key: cord-0857345-mqry4j39 authors: Firouzkouhi, Narjes; Amini, Abbas; Cheng, Chun; Zarrabi, Ali; Davvaz, Bijan title: Intuitionistic fuzzy set of [Formula: see text] ‐submodules and its application in modeling spread of viral diseases, mutated COVID‐n, via flights date: 2021-11-19 journal: International Journal of Intelligent Systems DOI: 10.1002/int.22754 sha: 2cb04bd200d9de947d5186aefc0703aa9d91f76f doc_id: 857345 cord_uid: mqry4j39 In this study, we generalize fuzzy [Formula: see text] ‐module, as intuitionistic fuzzy [Formula: see text] ‐submodule of [Formula: see text] ‐module (IF [Formula: see text] M), and utilize it for modeling the spread of coronavirus in air travels. Certain fundamental features of intuitionistic fuzzy [Formula: see text] ‐submodule are provided, and it is proved that IF [Formula: see text] M can be considered as a complete lattice. Some elucidatory examples are demonstrated to explain the properties of IF [Formula: see text] M. The relevance between the upper and lower [Formula: see text] ‐level cut and intuitionistic fuzzy [Formula: see text] ‐submodules are presented and the characteristics of upper and lower under image and inverse image of IF [Formula: see text] M are acquired. It is verified that the image and inverse image of intuitionistic fuzzy [Formula: see text] ‐submodule are preserved under the module homomorphism. The obtained IF [Formula: see text] M is used to model the aerial transition of viral diseases, that is, COVID‐n, via flights. period of time. 41 This viral disease has infected humanity worldwide with typical symptoms of fever, sore throat, cough, fatigue and dyspnea. Despite the capability of some countries on effective vaccination against coronavirus disease, the emergence of new infected cases is unpredictable and seriously worrying, as there is yet neither an adamant treatment against the mutated versions of COVID nor a prohibition methodology against the detrimental/deadly side effects of known vaccines. 42 As such, various countries implemented severe precautions to decelerate the diffusion of this disease after the World Health Organization (WHO) officially publicized the epidemic situation in mid-March 2019. 43 Due to the COVID-19 outbreak, many countries have faced case threats through inbound international and national flights. After identifying the first cases of coronavirus in different countries, strict rules were imposed on the airlines that yielded the disruption of global transportation. 44 In fact, to lessen the chances of proliferation of COVID-19, very strict protocols were issued by governments on aerial sectors. These restrictions included installing high-efficiency air filters in aircrafts, imposing C-reactive protein (CRP) tests and vaccinations for travelers, wearing protective masks, and keeping social distances during the aerial trips. 45 While the air travels are considered as an essential transportation service worldwide, the surveillance/modeling of the corresponding global factors (studied here) is necessary to resume safe aerial trips with reduced/controlled COVID threats 46 (Figure 1 ). The main contribution of this paper is the generalization of fuzzy Γ-module through the development of IFS, and the construction of new application for the spread of viral diseases, that is, coronavirus, among individuals in air travels. By using Γ-module, we expand the framework of IFS via the expression of some basic and significant characteristics with certain foundational traits. In Section 3, the intuitionistic fuzzy Γ-submodule (IFΓM) is established via the notion of Γ-modules to extend the fuzzy sets. Fundamental properties of intuitionistic fuzzy Γ-submodule are found, and it is verified that IFΓM can be regarded as a complete lattice. Furthermore, by considering the upper and lower α-level cut, we express the relationship between them and IFΓM, along with several traits of upper and lower via image and inverse image of IFΓM. It is shown that the image and inverse image of intuitionistic fuzzy Γ-submodule are preserved under the module homomorphism. In Section 4, the elucidatory examples address the application of IFΓM in the immunological transmission of COVID-n. The IFSs are the generalization of the fuzzy sets which were proposed by Atanassov. 3 An IFS A of a nonvoid set X is described by the formation B B be two IFS of X . Thus, the next statements are introduced   t X, as follows: Definition 2.1 (Barnes 16 ). Suppose R and Γ be additive abelian groups. R is considered as a Γ-ring if a mapping exists: . . (= ) 1 , the next circumstances hold: (i) r r α r r α r r α r ( + ) = + and  r r r R , , 1 2 the next implications are satisfied: x xγr x ϑ( ) ϑ( )(ϑ( ) ϑ( )).