Modeling the Effect of Environmental Factors on the Spread of Bacterial Disease in an Economically Structured Population

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1 vailable at ppl. ppl. Math. ISS: ol. 7 Iue June pp pplication an pplie Matheatic: n International Journal M Moeling the ffect of nvironental Factor on the Sprea of acterial Dieae in an conoicall Structure Population Ra areh an Surabhi Pane Departent of Matheatic Harcourt utler Technological Intitute Kanpur-8 IDI rantripathi@ahoo.co; purabhi_8@ahoo.co.in Receive: ugut 7 ; ccepte: Ma 3 btract We have propoe an analze a nonlinear atheatical oel for the prea of bacterial ieae in an econoicall tructure population rich an poor incluing the role of vaccination. It i aue that rich uceptible get infecte through irect contact with infective in the ae cla an with infective fro the poor cla who wor a ervice provier in the houe of rich people living in uch cleaner environent. The uceptible in the poor cla are aue to becoe infecte through irect contact with infective in the ae cla a well a b bacteria preent in their own environent egrae ue to unhgienic environental conition. It i further aue that the bacteria population affect onl the population in the egrae environent of the poor cla but oe not urvive in the clean environent of rich people. The enit of bacteria population i aue to be governe b a logitic oel an i epenent on environental icharge conucive to the growth of bacteria population. The cuulative enit of environental icharge epen upon the huan population relate factor of the poor cla. The oel anali how that the increae growth rate of environental icharge increae the bacteria population enit in the poor cla ue to unhgienic environental conition leaing to increae the infective in the poor cla i.e. ervice provier. a conequence ue to interaction with thee ervice provier the prea of ieae increae in the rich cla. The iprove environental conition of the region inhabite b ervice provier along with uitable vaccination trateg can be helpful in reucing the prea of the ieae. Kewor: acterial ieae econoicall tructure population ervice provier environental factor vaccination tabilit iulation MSC : 9D3 9D5 46

2 M: Intern. J. ol. 7 Iue June 47. Introuction Infectiou ieae are the worl bigget iller of people an account for illion of eath per ear. There are an infectiou ieae in which infection i tranitte b irect contact of uceptible with infective while there are oe ieae lie tuberculoi which are alo tranitte inirectl b the flow of bacteria fro infective into the environent. The poor environental conition exiting in the enel populate citie of the thir worl countrie have the greatet ipact on the prea of bacterial ieae. If the environent i conucive to the growth of bacterial population then it further help in the prea of infectiou ieae Ghoh et al. 5 6 areh et al lo the igration of population fro environentall egrae region to a cleaner region pla a vital role in the prea of infectiou ieae a infecte peron act a carrier/reervoir of infection. ariou oeling tuie of infectiou ieae have been one Ma an neron 979 neron an Ma Struchiner et al. 989 Mena-orca an Hethcote 99 Krib-Zaleta an elaco-hernanez Feng et al. Murph et al. Moghaa an lexaner 4 owan et. al. 5 areh an Tripathi 5 White an Coie 7 areh et al. 9 Pero an Tchuenche Muhaabaa et al.. In particular Ghoh et al. 5 oele the effect of ervice provier fro an environentall egrae region on the prea of bacterial ieae an conclue that the prea of the infectiou ieae increae when the growth of bacteria caue b conucive environental icharge ue to huan ource increae. lo the prea of the infectiou ieae in rich cla increae ue to the interaction with ervice provier. Thu unhgienic environental conition in the habitat caue b ervice provier becoe reponible for the fat prea of an infectiou ieae. The 6 alo forulate an SIS oel for the prea of a bacterial ieae auing logiticall growing huan population an conclue that the ieae prea i fater when bacterial growth increae ue to conucive environental icharge. i an Jin 5 analze a SIR oel having infectiou force in latent infecte an iune perio. The erive baic reprouction nuber R an conclue that if R the ieaefree equilibriu i globall table o that the ieae alwa ie out an if R the ieaefree equilibriu becoe untable while the eneic equilibriu eerge a the unique poitive equilibriu an i locall an globall table when ieae inuce eath rate i zero. Mcclue 6 propoe an analze oel for the prea of T which inclue fat an low progreion to the infecte cla an howe that when baic reprouction nuber i le than or equal to one the ieae-free equilibriu i globall aptoticall table an when it i greater than one there i an eneic equilibriu which i globall aptoticall table. Martcheva et al. 7 forulate an epieic oel to invetigate the coplexitie of the effect of vaccination on a ulti train ieae in the preence of utation. areh et al. 9 analze a nonlinear oel for the prea of HI/IDS in a population of varing ize with iigration of infective alo aue infectiou an all infective ultiatel eveloping IDS. The conclue that the prea of infection can be lowe own if irect inflow of infective i retricte into the population. Pero an Tchuenche tuie an HI/IDS oel b taing into account the ocial tructure of population rich an poor an foun that the prevalence of HI in rich counitie i higher than that in the poor but the ieae evelop fater in ipoverihe iniviual.

