The Study of HIV Diagnosis Using Complex Fuzzy Hypersoft Mapping and Proposing Appropriate Treatment

HIV is still a global epidemic more than 40 years after it was described initially, impacting mainly Sub-Saharan Africa, Southeast Asia, and Latin America. HIV is an RNA retrovirus that wreaks havoc on the immune system, making the infected individual vulnerable to opportunistic pathologies. For the diagnosis and therapy of infected patients, several models have been proposed in literature. The main goal of this paper is to present an innovative mathematical model for diagnosing and treating this pandemic based on a unique flexible fuzzy-like structure called the Complex Fuzzy Hypersoft (CFHS) set, which is a glued structure of complex fuzzy (CF) set and hypersoft sets (an extension of soft set). To address ambiguity and unclear data, the basic theory of CFHS set is created, which examines the amplitude term (A-term) and phase term (P-term) of complex numbers concurrently. In two aspects, this new fuzzy-like hybrid theory is adaptable. First, extending membership function values to the unit circle on an Argand plane and including an extra term, the P-term, to suit the recurring character of data that provide access to a wide range of membership function values. Second, it categorises the different attributes into matching disjoint attribute-valued sets for a more straightforward interpretation; it’s tough to identify which HIV is and how serious it is after looking at the HIV side effects. To deal with such problems, the CFHS set, and CFHS-mapping with its inverse mapping is utilised. These concepts are practical and required for adequately assessing the situation using mathematical modelling. This investigation shows a relationship between symptoms and medications, making the story easier to follow. A table is constructed for the HIV kinds based on a fuzzy interval of [0, 1]. The calculation is based on CFHS-mapping, which correctly diagnoses the condition and prescribes the best medicine. A generalised CFHS-mapping is also offered, which can assist a specialist in extracting the patient’s improvement record and estimating how long it will take to eradicate the infection.


I. INTRODUCTION
HIV or the Human Immunodeficiency virus is a significant threat to all communities around the globe. It attacks the body's immune system, makes it prone to infection, and may lead to AIDS (Acquired Immuno-Deficiency System). A mature virion has a lipid bilayer membrane that encapsulates a dense cone-shaped nucleocapsid contain-The associate editor coordinating the review of this manuscript and approving it for publication was Yiqi Liu. ing the genomic RNA molecules and associated enzymes like reverse transcriptase, integrase, and various cellular factors. The virion has a diameter of 100-120 nm with spherical morphology [1]. The virus is categorized into two types, namely HIV-1 and HIV-2. It was found that HIV-2 viremia was less frequently detected in the plasma viral load in the patients with relatively similar conditions [2]- [4].
Zadeh [14] was the first to establish the fuzzy set (FS) theoretical idea in 1965. With the assistance of FS, The term VOLUME 9, 2021 This work is licensed under a Creative Commons Attribution 4.0 License. For more information, see https://creativecommons.org/licenses/by/4.0/ ''membership function'' was defined, and the concept of ''uncertainty'' was discussed. The notion of Atanassov [15] generalized FS and included the degree of non-membership as a component, and came up with the idea of the intuitionistic fuzzy set (IFS). Molodtsov [22] presented the concept of soft set (SS) theory as another basic set theory in 1999. SS theory is now utilized in many science and technology sectors, and it has become one of the most popular branches of mathematics because of its wide range of applications in many fields of research. To tackle decision-making difficulties, Maji et al. [21] utilized soft sets. Yang et al. [32] stressed the importance of S-sets and their applications. The notion of fuzzy SS and its various properties were put forward by Maji et al. [20].
Broumi et al. [43] developed and characterized mappings between neutrosophic soft expert sets, as well as their images and inverse images. In [19], Karaaslan expands the idea of ''soft class'' and its accompanying methods. In 2009 and 2011, Kharal and Ahmad [16], [17] established the concept of mappings on fuzzy soft classes and soft classes, respectively. Alkhazaleh and Marei [34] explored the notion of a mapping on classes and categorized neutrosophic soft sets into neutrosophic soft classes. Sulaiman and Mohamad [36] conceptualized the mappings between collections of multi-aspect fuzzy soft sets. Bashir established the concept of mappings between intuitionistic fuzzy soft sets and Salleh [35]. The Fuzzy Hypersoft (FHS) and Hypersoft (HS) sets were suggested by Samarandache [18] in 2018 as extensions of fuzzy soft and soft sets, respectively. Crisp, fuzzy, intuitionistic fuzzy, neutrosophic, and plithogenic early worlds were the types of initial universes he distinguished at that time. He also demonstrated that an FHS set might be crisp, intuitionistic, neutrosophic, and plithogenic in this way. Saeed et al. [25], [48] explained the rudiments of Hypersoft set. Numerous applications of SS, neutrosophic set, neutrosophic Hypersoft set in the pattern recognition, medical diagnostics, MCDM and described mappings in a hypersoft set scenario were also explained by Saeed et al. [13], [24], [26], [28], [29], [33], [47]. Osgouie and Azizi [53] presented the use of fuzzy logic direct model reference adaptive control (DMRAC) for insulin infusion control for diabetic type 1 patients. Azizi and Seifipour [54] attempted to simulate the remodeling phase of dermal wound healing processing using neural networks as an intelligence technique. Wang et al. [55] presented the Pythagorean fuzzy interactive Hamacher power aggregation operators for assessment of express service quality with entropy weight. Liu et al. [56] developed the relation between hesitant fuzzy sets with application in medical diagnosis. Molla et al. [57] discussed the Pythagorean fuzzy Promethee method and apply it in medical diagnosis. Mahmood et al. [58], [59] extended the concept of fuzzy set to spherical fuzzy set and applied it for decision making purpose and medical diagnosis.
Ramot et al. [50] presented the idea of a thorough examination of the mathematical characteristics of the CF set. The CF complement, union, and intersection were explored as basic set-theoretic operations on CF sets. Thirunavukarasu et al. [45] investigated the intuitive comprehension of the aggregation process of a CF soft set. They also presented aggregation operations applications, proving that the method may be utilized successfully in various circumstances, including uncertainties and periodicities. Rahman et al. [46] developed the complex hypersoft (CHS) set and used a CF set, a complex intuitionistic fuzzy set, and a complex neutrosophic set to create hypersoft hybrids. Al-Qudah and Hassan [44] introduced the notion of a complex multi-fuzzy set, which is a mix of CF sets and multi-fuzzy sets. By simultaneously preserving the amplitude and periodic nature of the C-numbers, their proposed approach will be prepared to handle with two-dimensional multi-fuzzy data's uncertainties and ambiguity.

