View of Some Applications of Closed Setsin Second Wave Covid-19 Infections Parameters

(1)

4353 http://annalsofrscb.ro

Some Applications of Closed Setsin Second Wave Covid-19 Infections Parameters

1Dr Monika DasharathGorkhe, ²Dr. Minirani S, ³Dr. A.KRISHNARAJU,

4Devendra Singh, ⁵RAHUL KAR, ⁶Dr.Makarand Upadhyaya, ⁷A.PANDI,

8S.Balamuralitharan*

1Assistant Professor, Dr D Y Patil Institute of Management Studies (DYPIMS), Maharashtra Pune -411018

2Associate Professor, Department of BSH, MPSTME, SVKM's NMIMS Deemed to university, Mumbai- 400056

3PROFESSOR, Department of Mechanical and Automation Engineering PSN College of Engineering and Technology

TIRUNELVELI - 627152

4Department of Biotechnology, Motilal Nehru National Institute of Technology, Allahabad-211004, India

5Department of mathematics,KalyaniMahavidyalaya, Kalyani, Nadia, India Email: [email protected]

6Associate Professor-Marketing, University of Bahrain, Department of Management & Marketing, College of Business Administration, Bahrain- 32038

E-mail: [email protected]

7ASSISTANT PROFESSOR,

DEPARTMENT OF MATHEMATICS, RATHINAM TECHNICAL CAMPUS, POLLACHI MAIN ROAD, EACHANARI, COIMBATORE-21. TAMILNADU, INDIA.

8Department of Mathematics, College of Engineering and Technology, SRM Institute of Science and Technology, Kattankulathur-603 203,

Chengalpattu District, Tamilnadu, INDIA.

*Corresponding Author Email: [email protected]

Abstract:iWe connected current pandemic second wave COVID-19 infections parameters with algebraic structure. This parameter estimation compare to closed set in Nano

(2)

topological spaces under the structure of COVID-19.In thisarticle, we introduce a (i,j)*-- cld in BTPS. This sets lies between i,j-cld and the class of (i,j)*-g-cld.

2010iMathematicsiSubjectiClassification:i54E55

KeyiwordsiandiPhrases:second wave COVID-19 infections, (i,j)*-g-cldiset,i(i,j)*-- cldiset,i(i,j)*--openiseti

1.iINTRODUCTION

This paper deals the parameters of COVID-19 equation models ([1]- [4]) connected to closed sets. In this regard, we compare the sg-closedisets,igs-closedisets,iclosedisetsiandi

gs-closedisets to COVID-19 infection models. It is a connection for this infection model analysis to (i,j)*--cld in BTPS. This paper was fully analyzed the pure mathematics under the connection of closed sets in Nano topological spaces.

Recently,Several authors such as

BhattacharyaiandiLahiri,iAryaiandiNour,iiSheikiJohniandiRajamaniiandiViswanathaniiintr oducedisg-closedisets,igs-closedisets,i-closedisetsiandigs-closedisetsirespectively ([5]- [21]).iIn myarticle, we introduce a (i,j)*--cld in BTPS. This sets lies between i,j-cld and the class of (i,j)*-g-cld.

2.iPRELIMINARIES

Throughoutithisipaper(X,iτi,iτj)i(briefly,iX)willdenote BTPS.

Definitioni2.1

Let H X. Then H is said to be τi,j-open [12] if H = PQwhere Pτi and Qτj. Theicomplementiofiτi,j-openisetiisicallediτi,j-cld.

Definitioni2.2i[12]

LetiHiX.iThen

(i)i theiτi,j-closureiofiH,idenotedibyiτi,j-cl(H),iisidefinediasiii{Fi:iHiiFiandiFiisiτi,j- cld}.

(3)

(ii)i theiτi,j-interioriofiH,idenotedibyiτi,j-int(H),iisidefinediasii{Fi:iFiiHiandiFiisiτi,j- open}.

