CONSTRUCTION OF STOCHASTIC MODEL FOR TIME TO DENGUE VIRUS TRANSMISSION WITH EXPONENTIAL DISTRIBUTION

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1 Internatonal Journal of Pure and Appled Mathematcal Scences. ISSN Volume, Number (7), pp. 3- Research Inda Publcatons ONSTRUTION OF STOHASTI MODEL FOR TIME TO DENGUE VIRUS TRANSMISSION WITH EXPONENTIAL DISTRIBUTION. Nanthaumar and.r.vjayashree Assocate Professor & Head, Department of Statstcs, Salem Sowdeswar ollege, Salem 636, TamlNadu, Inda. Emal: Assstant Professor & Head, Department of Statstcs, Vysya ollege, Salem 6363, TamlNadu, Inda Emal: ABSTRAT In ths paper deals wth the study of a stochastc model for predctng the tme to Dengue vrus transmsson. As the mmune capactes of an ndvdual dffer and also have ts personal resstance, the antgenc dversty threshold s dssmlar for dfferent person. We construct a stochastc model to study the damage process actng on the mmune system that s non-lnear. The mean tme to Dengue vrus transmsson and ts varance are derved wth numercal example. Keywords: Antgenc dversty threshold, Alpha-Posson process, Mttag-Laffler dstrbuton, Dengue, Tme to Dengue vrus transmsson.. Introducton Dengue Fever (DF) and Dengue Hemorrhagc Fever (DHF), collectvely nown as dengue," are mosquto-borne, potentally mortal, flu-le vral dseases that affect humans worldwde. Transmtted to humans by the bte of an nfected mosquto Aedes aegypt, dengue s caused by any one of four serotypes are termed as DENV-, DENV-, DENV-3, and DENV-4, or antgen-specfc vruses; dengue vrus s part of the Flavvrdae famly. Dengue s one of the most rapdly spreadng mosquto-borne

2 onstructon Of Stochastc Model For Tme To Dengue Vrus 4 vral dseases n the world and nflcts sgnfcant health, economc and socal burdens on populatons. The severe form of dengue s medated by an ncrease n capllary permeablty that can cause severe bleedng (e.g. Gastrontestnal bleedng) and plasma leaage resultng n asctes and pleural effusons. Dengue dsease s a mosquto-borne condton that has become a major publc health concern. Dengue severty can be classfed nto mld Dengue fever (DF) and severe Dengue or Dengue hemorrhagc fever (DHF). Mathematcal and statstcal models descrbng the transmsson of dengue vruses appeared n the study of observatons related to pathogeness of dengue hemorrhagc fever (Fscher, 97) and provdng a better understandng of the nature and dynamcs of the transmsson of dengue nfecton, as well as predct outbreas and smulate the mpact of control strateges n dsease transmsson (Rco-Hesse, ). The tme to Dengue vrus transmsson by btng mosquto alone s the only mode of Dengue transmsson. The btes of mosqutoes are assumed between a seropostve person who s labeled as ndex case and seropostve state taes place over an ncubaton perod due to the contracton of Dengue to the partner from the ndex case by the btes of mosqutoes. In ths research paper constructng a stochastc model to learn the damage process performng on the mmune system that s non-lnear and also mean and varance of tme to Dengue vrus transmsson are derved wth numercal example.. Objectves Because of hardy assocated wth btes of mosqutoes, ths wor concentrates prmarly on estmatng the dynamcs of the Dengue transmsson by the btes of mosqutoes, as more than 95% of Dengue transmsson accounts for btes of mosqutoes. The objectves are: To realze the dynamcs of Dengue n an nfected ndvdual so that control measures can be adopted n order to slow down the severty of the dsease To learn the transmsson dynamcs of Dengue nfecton n a susceptble person so that the effcent strateges for controllng the spread of the epdemc can be mplemented To categorze the statstcal pattern of recognton of the tme to Dengue vrus transmsson To derve the dstrbuton of the tme to Dengue vrus transmsson To grant the nature and extent of uncertanty n tme to Dengue vrus transmsson wth respect to the vral load, bte rate and antgenc dversty threshold