A. MOTIVATION
Since it is difficult to determine the specific type of HIV from its severity using prior existing theories and methods [16], [17], [49], and [42] because these techniques are confined to complete models, the main goal of this study is to model a realistic scenario of HIV clinical diagnosis and appropriate and successful treatment. The techniques outlined in [16], [17], [49] and [42] are insufficient for a thorough examination of the data for improved comprehension and treatment. They are incapable of managing complex (two-dimensional) information. We extended these models to a complex system characterized by a merging of fuzzy set and Hypersoft (HS) set to address such issues. In two respects, this structure is more adaptable. First, its membership range expands to the unit circle in a complex plane by altering the CFHS to include an additional term, the P-term, to account for the data's periodic nature. Second, the attributes in CFHS may be subdivided into attributive values for better comprehension. A mapping is a relationship between two region regulated by some mathematical laws that convert an knotted parameter to its related fundamental parametric types of values based on structural and basis similarities. This mapping enables to deal with similar-type parameters in an individual basic parameter. The objective of the research is to describe HIV diagnoses in the area and the symptoms that accompany them. It's tough to identify which type of HIV is causing the issue and how serious it is after looking at the HIV side effects. To tackle this problem, the CFHS set is utilized and CFHS-mapping and its inverse mapping. This study shows a connection between symptoms and medicines, which helps to simplify the story. A table based on a fuzzy interval of [0, 1] is developed for the various kinds of HIV. The computation is based on CFHSmapping, which correctly identifies the condition and selects the optimum treatment for each patient's associated illness. Finally, a generalized CFHS-mapping is provided, which will aid a specialist in extracting the patient's improvement record and forecasting the length of therapies needed to clear the illness.

B. PAPER PRESENTATION
The following is how the rest of the article is organized. The concepts of CF soft set, HS set, CFHS set, HS mapping and HS inverse mapping are re-imagined in Section II. Section III describes mapping on CFHS classes, the CFHS image, inverse image of CFHS. The validity of the recommended approach is demonstrated in section IV by a practical implementation and a comparison study. Finally, the conclusion has been discussed in the last section.