Definitioni2.3

AisubsetiHiiXiisicalled:

(i) (i,j)*-semi-openiseti[11]iifiHiii,j-cl(i,j-int(H));

(ii) (i,j)*-preopeniseti[11]iifiHiiii,j-int(i,j-cl(H));

(iii) (i,j)*- -openiseti[8]iifiHiii,j-int(i,j-cl(i,j-int(H)));

(iv) (i,j)*-β-openiseti[13]i(i=i(i,j)*-semi-preopeni[13]i)iifiHiii,j-cl(i,j-int(i,j-cl(H)));

(v) regulari(i,j)*-openiseti[11]iifiHi=ii,j-int(i,j-cl(H)).

Theicomplementsiofitheiaboveimentionediopenisetsiareicalleditheirirespectiveiclos edisets.

Definitioni2.4

AiHiXiisicalled

(i)i (i,j)*-generalizediclosedi(briefly,i(i,j)*-g-cld)iseti[16]iifii,j- cl(H)iiUiwheneveriHiiUiandiUiisii,j-openiiniX.i

(ii)ii (i,j)*-semi-generalizediclosedi(briefly,i(i,j)*-sg-cld)iseti[11]iifi(i,j)*- scl(H)iiUiwheneveriHiiUiandiUiisi(i,j)*-semi-openiiniX.i

(iii)ii (i,j)*-generalizedisemi-closedi(briefly,i(i,j)*-gs-cld)iseti[11]iifi(i,j)*- scl(H)iiUiwheneveriHiiUiandiUiisii,j-openiiniX.i

(iv) i(i,j)*- -generalizediclosedi(briefly,i(i,j)*- g-cld)iseti[15]iifi(i,j)*- cl(H)iiUiiiwheneveriHiiUiandiUiisii,j-openiiniX.i

(v)i (i,j)*-generalizedisemi-preclosedi(briefly,i(i,j)*-gsp-cld)iseti[15]iifi(i,j)*- spcl(H)iiUiwheneveriHiiUiandiUiisii,j-openiiniX.i

(vi)i (i,j)*-ĝ-closediseti(i(i,j)*--cld))i[5]iiif ii,j-

cl(H)iiUiwheneveriHiiUiandiUiisi(i,j)*-semi-openiiniX.i

(4)

(vii)i (i,j)*-gs-cldiseti[15]iifi(i,j)*- cl(H)iiUiiwheneveriHiiUiandiUiisi(i,j)*-semi- openiiniX.i

(viii)i (i,j)*-g*s-cldiseti[11]iiifi(i,j)*-scl(H)iiUiiwheneveriHiiUiandiUiisi(i,j)*-gs- openiiniX.i

Theicomplementsiofitheiaboveimentionediclosedisetsiareicalleditheirirespectiveiop enisets.

Definitioni2.5i[16]

AisubsetiHiofiaiBTPSiXiisisaiditoibei(i,j)*-

locallyiclosediifiiHi=iUiiF,iwhereiUiisii,j-openiandiFiisii,j-cldiiniX.iii Remarki2.6

(1)i Everyii,j-openisetiisi(i,j)*-g*s-openi[16].

(2)i Everyi(i,j)*-semi-openisetiisi(i,j)*-g*s-openi[11].

(3)i Everyi(i,j)*-g*s-openisetiisi(i,j)*-sg-openi[16].

(4) Everyi(i,j)*-semi-cldisetiisi(i,j)*-gs-cldi[16].

(5) Everyii,j-cldisetiisi(i,j)*-gs-cldi[16].

**3.i(i,j)*--CLDiSETSiINiBTPS**

Weiintroduceitheifollowingidefinition.

Definitioni3.1

AisubsetiHiofiaiBTPSiXiisicallediai(i,j)*--cldisetiifii,j- cl(H)iiUiwheneveriHiiUiandiUiisi(i,j)*-gs-openiiniX.

Propositioni3.2

Everyi_i,j-cldisetiisi(i,j)*--cld.

