Stability Analysis of a delayed HIV/AIDS Epidemic Model with Saturated Incidence

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1 nrnaional Journal of Mahmaics Trnds Tchnology JMTT Volum 4 Numbr - March 7 abiliy nalysis of a dlayd HV/D Epidmic Modl wih aurad ncidnc Dbashis iswas, amars Pal Calcua Girls Collg, Kolkaa-, ndia. Dparmn of Mahmaics, Univrsiy of Kalyani, Kalyani- 745, ndia. bsrac n his papr, w invsiga h ffc of im dlay on an HV/D pidmic modl wih saurad incidnc ra. W accp ha h individuals ar bing rcruid ino sxually maurd ag group a a consan ra incorporas im dlay for on o bcom infcd h ohr bcom fullyblown. W assum ha h disas sprad only by sxually ransmission.th modl consiss wo quilibria, namly, a disas-fr quilibrium an ndmic quilibrium. W calcula h basic rproducion numbr by using h nx gnraion marix. Mahmaical analyss conscrad ha h global dynamics of h sprad of h HV/D infcious disas ar oally drmind by h basic rproducion numbr. For h basic rproducion numbr <, h disas-fr quilibrium is locally asympoically sabl. Whhr >, h ndmic quilibrium poin is locally asympoically sabl. Finally; w find h numrical soluion of h modl which jusify h analyical rsuls. Ky words: HV/D Epidmic modl, Tim Dlay, asic rproducion numbr, abiliy, aurad ncidnc..nroducion Th firs cas of cquird mmunodficincy yndrom D was rpord by Cnrs for Disas Conrol CDC of mrica in 98 [. Human mmunodficincy Virus HV ha causs D, is on of h World mos srious disas. Thr approximaly 6.7 million popl World wid living wih HV/D a h nd of 5. Human mmunodficincy Virus infcion acquird immunodficincy syndrom HV/D is a spcrum of condiions causd by infcion wih h human immunodficincy virus HV [-5. Diffrn obsrvaion hav bn conducd o sudy h dynamics of HV/D wihou im dlay [6, 8, 9,. Howvr a fw modling sudis hav bn conducd o dscrib h ffc of im dlays [-5. Mainly HV ransmission in human body has wo basic mods, horizonal ransmission vrical ransmission. Hr w assum ha h disas only sprad by horizonal ransmission mans sxual conac. n 986, h firs known cas of HV in ndia was diagnosd amongs fmal sx workrs in Chnnai. n h vry nx h disas sprad rapidly all ovr h ndia. Now ndia has h hird highs numbr of popl living wih HV in h world. Th disas sprad maximum by hrosxual ransmission. sx-srucurd mahmaical modl for hrosxual ransmission of HV/D wih xplici incubaion priod was sudid by Mukavir [6. Hr also w assum ha whn h suscpibl class ar conac wih h infcd class, hn h disas sprad. Hnc i rquirs som im dlay, say >, for i o b dcabl; ha is, for an infcd individuals o bcom infciv/infcious. Hr is a im lag >, ha is infciv o bcom fully blown wih D sympoms. Hr w proposd h mahmaical modl find ou h basic rproducion numbr by using nx gnraion marix sudy hir sabiliy. Th modl is solvd numrically by using an iraiv numrical chniqu, which jusify hhorical rsuls..th mahmaical modl inc h disas sprad only by sxually ransmission h individuals ar bing rcruid sxually maurd ag group a a consan ra. Hr w can dividd h hrosxual populaion ino hr comparmns namly, h sxually maur suscpibl a a im >, h numbr of individuals a a im who ar alrady infcd wih HV, h numbr of individuals who hav dvlopd full blown D sympoms a a im. Th paramrs, is h probabiliy of ging infcd from a romly chosn parnr, is h naural dah ra, is a posiiv consan, h ra a which HV infcd individuals progrss o D, d D rlad dah ra, is h im ha akn for an individuals o bcom infcious afr bing in conac N: -57 hp:// Pag