3 5. Nanthaumar and.r.vjayashree 3. Assumptons and Notatons Btes of mosquto are the only source of Dengue transmsson. Damages to ndvduals are caused by transmsson of Dengue at each bte and the nter arrval tme between the btes are ndependent, dentcally dstrbuted random varables. The damage process actng on the mmune system of an nfected ndvdual s non-lnear and cumulatve. The total damage caused exceeds a threshold level, whch tself s a random varable, the tme to Dengue vrus transmsson occurs and the person s recognzed as nfected. The process that generates the btes, the sequence of damages and threshold are mutually ndependent. and the model parameters are: Btes rate of the nfected person (a) Intensty of the Dengue of the nfected person (β) Antgenc dversty threshold () 4. Stochastc model for tme to Dengue vrus transmsson wth one parameter Exponental dstrbuton 4. Dstrbuton of tme to Dengue vrus transmsson Let us consder a susceptble populaton whose major mode of transmsson s through btes of mosqutoes. Assume that at tme t=, a new member tested Dengue negatve enters the populaton and maes btes of mosquto wth members of the susceptble populaton. Let the btes of mosquto occur at random tme ponts whch s assumed to follow the Alpha-Posson dstrbuton () wth parameters a and β whch s gven as ( n) n ( at) pa, ( n, t) ( ), a,, n,,... ( ( n) ( at) e ( at), a, t,, ( )

4 onstructon Of Stochastc Model For Tme To Dengue Vrus 6 Where a = No. of btes, β = ntensty of Dengue nfected person Let G(t) be the dstrbuton functon of the nter bte between the btes whch follows Mttag-Leffler dstrbuton. The dstrbuton functon of Mttag-Leffler dstrbuton () s gven by ( ) ( ) Ga, ( t) ( at), t, a and ( ) Let the tme to Dengue vrus transmsson of the ndvdual be represented by the random varable. We obtan the dstrbuton of tme to Dengue vrus transmsson by a stochastc model based on the assumptons wth lnear damage process actng on the mmune system, we have the followng theorem. If the number of btes s an Alpha-Posson process wth parameters a and β and nter bte tme s a Mttag-Leffler dstrbuton whle the threshold level s an exponental dstrbuton wth parameter, then the probablty densty functon of tme to Dengue vrus transmsson s a three parameter Webull dstrbuton. The densty functon of the dstrbuton of tme to Dengue vrus transmsson: onsder S(t) = P{no nfecton n (, t)} P T t P{ Number of Seroconverson before t gven exactly bte n, t wth ntensty } P exactly btes n, t wth ntensty V() t X Y where V ( t) Probablty of btes n, t wth ntensty p( x) e ( at). e., the Alpha Posson dstrbuton wth parameter a and ( at) e, a,,,,3... ( ) x x g ( ) x P( X Y) G( x) e dx G ( ) g ( ), whereg ( ) G( x) e dx P( X Y) g ( ) x P X X... X Y g( x) e dx g ( ) g ( ) g ( )... g ( ) g ( )

5 7. Nanthaumar and.r.vjayashree where g ( ) sthe LaplaceTransformatonof g( x) and g ( x) p. d. f. of X S( t) V ( t) g ( ) ( at) g ( ) ( at) g ( ) ( at) g ( ) ( at) g ( ) ( at) e... sn ce, hence St ( ) e ( at) g( ) 3 ( ) ( at) g ( ) ( at) g ( ) ( at) g ( ) ( at) g ( ) ( at) e... 3 ( ) L t S t s called Prevalence functon e x ( at) g( ) The probablty densty functon of tme to Dengue vrus transmssont s d f () t Lt dt d f() t e a t g ( ) e dt ( at) g ( ) ( at) g ( ) The probablty densty functon of tme to Dengue vrus transmsson t s ( at) g ( ) a t g ( ) e, t,, a, f() t, Otherwse Usng Mttag Leffler dstrbuton a a g ( ) g ( ) g ( ) a a a The above functon s n the form of three parameter Webull dstrbuton. Furthermore, we notced that f the number of btes s a Alpha-Posson process wth parameters a and the nter bte tme s a Mttag-Leffler dstrbuton whle the threshold level s also an exponental dstrbuton wth parameter (), then the probablty densty functon of tme to Dengue vrus transmsson s a three parameter Webull dstrbuton.