3 48 Ra areh an Surabhi Pane It i pointe out here that in oe of the above oel vaccination ha been tuie without coniering the effective role of variable bacteria population enit which epen on cuulative enit of environental icharge an inirectl on population enit of ervice provier an i reponible for preaing the bacterial ieae. In big citie of thir worl countrie where rich an poor people live in nearb neighborhoo then the poor people wor a ervice provier in the houe of rich people but o not ettle in the habitat of rich people. Thee ervice provier pla vital role in the prea of infectiou ieae a the carr pathogen in or on their boie an a alo tranport ieae vector. With increae in the population enit of ervice provier the effect of huan population relate factor lie icharge of houehol wate; open ewage rainage open water torage tan pon etc. lea to further growth in the enit of bacteria population thereb increaing the fat prea of bacterial ieae. It i therefore reaonable to aue the growth rate an the carring capacit of bacteria population enit to be the function of cuulative enit of environental icharge. The growth rate of environental icharge i alo aue to be the function of total population of ervice provier. In thi paper we therefore propoe a nonlinear oel to tu the prea of bacterial ieae in an econoicall tructure population rich an poor incluing the role of vaccination. We have taen into account the growth rate an carring capacit of bacteria population to be function of cuulative enit of environental icharge. The growth rate of environental icharge i alo taen to be epenent on the total population of ervice provier living in a egrae environent. Since our objective i to tu explicitl the role of environental factor conucive to the growth of bacteria population on the prea of ieae with the above population tructure we have coniere bilinear interaction to oel the traniion naic. The oel however can further be generalize b auing tanar incience or other nonlinear interaction Mena-orca an Hethcote 99 areh et al. 8.. Matheatical Moel We conier the prea of bacterial infectiou ieae in an econoicall tructure population rich an poor living in two ajoining habitat or neighborhoo with ifferent environental conition. The environent where the rich people live i uch cleaner wherea the environent where poor people live i not o clean an i ver conucive to the growth of bacteria population ue to unhgienic houehol icharge. Here the total population of rich cla i ivie into uceptible X infective an vaccinate iniviual. The total population of poor cla i ivie into uceptible X infective an vaccinate iniviual. It i aue that rich uceptible get infecte through irect contact with infective in the ae cla an with infective fro the poor cla who wor a ervice provier in the houe of the rich people. Thee ervice provier interact with people in the rich cla uring wor an then the return bac to their hoe. The ervice provier o not ettle in the habitat of rich people. Thu we have coniere interaction between rich an poor population but not the igration of poor population into rich population. It i further aue that the bacteria population affect onl the population in the egrae environent of the poor cla but oe not urvive in the