2 nrnaional Journal of Mahmaics Trnds Tchnology JMTT Volum 4 Numbr - March 7 wih an infcd h im ha akn for infcd individuals o bcom fully blown wih D afr bcoming infcious, rcruimn ra of suscpibl class ino a sxually aciv populaion. W chosn h sxually maurd individuals ha is suscpibl haphazardly uniformly from h populaion. f h chosn individuals is infcd, hn h suscpibl individuals is assumd o g h virus. Whn w sd h individuals afr a im hn i will b considrd as infcious. Wihou any awarnss drug inrvnion, h disas will akn progrss o fully blown ino h infcd individuals afr a im >. Hr w also assumd ha h following assumpions: a Th rcruimn ra of h populaionsxually maurd aduls is mainly by birh. b n individuals onc a a im bing infcd hn i bcom rmains infciv unil dah. c n his sudy h populaion is homognous; ha is, hr is idnical mixing. d Th forc of infcion dpnds on h numbr of infciv in h populaion by h produc. Th full blown D individuals ar asily acknowldgd in h populaion; ha is, hy do no ngag in h ransmission dynamics. nfcd individuals may di du o naural dah or progrss of fully blown individuals comparmn. fr progrssion of fully blown comparmn, individuals ar rmovd from his comparmn du o naural dahs or disas rlad dahs. Undr his assumpions our modl quaion ar givn by. d, d d, d d d. d uppos ha an individuals bing in conac wih infcd individuals, afr som im clinically infciv. Furhr l an infciv individuals bcom fully blown afr a som im considraion our modl rducs o d, d d, d d d. d, o b. Undr his L [, max, : [,,, [, sup whr., wih h norm of dfind as is h norm in. Thrfor h iniial condiion for quaion is : [,,, whr,,, for all [,. Equaion subjc o h abov assumpion has a uniqu soluion [7..asic Propris of h Modl..Posiiviy of h oluions inc h modl suggs human populaion w nd o show ha all h sa variabls rmain non-ngaiv for all ims. : Lmma... L [,, >, >, > hn h soluions of,, of h sysm ar posiiv for all. N: -57 hp:// Pag

3 nrnaional Journal of Mahmaics Trnds Tchnology JMTT Volum 4 Numbr - March 7 Proof:Taking h firs quaion of quaion, w hav > xp[ d, d in h rgion...nvarian gion d d >,, hr ingraion limi is o, similarly w can prov ha. Thrfor all soluions of h sysm ar posiiv for all Lmma...Th sysm has soluions which ar conaind, in h fasibl rgion. Proof: L,, b any soluion of h sysms wih non ngaiv iniial condiions hn adding h quaion of h sysm, w hav [ [, Hnc, limsup. This implis ha all soluions of modl vnually nr h aracing s. 4.Equilibrium poins asic rproducion numbr Th sysm convy h following quilibria: ar boundd a h disas fr quilibrium E,,, b h ndmic quilibrium E,, whr,, d is clar ha if > hn hr xiss a uniqu quilibria xiss if < hn hr is no posiiv quilibria. Th local sabiliy of E is govrnd by h basic rproducion numbr which can b found from h nx gnraion marix. Th non-ngaiv marix, f of h infcion rms h non singular marix, v of h ransiion rms ar f, v d T FV whr F Jacobian of f a disas fr quilibrium V Jacobian of v a disas fr quilibrium FV. pcral of h marix 5.Global sabiliy of ndmic quilibrium Thorm 5...f >, hn h sysm a ndmic quilibrium unsabl ohrwis, Proof: To proof his rsul, w consruc h scond addiiv compound marix E is globally asympoically sabl, [ J for h sysm a E is N: -57 hp:// Pag 4

4 nrnaional Journal of Mahmaics Trnds Tchnology JMTT Volum 4 Numbr - March 7 givn blow J [ whr, d d P P,, diag,, wih P,, Pf,,, so P P,, f [ G P f P PJ P L G G G G G, whr G, G, G G ˆ ˆ ˆ L v, v, rprsn h vcor in v h associad norms is., dfind as ˆ v, ˆ, ˆ ˆ, ˆ, ˆ v v max v v v. L G rprsns Lozinski masur wih h hlp of h abov dfind norm, so as dscribd in [7, w assum G sup g, g, whr g G, g G, G G marix norm wih rspc o h vcor G G ar h rprsn h Lozinski masur wih o his l norm, hn G, G max[,,, Now g G G g G G d max, N: -57 hp:// Pag 5

5 nrnaional Journal of Mahmaics Trnds Tchnology JMTT Volum 4 Numbr - March 7 Thrfor, G sup g, g hn p [ G ds [ ds lg, hr ingraion limi is o implis ha p < quilibrium poin E is globally asympoically sabl. 5.. Global sabiliy of disas fr quilibrium E. Hnc h rsul [8, conclud ha h ndmic Lmma 5...f <, hn h disas fr quilibrium poin E of h modl is globally asympoically sabl in unsabl if >. Proof: Dfin Lyapunov funcion: L d L d L d[ L d[ L d[, sinc d [ f, L bu if <, < asympoically sabl in. 6.abiliy nalysis of h dlayd modl 6.. Local abiliy L hrfor, h disas fr quilibrium is globally W find h local sabiliy of h disas fr ndmic sas in currn scion. W prsn h local sabiliy of disas fr quilibrium E,, in h following horm Thorm 6...Th disas fr quilibrium E is locally asympoically sabl if only if whnvr <. Proof. Th variaional marix abou h disas fr quilibrium poin E is givn by J d Th ignvalus associad o J ar, < d h roos of h quaion P Q N: -57 hp:// Pag 6