(5)

Proof

IfiHiisianyii,j-cldisetiiniXiandiGiisianyi(i,j)*-gs-

openiseticontainingiH,itheniiGiiHi=ii,j-cl(H).iHenceiHiisi(i,j)*--cld.

TheiconverseiofiPropositioni3.2ineedinotibeitrueiasiseenifromitheifollowingiexamp le.

Examplei3.3

LetiXi=i{p,iq,ir},iii=i{,iX,i{p,iq}}iandiiji=i{,iX}.iThenitheisetsiini{,i{p,iq},i X}iareicalledii,j-openianditheisetsiini{,iX,i{r}}iareicalledii,j-closed.iTheni(i,j)*-- C(X)i=i{,i{r},i{p,ir},i{q,ir},iX}.iHere,iHi=i{p,ir}iisi(i,j)*--cldisetibutinoti_i,j-cld.

Propositioni3.4

Everyi(i,j)*--cldisetiisi(i,j)*-g*s-cld.

Proof

IfiHiisiai(i,j)*--cldisubsetiofiXiandiGiisianyi(i,j)*-gs-

openiseticontainingiH,itheniGiii,j-cl(H)ii(i,j)*-scl(H).iHenceiHiisi(i,j)*-g*s-cldiiniX.

Examplei3.5

IniExamplei3.3,iHere,i(i,j)*G*SC(X)i=ii{,i{p},i{r},i{p,ir},iX}.iHere,iHi=i{r}iisi(i ,j)*-g*s-cldibutinoti(i,j)*--cldisetiiniX.

Propositioni3.6

Everyi(i,j)*--cldisetiisii(i,j)*--cld.

Proof

SupposeithatiHiiGiandiGiisi(i,j)*-semi-openiiniX.iSinceieveryi(i,j)*-semi- openisetiisi(i,j)*-gs-openiandiHiisi(i,j)*--cld,Ithereforeii,j-cl(H)iiG.iHenceiHiisi(i,j)*-

-cldiiniX.i

TheiconverseiofiPropositioni3.6ineedinotibeitrueiasiseenifromitheifollowingiexample.

Examplei3.7

(6)

LetiXi=i{p,iq,ir},iii=i{,iX,i{p}}iandiiji=i{,iX,i{q,ir}}.iThenitheisetsiini{,i{p}

,i{q,ir},iX}iareicalledii,j-openianditheisetsiini{,iX,i{p},i{q,ir}}iareicalledii,j- closed.iiTheni(i,j)*--C(X)i=i{,i{p},i{q,ir},iX}iandi(i,j)*-

C(X)i=iiP(X).iHere,iHi=i{p,ir}iisi(i,j)*--cldibutinoti(i,j)*--cldisetiiniX.

i

Propositioni3.8

Everyi(i,j)*-g*s-cldisetiisi(i,j)*-sg-cld.

Proof

SupposeithatiHiiGiandiGiisi(i,j)*-semi-openiiniX.iSinceieveryi(i,j)*-semi- openisetiisi(i,j)*-gs-openiandiHiisi(i,j)*-g*s-cld,ithereforei(i,j)*-

scl(H)iiG.iHenceiHiisi(i,j)*-sg-cldiiniX.i

Examplei3.9

,i{q,ir},iX}iareicalledii,j-openianditheisetsiini{,iX,i{p},i{q,ir}}iareicalledii,j- closed.iTheni(i,j)*-G*SC(X)i=i{,i{p},i{q,ir},iX}iandi(i,j)*-

SGC(X)i=iP(X).iHere,iHi=i{p,iq}iisi(i,j)*-sg-cldiibutinoti(i,j)*-g*s-cldisetiiniX.

Propositioni3.10

Everyi(i,j)*--cldisetiisi(i,j)*-gs-cld.

Proof

IfiHiisiai(i,j)*--cldisubsetiofiXiandiGiisianyi(i,j)*-semi-

openiseticontainingiH,itheniGiii,j-cl(H)ii(i,j)*- cl(H).iHenceiHiisi(i,j)*-gs- cldiiniX.