4 M: Intern. J. ol. 7 Iue June 49 clean environent of rich people. It i alo aue that uceptible in the poor cla get infecte through irect contact with infective in the ae cla an inirectl b bacteria preent in their egrae environent. The uceptible in the poor cla a alo get infecte b the infecte rich people an the heterogeneit of the ixing pattern between the rich an poor a alo influence the infection naic of bacterial ieae hunu et al.. However in the oeling proce we have coniere the interaction of infecte rich people onl with uceptible in the ae cla but not with the uceptible of the poor cla a our purpoe i to tu the prea of bacterial ieae through ervice provier poor people living in unhgienic environental conition conucive to the growth of bacteria Ghoh et al 5. The enit of bacteria population i aue to grow logiticall with the growth rate an carring capacit epenent on the cuulative enit of environental icharge in the poor cla which further epen on the huan population relate factor in the poor cla. The bacteria population enit i alo enhance b the releae of bacteria fro infecte poor population. We have alo aue that the rich an poor uceptible X an X repectivel are vaccinate at a contant rate an oe of the a again becoe uceptible ue to inefficac of vaccine. lo a fraction of infective in both the clae after recover a join the repective uceptible clae. In view of the above auption an conieration the oel naic i governe b the following te of nonlinear orinar ifferential equation: X X X X t X X t X t 3 X X X X t 4 X X t 5 X t 6 t 7 t 8 X X X X

5 43 Ra areh an Surabhi Pane where an are the contant iigration rate of huan population into the rich an poor population repectivel an are traniion coefficient in the rich population ue to infective of the rich an poor cla repectivel an are traniion coefficient in the poor population ue to infective of the poor cla an bacteria repectivel an are the natural eath rate correponing to rich an poor clae an are the ieae inuce eath rate correponing to rich an poor clae repectivel. The paraeter an repreent the vaccination coverage of uceptible of rich an poor population repectivel an are therapeutic treatent coverage of infecte iniviual of rich an poor clae repectivel an enote the rate at which vaccinate iniviual of rich an poor population again becoe uceptible ue to inefficac of vaccine. Here t enote the cuulative enit of environental icharge conucive to the growth of bacteria population an i the intrinic growth rate of the bacteria population enit i the carring capacit of the bacteria population in the natural environent i the eca rate of bacteria population enit ue to natural factor a well a b control eaure i the rate of releae of bacteria fro the infective poor population i the rate of cuulative environental icharge conucive to the growth of bacteria into the poor population which epen on the enit of the poor population an i the epletion rate coefficient of the cuulative environental icharge. the huan population increae the effect of huan population relate factor/activitie enhance the cuulative enit of environental icharge which further intenif the growth of bacteria population. Thu in the oel an are taen to be function of cuulative enit of environental icharge. Since we aue that the growth rate per capita increae a the cuulative enit of environental icharge we have an 9 where i the value of at = an enote the erivative of the function with repect to it arguent. We aue that the oifie carring capacit increae with the cuulative enit of environental icharge o that an where i the value of when =. We alo aue that rate of cuulative environental icharge increae with the huan population enit o that an where i the value of when =.

6 M: Intern. J. ol. 7 Iue June quilibriu nali It i ufficient to conier the following reuce te of oel -8 ince X an X a follow: t t 3 t 4 t 5 t 6 t 7 8 t 9 ea. The region of attraction for the te -9 i given b / / / / : i poitivel invariant an all olution tarting in the region ta in where 4 ] [ ] [ an /. The equilibriu anali of the oel te -9 ha been carrie out an the reult are given a follow. There exit following three non-negative equilibria of the te -9 / W. t

7 43 Ra areh an Surabhi Pane Thi i the ieae-free equilibriu DF which exit without an conition. The exitence of W i obviou. W. Thi i the bacteria-free equilibriu F an it exit uner the following conition where /. In thi cae ieae onl prea through irect contact of uceptible with infective. 3 neic equilibriu W We prove the exitence of eneic equilibriu b the iocline etho. Setting the right han ie of the equation in oel -9 to zero we get following algebraic equation ] [ ] [ 3

8 M: Intern. J. ol. 7 Iue June ] [. 5 ow we how the exitence of an fro quation 4 an 5 an the correponing value of an can be obtaine fro quation an 3. Fro quation 4 we have i For = = an ~ which i poitive if an negative otherwie. ii t there i a horizontal aptote. iii t the lope of eq. 4 i given b which i poitive or negative epening upon ~ being negative or poitive repectivel. iv t ~ the lope of quation 4 i given b which i poitive or negative epening upon ~ being poitive or negative repectivel. Fro quation 5 we oberve the following point

9 434 Ra areh an Surabhi Pane i When = = an [ ] ~. ii t the lope of quation 5 i given b [ ]. iii t ~ the lope of quation 5 i given b [ ] [ ] which i poitive if. Thu after plotting an correponing to quation 4 an 5 in Figure we ee that there are two interecting point an. fter fining an we can calculate an uing eq. an 3. Figure a. xitence of eneic equilibriu when >.

