Analysis of a dengue disease transmission model with vaccination

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1 Aailabl onlin at wwwplagiarsarchlibrarycom Adancs in Applid Scinc sarch 4 5(3):37-4 ISSN: CODEN (USA): AASFC Analysis of a dngu disas transmission modl with accination B Singh¹ S Jain² Khandlwal¹ Snha Porwal¹ * and G Ujjainkar 3 ¹School of Studis in Mathmatics Vikram Unirsity Ujjain (MP) India ²Got Collg Kalapipal Distt Shajapur (MP) India 3 Got PG Collg Dhar (MP) India ABSTACT Sowono and Supriatna [9] studid a simpl SI dngu disas transmission modl with accination In th prsnt papr w ha modifid th modl with assumption that a random fraction of th rcord host population can loss th immunity and bcoms suscptibl again Th dynamics of th disas is studid by a compartmntal modl inoling ordinary diffrntial quations for th human and th mosquito populations stricting th dynamics for th constant host and ctor populations th modl is rducd to a thr-dimnsional planar quation Two stats of quilibrium ar studid on disas-fr and othr ndmic Th basic rproduction numbr is obtaind In this modl th disas-fr quilibrium stat is stabl if and if > th stabl ndmic quilibrium appars Numrical simulation and graphical prsntation ar also proidd to justify th stability Kywords: Epidmiology Dngu disas Vctor-host modl Stability production numbr Vaccination AMS Classification: 9D3 INTODUCTION Dngu fr is high on th list of mosquito-born disass that may worsn with global warming It is a globally rmrging iral disas transmittd to humans by th bit of an infctd Ads Agypti mosquito and xists in two forms: th Dngu Fr (DF) and th Dngu amorrhagic Fr (DF) Th symptoms of th disas includ high fr rash and sr hadach with aching bons joints and muscls Dngu and its dadly complications dngu hmorrhagic fr and dngu shock syndrom ha incrasd or th past sral dcads Global warming could substantially incras th numbr of popl at risk of dngu pidmics as warmr tmpraturs and changing rainfall conditions xpand both th ara suitabl for th mosquito ctors and th lngth of th dngu transmission sason in tmprat aras Dngu fr is causd by a mmbr of th sam family of iruss that caus yllow fr Wst Nil and Japans ncphalitis It is possibl to bcom infctd by dngu multipl tims bcaus th irus has four diffrnt srotyps known as DEN DEN DEN3 and DEN4 A prson infctd by on of th four srotyps will nr b infctd again by th sam srotyp but h loss immunity to th othr thr srotyps in about wks and thn bcoms mor suscptibl to dloping dngu hamorrhagic fr Th stratgis of mosquito control by inscticids or similar tchniqus prod to b infficint A standard program usd in many countris to control th sprad of th disas is th control of th main disas ctor by fuming or fogging Many studis show that this program was not fully ffcti A simpl SI modl for dngu disas transmission has bn studid by many rsarchrs [ ] Now a day s rsarchrs ar going on towards th inntion of accin for dngu disas Th ffcts of accination on th transmission of infctious disas ar studid by som of th rsarchrs [ 9 ] 37