ϑ is considered as a fuzzy Γ-module of M supposing ϑ is a fuzzy left Γ-module and also fuzzy right Γ-module of M. Example 3.2. Assume M = n for prime integer n, and R = and Γ = . Define Moreover, introduce the fuzzy set ϑ of M as follows: Thus, ϑ is a fuzzy Γ-module of M. Example 3.3. Suppose M = and R = and Γ be a subring of  ( , +, ) . Hence, R is a Γ-ring and M ( , +) is an abelian group. Define . Therefore, M is a Γ-module. Now, describe ϑ in the following way: Hence, ϑ is a fuzzy Γ-module of M. Γ the next statements is satisfied: Denote that IFΓM is intuitionistic fuzzy Γ-submodule. Also, it is defined for right Γ-submodule, the IFS of A A of M is considered an IFΓM provided it is left and right IFΓM. Thus, M is a Γ-module. Describe two fuzzy sets ϑ and ζ of M, in the following way: This completes the proof. , and it is the least intuitionistic fuzzy Γ-submodule containing A i . So, We verify the next statements. Proof. We have In addition, we have The proof is completed. □ On the contrary, assume that the fuzzy sets ϑ A and ζ A c are fuzzy left (right) Γ-module. Therefore, we have It implies that □A is an IFΓM of M. Similarly, we can verify for ♢A. □ Remark 3.14. For a proper IFS of A, we have U α (ϑ ; ) A is described as an upper bound α-level cut of ϑ, and written by U α , and also L α (ϑ ; ) A is considered as lower bound α-level cut of ϑ, and written by , for any fuzzy set ϑ of M and  α [0, 1]. On the contrary, assume that the subsets U α (ϑ ; ) be a map. Hence, we have (i) The image of A under the map π is signified by π A ( ), that is written π A The image and inverse image are depicted in Figure 3 . Proof. We prove (i), also, we prove (ii) in the following: Proof. We verify (i) in the following: Moreover, we prove (ii) as follows: exist. π φ ψ ( , , ) is called a homomorphism of M to M , provided for all   x y M γ , , Γ, we attain π x y π x π y ( + ) = ( ) + ( ), Moreover, if π be a bijection, then we call π φ ψ ( , , ) is an isomorphism. Theorem 3.20. Assume M be Γ-module, and M be Γ-module. Let π φ ψ ( , , ) be homomorphism from M to M . Hence, Assume Γ be important airlines which operate in different countries. Consider Γ = Qatar Airline Delta Airline United Airline { , , }with the operation "+" that is defined as follows: x y x y + = The airline which plays a role in disease transmission to and The set Γ with the operation + is shown in Table 1 . Thus, (Γ, +) is an abelian group. Suppose R be the countries that participated in our model. Let R China Canada USA = { , , } and the operation  determined in the following manner: The set R via the operation  is given in Table 2 . Therefore,  R ( , ) is an abelian group. Now, we introduce the operation "" in the next way: which   r γ r′ means the country infected by COVID-19 in relation with the airlines. Hence, Consider the set M as the family members who travel to countries R with airlines Γ. Let M Bob Jack Sara Nancy = { , , , }. Describe the operation "" as follows:  t s = The person who transmits the disease to t and s In Table 3 . Therefore, The IFS A of M is determined as follows. The degree of membership can be interpreted as a percentage of dependence. Table 4 depicts that the disease transmission power of Bob is more than the others, Jack is in the second rank and so on. To verify that A is IFΓM of M, we pursue the following procedure for all elements of A. For example, . Therefore, an IFS In this paper, a framework for the IFS associated to Γ-submodule was constructed to generalize the fuzzy set. Certain features of IFS of Γ-modules were expressed along with illustrative examples, and a link between upper and lower α-level cut and intuitionistic fuzzy Γ-submodules was also presented. By applying the module homomorphism, the image and inverse image of intuitionistic fuzzy Γ-submodule were preserved under the homomorphism. In addition, the convenient circumstance was carried out to create the t-IFS of Γ-modules, α β ( , )-IFS of Γ-modules, homomorphism and direct product of IFS of Γ-modules which were the main characteristics of the intuitionistic fuzzy Γ-submodules. The effective application of this survey was demonstrated in modeling the spread of COVID-19 via air travels. The results rationalized the immunological case by using the developed intuitionistic fuzzy