10 M: Intern. J. ol. 7 Iue June 435 Figure b. xitence of eneic equilibriu when <. 4. Stabilit nali ow we analze the tabilit of equilibria W W an W. The local tabilit reult of thee equilibria are tate in the following theore. Theore 4.. The equilibriu W an W are untable an the eneic equilibriu W i locall aptoticall table uner the following conition: in. 8 9 in. 9 For proof ee ppenix - I

11 436 Ra areh an Surabhi Pane Theore 4.. In aition to auption 9 an let an atif p q an r for oe poitive contant p q an r in then eneic equilibriu W i nonlinearl aptoticall table in the region provie the following inequalitie are atifie r p q 3 34 / in. / in./ 5 / For proof ee ppenix-ii Rear. If the contact rate of the uceptible of the rich cla with the infective of the poor cla i ver all i.e. then inequalitie 6 3 an 3 are autoaticall atifie. If the inirect contact rate of the uceptible of the poor cla with bacteria i ver all i.e. then inequalitie 7 3 an 33 are autoaticall atifie. It i clear fro inequalit 34 that in the abence of environental an huan population relate factor i.e. p = q = r = the inequalit i autoaticall atifie. Thi iplie that the environental an huan population relate factor conucive to the growth of bacteria population have a etabilizing effect on the te. We alo note that ue to the preence of vaccinate cla conition 8 9 an 35 an 36 are require for the local an nonlinear tabilit which further etabilize the te. 5. uerical Siulation In orer to tu the naical behavior of the oel -9 an to prove feaibilit of tabilit conition we have conucte nuerical iulation for the et of paraeter given in Table uing MP 7. Feng et al. owan et. al. 5 Ghoh et al. 5. In the oel an are the growth rate an oifie carring capacit of the bacteria population an are function of the cuulative enit of the environental icharge. The rate of the cuulative environental icharge i alo a function of the population enit of

12 M: Intern. J. ol. 7 Iue June 437 the poor cla. Thu for nuerical iulation it i aue that an are linear function of i.e. = + a an = +b atifing conition 9 an. We have alo aue to be a linear function of uch a = + l atifing conition. Table. Paraeter alue Paraeter Sbol Paraeter value Recruitent rate of uceptible in rich cla Rrecruitent rate of uceptible in poor cla Traniion coefficient ue to contact of uceptible with. infective in rich cla Traniion coefficient ue to contact of uceptible with.3 infective in poor cla Traniion coefficient ue to contact of uceptible in rich cla.5 with infective in poor cla Traniion coefficient ue to contact of uceptible with. bacteria in poor cla Recover rate of infecte iniviual in rich cla. Recover rate of infecte iniviual in poor cla. atural eath rate of inviual in rich cla.5 atural eath rate of inviual in poor cla.3 Dieae-inuce eath rate in rich cla. Dieae-inuce eath rate in poor cla.5 accination coverage of uceptible of rich population.5 accination coverage of uceptible of poor population.4 Rate at which vaccinate iniviual of rich population again. becoe uceptible Rate at which vaccinate iniviual of poor population again becoe uceptible Growth rate of bacteria population = + a.85 a. Rate of releae of bacteria fro infecte iniviual of poor cla. Deca rate of bacteria in the environent.3 Carring capacit of the bacterial population in the natural environent = + b Rate of cuulative environental icharge conucive to the growth of bacteria into the poor population = + l Depletion rate coefficient of the cuulative environental icharge. b. 5 l.. The equilibriu value for the oel te -9 are copute a follow

