Chahrazed L Journal of Scientific and Engineering Research, 2018, 5(4): and

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1 vilbl onlin Journl of cinific n nginring srch 8 54:- srch ricl N: 94-6 CODNU: JB Mhmicl nlysis of wo pimic mols wih mporry immuniy Li Chhrz Dprmn of Mhmics Fculy of xc scincs Univrsiy frrs mnouri Consnin lgri bsrc h prsn ppr prsn nonlinr mhmicl mol which nlyzs h spr n sbiliy of h mol pimic. n h firs mol populion of siz N im is ivi ino hr subclsss whr n no h sizs of h populion suscpibl o iss n infcious mmbrs qurnin mmbrs hn h scon mol w inrouc wo clsss his ppr ls wih h quilibrium n sbiliy for h wo mols. ubjc clssificion: 4D 9D n. Kywors Bsic rproucion numbr nmic quilibrium pimic mol mporry immuniy. nroucion his ppr iscuss h quilibrium n sbiliy of wo non linr mols wih mporry immuniy n h iffrn posiivs prmrs. W hv m h following conribuions: h quilibrium n sbiliy of h firs mol w obin iss-fr quilibrium in h bsnc of infcion bu in h prsnc of infcion i ws uniqu posiiv nmic quilibrium n w fin h bsic rproucion numbr of h infcion. h quilibrium n sbiliy of h firs mol wih g w obin h uniqu posiiv quilibrium poin. Nx w moifi h prvious mol hn w hv h populion is ivi ino fiv subclsss w suy h quilibrium n sbiliy of h mol n w fin h bsic rproucion numbr of h. infcion Finlly w fin h rlionship bwn h bsic rproucion numbr of boh pimic mols.. Mol his ppr consirs h following pimic mol wih mporry immuniy: Consir populion of siz N ; whr n N im his populion is ivi ino hr subclsss wih no h sizs of h populion suscpibl o iss n infcious mmbrs qurnin mmbrs wih h possibiliy of infcion hrough mporry. Journl of cinific n nginring srch

2 Chhrz L Journl of cinific n nginring srch 8 54:- immuniy rspcivly. h posiiv consns n rprsn h h rs of suscpibl infcious n qurnin. Biologiclly i is nurl o ssum h min. h posiiv consn is nurl morliy r. h posiiv consns rprsn r of insinc. h posiiv consn rprsn h rcovry r of infcion. h posiiv consn is h vrg numbrs of concs infciv for n. h r of unnown prsons infc wih r c by h sysm. h posiiv consn is h prmr of immigrion. h posiiv consn is h prmr of migrion. h rm rflcs h fc h n iniviul hs rcovr from infcion n sill r liv fr infcious prio whr is h lngh of immuniy prio. h iniil coniion of. is givn s. Whr. C such h. L C no h Bnch spc C [ ] of coninuous funcions mpping h inrvl [ ]. Wih biologicl mning w furhr ssum h i i for i. ino Wih h iniil coniions..4 Whr. h rgion {.. quilibrium n sbiliy n quilibrium poin of sysm. sisfis N } is posiivly invrin s of. W clcul h poins of quilibrium in h bsnc n prsnc of infcion. n h bsnc of infcion h sysm.5 hs iss-fr quilibrium : ˆ ˆ ˆ. h ignvlus cn b rmin by solving h chrcrisic quion of h linrizion of. nr is ccc o h ignvlus r..5.6 Journl of cinific n nginring srch

3 Chhrz L Journl of cinific n nginring srch 8 54:- Journl of cinific n nginring srch. n orr for o b ngiv i is rquir h..7 hn h bsic rproucion numbr of h infcion s follows..8 n h prsnc of infcion subsiuing in h sysm lso conins uniqu posiiv nmic quilibrium whr.9 o is h uniqu posiiv nmic quilibrium poin which xiss if. horm h iss-fr quilibrium is loclly sympoiclly sbl if n unsbl if. horm Wih sysm. hs uniqu non-rivil quilibrium is loclly sympoiclly sbl.. Mol wih g h g isribuions of h numbrs in h clsss r no by n no h sizs of h populion suscpibl o iss n infcious mmbrs qurnin mmbrs wih h possibiliy of infcion hrough mporry immuniy rspcivly of g im is h g-spcific h r h sysm of pril quions for h g isribuions is. Wih... quilibrium n sbiliy ssum h sub populion os no pn on h im whn h sysm. is wrin s follows