6 onstructon Of Stochastc Model For Tme To Dengue Vrus 8 4. Probablty of tme to Dengue vrus transmsson The probablty of tme to Dengue vrus transmsson s calculated for varous ntervals by defnng p t t t t f ( t) dt for,,3... a t a t a a t a t a where f ( t) e, t,, a and a p e dt a a a t t a t a t e dt for,,3... a t a t dz Let z then dz dt and t dt a a a a where then t dt, lmt a z t t t a dz t t t z z e p e dz for,,3... t t t The probablty of tme to Dengue vrus transmsson s calculated for varous ntervals s gven as a a t t t t a a p e e e e where t and t + has one wdth of class nterval wdth

7 9. Nanthaumar and.r.vjayashree 4.3 Performance of measures of tme to Dengue vrus transmsson The expected tme to Dengue vrus transmsson and ts varance are obtaned below: Mean E( T ) t f ( t) dt ' a t a t ' a t a ' a a a a t e dt t e dt a t a y Let y where y t t a a dy dy t dy y t dt dt t t dt t ' ( ) ' ( ) ( ) E T t f t dt ' ' a ( ) at at a t a a a a a t e dt t t e dt at a a t e dt t dt, lmt t y t y a t a y y Let y where y t t t a a ( ) dy dy t dy y dy t y t dt dt t t dt t dt, lmt t t y ' y e ( ) dy ( ) ' e y dy ( ) ( )

8 onstructon Of Stochastc Model For Tme To Dengue Vrus Varance of tme to Dengue vrus transmsson s V ( T ) E( T ) E( T) VT ( ) ( ) ( ) Partcular case f = then ' ET ( ) ( ) VT ( ) ( ) a a a VT ( ) a 5. oncluson In the study of Dengue epdemc the seroconverson tme of Dengue transmsson s an nevtable component. Snce the bte rate s non observable n most cases one mosquto would expect that the spread of Dengue would have an mpact on the human lfe. In ths research artcle we constructed a stochastc model for tme to Dengue transmsson and t s noted that mean tme to Dengue transmsson to s less than the varance of tme to Dengue transmsson. REFERENES [] Anl, V.,, A generalzed Posson dstrbuton and ts applcaton, J. Kerala Stat. Asso., Volume, pp.-. [] Fscher, D.B., and Halstead, S.B., 97, Observatons related to pathogeness of dengue hemorrhagc fever, V. Examnaton of age specfc sequental nfecton rates usng a mathematcal model, J. of Boogy and Medcne, Volume 4, pp [3] Nanthaumar and Vjayashree R (7). onstructon of Stochastc model for tme to Dengue vrus transmsson wth three parameters exponental

9 . Nanthaumar and.r.vjayashree dstrbuton, Bulletn of Mathematcs and Statstcs Research, Vol.5, Issue, pp [4] Plla, R.N., 99, On Mttag-Lffler functon and related dstrbutons, Annals of the Insttute of Statstcal Mathematcs, Volume, pp [5] Rco-Hesse, R.,, Dengue vrus vrulence and transmsson determnants, urr. Topcs n Mcrobology and Immunology, Volume 338, pp [6] Santhaumaran, A., Nanthaumar,., Venaatesan, P., and Kannan, R., 3, A Stochastc model for seroconverson tmes of HIV transmsson, Bo- Scence research Bulletn, Volume 9, pp [7] Sathyamoorthy, R. and Kannan, R.,, A Stochastc model for tme to seroconverson of HIV transmsson, J. of the Kerala Stat. Asso., Volume, pp.3-8. [8] Venatesan, P. and Nanthaumar,., 4, Some contrbutons to the modelng of seroconverson tme of human mmunodefcency vrus, Ph.D Thess, Unversty of Madras, henna. Webste:

10 onstructon Of Stochastc Model For Tme To Dengue Vrus