13 438 Ra areh an Surabhi Pane = = = = = = =.87 = The eigenvalue of variational atrix correponing to the eneic equilibriu for the oel te -9 are i i i i Since all the eigenvalue are negative or have negative real part it iplie that the eneic equilibriu W i locall aptoticall table for the above et of paraeter value. The coputer iulation i perfore for ifferent initial tart in the following four cae an iplae graphicall in fig. an 3. In thee figure the variation of infective with the total population of rich an poor clae i hown repectivel. The trajectorie tarting with ifferent initial tart reach the equilibriu point. Thu the te -9 i nonlinearl aptoticall table for the above et of paraeter value.. = = 6 = 3 = 5 = = = 7 = 5.. = 3 = 6 = 6 = 5 = = 6 = 7 = = 3 = 6 = 35 = 5 = = 5 = 7 = = = 6 = 55 = 7 = = 3 = 7 = 5. The reult of nuerical iulation are iplae graphicall in Figure 4-9. In Figure 4 an 5 the variation of bacteria population an infective population of the rich an poor clae i hown with tie repectivel for ifferent growth rate of cuulative enit of environental icharge. It i een that a the growth rate of environental icharge increae bacteria population increae. With the increae in bacteria population the prea of ieae alo increae in infective population of poor cla. When thi infective population of the poor cla i.e. ervice provier interact with the uceptible of the rich population at a higher rate their infective population alo increae. Thi iplie that the increae bacteria enit in poor cla ue to growth of unhgienic environental icharge reult in increaing the infective in poor cla an a uch ieae prea i fater in thi cla. Conequentl the population in rich cla living in uch cleaner environent i irectl affecte b the higher nuber of infective available fro poor cla who wor a ervice provier. In Figure 6 we have hown the eca of bacterial population enit ue to natural

14 M: Intern. J. ol. 7 Iue June 439 factor or control eaure with tie. It i oberve that a the ipact of control eaure ecreae the nuber of infective in poor cla increae leaing to higher nuber of infective in rich cla. Figure 7 epict the role of bacteria releae fro the infective of poor cla who wor a ervice provier. Thi aitional loa of bacteria population enit further increae the infective population in the poor cla which ultiatel lea to enhance the ieae prea in rich population with increae interaction rate with ervice provier. Here it a be note that ince the environent of rich cla i coparativel clean an hgienic the increae ieae prea in the cla of ervice provier will ipact the rich cla onl if the higher nuber of infecte ervice provier eliver ervice to the. It i therefore peculate that not onl the growth of the bacteria population ue to environental conieration or it releae fro the infective nee to be curbe uing effective control echani but the irect interaction of the uceptible of the rich cla with infecte ervice provier houl alo be retricte. In Figure 8-9 the effect of vaccination i hown on the vaccinate an the infective population of the poor cla. It i foun that a the vaccination rate increae the vaccinate population increae treenoul an conequentl the infective population ecline. iilar obervation i ae for vaccination rate in the rich cla. Fro the above icuion we infer that in orer to control the prea of bacterial infection in a population where ervant an ai woring a ervice provier act a carrier of infection apart fro the introuction of proper vaccination trateg in the population the environental conition in which the ervice provier live be iprove o that the bacteria o not get a conucive environent to grow or accuulate in the atophere. Moreover people fro the poor cla who are infecte with the ieae be retricte to act a ervice provier in the houe of rich people in orer to eep the ieae prea at iniu. Figure. ariation of total population with infective population of rich cla

15 44 Ra areh an Surabhi Pane Figure 3. ariation of total population with infective population of poor cla Figure 4. ariation of bacteria population with tie for ifferent value of i.e. + l

16 M: Intern. J. ol. 7 Iue June 44 Figure 5. ariation of infective population with tie for ifferent value of i.e. + l Figure 6. ariation of infective population with tie for ifferent value of

17 44 Ra areh an Surabhi Pane Figure 7. ariation of infective population with tie for ifferent value of Figure 8. ariation of vaccinate poor population with tie for ifferent value of

