Europan Journal of Biological Scincs 9 (1): 52-57, 2017 ISSN 2079-2085 IDOSI Publicaions, 2017 DOI: 10.5829/idosi.jbs.2017.52.57 Smoking Tobacco Exprincing wih Inducd Dah Gachw Abiy Salilw Dparmn of Mahmaics, Collg of Naural and Compuaional Scinc, Madda Walabu Univrsiy, Bal Rob, Ehiopia Absrac: Smoking obacco is a sris problm in h nir world. I is a bad habi globally sprad and socially accpd. Many popl sar smoking during hir adolscnc or arly adulhood sag. In his sudy, w considr a drminisic comparmnal mahmaical modl. Comparmnal modls in pidmiology ar widly usd as a mans o modl disas sprad mchanisms and undrsand how on can bs conrol h disas in cas an oubrak of a widsprad pidmic occurs. W classify h populaion in o four comparmn namd as ponial smokr, smokr, mporarily qui smoking and prmannly qui smoking. Thos in his comparmn ar no abl o b infcd again. W prsn sabiliy analysis for h smoking fr and smoking prsn quilibria. W idnify h conrol paramr giv insigh o giv up smoking wih hir numrical simulaion. Ky words: Smoking Tobacco Rproducion Numbr Sabiliy Analysis INTRODUCTION papr using h work don [2, 3], w prsnd smoking obacco xprincd by inducd dah ras by Smoking obacco is h pracic of burning obacco classifying h oal populaion in o four comparmns and inhaling h smok which consising of paricls and namly ponial smokrs, smokrs, mporarily qui gasous phass. Smoking is h mos common mhod of smoking and prmannly qui smoking. Th dynamic consuming obacco and obacco is h mos common mahmaical modls s on h background of biology subsanc smokd. Th agriculural produc is ofn mixd and pidmiological knowldg. wih addiiv and hn combusd. Th rsuling smok is hn inhald and h aciv subsancs absorbd hrough Th Proposd Mahmaical Modl: L us considr h h alvoli in h lungs. Subsancs riggr chmical oal populaion siz a im, N(), is sub dividd ino four racions in nrv ndings, which highn har ra, subclasss: P() b numbr of ponial smokrs alrnss and racion im. Cigar smok is rsponsibl (non-smokr), S() b numbr of smokrs, Q () b numbr for a gra proporion of dahs wihin obacco smok. of smokrs who mporarily qui smoking and Q p() b Th liklihood ha a smokr will dvlop lung cancr from numbr of smokrs who prmannly qui smoking a im cigar smok dpnds on many aspcs; such as h. Thos in his cagory ar no abl o b infcd again. ag a which smoking bgan, how long h prson has Th mahmaical rprsnaion of h modl consiss of smokd, h numbr of cigars smokd pr day and a sysm of diffrnial quaions wih four sa how dply h smokr inhals [1]. Many smokrs bgin variabls. Th flow char of his modl considrd as during adolscnc or arly adulhood. Afr an follows. individual, has smokd for som yars, h avoidanc of wihdrawal sympoms and ngaiv rinforcmn bcom h ky moivaions o coninu. Lik many infcious disass [2], mahmaical modls can b usd o undrsand h sprad of smoking and o prdic h impac of smokrs on h communiy in ordr o hlp rducing h numbr of smokrs. In his Corrsponding Auhor: Gachw Abiy Salilw, Dparmn of Mahmaics, Collg of Naural and Compuaional Scinc Madda Walabu Univrsiy, Bal Rob, Ehiopia. E-mail: gachwaby@gmail.com. 52
W considr ha, h populaion is no consan siz N, which is sufficinly larg so ha h sizs in ach class can b considrd as coninuous variabls. Th corrsponding mahmaical modl aks h form of h following drminisic sysms of nonlinar diffrnial quaions: (1) whr b is h naural birh ra, µ is h naural dah ra, is ransmission ra of smoking obacco, is h rcovry ra from infcion (smokrs), (1 ) is h fracion of smokrs who mporarily qui smoking (a a ra ), is h rmaining fracion of smokrs who prmannly qui smoking (a a ra ), is h ra of quiing smoking of mporarily qui smoking, d 1, d 2, d 3 and d 4 rprsns h dah ra of ponial smokrs, smokrs, mporarily qui smoking and prmannly qui smoking rlad o smoking obacco rspcivly which all ar posiiv. Th oal populaion of h dynamical sysm (1) is govrnd by h diffrnial quaion (2) If dah ras rlad o smoking obacco ar qual, hn h oal populaion bcoms Th oal populaion rmains consan [4], whnvr µ + d = b. W considr in his sudy h oal populaion is no consan, hus w assum ha b < µ + d. L us considr h rgion and iniial condiions for h dynamical sysm (1) which is givn by: (3) Lmma 1: Soluions of sysm (1) wih iniial condiion (4) ar non- ngaiv for all > 0. (4) Proof: Muliplying boh sids of h firs quaion of sysm (1) by ( + d1), w hav: Using fundamnal horm of calculus and aking ingraion from 0 o, w obain: Sinc PS bn and P() = P(0) 0, so w obain P() 0 for all > 0. Muliply boh sids of scond quaion of sysm (1) by, w hav: ( + d2) 53
