Soft ζ -Rough Set and Its Applications in Decision Making of Coronavirus

: In this paper, we present a proposed method for generating a soft rough approximation as a modification and generalization of Zhaowen et al. approach. Comparisons were obtained between our approach and the previous study and also. Eventually, an application on Coronavirus (COVID-19) has been presented, illustrated using our proposed concept, and some influ-encing results for symptoms of Coronavirus patients have been deduced. Moreover, following these concepts, we construct an algorithm and apply it to a decision-making problem to demonstrate the applicability of our proposed approach. Finally, a proposed approach that competes with others has been obtained, as well as realistic results for patients with Coronavirus. Moreover, we used MATLAB programming to obtain the results; these results are consistent with those of the World Health Organization and an accurate proposal competing with the method of Zhaowen et al. has been studied. Therefore, it is recommended that our proposed concept be used in future decision making.

Coronavirus emerged in 2019, in Wuhan, China. This virus is a new strain that has not been previously identified in humans. It was believed that Coronaviruses spread from dirty, dry surfaces, like automatic mucous membrane pollination in the nose, eyes, or mouth, reinforcing the importance of a clear understanding of the persistence of coronaviruses on inanimate surfaces [16]. Therefore, two factors which are in contact with infected surfaces and encounters with infected viruses, affect the transmission. As a result, many scientific papers have been published and many researchers have studied this virus, such as ( [16][17][18][19][20][21][22]).
The main objective of our belief is to have a certain influence on the continuous approximation of such basic mathematical principles and to provide a modern method for computational mathematics of real-life problems. In fact, it considers latest generalized soft, rough approximations, called soft ζ -rough approximations, are defined as a generalization to Zhaowen et al. [15] approximations and their properties are studied. We will prove that our approaches are more accurate and general from Zhaowen et al. approaches. The importance of the current approximations is not only that it is reducing or deleting the boundary regions, but also, it's satisfying all properties of Pawlak's rough sets without any restrictions. Comparisons between our method and the method of Zhaowen et al. are obtained.
Several examples are provided to illustrate the links between topologies and relationships of the soft set. Finally, we are added three applications. in making decisions regarding our strategy. One of them represents a beginning point for apply soft rough approach to solve the problem of Coronavirus contagion. At the end of the paper, we give two an algorithm which can be used to have a decision making for information system in terms of soft ζ -rough approximations.
The main programming for this paper is as follows: Step 1: Input the setŴ and the set of features represent the data as an information table, rows of which are labeled by features A, columns by objects and entries of the table are features values.
Step 2: Compute the rough neighborhood from the information table.
Step 3: Compute the soft ζ -upper approximation, ζ -lower approximation and ζ -boundary for the decision set M ⊆Ŵ .
Step 4: Remove a feature a 1 from the condition's features (A) and then find the rough neighborhood A − {a 1 }.
Step 5: Comparing ζ -boundary for the decision set M ⊆Ŵ on A − {a i } with Step 3.
Step 6: Repeat Steps 4 and 5 for all attributes in A.
Step 7: Those attributes in A for which BND ε Finally, we explain the importance of the proposed method in the medical sciences for application in decision-making problems. In fact, a medical application has been introduced in the decision-making process of COVID-19 Medical Diagnostic Information System with the algorithm. This application may help the world to reduce the spread of Coronavirus.
The paper is structured as follows: The basic concepts of the rough set and soft set were explored in section two and three. The implementation of COVID-19 for each subclass of attributes in the information systems and comparative analysis was presented in section four and five. Section six concludes and highlights future scope.

Preliminaries
In this section, we give some basic definitions and results that used in sequel are mentioned.

Pawlak Rough Set Theory
In 1982, Pawlak [23] introduced the theory of rough set as a new mathematical methodology or easy tools in order to deal with the vagueness in knowledge-based systems, information systems and data dissection. This theory has many applications in many fields that are used to process control, economics, such as medical diagnosis, chemistry, psychology, finance, marketing, biochemistry, environmental science, intelligent agents, image analysis, biology, conflict analysis, telecommunication, and other fields (See: [23][24][25][26][27], and the bibliography in these papers). Proposition 2.1 [23] Let φ be the empty set and M c be the complement of M ⊆Ŵ . Pawlak's rough sets have the next characteristic:

Soft Set Theory and Soft Rough Set
Let us recall now the soft set notion, which is a newly-emerging mathematical approach to vagueness. LetŴ be an initial universe of objects and E W (simply denoted by E) the set of certain parameters in relation to the objects inŴ . Parameters are often attributing, characteristics, or properties of the objects inŴ . Let P Ŵ denote the power set ofŴ . Following the Definition 2.1 gives the concept of soft sets as follows. [12] LetŜ = (F, A) be soft set overŴ , then we define a binary relation on W by
Proposition 2.2 [15] Assuming thatŜ = (F, A) be a soft set uponŴ and AŜ = Ŵ ,Ŝ a soft approximation space. Then the soft AŜ-lower and AŜ-upper approximations of M ⊆Ŵ : It is clear that ifŜ is a full soft set, then ∀ x ∈Ŵ , ∃ e ∈ A such that x ∈ F (e).
Proposition 2.5 [15] LetŜ = (F, A) be a full soft set overŴ and AŜ = Ŵ ,Ŝ a soft approximation space. Then, the following conditions are true:

