A study of Dengue Disease Model with Vaccination Strategy

Transcription

1 A su of Dengue Disease Model wih accinaion Sraegy Pradeep Porwal,.. Badshah. School of Sudies in Mahemaics, ikram Universiy, Ujjain (M.P.), India Absrac In his paper, we proposed and analyzed he effecs of vaccinaion sraegy on he ransmission of he Dengue diseases. We propose SIR model wih logisic recruimen rae, and analyzed he Sea sae and sabiliy of he equilibrium poins. If R hen he non- infeced sea sae P will be sable. Also if R hen he endemic equilibrium P is sable. umerical simulaions show ha he effec of newborn vaccinaion is significanly less effecive han vaccinaing suscepible populaion. Also he effec of vaccinaion is o replace muliple oubreaks wih a single oubreak. Keywords Dengue disease model, accinaion sraegy, Sabiliy analysis, umerical analysis. Mahemaics Subjec Classificaion - 93D2, 92D3, 65L7.. Inroducion Dengue is he mos imporan human viral disease ransmied by arhropod vecors. Annually here are an esimaed 5 million cases of dengue fever (DF), and 25 o 5 cases of dengue haemorrhagic fever (DF) in he world. Dengue infecion is classified ino hree caegories: Dengue fever (DF), Dengue hemorrhagic fever (DF) and Dengue shock syndrome (DSS). DF, DF and DSS are caused by he four dengue viruses DE, 2, 3, and 4. Infecion in humans wih one seroype provides life-long immuniy o ha virus bu no o he ohers. Dengue viruses are mainained in an urban ransmission cycle in ropical and subropical areas by he mosquio Aedes aegypi, a species closely associaed wih human habiaion. In some regions oher Aedes species, such as Ae albopicus and Ae polynesiensis are also involved. There are some epidemiological and demographical facors ha conribue o he ransmission of he disease. From a pracical poin of view, many counries organized vaccinaion programs o conrol he spread of he disease. The effecs of vaccinaion on he ransmission of infecious diseases are sudied by some researchers [2, 3, 4. These researchers sudied he direc ransmied disease. In our second model we sudied he effec of vaccinaion on an indirec ransmied disease. Recenly many researchers [4, 5, 6, 7 sudied he vaccinaion sraegy for all ype of dengue viruses bu i is no perfec. The mahemaical models for Dengue Fever found in lieraure [2, 7, 8 are based on comparmenal namics. We have also used he comparmenal namics. 2. Formulaion of he model In his paper we consider he effecs of vaccinaion sraegy on he ransmission of he Dengue diseases. The model assumes ha he hos populaion grows logisically, and has a consan disease deah rae. We also assume ha he vecor populaion has consan size wih birh and deah rae equal o. The hos p o p u l a i o n is subdivided ino h e suscepible recovered R classes. The ecor populaion, due o a sho r life period, is subdivided ino he suscepible The ransmission model for he dengue disease is as follows: ds I S S K 2 S, infecive I, and S, infecive I. di I S ( )I (2.) 7

2 dr I R For ecor populaion: ds di I S S I S I (2.2) The iniial condiions S I R and S I d I (2.3) K I is convenien o reformulae he model (6.2.4) in erms of populaion proporion fracions of he suscepible, invecives and removals, respecively. ence he sysem can be wrien as d z K S, I y and I z z My (2.4), which are he dz ( z) y z where M We consider wo ypes of vaccinaion, one ha is being adminisered o a porion of new born hos and anoher one is being adminisered o a porion of suscepible hos. The main quesion here is wheher i is enough o vaccine only new born hos in order o conrol he spread of he disease or i is necessary o vaccinae he larger suscepible hos [4. Le a porion,, of newborn hos be vaccinaed. Assume ha h e vaccine is no perfec and assume ha he effeciveness of he vaccine is s, and hen ( ) s newborns remain suscepible, and s K K being removed o R The corresponding namic equaion for is given by direcly 7

3 d ( ) s z (2.5) K and he oher wo equaions in (2.4) remain he same. On he oher hand, le a porion, vaccinaed. Then he namical equaion of and y is as follows: d z( s) K z( s) My (2.6) These wo cases ( 2.5) a n d (2.6) are wrien on one sysem as follows: d ( s) ( s) z K z( s) My (2.7) of suscepible hos be dz ( z) y z In succession by puing or, hese wo equaions (2.6) and (2.7) can be generaed. Rescale by he sysem of equaions (2.7) simplified o d b( r) z b z y (2.8) dz ( z) y z where ( s),, 3. Sea sae and Sabiliy Analysis M,, r s, b K Equilibrium poins are obained by seing ime derivaives of, y and z equal o zero hen he sysem(2.8) has wo possible equilibria, i.e., he non-endemic equilibrium P r and he endemic equilibrium P (, y, z ) 2, where ( ),, 72