By using fundamnal horm of calculus and aking ingraion from 0 o, w g: Sinc S PS and S() = S(0) 0, so w obain S() 0 for all > 0. Muliply boh sids of hird quaion of sysm (1) by, w hav: ( + d3) Using fundamnal horm of calculus and aking ingraion from 0 o, w obain: Sinc Q + S S and Q () = Q (0) 0, so w obain Q () 0, for all > 0. Muliply boh sids of fourh quaion of h dynamical sysm (1) by ( + d4), w hav: Using fundamnal horm of calculus and aking ingraion from 0 o, w obain: Sinc S() 0 Q () 0 and Q () = Q (0), 0 so w obain Q () 0 for all > 0. p p p Hnc, w provd ha h soluions of all individuals ar non-ngaiv for all > 0. Lmma 2: All fasibl soluions of h dynamical sysm (1) ar boundd. Proof: W hav shown ha all individuals ar non-ngaiv for h posiiv paramrs. Th dynamics of h oal populaion ar govrnd by: 54
K whr K = µ + d n b, d n = min{d 1,d 2,d 3,d 4} and b>µ + d n. Thus w hav 0 N() N(0) as nds o infiniy. Thrfor, all fasibl soluions of h sysm (1) ar boundd. Smoking Gnraion Numbr and Equilibrium: In sudying any pidmiological modl idnifying h smoking gnraion numbr (hrshold valu) is xrmly imporan. This hrshold quaniy which drmins whhr an pidmic occurs or h disas simply dis ou. Th smoking gnraion numbr dnod by R s which can b dfind as h numbr of scondary infcions causd by a singl infciv inroducd ino a hos populaion. Th dynamical sysm (1) has hr quilibria: h rivial quilibrium E 0 = (0,0,0,0), h smoking-fr quilibrium E f= (N,0,0,0) and h ndmic quilibrium E = (P,S,Q,Q p ). Whr,, and. W will find smoking gnraion numbr, R, by h mhod [5]. L X = (S*,Q*,Qp*). hn h dynamical sysm s (1) can b rwrin as: such ha, By calculaing h Jacoban marics a smoking-fr quilibrium, E, w g: f and Whr and Thus, h nx gnraion marix is 1 Smoking gnraion numbr, R s, is h spcral radius (FV ) = R s = N/( + µ + d 2). I masurs h avrag numbr of nw smokrs gnrad by singl smokr in a populaion of ponial smokrs. Sinc h oal populaion of sysm (1) N() = P() + S() + Q (), + Q p(), hn h smoking-fr quilibrium poin E f = (N,0,0,0) xiss for all posiiv paramr valus. Th only posiiv smoking-prsn quilibrium poin o sysm (1) is E = (P,S,Q,Q p ). whnvr R s > 1. whr, 55
Hnc, hr is no posiiv ndmic quilibrium poin whnvr R < 1. s Dynamical Bhavior of Equilibrium: Th dynamical bhavior of quilibria can b sudid by compuing h Jacoban marics corrsponding o ach quilibrium poin and finding h ignvalus valuad a ach quilibrium poin. For sabiliy of h quilibrium poins, h ral pars of h ignvalus of h Jacoban marix mus b ngaiv. Thorm 1: Th rivial quilibrium E = (0,0,0,0) is locally asympoically sabl. 0 Proof: Evaluaing h Jacoban marix of sysm (1) around E = (0,0,0,0), w hav: 0 Thus h ignvalus of J(E ) ar a = µ d, a = µ d, a = µ d, a = µ d. Hnc, all h roos 0 1 1 2 2 3 3 4 4 hav ngaiv ral par which implis ha E is locally asympoically sabl. 0 Thorm 2: Smoking-fr quilibrium E f= (0,0,0,0) for dynamical sysm (1) is locally asympoically sabl if R s< 1. If R s = 1, E f is locally sabl. If R s > 1, E f is unsabl. Proof: Th local sabiliy of his quilibrium soluion can b xamind by linarizing sysm (1) around E = (N,0,0,0). This f quilibrium poin givs us h Jacoban marix: Thus h ignvalus of J(E) ar = (µ + d ), = ( + µ + d ), = (µ + d ) and = (R 1)/( + µ + d ). Clarly f 1 1 2 3 3 4 4 s 2 is ngaiv if R < 1. Thrfor, all ignvalus ar ngaiv if R < 1 and hnc E is locally asympoically sabl. If 4 s s f R < 1, hn = 0 and E is locally