II. PRELIMINARIES
A few fundamental definitions for the universe S are offered in this section.
Definition 1 ( [45]): Let S and H be initial universal set and parameters set respectively, let A ⊆ H and ϕ A be a CF soft set over S for all y ∈ H . Then, an CF soft set ϕ A over S defined by a function µ A representing a mapping ϕ A : A → C(S).
Here, µ A is called CF approximate function of the CF soft set ϕ A , and the value µ A is a CF set called y-element of the CF soft set for all y ∈ A. Thus, a CF soft set ϕ A over S can be represented by the set of ordered pairs ϕ A = {(y, µ A (y)) : y ∈ E, µ A (y) ∈ C(S)}. Note that the set of all the CF sets over S will be denoted by C(S). Definition 2 ( [18]): Let δ 1 , δ 2 , δ 3 , . . . , δ n be attribute valued sets of distinct attributes t 1 , t 2 , t 3 , . . . , t n respectively, where δ l ∩ δ m = ∅, for l = m. Then ( , N ) is said to be HS set over S, where is the mapping from N to the power set of S and N = δ 1 × δ 2 × δ 3 × · · · × δ n . For more detail please see [23], [25], [30], [31], [52].

III. COMPLEX FUZZY HYPERSOFT CLASSES MAPPINGS
This portion introduces the idea of mapping on CFHS classes. CFHS sets are gathered in CFHS sets. We also characterise the features of CFHS, such as CFHS images and CFHS inverse images of CFHS sets. Consider the following points as you read through this section, and (Y , Q) be two classes of CFHS over X and Y respectively. Let η : X → Y and θ : D → Q be mappings, and let (χ , D) ∈ (X , D) and where ϕ is a mapping ϕ : is an CFHS in (X , D), and is defined as for ν ∈ θ −1 (τ ) and u ∈ X . VOLUME 9, 2021 Definition 7: Let ϕ : (X , D) → (Y , Q) be a mapping and (χ, D) and (ψ, Q) be the two CFHS sets in (X , D). Then, for τ ∈ Q, CFHS union and intersection of CFHS images of (χ, D) and (ψ, Q) in (X , D) are given as; Definition 8: Let ϕ : (X , D) → (Y , Q) be a mapping and (χ, D) and (ψ, Q) be the two CFHS sets in (X , D). Then for ν ∈ D, CFHS union and intersection of CFHS inverse images of (χ, D) and (ψ, Q) in (X , D) are given as; The properties of CFHS mapping are illustrated in the following manner.
Following theorem provides some important properties of inverse CFHS mapping under two sets.

IV. APPLICATIONS OF CFHS SET IN HIV
In this section, HIV and its relevant problems are analyzed. The factors taken into consideration are reasons, symptoms, diagnosis, and treatment of HIV patients. We go through the CFHS set's overall notion and its relative mapping, and inverse mapping. Then it is demonstrated how to build a treatment plan using the suggested mathematical model for HIV patients.

A. HIV AND ITS PROPERTIES ARE BEING INVESTIGATED
The importance of HIV analytical research and mathematical modeling in the medical profession is immeasurable. There are two types of HIV which are under consideration.
The HIV-1 virus's genome comprises two similar 9.2 kb single-stranded RNA chains encapsulated within the virion.  The consistent for the virus presents itself as proviral double-stranded DNA when tested inside an infected cell. One of the significant signs of progression of AIDS in an HIV-1 infected patient is the respective decline of CD4 lymphocytes that serve as the primary attachment site for HIV and the induction of specific cell-mediated responses [1]. Also, it was found that high levels of plasma viral RNA can cause an increased progression rate; a great deal of emphasis has been put on ways to reduce plasma viremia. Many antiretroviral drugs have been designed to alleviate the issue of stopping the life cycle of the virus [5], [6]. For more detail see Fig 1, 2.

