MATHEMATICAL MODELLING AND ANALYSIS OF THE SPREAD OF AIDS CAUSED BY CO- INFECTION OF HIV WITH TB

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1 Itertol jourl of setf proesses reserh pplto IJSPR ISS -76 wwwjspro Volue Issue r-pr 8 pp- ICL ODLLIG D LSIS OF SPRD OF IDS CUSD B CO- IFCIO OF IV WI B h v Deprtet of pple See utes KI Sultpur Rghw Shukl Deprtet of pple thet PSI Kpur hhek wr Deprtet of Phs SRGI Lukow BSRC oe SIS-SIRS oel wth grto for the ufol of IDS use o- feto volvg B IV ple urg th usrpt I thetl oelg the tter ssue tht IV fete vul tert wth B fete per the forer to possess IV &p; B t lutel ws ssue tol tht frto of th IV tegor eoes IDS ptets t's toll thought of tht pree frto of B teree per tpe reservor tegor eghorhoo of whh reover eoes suseptle oe ore he oel le vtto stlt theor of equto t lutel ws fou tht IDS epe ufol quker euse of o- feto of IV wthb KWORDS: CO-IFCIO IV B IRODUCIO he resg ee of vrous fetous eses suh s goorrhebividset resg jor pul helth prole ost evelope evelopg outres he trsso spre of suh eses hu populto epes upo vrous ftors suh s the uer of fetves suseptles oes of trsso leologlgeogrphlotosout of oelg stu of epe eses ws reporte erler otes ooks fetous eses re spre ret ott etwee suseptles fetveswhle e others re spre retl rrers ter or vetors uerulosb fetous ese use ter lle oteru tuerulos ffets lug other prts of the o I evelopg outres pulor tuerulos e B the lug outs ore th sevet peret of ses the worl B whh h ee otrolle to sgft extet erler hs re-eerge reet ers s oe of the leg uses of eth out three llo people e ever ers ue to B oe eoe fete wth Bter ut people wth IV fetos ese greter rk of gettg fete wth th ese B trstte l roplet ule whe the per sufferg fro tve B rethes seees or oughs the rids the qure uo-efe sroe ftl ese use retrovrus kow s the hu uo efe vrus IV whh lthough ttks loo ells t uses hvo o the -ell the loo estrog eresg ther uer leg to ele o s ut to fght feto he ter IDS refers to ol the lst stge of the IV feto fter whh eth ours he IDS epe sweepg the worl oth evelope evelopg tos s IV hs fete llos of people ll over the worl wthout restrto of tolt olour ste or relogo et IV feto ults 7-8^% hs ee trstte to oe prter through uprotete sexul ter ourse whe the other prter fete wth IV other to hl trsso vertl feto outs ore th ^% of glol feto to fts hlre prt fro these other possle ure of feto re loo trsfuso trveous rug to et I geerl sexull trstte esesd suh s goorrhe ue lttle or o qure ut upo reover ut se of IV ut ereses 6 he trsso of IV feto ffete vrous ftors suh s ltet pero the uer of fete/fetous per the populto hvg IDS tpe of sexul tvt tpe of xg hoogeeous or heterogeeous vrous hgh rk groups et It ote tht se of IV fete pers ut ereses the erl fete other eses suh s B forg lss IV o fete wth B Soe efforts hve ee tke to uerst the trsso of o feto of IV wth B fetos thetl oels kg to out fxe vlues for the trsso oeffets these oels elt wth the sultos the equlru pot ls owever f the equlru pot les oul e heve for ll fte the trsso oeffets the t woul proves us fferet seros of the terto etwee IV B fetos I th usrpt oel for the trsso of o feto of IV wth B fetos exe e of the epeologl pltos COBID SIS-SIRS ODL WI COS IIGRIO I th seto oe SIS-SIRS oel wth ostt grto of hu populto propose le I proposg the oel the followg ssuptos re e P g e