2 Snha Porwal t al Ad Appl Sci s 4 5(3):37-4 In th rcnt communication Sowono and Supriatna [9] considrd two typs of accination in a host transmission modl for dngu fr In th modl considrd by thm it has bn assumd that th accin prnts accinatd popl by all typs of dngu iruss but it is not prfct En aftr accination th host may suffr from th disas with crtain probability In th prsnt papr w ha modifid th modl of Sowono and Supriatna [9] with assumption that a random fraction of th rcord host population can loss th immunity and bcoms suscptibl again Formulation of th Modl for Dngu Disas Transmission Lt and V b th host and ctor population sizs rspctily It is assumd that th host and ctor population has constant siz with birth and dath rat qual to µ and µ V Th host population is subdiidd into th suscptibl S th infcti I and th rcord (immun) Th ctor population du to a short lif priod is subdiidd into th suscptibl S and th infcti I W considr hr two typs of accination in a hostctor modl for th dngu disas transmission On is bing administrd to a portion of nw born host and anothr on is bing administrd to a portion of suscptibl host Lt a portion ρ ρ of nwborn host b accinatd Assum that th accin is not prfct and lt th ffctinss of th accin is s thn ( ρs ) µ nwborns rmain suscptibl and ρ sµ dirctly bing rmod to On th othr hand whn a portion σ σ of suscptibl dynamics of both S and I S ar accinatd thn th ar affctd Anothr assumption for this modl is that a random fraction of th rcord host population can loss th immunity and bcoms suscptibl again Th intraction modl is gornd by th following mathmatical quations For human population th quations ar ds bpis ( σs) = µ ( ρs) µ S di bpis ( σs) = µ I γi d = µ ρ s + γi µ φ () and for ctor population ds bpisv = µ V µ S di bpisv = µ I () whr b is th biting rat of th ctor p is th transmission rat from infctd ctor to suscptibl host p is th transmission rat from infctd host to suscptibl ctor γ is th rcory rat of th host population Φ is th loss of immunity rat in th host population Using S + I + = and S + I = V th systms () and () bcom ds bpis ( σs) = µ ( ρs) µ S ( S I ) di bpis ( σs) = ( µ + γ)i di bpi (V I ) = µ I (3) 38

3 Snha Porwal t al Ad Appl Sci s 4 5(3):37-4 Writing th dynamics (3) in population proportion S I I S h = Ih = and I = w ha V dsh = µ ( ρs) bpish ( σs) µ S h φ (Sh + I h ) dih = bpis h ( σs) ( µ + γ)ih di = bpi h ( I ) µ I V = is th ratio of host and ctor population S = x I = y I = z and rscaling t by ; ths quations can b simplifid to whr Stting h h dx = µ ( r) αx η xz φy dih = ηxz βy di = y( z) δz µ φ µ = r = ρs µ φ = p( σs ) µ α = + γ µ η = β = and δ = p bp whr (4) (5) 3 Stability Analysis of th Equilibrium Points Equilibrium points of systm (5) ar obtaind by stting tim driatis of x y z to zro Th systm of µ ( r) quations (3) possss two quilibrium points; on is disas-fr quilibrium E = α and othr is ndmic quilibrium E = (x y z ) whr ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) β µ r + β ( µ + γ ) bp { µ ( ρ s) } + ( µ + γ ) µ x = = η β + αβ bp [( σ s + γ ) + ( µ + γ)] η µ r αβδ b pp ( σ s) µ ( ρ s) ( µ + γ ) µ y = = η β + αβ bp [( σ s + γ ) + ( µ + γ)] { } η µ r αβδ b pp ( σ s) µ ( ρ s) ( µ + γ ) µ z = = η µ r + β ( σ s) bp µ ( ρ s) + ( µ + γ ) µ b pp ( σ s) µ ( ρ s) = ( µ + γ ) µ This proids a rproduction numbr E is stabl whn ( ) Th ndmic quilibrium η µ r αβδ > b pp ( σ s) µ ( ρ s) i = > ( µ + γ ) µ Now w shall discuss th local stability of th quilibrium points Th ariation matrix of th systm (5) is gin by 39