Generalized Soft Rough Approximations
In this section, we define new generalized soft, rough approximations so-called soft ξ -rough approximations. The properties of the suggested approaches are superimposed. Relationship among our approaches and the previous one in Li et al. [15] are obtained. Many examples and counter examples are introduced. We will prove that our approach is a generalization to Pawlak [23] and Feng et al. [2] approaches.  The main goal of the following results is to introduce and studied the basic properties of soft ξ -rough approximationsŜ ξ andŜ ξ . = (F, A) be a soft set overŴ and AŜ = Ŵ ,Ŝ a soft approximation   (N) . v. By similar way as (iv). vi. By using (iv)-(v), the proof is obvious Remark 3. 1 The inclusion in the above Proposition part (iv) is not instead of to equal the following example shows this remark.

Relationship Between Our Method and the Pawlak Approximation
In this section, we shall compare between current method and the method of Pawlak.   [15] approach. By this proposition, our approximations satisfied most of Pawlak's properties and then Tab. 3, summarize these properties and give first comparison among our method and [15] method. We then list codes in Tab. 3 to show whether these approximations satisfy the properties (L1) to (U9). In Tab. 3, the number 1 denotes yes and 0 denotes not.
The main goal of the following results is to illustrate the relationship between soft rough approximations (that given by Wang et al. [16]) and soft pre-rough approximations (that given by our approach in the present paper).
The intuitive meaning of this classification is as follows: -If M is roughly soft ξ -definable, this suggests that we are able to decide about some elements ofŴ that they belong to M, and for some U elements, while, we can decide that they belong to M c , by using the knowledge available of the soft approximation space AŜ.
-If M is internally soft ξ -indefinable, this suggests that we are able to decide about some elements ofŴ that they belong to M c , but we are incapable to decide for any element ofŴ that it belongs to M, by employing AŜ.
-If M is externally soft ξ -indefinable, this suggests that we are able to decide about some elements ofŴ which they belong to M, but we are incapable to decide, for any element ofŴ that it belongs to M c , by employing AŜ.
-If M is totally soft ξ -indefinable, we are incapable to decide for any element ofŴ , whether it belongs to M or M c , by employing AŜ.
iii. If M is externally soft ξ -definable then M is externally soft AŜ-indefinable. iv. If M is totally soft ξ -indefinable then M is totally soft AŜ-indefinable.
Proof: By Proposition 3.5, the proof is obvious. Remark 4.2 Theorem 4.2 represents a one of differences between soft rough approximations (that given by [15]) and soft ξ -rough approximations (that given by the present paper). Moreover, it illustrates the importance of our approaches in defining the sets, for example: if M is totally soft AŜ-indefinable which impliesŜ (M) = φ andŜ (M) =Ŵ that is, we are incapable to decide for any element ofŴ whether it belongs to M or M c . But, by using soft ξ -rough approximations, S ξ (M) = φ andŜ ξ (M) =Ŵ and then M can be roughly soft ξ -definable Which implies that we can decide on certain elements ofŴ which they belong to M, and this meant while for some elements ofŴ , we able should decide that they are belong of M c , Through using the information obtainable from the soft approximation space AŜ.

Medical Application via in Decision Making of Covid-19
In this section, we introduce a practical example as an application of our approaches in decision making for information system about infections of Coronavirus . In fact, we identify deciding factors of infections for COVID-19 in humans. In this model, we find gatherings, contact with injured people, and work in hospitals is the only deciding factors for infection transmission. We conclude that staying at home and not being in contact with humans protect and against viral infection with Coronavirus. According to [18] (Human-to-Human transmissions have been described with incubation times between 2-10 days, facilitating its spread via droplets, contaminated hands or surfaces). Now, we introduce the proposed method; the application can be described as follows, where the objects as in [18]: U = x 1 , x 2 , . . . , x 10 denotes 10 listed patients, the features as A = {a 1 , a 2 , . . . , a 6 } = {Difficulty breathing, Chest pain, Temperature, Dry cough, Headache, Loss of taste or smell} and Decision Coronavirus {d}, as follows in information was collected by the World Health Organization as well as through medical groups specializing in Coronavirus (COVID-19). Considering the following information system.  Then, we get the removal of attributes as the next Tab. 6,  Now, we can generate the following relation:

Algorithm-ii
Step 1: Input the soft set (F, E).
Step 2: Compute the right neighborhood for all elements ofŴ .

Conflicts of Interest:
The authors declare that they have no conflicts of interest to report regarding the present study.