4 Chhrz L Journl of cinific n nginring srch 8 54:- h iniil coniion of. is givn s Diffrnil quions of h sysm so Whr h sysm... r solv wih iffrn mhos of rsoluions n wih hs h uniqu posiiv quilibrium poin P.6 P xp.7 ˆ ˆ ˆ. W clcul h Jcobin mrix ccoring o h sysm wih JP ccc h pimic is loclly sympoiclly sbl if n only if ll ignvlus of h Jcobin mrix P J hv ngiv rl pr. h ignvlus cn b rmin by solving h chrcrisic quion of h linrizion of nr P is o h ignvlus r ccc.8 n orr o n will b ngiv i is rquir h h bsic rproucion numbr is fin s h ol numbr of infc populion in h rsuling subinfc populion whr lmos ll of h uninfc. h bsic rproucion numbr of h infcion fin s follows: h im uring which popl rmin infciv is fin s P is.9 Journl of cinific n nginring srch 4

5 Chhrz L Journl of cinific n nginring srch 8 54:- h oubling im of h pimic cn b obin s horm h iss-fr quilibrium Proof L 4 so if hn so L 5 so ln. P is loclly sympoiclly sbl if n unsbl if. convrgs o zro. m m min convrgs o zro. f L 7 so m m min convrgs o zro. f.. 4. Moifi Mol his ppr consirs h moifi pimic mol wih mporry immuniy: n h moifi pimic mol is ivi ino fiv subclsss no h sizs of h populion suscpibl o iss n infcious mmbrs qurnin mmbrs wih h possibiliy of infcion hrough mporry immuniy rspcivly. consn r from o ; n consn r from o. h posiiv consns ; min Whr h iniil coniion of. is givn s. rprsn h h rs of C such h L C no h Bnch spc C [ ] of coninuous funcions mpping h inrvl [ ] 5. Wih biologicl mning w furhr ssum h i i for i 45. ino. Journl of cinific n nginring srch 5

6 Chhrz L Journl of cinific n nginring srch 8 54:- Journl of cinific n nginring srch 6. Wih h iniil coniions Whr. 5 4 h rgion { 5 } N is posiivly invrin s of quilibrium n sbiliy n quilibrium poin of sysm. sisfis.5 W clcul h poins of quilibrium in h bsnc n prsnc of infcion. n h bsnc of infcion h sysm.5 hs iss-fr quilibrium :. ˆ ˆ ˆ ˆ ˆ.6 h ignvlus cn b rmin by solving h chrcrisic quion of h linrizion of. nr. o h ignvlus r n orr for 4 o b ngiv hn h bsic rproucion numbr of h infcion s follows..7 n h prsnc of infcion subsiuing in h sysm lso conins uniqu posiiv nmic quilibrium whr

7 Chhrz L Journl of cinific n nginring srch 8 54:- Journl of cinific n nginring srch 7.8 o is h uniqu posiiv nmic quilibrium poin which xiss if. horm is h bsic rproucion numbr of sysm. n is h bsic rproucion numbr of sysm. hn.9 Proof W hv.8 hn. W rmplc. in.7 hn.

8 Chhrz L Journl of cinific n nginring srch 8 54:- Journl of cinific n nginring srch 8 horm 4 h iss-fr quilibrium is loclly sympoiclly sbl if n if sysm. hs uniqu non-rivil quilibrium is loclly sympoiclly sbl. Proof. W hv. Wih. ; hn. W hv. Wih. ; hn