sabl. If R > 1, hn > 0 which mans ha hr xiss a posiiv ign valu. So, s 4 f s 4 E is unsabl. Hnc, w compld h proof. f Thorm 3: Smoking-prsn quilibrium E = (P,S,Q,Q p). for h dynamical sysm (1) is locally asympoically sabl. Proof: Th Jacoban marix of sysm (1) around h ndmic quilibrium E = (P,S,Q,Q p ). rjc: Th ign valus of h Jacoban marix around h smoking-prsn quilibrium E = (P,S,Q,Q p ). ar 1 = ( + 2 * * µ + d 3), 2= (µ + d 4), and + ( S + µ + d 1) + S ( + µ + d 2) = 0, which is clarly a quadraic quaion wih posiiv 2 * * cofficins. All roos of quadraic quaion + ( S + µ + d 1) + S ( + µ + d 2) = 0, hav ngaiv ral par. All ign valus of h Jacoban marix around h quilibrium E ar ngaiv. Hnc h smoking-prsn quilibrium E is locally asympoically sabl. 56
Fig. 1: Tabl 1: Paramrs usd for numrical simulaion Paramr Dscripion Valus b Birh ra 0.00091 µ Naural dah ra 0.001467 Ra of ransmission nonsmokr 0.5925926 ino smokr class Ra of quiing smoking 0.1666667 Th probabiliy convr ino Q and Qp 0.2222222 Ra of rmoval from Q o Qp 0.037037 d1 Disass dah ra of P 0.00004 d2 Smoking caus dah ra 0.00037 d3 Disass dah ra of Q 0.00035 d4 Disass dah ra of Qp 0.00003 Paramr Esimaion and Numrical Simulaion: Using ral daa [3] numrical simulaions of h modl (1) ar carrid ou by a s of rasonabl paramr valus givn in Tabl 1. Th aiud owards smoking is saring from junior or high school im. Thus 1/b is an avrag im for a prson in h sysm which s o b 1095 days. Dah from lung cancr was h lading caus of smoking, wih a ra of 37 pr 100,000 individuals [6]. Th dah ra of ach individual is diffrn from ach- ohrs and dpnds on h ral lif siuaion. Hr, w assum ha d4 d1 d3 d 2. W carrid ou numrical simulaions using WINPOLT sofwar o illusra h bhavior of basic smoking gnraion numbr, R s = N()/( + µ + d 2), as a funcion of on paramr (variabl) which is slcd from Tabl 1 wih fixing ohr paramrs. From figur 1 smoking gnraion numbr incrass as h ransmission ra. Th smoking gnraion numbr dcrass whn smoking caus dah ra d 2 and h ra of quiing smoking incrass for h oal populaion N>0. Thus smoking gnraion numbr is lss han on as h inducd dah ra d 2 and h ra of quiing smoking bcoms largr which shows in principl hr is no sprad of obacco smoking. CONCLUSIONS This papr prsns smoking obacco modl xprincing wih disas inducd dah which provids a mor ralisic modl. Th main focus of his papr was o inroduc comparmnal modls in pidmiology and chniqus for hir analysis. W sablishd som nw rsuls such as h xisnc of sabl or unsabl quilibrium poins undr suiabl valus of paramrs in h modls. And also using daa [3] in modl (1) w found ha R s = 3.5. Smoking gnraion numbr is grar han on shows ha smoking obacco sprads. Smoking gnraion numbr R s as a funcion of smoking caus dah ra d 2 and h ra of quiing smoking w found ha h basic smoking gnraion numbr R s 0 as d 2 and bcoms largr and largr rspcivly. Tha is smoking gnraion numbr is lss han on which shows in principl is ha hr is no sprad of smoking obacco. To conrol h sprad of smoking obacco w can conrol h paramric valu of h ransmission ra no o b incras and also incras h ra of quiing smoking. Finally w rcommnd his modl may b usd o ohr pidmiological modls. REFERENCES 1. hp:/ /cancrn.nci.nih.gov/wynk-pubs/lung.hm#5 Lung Cancr: Who's a Risk'? pp: 1-2. 2. Gachw Abiy Salilw, 2016. A Mahmaical Analysis for Conrolling h Sprad of HIV/AIDS wih Inducd Dah, TJPRC: IJHMR, 1(1): 1-14. 3. Gachw Abiy Salilw, 2016. Mahmaical Analysis on h Sprad and Conrol of Global Dynamics of Tobacco Smoking wih Inducd Dah, TJPRC Pv. Ld., JMCAR, 3(1): 15-26. 4. Braur, F., 2001. Casillo Chavz, Mahmaical Modls in Populaion Biology and Epidmiology, Springr-Vrlag, Nw York. 5. van dn Drissch, P. and J. Wamough, 2002. Rproducion numbrs and sub-hrshhold ndmic quilibria for comparmnal modls of disas ransmission, Mahmaical Bioscincs, 180: 29-48. 6. Naional Cancr Informaion Cnr, 2002. Dah saisics basd on vial rgisraion, R- rivd Fbruary 17, 2006, from hp://www.ncc.r.kr/,(2002). 57