TheiconverseiofiPropositioni3.10ineedinotibeitrueiasiseenifromitheifollowingiexam ple.

Examplei3.11

LetiXi=i{p,iq,ir},i_ii=i{,iX,i{p}}iandii_ji=i{,iX}.iThenitheisetsiini{,i{p},iX}iar eicalledii,j-openianditheisetsiini{,iX,i{q,ir}}iareicalledii,j-closed.iTheni(i,j)*-

C(X)i=i{,i{q,ir},iX}iandi(i,j)*-GS

(7)

C(X)i=ii{,i{q},i{r},i{q,ir},iX}.iHere,iHi=i{q}iisi(i,j)*-gs-cldibutinoti(i,j)*-- cldisetiiniX.

Everyi(i,j)*--cldisetiisi(i,j)*-g-cld.

Proof

IfiHiisiai(i,j)*--

closedisubsetiofiXiandiGiisianyiopeniseticontainingiH,isinceieveryii,j-openisetiisi(i,j)*-gs- open,iweihaveiGiii,j-cl(H).iHenceiHiisi(i,j)*-g-cldiiniX.

Examplei3.13

,i{q,ir},iX}iareicalledi_i,j-openianditheisetsiini{,iX,i{p},i{q,ir}}iareicalledi_i,j- closed.iTheni(i,j)*--C(X)i=i{,i{p},i{q,ir},iX}iandi(i,j)*-G

C(X)i=iP(X).iHere,iHi=i{p,iq}iisi(i,j)*-g-cldibutinoti(i,j)*--cldisetiiniX.

Everyi(i,j)*--cldisetiisi(i,j)*-gs-cld.

Proof

IfiHiisiai(i,j)*--closedisubsetiofiXiandiGiisianyi(i,j)*-semi-

openiseticontainingiH,isinceieveryi(i,j)*-semi-openisetiisi(i,j)*-gs-open,iweihaveiGiii,j- cl(H)ii(i,j)*- cl(H).iHenceiHiisi(i,j)*-gs-cldiiniX.

Examplei3.15

,i{q,ir},iX}iareicalledii,j-openianditheisetsiini{,iX,i{p},i{q,ir}}iareicalledii,j- closed.iTheni(i,j)*-i-C(X)i=i{,i{p},i{q,ir},iX}iandi(i,j)*-GS

C(X)i=iiP(X).iHere,iHi=i{p,ir}iisi(i,j)*-gs-cldibutinoti(i,j)*--cldisetiiniX.

(8)

Everyi(i,j)*--cldisetiisi(i,j)*- g-cld.

Proof

IfiHiisiai(i,j)*--cldisubsetiofiXiiandiGiisianyii,j-

openiseticontainingiH,iisinceieveryii,j-openisetiisi(i,j)*-gs-open,iweihaveiGiii,j- cl(H)ii(i,j)*- cl(H).iHenceiHiisi(i,j)*- g-cldiiniX.

Examplei3.17

LetiXi=i{p,iq,ir},iii=i{,iX,i{r}}iandiiji=i{,iX,i{p,iq}}.iThenitheisetsiini{,i{r}, i{p,iq},iX}iareicalledii,j-openianditheisetsiini{,iX,i{r},i{p,iq}}iareicalledii,j-

closed.iTheni(i,j)*-i-C(X)i=i{,i{r},i{p,iq},iX}iandii(i,j)*-g

C(X)i=iiP(X).iHere,iHi=i{p,ir}iisi(i,j)*- g-cldibutinoti(i,j)*--cldisetiiniX.

Everyi(i,j)*-i-cldisetiisi(i,j)*-gs-cld.

Proof

IfiHiisiai(i,j)*--cldisubsetiofiXiandiGiisianyii,j-

openiseticontainingiH,isinceieveryii,j-openisetiisi(i,j)*-gs-open,iweihaveiGiii,j- cl(H)ii(i,j)*-scl(H).iHenceiHiisi(i,j)*-gs-cldiiniX.