2 Itertol jourl of setf proesses reserh pplto IJSPR ISS -76 wwwjspro Volue Issue r-pr 8 pp- ll IV fete vuls re osere suseptles to B feto t the se er ll B fete vuls re osere suseptles to IV Due to loss of ut IV fete pers get fete wth B fter terto here o seprte lss of per fete oth wth IV Bhe IV lss l lues pers hvg o feto Bse o the ove ssuptos we propose thetl oel to ssess the spre of o feto IV wth B fetos It ssue tht the populto est t ve to fve lsses the suseptle est Xtthe est of populto fete ol wth B the est of populto fete wth IV frto of whh e l fete wth B the est of IDS lss the est of reove lss t he oels propose s follows X t t t X X X X t t o lse the oel we oser the followg reue sste Se X t t t t t he results of the equlru ls re gve the followg theores; OR-here exts followg four equlr of the oel usse efore Whh exts f Whh exts f X t th oel re the ott-rtes whh suseptles erese followg ott wth pers hvg of B IDS e the o-feto ostt the ll IDS evelope t the rte µ per er the reover rte t whh B fetves eoes suseptles g the rte whh fetves goes to reove lss the eth rte ue to θ the reover rte reove lss the turl eth rte α e eth rte ue to IDS v o ero equlru s follows; for O we oser three ses of th QUILIBRIU LSIS P g e

3 Itertol jourl of setf proesses reserh pplto IJSPR ISS -76 wwwjspro Volue Issue r-pr 8 pp- P g e O t exts ol whe } { stfe re otos followg prove the exts t for for CS GRL e tht prove exts FOR PROOF S PPDIX SBILI LSIS- ow we stu the lol stlt of equlr the o-ler stlt of o trvl equlru he results re stte the for of the followg theores OR - ; he equlru ustle prove the followg otos re stfe FOR PROOF S PPDIX B OR -; he equlru loll sptotll stle prove tht followg otos re stfe v v v v v

4 Itertol jourl of setf proesses reserh pplto IJSPR ISS -76 wwwjspro Volue Issue r-pr 8 pp- P g e x x FOR PROOF S PPDIX C OR;- he equlru o ler stle prove the followg otos re stfe v v v v v x to prove the ove theore we ee the followg lee the rego of ttrto gve the followg set FOR PROOF S PPDIX-D PRICULR CS SIS ODL WI COS IIGRIO; If the oel f we put θ= t eoes s oel X X t X X t X t t X t ll the ls hve ther se eg s the prevous oel QUILIBRIU LSIS; I th se there exts four equlr of the oel ; ; whh exts f ; Whh exts f ; h efe three prts ; O t exts ol whe } {

5 Itertol jourl of setf proesses reserh pplto IJSPR ISS -76 wwwjspro Volue Issue r-pr 8 pp- P g e stfe re otos followg prove the exts t CS GRL e tht prove exts t FOR PROOF S PPDIX SBILI LSIS;- ow we stu the lol stlt of equlr the o-ler stlt of o trvl equlru he results re stte the for of the followg theores OR - ; he equlru ustle prove the followg otos re stfe FOR PROOF S PPDIX B OR -; he equlru loll sptotll stle prove tht followg otos re stfe v v v FOR PROOF S PPDIX-C OR- he equlru o lerl stle prove the followg otos re stfe v v v o prove the ove theore we ee the followg le he rego of ttrto

6 Itertol jourl of setf proesses reserh pplto IJSPR ISS -76 wwwjspro Volue Issue r-pr 8 pp- t FOR PROOF S PPDIX-D COCLUSIO t I th pper o ler oel propose le to stu the o- feto of B IDS It show tht the t spre of IDS reses ue to o-feto of B IDS the ese eoes ee It l fou tht ue to grto of IV feto the spre of IDS reses further t PPDIX- he gve oel X X t X t X XISC OF t It ovous fro the oel 7 we f f X t So h ese free equlru XISC OF t Se Fro we get Fro t we get t X 6 the reue sste gve elow fro we get 7 Se So o lvg we get Whh exts f XISC OF Se 8 So o lvg We get 6 P g e

7 Itertol jourl of setf proesses reserh pplto IJSPR ISS -76 wwwjspro Volue Issue r-pr 8 pp- 7 P g e Whh exts f XISC OF Fro Fro get We fro we get So Beoes 8 7 Beoes ow here re the three ses ;f equlru the O O O t exts ol whe } { he qulru < f followg prove the exts t f he qulru CS GRL e prov exts t PPDIX-B Usg the oel 7 he geerl jo trx e wrtte s follows;