18 M: Intern. J. ol. 7 Iue June Concluion Figure 9. ariation of infective poor population with tie for ifferent value of In thi paper a nonlinear atheatical oel i propoe an analze to tu the effect of environental factor on the prea of bacterial ieae in an econoicall tructure population rich an poor where people fro the poor population wor a ervice provier in the houe of the rich people. It i aue that the uceptible fro the rich cla get infecte through irect contact with infective in ae cla an with infective fro the poor cla who wor a ervice provier. The uceptible in the poor cla get infecte through irect contact with the infective in their own cla a well a b bacteria preent in their unhgienic environent. The enit of the bacteria population i aue to be governe b a logitic oel an i epenent on the environental factor which are conucive to the growth of the bacteria population. The cuulative enit of the environental icharge epen upon the huan population enit relate factor in the poor cla. The oel i analze uing tabilit theor of ifferential equation an nuerical iulation. The anali how that a ieae-free equilibriu DF an bacteria-free equilibriu F are alwa untable wherea eneic equilibriu i locall a well a nonlinearl aptoticall table uner certain conition. Further the environental a well a huan population relate factor conucive to the growth of bacteria population have a etabilizing effect on the te. It i foun that increae growth rate of environental icharge increae the bacteria population enit. Thi increae of bacteria enit in the poor cla ue to unhgienic environental icharge reult in increaing the infective in the poor cla i.e. ervice provier. a conequence thee ervice provier further ecalate the ieae in the rich cla living in uch cleaner environent. It i uggete that along with a uitable vaccination trateg the environental conition of the region inhabite b ervice provier be iprove o that the bacteria o not get a conucive environent to grow an in orer to ignificantl reuce the ieae prea.

19 444 Ra areh an Surabhi Pane cnowlegeent: uthor are thanful to the anonou reviewer for their contructive coent an uggetion which helpe u to iprove an finalize the anucript. RFRCS neron R. M. an Ma R. M accination againt rubella an eale: qualitative invetigation of ifferent policie. J. Hg. Cab. 9 pp neron R. M. an Ma R. M. 98. Population iolog of Infectiou Dieae Part I ature 8 pp hunu C.P. Muhaabaa S. an Sith R.J.. eing the effect of povert in tuberculoi traniion naic ppl. Math. Moel. 36 pp owan C. Guel.. an en Drieche P. Wu J. an Zhu H. 5. atheatical oel for aeing control trategie againt Wet ile viru ull. Math. iol. 67 pp Feng Z. Iannelli M. an Milner F... Two train Tuberculoi oel with age of infection Sia J. ppl. Math. 6 pp Ghoh M. Chanra P. Sinha P. an Shula J.. 5. Moeling the prea of bacterial ieae: effect of ervice provier fro an environentall egrae region ppl. Math. Cop. 6 pp Ghoh M. Chanra P. Sinha P. an Shula J.. 6. Moeling the prea of bacterial infectiou ieae with environental effect in a logiticall growing huan population onlinear nali: RW 73 pp Krib-Zaleta C. M. an elaco-hernanez J. X.. iple vaccination oel with ultiple eneic tate. Math. ioc. 64 pp i G. an Jin Z. 5. Global tabilit of a SIR epieic oel with infectiou force in latent infecte an iune perio Chao Solition an Fractal 5 pp Martcheva M. Iannelli M. i Xue-Zhe 7. Subthrehol coexitence of train: the ipact of vaccination an utation Math. ioc. ngg. 4 pp Ma R. M. an neron R. M Population iolog of Infectiou Dieae Part I ature 8 pp Mcclue C. C. 6. apunov function for Tuberculoi oel with fat an low progreion Math. ioc. ngg. 34 pp Mena-orca J. an Hethcote H. W. 99. Dnaic oel of infectiou ieae a regulator of population ize J. Math. iol. 3 pp Moghaa S. M. an lexaner M.. 4. xogenou reinfection an reurgence of Tuberculoi: theoretical fraewor J. iol. S. pp Murph. M. Singer. H. neron S. an Kirchner D.. Coparing epieic tuberculoi in eographicall itinct heterogeneou population Math. ioc. 8 pp Muhaabaa S. hunu C.P. Schwartz.J. Magobeze G. an Tchuenche J.M.. Socio-econoic tatu an HI/IDS naic: a oeling approach Worl J. Moel. Siul. 74 pp