Examplei3.19

LetiXi=i{p,iq,ir},iii=i{,iX,i{p}}iandiiji=i{,iX}.iThenitheisetsiini{,i{p},iX}I areicalledii,j-openianditheisetsiini{,iX,i{q,ir}}iareicalledii,j-closed.iTheni(i,j)*-- C(X)i=i{,i{q,ir},iX}iandi(i,j)*-GS

C(X)i=ii{,i{q},i{r},i{p,iq},i{p,ir},i{q,ir},iX}.iHere,iHi=i{r}iisi(i,j)*-gs- closedibutinoti(i,j)*--cldisetiiniX.

(9)

Everyi(i,j)*--cldisetiisi(i,j)*-gsp-cld.

Proof

IfiHiisiai(i,j)*--cldisubsetiofiXiiandiGiisianyii,j-openiseticontainingiH,ieveryii,j- openisetiisi(i,j)*-gs-open,iweihaveiGiii,j-cl(H)ii(i,j)*-spcl(H).iHenceiHiisi(i,j)*-gsp- cldiiniX.

Examplei3.21

IniExamplei3.19,iHere,i(i,j)*-GSP

C(X)i=ii{,i{q},i{r},i{p,iq},i{p,ir},i{q,ir},iX}.iHere,iHi=i{r}iisi(i,j)*-gsp- cldibutinoti(i,j)*--cldisetiiniX.

Remarki3.22

Theifollowingiexampleishowsithati(i,j)*--cldisetsiareiindependentiofi(i,j)*- - cldisetsiandi(i,j)*-semi-cldisets.

Examplei3.23

LetiXi=i{p,iq,ir},iii=i{,iX,i{p,iq}}iandiiji=i{,iX}.iThenitheisetsiini{,i{p,iq},i X}iareicalledii,j-openianditheisetsiini{,iX,i{r}}iareicalledii,j-closed.iThen(i,j)*-- C(X)i=i{,i{r},i{p,ir},i{q,ir},iX}iandi(i,j)*- C(X)i=i(i,j)*-S

C(X)i=ii{,i{r},iX}.iHere,iHi=i{p,ir}iisi(i,j)*--cldibutiitiisineitherii(i,j)*- - cldiinori(i,j)*-semi-cldiiniX.

Examplei3.24

LetiXi=i{p,iq,ir},iii=i{,iX,i{p}}iandiiji=i{,iX}.iThenitheisetsiini{,i{p},iX}iar eicalledii,j-openianditheisetsiini{,iX,i{q,ir}}iareicalledii,j-closed.iTheni(i,j)*--

C(X)i=i{,i{q,ir},iX}iandi(i,j)*- C(X)i=i(i,j)*-S

C(X)i=ii{,i{q},i{r},i{q,ir},iX}.iHere,iHi=i{q}iisi(i,j)*- -cldiasiwelliasi(i,j)*-semi- cldiiniiXibutiitiisinotii(i,j)*--cldiiniX.

(10)

Remarki3.25

(i,j)*- -cld(i,j)*-gs-cld(i,j)*- g-cld

i,j-cld(i,j)*--cld(i,j)*--cld(i,j)*- g-cld

(i,j)*-semi-cld(i,j)*-g*s-cld(i,j)*-sg-cld(i,j)*-gs-cld

Noneiofitheiaboveiimplicationsiisireversibleiasishowniinitheiremainingiexamplesiandiinith eirelatedipapersi[15,16].

REFERENCES

1. M. Radha and S. Balamuralitharan, A study on COVID-19 transmission dynamics:

stability analysis of SEIR model with Hopf bifurcation for effect of time delay, Advances in Difference Equations, (2020) 2020:523 https://doi.org/10.1186/s13662-020-02958-6

2. T. Sundaresan, A. Govindarajan, S. Balamuralitharan, P. Venkataraman, and IqraLiaqat, A classical SEIR model of transmission dynamics and clinical dynamics in controlling of coronavirus disease 2019 (COVID-19) with reproduction number, AIP Conference Proceedings 2277, 120008 (2020);

https://doi.org/10.1063/5.0025236, Published Online: 06 November 2020.