4 Snha Porwal t al Ad Appl Sci s 4 5(3):37-4 α ηz φ ηx J = ηz β ηx z y δ µ ( r) For th disas-fr quilibrium point E = th ariation matrix will b α ( r) η µ α φ α η µ ( r) J(E ) = β α δ Its charactristic quation will b { ( ) } η µ r ( λ + α) λ + ( β + δ) λ + βδ α By looking at ign alus on can asily sn that disas-fr quilibrium E is locally stabl if { ( r) } η µ βδ > α i < Now w can turn to an ndmic quilibrium and study about its stability For th ndmic quilibrium point E = (x y z ) th ariation matrix will b α ηz φ ηx J(E ) = ηz β ηx z y δ ( η + α ) µ ( r) αδφ ηβ µ ( r) + ( β ) φ µ ( r) + ( β ) η( β ) + αβ η µ ( r) αβδ ηβ µ ( r) + ( β ) J(E ) = β µ ( r) + ( β ) η( β ) + αβ η( β ) δ + αβδ η µ ( r) + ( β ) η µ ( r) + ( β ) η( β ) + αβ Its charactristic quation will b a λ + a λ + a λ + a = 3 3 whr a3 = β( η + α ) + ηφ µ ( r) ( + δ ) + βδ > a = β{ β( η + α ) + ηφ } + η { µ ( r) ( + δ ) + βδ } µ ( r) ( + δ ) + β ( ) ( ){ ( r) } + β η + α + ηφ η + α µ + αδφ > 4

5 Snha Porwal t al Ad Appl Sci s 4 5(3):37-4 { ( ) } ( ) ( ) { } { ( ) } { r } { ( ) } { ( r) ( ) } { ( ) } ( ){ ( ) } { ( ) ( ) } ( ) > if η { µ ( r) } αβδ > i > a = η µ r αβ β µ r + δ +βδ β η+ α + ηφ + ( ) ( ) + η+ α µ + αδφ β β η+ α + ηφ + η µ + δ + βδ a = η µ r αβδ β η + α µ r + βαδφ + ηφ µ r + δ + βδ + β δ η + α W s that a a nc th ndmic quilibrium point E is locally stabl if > W conclud this in th following thorm: Thorm 3: If < thn th disas-fr quilibrium E is locally stabl; if = E is stabl and if > th stabl ndmic quilibrium E will appars 4 Numrical simulation In this sction w gi an xampl to illustrat th main thortical rsults prsntd abo In systm (5) lt µ = 6 r = 8 α = 6 η = 76 φ = 8 β = 36 and δ = Computation gis th following alu for th basic rproduction numbr = 6398 > and systm (5) has a uniqu ndmic quilibrium point E = (x = 9 y = 8756 z = ) By thorm 3 w s that ndmic quilibrium E of systm (5) is globally asymptotically stabl if > Numrical simulation illustrats th abo rsult (s figur ) Stability graph for th point(x=9 y=8756 z=465687) Population Tim Figur CONCLUSION X Y Z In this papr w ha discussd th ffcts of accination stratgis on th dynamic of th dngu disas transmission modl with assumption that a random fraction of th rcord host population can loss th immunity and bcoms suscptibl again Dynamic of th modl is compltly dtrmins by th basic rproduction numbr W ha prod that th modl has a disas-fr quilibrium E if th rproduction numbr is lss than or qual on and has a uniqu positi ndmic quilibrium E if th rproduction numbr is gratr than on Stability conditions ar gin which would b a usful tool for th disas control stratgis For φ = th modl coincids with that of Sowono and Supriatna [9] EFEENCES [] Abual-ub M S Intrnat J Math & Math Sci

6 Snha Porwal t al Ad Appl Sci s 4 5(3):37-4 [] Drouich M Boutayb A Twizll E Biomdical Cntral Ltd 3 [3] Esta L Vargas C Mathmatical Bioscincs [4] Esta L Vargas C J Math Biol [5] Esta L Vargas C Mathmatical Bioscincs [6] Mhta Singh B Tridi N Khandlwal Adancs in Applid scinc sarch 3(4) 978 [7] Singh B Jain S Khandlwal Porwal S and Ujjainkar G Adancs in Applid scinc sarch 4 5() 8-6 [8] Sowono E Supriatna A K Bull Malay Math Sci Soc [9] Sowono E Supriatna A K Narosa Pub ous Nw Dlhi [] Tridi N Singh B Bulltin of Pur and Appl Math 7 () -7 [] Ujjainkar G Gupta V K Singh B Khandlwal Tridi N Adancs in Applid scinc sarch 3(5) 3 [] Yaacob Y MATEMATIKA 7 3()