20 M: Intern. J. ol. 7 Iue June 445 areh R. Pane S. an Mira. K. 8. nali of a accination oel for carrier epenent infectiou ieae with environental effect onlinear nali: Moelling an Control 33 pp areh R. Pane S. an Shula J.. 9. Moeling the cuulative effect of ecological factor in the habitat on the prea of tuberculoi Int. J. ioath. 3 pp areh R. an Tripathi. 5. Moeling an anali of HI-T coinfection in a variable ize population Math. Moel. nal. pp areh R. Tripathi. an Shara D. 9. Moelling an anali of the prea of IDS epieic with iigration of HI infective Math. Cop. Moel. 49 pp Pero S.. an Tchuenche J.M.. HI/IDS naic: ipact of econoic clae with traniion fro poor clinical etting J. Theor. iol. 67 pp Struchiner C. J. Halloran M.. an Spielan Moeling alaria accine I & II: ew ue for ol iea. Math. ioc. 94 pp White. an Coie C. 7. Heroin epieic treatent an OD oelling Math. ioc. 8 pp

21 446 Ra areh an Surabhi Pane PPDIX - I Proof of the Theore 4.. i The variational atrix M of oel -9 correponing to equilibriu W i given b 44 M where 44. The characteritic equation correponing to above atrix i given b h h where 44 / h / 44 h

22 M: Intern. J. ol. 7 Iue June 447 Uing Routh-Hurwitz criteria thi equilibriu i untable becaue in the above quaratic the coefficient of an the contant ter are not poitive iultaneoul. ii The variational atrix M of oel -9 correponing to equilibriu W i given b 44 M O M M M where M M an / M. The partition atrix will give rie to following characteritic equation

23 448 Ra areh an Surabhi Pane / 44 The characteritic equation correponing to M i given b 3 3 a a a h h where a a 3 a 44 / h / 44 h. Uing Routh-Hurwitz criteria thi equilibriu i untable becaue in the above quaratic the coefficient of an contant ter are not poitive iultaneoul although a a - a 3 >. To etablih the local tabilit of eneic equilibriu W we conier the following poitive efinite function

24 M: Intern. J. ol. 7 Iue June 449 U = e b n v n v where i i = - 7 are poitive contant to be choen appropriatel an v n v n b an e are all perturbation about W efine a follow = + = + v = + n = + = + v = + n = + b an = + e. Differentiating above equation with repect to t an uing the linearize te of oel equation -9 correponing to W we get be e n n v b v n n v v n e b n v n v t U ] [ ] [ ] [ ] [ ] [ uing = 3 = 5 an 6 the above equation reuce to the for

25 45 Ra areh an Surabhi Pane be e n n v b v n v v e b n v n v t U ] [ ] [ ow we chooe 4 7 uch that 9 4 in in. 3 3 n 7 3. Thu t U i negative efinite function uner the following conition 3

26 M: Intern. J. ol. 7 Iue June in 9 in howing that U i a apunov function with repect to W proving the theore. Proof of the Theore 4. PPDIX II Conier the following poitive efinite function correponing to the oel te -9 about W U ln 3 ln ln where the coefficient i i = - 7 are poitive contant to be choen appropriatel. Differentiating the above equation with repect to t an uing -9 we get

27 45 Ra areh an Surabhi Pane U t f 6[ f g ] 6 7h where f g an h are efine a follow f a g b h. c Then b coniering the auption of the theore an the ean value theore we have

28 M: Intern. J. ol. 7 Iue June 453 q f p g an h r. ow auing = 3 = the for 5 an 6 the above equation reuce to U t p q 7r. ow we chooe 4 7 uch that 3 in in. 5 3 p q 7 4 r 3

29 454 Ra areh an Surabhi Pane U Thu t conition i negative efinite function inie the region of attraction uner following r p q 3 / in / 9 6 in/5 /3 45 howing that U i a apunov function with repect to W proving the theore.