8 Itertol jourl of setf proesses reserh pplto IJSPR ISS -76 wwwjspro Volue Issue r-pr 8 pp- 8 P g e - PROOF OF OR- stlt of he jo trx for gve the ege vlues re - se the equlru ustle Stlt for he jo trx for gve s Se ege vlue re Se ustle stlt of he jo for = Oe of the ege vlues whh > e ustle PPDIX-C Lol stlt of qulru etho B Lpuov he oel 7 e lere kg of followg ssuptos we get t t

9 Itertol jourl of setf proesses reserh pplto IJSPR ISS -76 wwwjspro Volue Issue r-pr 8 pp- P g e t t t we use followg Lpuov futo ; V re to e etere So t t t t t V = + s stfe re otos V egtve re ters the oeffe ts ofsqure ll e the equlru for v v v v v x x

10 Itertol jourl of setf proesses reserh pplto IJSPR ISS -76 wwwjspro Volue Issue r-pr 8 pp- P g e PPDIX-D O LIR SBILI PROOF OF L Fro oel 7 we ot s follows; t t slrl t t the rego of ttrto PROOF OF OR I th se we use the followg Lpuov futo; o log log fro vlues the susttutg tg fferet o W COSIDR W egtve efte f the followg otos re stfe ; v

11 v v v Itertol jourl of setf proesses reserh pplto IJSPR ISS -76 wwwjspro Volue Issue r-pr 8 pp- t t t t XISC OF v x he gve oel X t t t X FOR SIS ODL; PPDIX- X t t X X X the reue sste gve elow It ovous fro the oel So XISC OF Se Fro 6 h ese free equlru we get fro we get 6 Se So o lvg we get Whh exts f XISC OF 7 Se So o lvg 8 We get we f f Whh exts P g e

12 Itertol jourl of setf proesses reserh pplto IJSPR ISS -76 wwwjspro Volue Issue r-pr 8 pp- P g e f XISC OF Fro 8 Fro we get So Beoes 7 6 Beoes ow here re the three ses ;f equlru the O O O t exts ol whe } { he qulru < f followg prove the exts t W f he qulru CS GRL e tht prove exts t PPDIX-B Usg the oel 6 he geerl jo trx e wrtte s follows;

13 Itertol jourl of setf proesses reserh pplto IJSPR ISS -76 wwwjspro Volue Issue r-pr 8 pp- P g e PROOF OF OR- stlt of he jo trx for gve the ege vlues re - se the equlru ustle Stlt for he jo trx for gve s Se ege vlue re Se ustle stlt of he jo for Oe of the ege vlues whh > e ustle PPDIX-C Lol stlt of qulru etho B Lpuov he oel 6 e lere kg of followg ssuptos we get t t t t

14 Itertol jourl of setf proesses reserh pplto IJSPR ISS -76 wwwjspro Volue Issue r-pr 8 pp- P g e we use followg Lpuov futo ; V re to e etere So t t t t V = + s stfe re otos followg the prove efte egtve V WILLe egtve re ters the oeffe ts ofsqure ll e the equlru for v v v PPDIX-D O LIR SBILI PROOF OF L Fro oel 6 we ot s follows; t t slrl t the rego of ttrto PROOF OF OR I th se we use the followg Lpuov futo;

15 Itertol jourl of setf proesses reserh pplto IJSPR ISS -76 wwwjspro Volue Issue r-pr 8 pp- P g e ; o log log get we el fro vlues the susttutg fferettg o W COS ID W egtve efte f the followg otos re stfe ; v v v RFRRCS; Fuetl67 p 7-7 Dufr87- Wlt ethote7 8-8 Shukl JB87- Cooke7r ethote76golegu8 Rkrh Chrshekhr p -8 6 er 87 p-6 7erR Rture78 p er R R rsro PhlSoSeres8B p - BleJ 88p-6 CookeKL orke 7 p- ethotew7 p67-6 Blower Sll le gel Poroo rv C Peter poewell Phlp C she els &oss rew Ross- vol p - Blower S SllP &PC opewellpc-6 vol p- Dle sll Blower- vol67p 8-8 err& R-86 vol p-7 6 err 88 vol 66-7 err 88 volp -6 8 ethotew87 vol 8p -78 R88volp66-66 Pkerg JWle-J Po Set l 86vol7p67-68