9 Chhrz L Journl of cinific n nginring srch 8 54:- 5. Conclusion his ppr rsss h quilibrium n sbiliy of h firs pimic mol wih mporry immuniy in h bsnc of infcion h sysm hs iss-fr quilibrium in h prsnc of infcion h sysm hs uniqu posiiv nmic quilibrium. hn w suy quilibrium of h firs mol wih âg. Boh sysms hv h uniqu posiiv quilibrium poin loclly sympoiclly sbl if n hs uniqu non-rivil quilibrium is loclly sympoiclly sbl if. frncs []. nrson. M n Mly M n Jhonson K Prliminry uy of h rnsmission Dynmics of h Humn mmunoficincy Virus HV h Cusiv gn of D. M. J. Mh. ppl. M. Biol 9-6. []. b n Kr n libi H... Globl biliy for Dly n pimic Mols Wih ur ncinc s. lcronic Journl of Diffrnil quions -. []. Billy. N..J om ochsic Mols for mll pimics in Lrg Populion. ppl. is [4]. Billy. N..J h Mhmicl hory of nfcion Disss n is pplicion. ppli isics 6 N [5]. Bih M.. M. Noorni n. Hshim. 8. Numricl soluions of h nonlinr ingroiffrnil quions n. J. Opn Probl. Comp. Mh 4-4. [6]. Bcr. N.G h Uss of pimic Mols. Biomrics [7]. Billr.L ochsic Gnrl pimic in m ub-populion. J. ppl. Prob [8]. Jinling W Xinxin in.. Globl bily of Dly Diffrnil quion Of Hpiis B Virus nfcion Wih mmun spons. lcronic Journl of Diffrnil quions94-. [9]. Jin. Z Zhin. M n Mon. H. 6. Globl sbiliy of n pimic mol wih ly c Mimic cini. 6 B []. Kung Y. 99. Dly-Diffrnil quions wih pplicion in populion biology. cmic Prss nw yor. []. Louns. n rzoz. H.. Moling HV pimic Unr Conc rcing. h Cubn Cs. Journl of horiicl Mcin Vol []. Louns. rzoz.. H. Non-Linr Mol for xully rnsmi Diss wih conc rcing. M. J. MJh. ppl. M. Biol.9-4. []. Lhrouz n l Mroufy H.. uliiv Bhviour of Mol of n pimic:biliy n Prmnnc. ppli Mhmics & nformion cincs. n nrnionl Journl 5-8. [4]. Luo n Mo X. 7. ochsic populion ynmics unr rgim swiching. J. Mh. nl. ppl [5]. Michl l J.. ochsic clculus n finnil pplicions. pringr-vrlg. [6]. Nrsh n Omr.. n pimic mol for h rnsmission ynmics of HV/D n nohr infcion. nrnionl Journl of Mhmicl rchiv [7]. Pro. L Diffrnil quions n Dynmicl ysms. n iion pringr Nw Yor. [8]. y Wson. 98. usful nom im-l rnsformion For h nr pimic Mol. J. ppl. Prob [9]. y Wson 98. On h iz Disribuion For om pimic Mols. J. ppl. Prob []. obr N n My. 98 Populion Biology of infcious isss. nrnionl cnr of horiicl physics.-9. []. uoyn un.. Globl sbiliy of h nmic quilibrium of muligroup mols wih nonlinr incinc. Compurs n Mhmics wih pplicions Journl of cinific n nginring srch 9

10 Chhrz L Journl of cinific n nginring srch 8 54:- [].. ighi Chhrborj M.. bu Br. Fuzih.. Noor m. H. Mli V. lli.. Bhvior biliy in wo -yl Mols for HV. n. Journl of Mh. nlysis Vol. 4 no []. uchi n W. M biliy nlysis on ly pimic mol wih nsiy pnn birh procss Dy-nm. Conin. Discr mpuls. ysms [4]. Voloymyr Mrov Dnis Drgunov.. numric-nlyicl mho for solving h Cuchy problm for orinry ifrnil quions. ppli Mhmics n Compuion-6. [5]. W. M Y. uchi. Hr n. Br.. Prmnnc of r pimic mol wih isribu im lys ohou Mh. J [6]. W. Wng.. Globl bhvior of n pimic mol wih im ly ppl. Mh. Lrs [7]. Wn L n Yng X. 8. Globl sbiliy of ly mol wih mporry immuniy. Chos olions n Frcls 8-6. [8]. Xio L Chn n F. vn n Bosch.. Dynmicl bhvior for sg-srucur infcious iss mol Nonlinr nl. l Worl ppl [9]. Z. M J. Liu n J. Li.. biliy nlysis for iffrnil infciviy pimic mols Nonlinr nl. l Worl ppl []. Zhng F n Zhn Li n Zhng F. 8. Globl sbiliy of n pimic mol wih consn infcious prio. ppli Mhmics n Compuion Journl of cinific n nginring srch