2) HIV-2
HIV-2 is regarded as the second type of virus that leads to AIDS. It has been found that the HIV-2 virus has been tested positive in heterosexual communities in West Africa, but its spread is very limited [7], [8]. In addition, it has been reported that the infection of the HIV-2 virus has milder effects in comparison to the HIV-1 virus [9]. Also, the isolation of asymptomatic HIV-2 virus from the peripheral blood mononuclear cells is far less than the HIV-1 virus infection, but with the decline in the CD4 levels, the virus equally efficient leading to a higher rate of disease spread [10], [11]. For more detail see Fig 3 and    Because of the same symptoms of HIV, a specialist has a few challenges while examining an HIV patient. The contrast between the different classification of HIV is complicated to grasp. It suggests that these kinds of barriers are made up of ambiguity and vagueness. For this purpose, the CFHS set is suited for dealing with such data. A chart between 0 and 1 has been created to assess the genuine kind of HIV from its different types; see Table 1. Since each issue becomes more concentrated over time, the paper will collect more helpful information about a patient's history, and each physician will compare the appearance of side effects to at least 2-3 days of data after monitoring side effects, if there is any, of disease. To examine the HIV patient's condition, additional graphs of conditions and their day-by-day fixation have been created. This may be found in Table 2 and Figure 5. For each type of HIV, there are three stages: serious, moderate, and low. The flow chart in Figure 5 also shows the different ranges that have been allocated to these constraints.

2) ALGORITHM
Step 1: To classify the HIV family. Let P = {p 1 , p 2 , p 3 , . . . , p n } be group of patients suffering from HIV and A = {d 1 , d 2 , d 3 , . . . , d v } be a collection of clinical symptoms of HIV whose parametric values links to sets With the help of a mathematician, the administration creates the number of daily diagnostic charts denoted as ''t.'' (which may be fitted up as a CFHS set). This chart will aid us in determining the patient's proper infection. After an important assessment at εth times, the specialist's CFHS set chart might be fitted up as follows: are CFHS membership of HIV-1, HIV-2 for kth symptoms and lth patients respectively and (l = 1, 2, 3, . . . , n, k = 1, 2, 3, . . . , |D|, ε = 1, 2, 3, . . . , t).
To get maximum information union of CFHS sets is used.
Step 2: Assume B = {d 1 , d 2 , d 3 , . . . , d w } to represent the collection of linked symptoms to A whose attribute values correspond to sets D i 's, where D = w i=1 D i and evaluate a CFHS set (keep in mind the patients ε number of daily assessments of basic symptoms specialists allots the weights).
Let CFHS-mapping = (θ, σ ) : CFHS(P) → CFHS(P) defined as; where T s k corresponds to associated weights from z D . Get the z ε D image under the described mapping ,, which may be constructed as z D .
Step 4: Convert CFHS set to fuzzy hypersoft set to get weighted aggregation values by using the formula, T z (s ) (p) = w 1 µ z (s ) (p) + w 2 ( 1 2π )ω z (s ) (p) [51], where µ z (s ) (p) and ω z (s ) (p) are the amplitude and periodic terms in the CFHS set respectively, T z (s ) (p) is the membership function in the fuzzy Hypersoft set for w 1 , w 2 ∈ [0, 1].
Step 5: Compare these results obtained from Step 4 with the Table 2 and get basic diagnosis. These results will later be assessed to verify the reliability of the obtained results.   Step 6: Calculate the average of each individual patient corresponding to their connected symptoms. Now, the results obtained are compared with the diagnosis Table 1.
Step 7: Consider B = {d 1 , d 2 , d 3 , . . . , d w } to represent the collection of related symptoms, C = {g 1 , g 2 , g 3 , . . . , g x } is a collection of potential treatments and to create χ D , where |D i | is an accumulation of potential medications and χ = D → P(C) is a CFHS function from which is the bundle of doctor's recommendations accompanied with the right treatment for HIV symptoms.
Step 8: Calculate CFHS set union among z D , χ D and get P 1 C .
Step 9: Choose medications (treatments) that offer extra benefits and have fewer negative impacts. The following procedures can be followed for the patient's improvement record.
Step 11: Repeat step 10 as needed till we achieve the desired results. The flow chart for the proposed algorithm is shown in 6

3) METHODOLOGICAL LIMITATIONS
Before performing the algorithm, it is necessary to check that the following limits of the proposed technique are met: 1) Because both parameters have the same structure and base, there must have been a mapping that converts the linked parameter into its basic parametric value. 2) Both sets of mapping or composition specifications should be independent of one another and belong to the same structural class. 3) According to a doctor's advice, the right medication for the disease's symptoms should be understood based on the illness's supporting evidence. 4) With the assistance of a doctor, the various ranges of concerning sorts of sickness must be identified.