3. M. Suba, R.Shanmugapriya, S.Balamuralitharan, G. Arul Joseph, Current Mathematical Models and Numerical Simulation of SIR Model for Coronavirus Disease - 2019 (COVID-19), European Journal of Molecular & Clinical Medicine, Volume 07, Issue 05, 2020, 41-54, ISSN 2515-8260.

4. R.Ramesh, S.Balamuralitharan, Indian Second Wave Common COVID-19 Equation Analysis with SEIR Model and Effect of Time Delay, Annals of R.S.C.B., ISSN:1583-6258, Vol. 25, Issue 4, 2021, Pages. 6512 – 6523.

5. Abd El-Monsef, M. E., El-Deeb, S. N. and Mahmoud, R. A.: -open sets and - continuous mapping, Bull. Fac. Sci. AssiutUniv, 12(1983), 77-90.

6. Andrijevic, D.: Semi-preopen sets, Mat. Vesnik, 38(1986), 24-32.

(11)

7. Arya, S. P. and Nour, T.: Characterization of s-normal spaces, Indian J. Pure. Appl.

Math., 21(8)(1990), 717-719.

8. Bhattacharya, P. and Lahiri, B. K.: Semi-generalized closed sets in topology, Indian J. Math., 29(3)(1987), 375-382.

9. Duszynski, Z., Jeyaraman, M., Joseph Israel, M. and Ravi, O.: A new generalization of closed sets in bitopology, South Asian Journal of Mathematics, 4(5)(2014), 215- 224.

10. Kelly, J.C. :Bitopological spaces, Proc.London Math.Soc.,3(13)(1963), 71-89.

11. Levine, N.: Generalized closed sets in topology, Rend. Circ. Math. Palermo, 19(2)(1970), 89-96.

12. LellisThivagar, M., Ravi, O. and Abd El-Monsef, M. E.: Remarks on bitopological (1,2)*-quotient mappings, J. Egypt Math. Soc., 16(1) (2008), 17-25.

13. Rajamani, M., and Viswanathan, K.: On gs-closed sets in topological spaces, ActaCienciaIndica, XXXM (3)(2004), 21-25.

14. Ravi, O., Thivagar, M. L. and Hatir, E.: Decomposition of (1,2)*-continuity and (1,2)*- -continuity, Miskolc Mathematical Notes., 10(2) (2009), 163-171.

15. Ravi, O. and LellisThivagar, M.: A bitopological (1,2)*-semi-generalized continuous maps, Bull. Malays.Math. Sci. Soc., (2), 29(1) (2006), 79-88.

16. Ravi, O. and LellisThivagar, M.: On stronger forms of (1,2)*-quotient mappings in bitopological spaces, Internat. J. Math. Game Theory and Algebra., 14(6) (2004), 481-492.

17. Ravi, O. and Thivagar, M. L.: Remarks on -irresolute functions via (1,2)*-sets, Advances in App. Math. Analysis, 5(1) (2010), 1-15.

18. Ravi, O., Pious Missier, S. and SalaiParkunan, T.: On bitopological (1,2)*- generalized homeomorphisms, Int J. Contemp. Math. Sciences., 5(11) (2010), 543- 557.

19. Ravi, O., Pandi, A., Pious Missier, S. and SalaiParkunan, T.: Remarks onbitopological (1,2)*-rω-Homeomorphisms, International Journal of Mathematical Archive, 2(4) (2011), 465-475.

20. Ravi, O., Thivagar, M. L. and Jinjinli.: Remarks on extensions of (1,2)*-g-closed maps, Archimedes J. Math., 1(2) (2011), 177-187.

21. Sheik John, M.: A study on generalizations of closed sets and continuous maps in topological and bitopological spaces, Ph.D Thesis, Bharathiar University, Coimbatore, September 2002.