6 nrnaional Journal of Mahmaics Trnds Tchnology JMTT Volum 4 Numbr - March 7, whr P, Q For h abov quaions bcoms 4 f < hn h roos of h quaion 4 hav ngaiv ral pars. Thrfor E is locally asympoically sabl if only if For, by corollary.4 in uan Wi [, i follows ha if insabiliy occurs for a paricular valu of h dlay i, > is h roos of. Puing h valu i, a characrisic roo of mus inrsc h imaginary axis. L ino h quaion w g cos sin i sin cos 5 [ paraing h ral imaginary par, givs sin cos 6 cos sin 7 quaring adding h quaion 6 7, givs 4 [ 8 Thrfor i is clar ha if <, hn h roos of 8 hav ngaiv ral pars. uch ha quaion canno hav purly imaginary soluions. Thrfor w conclud ha h disas fr quilibrium E is locally asympoically sabl if only if whnvr <. 6..Local abiliy of Endmic Equilibrium : Thorm 6... For >, h ndmic quilibrium poin. Proof.Th Jacobian marix J abou h ndmic quilibrium poin E is locally asympoically sabl, if only if E is givn by, J d Th ignvalus associad o J ar, < d h roos of h quaion p, p, 9 N: -57 hp:// Pag 7

7 nrnaional Journal of Mahmaics Trnds Tchnology JMTT Volum 4 Numbr - March 7 q q. For, quaion 9 bcoms a a, whr a p q. a p q is clar ha, for >, a, > a. Thrfor, using h ouh-hurwiz cririon [, h roos of h quaion hav ngaiv ral pars. Thn w conclud ha h ndmic quilibrium E is locally asympoically sabl for. For, afr puing i, > ino h quaion 9 sparaing h ral imaginary pars w g q cos q sin p cos q sin q p quaring adding quaion no, w hav 4 p p q p q L u p p p q [ [ q p q [ [, hn quaion can b rwrin as [ u pu, whr 4 q [ [ [ Thrfor i is clar ha if >, hn p, q > ha if >, h ndmic quilibrium 7.Numrical imulaions [ [, which saisfy h Lmma.. in [. Thn, w conclud E is locally asympoically sabl for. W now prsn numrical simulaions for h sabiliy analysis of a dlayd HV/D pidmic modl wih saurad incidnc o illusra our rsuls. Now for.,.,., d.,.,.,, Fig. shows h disas fr quilibrium is locally asympoically sabl whn.9999 <. W choos,.,, d.,.7,.,. N: -57 hp:// Pag 8

8 nrnaional Journal of Mahmaics Trnds Tchnology JMTT Volum 4 Numbr - March 7 whil.7979 > hn Fig. shows h ndmic quilibrium is locally asympoically sabl. 8.Conclusion n his papr, w hav proposd analyzd a non-linar mahmaical modl o sudy h ffc of im dlay in h rcruimn of infcd prsons on h ransmission dynamics of HV/D. y analyzing h modl, w hav found a basic rproducion numbr. is nod ha whn < hn disas dis ou whn > hn disas bcom ndmic. Th modl has wo non-ngaiv quilibria namly, E,,, h disas fr quilibrium E,,, h ndmic quilibrium. is found ha h quilibrium sa E corrsponding o disapparanc of disas is locally asympoically sabl if <, unsabl if >. Th ndmic quilibrium > is always asympoically sabl by E, which xiss only whn using Jacobian marix globally asympoically sabl by using scond addiiv compound marix whil. Thr is no ffc of doubl dlay. W know hr is no cur for HV or D hr ar mdicins ramn ha figh HV hlp popl wih HV D liv longr. Figur : Th quilibrium poin E is asympoically sabl whil <. N: -57 hp:// Pag 9

9 nrnaional Journal of Mahmaics Trnds Tchnology JMTT Volum 4 Numbr - March 7 Figur : Th quilibrium poin E is locally asympoically sabl whil >. frncs [ Cnrs for Disas Conrol, 98 Pnumocysis Pnumonia-Los ngls Morbidiy Moraliy Wkly por.,, 5-5. [ H. Wi, X. Li, M. Marchv, " n pidmic modl of a vcor-born disas wih dirc ransmission im dlay, " J. Mah. nal. ppl., 4, , 8. [ pkowiz K, D-h firs yars. N. Engl. J. Md., 44,, [4 lxr Kramr, Mirjam Krzschmar, Klaus Krickbrg, Modrn infcion disas pidmiology concps, mhods,mahmaical modls, publichalh. Nw. York. pringr.,, p. 88. [5 Wilhlm. Kirch. Encyclopdia of public halh. Nw. York. pringr., 8, pp [6 Y. H. Hsih, C. H. Chn, Modling h social dynamics of a sx indusry: is implicaions for sprad of HV/D, ull. Mah. iol., 66, 4, [7 Jack, H., Hal, K., Thory of Funcional Diffrnial Equaions.nd d., pingr Vrlag, Nw York., 977. [8 C. M. Kribs-Zala, M. L, C. oman,. Wily, C. M. Hrnz-uarz, Th ffc of h HV/D pidmic on frica s ruck drivrs, Mah. ios. Eng. 4, 5, N: -57 hp:// Pag

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