4 y z ( s) M b ( r) K b [ ( s) ( s)( s) M b[ ( r) K ( b) M [ ( s) ( s)( s) M b[ ( r) K [ b ( r) ( s) ( s) M K I is clear ha he non-endemic equilibrium poin P ( r),, will be sable if R. A he endemic equilibrium poin P y z he variaion mari becomes (,, ) 2 z b Z2 z ( ) z y is characerisics equaion 3 2 A B C where b ( r)( b) [ b ( r) A [ b ( r) ( b) b( r)( b) b ( r) b[ ( r) B [ b ( r) ( b) C b[ ( r) I is clear ha all he coefficien of he characerisics polynomial are posiive. Direc calculaion shows ha A >, C > and AB > C by Rouh-urwiz crieria. ence he endemic equilibrium will be sable if R where 4. Resuls and Discussions. The sysem of equaion (2.8) was solved numerically using mahemaical sofware. The values of he parameers are aken from [. Afer simulaion, we observed ha he namic of infeced human and infeced vecor are no affeced by newborn vaccinaion. In Figure, (a) and (b) show he proporion of he infecive human populaion wihou vaccinaion. We see ha firs oubreak occurs a approimaely =6. Second oubreak begin from approimaely =24. If he newborn vaccinaion is applied, Figure (c) shows ha sill one oubreak occurs followed by eponenial decay. owever, if he suscepible vaccinaion is applied, Figure (d) shows ha here is almos no oubreak and he numbers of subsequen cases eponenially decay R ( )( ) s s K M

5 Infeced uman Infecive uman Infeced uman Infeced uman Fig. 2 presens he namics of S, I and I in days afer one infeced enered he populaion. We found ha S drops significanly in a relaively small period of ime. Infeced human I and infeced vecor I increases significanly during he period of 2 days and hen oscillae around he endemic equilibrium sae (a) (b) (c) (d) Fig. : Dynamics of I populaion wihou vaccinaion and he I populaion wih newborn vaccinaion and suscepible vaccinaion. 74

6 Infeced uman Populaion S I I Fig. 2: Dynamics of S, I and I in days wih he iniial condiion (.73,.35,.25) for =.25, =/(6365), =.428, =.75, =,, 5 K 4, =, 64. Wih hese parameers, R =.5>. Fig. 3 shows simulaions wih differen proporions of he suscepible vaccinaion in endemic equilibrium sae for =,.25,.5,.75,. We observed ha vaccinaion decreases he number of infecives. Tha is, he dose of suscepible vaccinaion is increases hen he number of infecives wih respec o ime is eponenial decreases σ= σ=.25. σ= σ=.75 σ= Fig. 3: Dynamics of infeced uman wih differen proporions of he suscepible accinaion in endemic equilibrium sae for =,.25,.5,.75,. 5. Conclusion. In his paper we consider wo ypes of vaccinaion, one ha is being adminisered o a porion of new born hos and anoher one is being adminisered o a porion of suscepible hos. If R hen he non- infeced sea sae P will be sable and if R hen he endemic equilibrium 2 P is sable. umerical simulaions show ha he effec of newborn vaccinaion is significanly less effecive han vaccinaing suscepible populaion. Also he effec of vaccinaion is o replace muliple oubreaks wih a single 75

7 oubreak. owever, if we apply he suscepible vaccinaion here is almos no oubreak occurs and he cases eponenially decay approaching he disease-free equilibrium (Figure (d)). REFERECES: [ enchal, E. A. and Punak, J.R., The dengue viruses, Clin Microbiol Rev., vol. 3, pp , 99. [2 Abual-Rub, M.S.: accinaion in a Model of an Epidemic, Inerne, J. Mah. And Mah. Sci., 23, (928), [3 Soewono, E and Supriana, Asep K.: A Two Dimensional Model for Transmission of Dengue Fever Disease, Bull. Malay Mah, Sci. Soc. 24, (2), [4 Soewono, E and Supriana, Asep K.: Parado of accinaion Prediced by a simple Dengue Disease model, Indusrial Mahemaics, arosa Pub. ouse, ew Delhi, (26), [5 Scherer, A., McLean, A., Mahemaical models of vaccinaion, Briish Medical Bullein 62 () (22) 87{99. [6 Murrell, S., Wu, S. C., Buler, M., Review of dengue virus and he developmen of a vaccine, Bioechnology Advances 29 (2) (2) [7 Rodrigues,. S., Moneiro, M. T. T., Torres, D. F. M., Zinober, A., Dengue disease, basic reproducion number and conrol, In. J. Compu. Mah. 89 (3) (22) [8 Eseva, L., and argas, C., A model for dengue disease wih variable human populaion, J. Mah. Biol., 38(3), (999), [9 Anderson, R.M. and R.M. May., 992. Infecious Diseases of umans: Dynamics and Conrol. s Edn., Oford Universiy Press, Oford, ISB: 98544X, pp: 768. [ Eseva, L., argas, C., Analysis of a dengue disease ransmission model, Mah. Biosciences 5 (998),