5)
If the proposed medicine causes the patient harm, inverse CFHS-mapping can be utilized to restore him to his proper level, and then we must restart medicine from the beginning.

C. STUDY PROPOSAL AND NUMERICAL EXAMPLE
This portion focuses on applying the recommended algorithm to a medical context. First, the input samples are translated and gathered in mathematical language with the help of medical personnel. The next stage is to choose a group of patients with HIV symptoms as identified by the doctor. A diagnostic map for the conditions of distinct HIV (Table 1) and their dayto-day circumstances (Table 2) about the diagnosis was developed under the supervision of the doctor. These tables can be used to compare HIV symptoms to the severity of the disease. The technique's most significant benefit is that it lets you utilize the initial data from our suggested model to figure out   what kind of sickness you have. In addition, the algorithm will suggest the optimal treatment choice for the identified condition's specific form. A generalised mapping of the patient's rehabilitation and optimum recovery for patients will help the approach expand in the future. Four patients had a specific type of HIV that required a doctor's diagnosis. However, it's challenging to pick the proper one because there are several overlapping symptoms for a comparable ailment. Numerous dynamics are ruled out the health of patient's, current and previous skin colour changes, history, hereditary variables, and other factors. Following the first evaluation of the patient, the doctor will recommend a treatment and rehabilitation plan. To execute the given method, we used hypothetical data. It's a methodology and a technique that demonstrates how to use mathematical calculations to compute the proposed model. However, if this model is created for real-time data, the required result can be obtained by analyzing appropriate information.
Step 1: Let P = {p 1 , p 2 , p 3 , p 4 } be collection of four patients. Let d 1 = Overexertion, d 2 = Headache, d 3 = Pneumonia, be symptoms with distinct attributes whose attribute values are related to the sets D 1 , D 2 and D 3 respectively. Let D 1 = {d 11 = feeling faint, d 12 = strains}, D 2 = {d 21 = Migraines}, D 3 = {d 31 = bacterial infections, d 32 = Chills}, which can be evaluated by a doctor after a thorough examination. Based on the first information provided by patients with the symptoms described above, we can generate a chart of two (ε = 2) days with the data gathered by the doctor supplied as z ε D ∈ CFHS(P) for the 1st and 2nd-day data given as Table 3 and Table 4 separately, both in CFHS. Next, we take CFHSunion over the z 1 D and z 2 D . The resultant CFHS z ε D is given as Table 5.
Step 2: Let D 1 = {d 11 = dizziness, d 12 = redness and swelling}, D 2 = {d 21 = sore or aching muscles},  D 3 = {d 31 = intermittent, d 32 = remittent} be three sets of attribute values corresponding to three distinct attributes d 1 = Muscle aches, d 2 = Fatigue, d 3 = Fever respectively for HIV related symptoms. Doctors assign the weight to the related symptoms based on patient's information and transform verbally information to the numerical notation to create the CFHS z D type shown in Table 6.
Step 5: Comparing the Tables 7 and 2 to acquire initial diagnosis table (8). This table will be used after analyzing the precision of our outcomes.
Step 6: Calculate average of all the attributes corresponding to every individual's symptoms from Table 7. These values are given in Table 9. HIV's diagnosis Table 1 is now being compared to the results obtained in Table 9. The comparison revealed that patients p 1 , p 3 are diagnosed with HIV-1 and patients p 2 , p 4 are diagnosed with HIV-2.
Step 7: After determining the true nature of each patient's disease, the doctor will recommend medication to them and to create the CFHS set based on the suggestions of specialists, including the appropriate treatment for the different sorts of HIV. Consider C = {g 1 = emtricitabine, g 2 = amivudine, g 3 = tenofovir DF, g 4 = zidovudine} be different viable medications (treatments), then construct χ D , that is a set of doctor's suggestions for the best treatment for HIV symptoms and convert CFHS into fuzzy hypersoft set by using the formula T z (s ) (p) = w 1 µ z (s ) (p) + w 2 ( 1 2π )ω z (s ) (p) [51], with weights w 1 = 0.2, w 2 = 0.4 to get weighted aggregation. The set χ D ∈ CFHS(P) given as Table 10. In table 10,  the assessments are given in accordance with each patient's medical history.
Step 8: Calculate CFHS set union among χ D and z D and get the relationship between proposed medications and patients as CFHS set χ D z D = P 1 C , see Table 11.
Step 9: The recommended medicine is appropriate for those patients which shows improvement and less side effects. The maximum values relating to each patient's medication are given in (Table 12). In Table 12, it is obvious that treatments g 4 , g 1 or g 3 or g 4 , g 1 , g 4 is fit for the patient of p 1 , p 2 , p 3 and p 4 respectively. The final choice is made depending on the patient's present state, clinical history, and kind of sickness.
The CFHS-mapping may then be written in this manner The CFHS-mapping is given as where g ∈ σ (C) ⊆ C, p ∈ P q , π ∈ P q−1 , ϑ ∈ C q−1 denotes the number of treatments and treatment episodes, see Tables 13, 14, 15 and 16 for values of q.
Step 11: Step 10 is repeated until the patients' outcomes are achieved. Figures 7, 8, 9 and 10 shows the progress report for each patient.

