases of this variant escalated so much that WHO upgraded its category from VM to VC in just two days. Within a short while, this variant spread to more than 35 nations across Europe, Africa and America. A lot of work is to be performed on research into how and where this variation initially developed. Among three major waves of COVID-19 from June 2020 to December 2021, β and γ variations are responsible for two waves in South Africa. Research [38] showed that the spread the of β variant was almost 50% of day-to-day diseases and increased up to 80% for the δ variant; however, in the case of Omicron, the level increased up to 90% in just the last month of 2021. These results show the dominance of Omicron over other variants.

Optimised Effect of Medication on Treatment of Omicron Patient
Suppose there are five patients diagnosed with Omicron who form the set of universe Z = {z 1 , z 2 , . . . , z 5 }. Among all the symptoms of Omicron, some symptoms, such as fever, tiredness, cough and sensory loss, are most common, but there are other symptoms which are less common in patients, such as rashes on skin, sore throat, irritated eyes, headache, discolouration of fingers, diarrhoea, aches and pains. A team of health-care professionals was assigned the duty of collection and interpretation of data. The team considered only the most common symptoms, so the set of attributes consists of the most common symptoms, i.e., Θ 1 = fever, Θ 2 = cough, Θ 3 = tiredness and Θ 4 = sensory loss. Therefore, E = {Θ 1 , Θ 2 , Θ 3 , Θ 4 }. A person with a body temperature between 99.5 F to 100.4 F is considered to have a low-grade fever. According to U.S. Centers for Disease Control and Prevention (CDC), a temperature at or above 100.4 F is considered as a high fever, so Θ 1 = { low fever, high fever } = {θ 11 , θ 12 }. According to a research by Hsu et al. [39], 0 to 16 coughs per day were recorded for a healthy individual, whereas above this range was considered as high coughing, so Θ 2 = { low coughing, high coughing } = {θ 21 , θ 22 }. Θ 3 = { tiredness } = {θ 31 }. Θ 4 = { loss of smell, loss of taste } = {θ 41 , θ 42 }. To construct iv-FHSSs, the cartesian product of disjoint attributive sets is needed. Therefore, In Stage I, data is collected from patients which shows how they feel and they are given treatment against the virus for five days. The Omicron test is taken and the same form is completed by patients whose tests result is positive. In Stage III, patients are treated again for five days and an Omicron test is taken. The patients having a positive report of the virus are treated for five days after completing the form. At the end of 15 days, a final Omicron test is taken and form completed. The patients having a negative test report at any stage are discharged. All those patients who still have symptoms of the virus are re-medicated from Stage I. The complete process of treatment is described a follows: A form was completed by each patient before medication during their treatment which shows how they feel. According to the information provided by them, iv-FHSSF iv f HS (Θ) is constructed and given asF iv f

Comparative Study
The proposed model (iv-FHSS) is more flexible and more general as compared to existing structures in following manner: 1.
If sub-attributes are replaced with attributes, the model will represent iv f s-set.

2.
If intervals are replaced with fuzzy values, the model will represent f hs-set.

3.
If the parametrization tool is neglected with attributive sets instead of disjoint attributive valued sets, the model will represent iv f -set.
The proposed model is compared with existing models and illustrated in Table 2: The following abbreviations are used in Table 2 MF = membership function, SAAF = single argument approximate function, MAAF = multi-argument approximate function, SP = set of parameters, Z = universal set, P(Z ) = power set of universe, C(Z ) = collection of fuzzy sets, I([0, 1]) = set of all sub-intervals of [0, 1], CP =Cartesian product of disjoint-attributive-valued sets.
Proposed structure MAAF CP P(Z ) Sufficient

Conclusions
In this study, the concept of iv-FHSS was developed and some new operations such as addition, multiplication, union, intersection, partial membership and partial nonmembership for iv-FHSSs were discussed. Euclidean, Hamming and Hausdorff distances for iv-FHSS are discussed. A decision-making algorithm with the support of distance measures based on a real-world application for the treatment of Omicron patients was discussed. Improvement in health of Omicron patients by three different distance measures with pictorial representation was also carried out. Future work for multi-argument approximate function under soft set environments may include hybridized study of structures such as the intuitionistic fuzzy set, neutrosophic set, picture fuzzy set, refined fuzzy set, and pythagorean fuzzy set with interval-valued fuzzy hypersoft set, and their applications in decision making.