Γ-submodules. There is a potential to exploit the capability of IFS of Γ-subrings and IFS of Γ-subgroups in other fields. Fuzzy sets, fuzzy logic, and fuzzy systems: selected papers by Lotfi A Zadeh Fuzzy groups Intuitionistic fuzzy sets Intuitionistic fuzzy subgroups Intuitionistic fuzzy ideals of semigroups The universal coefficient theorem in the category of intuitionistic fuzzy modules t -Intuitionistic fuzzy subgroups t -Intuitionistic fuzzy quotient group Cut of intuitionistic fuzzy groups Homomorphism of Intuitionistic fuzzy groups On the direct product of Intuitionistic fuzzy subgroups Quotient structures of intuitionistic fuzzy finite state machines Intuitionistic nil radicals of intuitionistic fuzzy ideals and Euclidean intuitionistic fuzzy ideals in rings On characterization of intuitionistic fuzzy rough sets based on intuitionistic fuzzy implicators On a generalization of the ring theory On the Γ-rings of Nobusawa On the theory of simple Γ-rings Γ-semihypergroups and their properties Gamma modules. Ratio math On fuzzy gamma hypermodules Fuzzy Γ-hyperrings and fuzzy Γ-hypermodules Intuitionistic fuzzy sets in Γ-semigroups Structure of intuitionistic fuzzy sets in Γ -semihyperrings Atanassovas intuitionistic fuzzy grade of hypergroups Atanassovs intuitionistic S T ( , )-fuzzy n-ary subhypergroups and their properties Atanassov's intuitionistic fuzzy interior ideals of Γ-semigroups On fundamental theorems of t -intuitionistic fuzzy isomorphism of t -intuitionistic fuzzy subgroups A certain class of t-intuitionistic fuzzy subgroups A novel applications of complex intuitionistic fuzzy sets in group theory An improved correlation coefficient between intuitionistic fuzzy sets and its applications to real-life decision-making problems Aggregation of infinite chains of intuitionistic fuzzy sets and their application to choices with temporal intuitionistic fuzzy information Representing complex intuitionistic fuzzy set by quaternion numbers and applications to decision making An information-based score function of interval-valued intuitionistic fuzzy sets and its application in multiattribute decision making A dynamic group MCDM model with intuitionistic fuzzy set: Perspective of alternative queuing method An application of intuitionistic fuzzy sets in medical diagnosis An application of intuitionistic fuzzy sets in medicine Application of intuitionistic fuzzy sets in electoral system A novel distance measure for intuitionistic fuzzy sets with diverse applications Width-based distance measures on interval-valued intuitionistic fuzzy sets Features, evaluation, and treatment of coronavirus (COVID-19) Treatment for COVID-19: an overview Emerging treatment strategies for COVID-19 infection Risk of symptomatic COVID-19 due to aircraft transmission: a retrospective cohort study of contact-traced flights during Englands containment phase. Influenza Other Respirat Viruses Measuring imported case risk of COVID-19 from inbound international flights-a case study on China A large national outbreak of COVID-19 linked to air travel, Ireland, summer 2020 In-flight transmission cluster of COVID-19: a retrospective case series Intuitionistic fuzzy set of Γ-submodules and its application in modelling spread of viral diseases, mutated COVID-n, via flights The proof of (ii) is analogous to (i). □ The application of an IFS on-submodules is expressed for the diffusion of coronavirus disease 2019 (COVID-19) via flights. COVID-19 is the most recent epidemic disease which has affected all over the world yielding nearly 4 million deaths till July 2021. This viral disease was first emerged in Wuhan, China, and quickly spread across the world in a short period of time, entangling all the countries and devastating numerous infrastructures. 41 Air travels have negatively assisted the global epidemic of viral diseases, specifically those highly infectious diseases, that is, COVID-n. 46 It was reported that after a major flight, there have been some new patients infected with coronavirus. 47 Here, we utilize the developed IFΓM to model the dispersion of coronavirus disease between individuals who traveled to different countries via different airlines. In this transition, we appoint Γ as the set of airlines, R as the set of countries, and M as the set of family members (Figure 4 ). The Australian College of ACK is highly acknowledged for providing the Research Grants No: IRC-2020/2021-SOE-ME-PR05 and PR06. BioRedner, Toronto, Canada, is acknowledged for providing drawing modules.