D. ADVANTAGES OF THE PROPOSED ALGORITHM
This model tries to find the closest diagnosis of any condition, as well as the symptoms that go with it. These ideas are capable and required for properly assessing an issue when VOLUME 9, 2021      combined with scientific modeling. This investigation shows a relationship between symptoms and medications, making the story easier to follow. To accurately identify the condition and determine the optimum therapy for each patient's ailment, the computation depends on CFHS-mapping.  The generalized CFHS-mapping will help a specialist (Doctor) forecast the patient's progress and estimate the time of therapy until the infection is relieved.

E. COMPARATIVE ANALYSIS
CFHS mapping is a comprehensive idea that may be used for a wide range of diseases. Existing theories cannot be utilised to address or investigate the issues; nevertheless, they do have limitations (see Table 17). Because of these restrictions, they are unable to collect all of a patient's initial data. The proposed methodology can convert a patient's medical history into a mathematical format without losing any data, allowing us to achieve the most refined diagnosis and treatment outcomes. In Table 17, our suggested model is compared to current methodologies. When the attributes have been further split into attribute values, and the concerns include complex (2D) information, all previous methods fail to execute. The proposed CFHS-mapping addresses this shortcoming. It demonstrates that, in comparison to existing methods, the proposed structure is sound and capable of successfully dealing with such challenges. Now, we talk about our proposed strategy and how precise it is.
• Add several days to this estimate since the HIV patient cannot analyze thoroughly after the initial checkup. The CFHS set contains all of the patient's information, and symptoms may be connected to its union and severity.
• The connection between related and essential indications, as well as the weights allocated to them, is vital in every patient study. Assume the findings will be non-specific if we choose early symptoms at that time.
• In the second stage, to determine therapy for the patients depending on their HIV type.
• Follow the patients' development in the third stage using a generalized form of CFHS-mapping. All memberships drop with each scene until they approach zero, suggesting that HIV symptoms, treatment neutral outcomes, and side effects are all reducing. Thus, this model depicts the evolution of patients throughout period.
• Suppose a patient fails to gain improvement, inverse mapping can be used to restore him to his correct level, and then medicine must be started over.
• The suggested technique is beneficial for many patients with varied diseases and multiform criteria when parameterizations are used. In dealing with difficulties in the medical profession and MCDM, this study is solid and consistent.
• The decision-making committee (Doctors) will evaluate the data in the form of CFHS by considering the degree of the influence and the total time of the influence as a complex number; along with the in-depth evaluation of the information by taking sub parametric values of assigned attributes as hypersoft structure; where all the data can be taken in a numeric value between 0 (degree of zero per cent match) and 1 (degree of hundred per cent match).

V. CONCLUSION
In this study, HIV and the problems that come with it are discussed. A technique for identifying the patient's significant symptoms and evaluating their HIV infection is proposed. Consequently, the CFHS-mapping and its inverse mapping are presented, as well as some actual work with connected characteristics. The calculation that has been established has three steps. First, CFHS-mapping was used in the second phase to find appropriate medicines for individuals based on their HIV severity. Finally, a generalised CFHS-mapping is created to follow the patient's recovery and predict when he would return to his normal range.