Dynamical Transmission Model of Chikungunya in Thailand

Transcription

1 World Academy of cience ngineering and Tecnology nternational Journal of Matematical and Computational ciences Vol: o:8 00 Dynamical Transmission Model of Cikungunya in Tailand P Pongsumpun nternational cience ndex Matematical and Computational ciences Vol: o:8 00 wasetorg/publication/5 Abstract One of te important tropical diseases is Cikunkunya Tis disease is transmitted between te uman by te insect-borne irus of te genus Alpairus t occurs in Africa Asia and te ndian subcontinent n Tailand te incidences due to tis disease are increasing eery year n tis study te transmission of tis disease is studied troug dynamical model analysis Keywords Cikunkunya dynamical model ndemic region out-hurwitz criteria M ODUCTO ATHMATC ae become a major tool for studying te eolution of ector-borne diseases [-] For instance seeral temporal deterministic models ae been proposed for dengue [-6] but to our knowledge only ery few for Cikungunya [7] Cikungunya is caused by te cikungunya irus wic is classified in te family Togairidae genus Alpairus t is spread by te biting of te infected Aedes mosquito Most commonly te mosquitoes inoled are Aedes aegypti and Aedes albopictus Humans are tougt to be te major source or reseroir of cikungunya irus for mosquitoes Terefore te mosquito usually transmits te disease by biting an infected person and ten biting someone else An infected person cannot spread te infection directly to oter persons Cikungunya is caracterized by an abrupt onset of feer frequently accompanied by joint pain Oter common signs and symptoms include muscle pain eadace nausea fatigue and ras Te joint pain is often ery debilitating but usually ends witin a few days or weeks Most patients recoer fully but in some cases joint pain may persist for seeral monts or een years Tere is no accine or specific antiiral treatment currently aailable for cikungunya feer Treatment is applied to te symptomatic patients and can include rest fluids and medicines to reliee symptoms of feer and acing suc as ibuprofen naproxen acetaminopen or paracetamol Cikungunya irus was first isolated in Tanzania during a 95 to 95 epidemic [8] Te first appearance of te irus in outeast Asia by isolation during an intense epidemic of dengue feer and dengue emorragic feer in Bangkok Tailand in 958 [9] Clinically te disease can resemble P Pongsumpun is wit te Department of Matematics Faculty of cience King Mongkut s nstitute of Tecnology Ladkrabang Calongkrung road Ladkrabang Bangkok 050 Tailand (corresponding autor to proide pone: ; fax: ; kppuntan@ kmitlact) classical dengue feer and in dengue endemic countries tis can gie rise to confusion and misdiagnosis Te irus continued to be actie in Tailand until te 970s after wic it almost disappeared n 988 eidence of cikungunya transmission in Tailand re-emerged but te pattern was one of occasional outbreaks rater tan seere epidemic disease n 008 a total of cases of cikungunya feer were reported from proinces of soutern Tailand aratiwat Pattani Yala and ongkla Te reported cases of cikungunya feer increased gradually week by week until te end of 008 wit te igest incidence of about 00 cases per week [0] tudying te matematical model for Cikunkunya began in 970 [] After tat tere is no study about te Cikunkunya model Because tere is te re-emergence of tis disease in 007 te study of te model for tis disease is started again Bacaer [] studied te basic reproductie number of tis disease by considering te mosquito population n 008 Y Dumont and F Ciroleu [] formulated te matematical model for uman and mosquito population and find te stability condition for te disease free and endemic state Te abundance of te mosquitoes in Tailand depends on te season and te temperature of te enironment n tis paper we formulate and analyze te Cikunkunya model by incorporating te effect of te season wic effect to te number of te mosquitoes in Tailand MATHMATCAL MODL Our model classifies te uman into tree epidemiological states: is te number of susceptible uman is te number of infectious uman is te number of recoered uman We assume tat te total uman population ( T ) is constant and = Te mosquitoes are classified in tree classes: i is te number of susceptible ector in i season i is te number of exposed ector in i season i is te number of infectious ector in i season wen i= if i= means winter season nternational colarly and cientific esearc & nnoation (8) 00 9 scolarwasetorg/07-689/5

2 World Academy of cience ngineering and Tecnology nternational Journal of Matematical and Computational ciences Vol: o:8 00 nternational cience ndex Matematical and Computational ciences Vol: o:8 00 wasetorg/publication/5 i= means rainy season i= means summer season We obtain te following system equations d = T ( ) d = ( r) d = r () di = A i ( )i di =i ( q)i di = qi i = were wit is te birt rate of te uman population is te deat rate of te uman population is te transmission rate from ector to uman population r is te recoerey rate of te uman population A is te constant recruitment rate of te ector population i at te i t season is te deat rate of te ector population is te transmission rate from uman to ector population q is te rate at wic te exposed ector cange to be te infectious ector Using te fact = = = and = are constants We let = = = = = = = = = = = and = () Te total uman population and te numbers of ector in i t season are assumed to be constant ten te rates of cange for all population equal to zero From setting te rate of cange for eac population equals to zero we obtained = = A / = A / and = A / Ten te normalized equations become: d = ( ( )) d = ( ) ( r) d =( ) T ( q) d = q () d =( ) T ( q) d = q d =( ) T ( q) d = q were = = = and = () AALY OF TH MODL A Analytical olutions Te equilibrium states ( ) are found by setting te rigt and side of () to zero ten we obtain: = ( ) as te disease free equilibrium state and = ( ) as te endemic equilibrium state: were = ( ) = σ ( ) ( )( ) = ( )( σ ) σ q( )( ) ( )( ) = q( )( ) = (( )( σ ) σ ) ( )( σ ) σ = (( )( σ ) σ ) nternational colarly and cientific esearc & nnoation (8) 00 9 scolarwasetorg/07-689/5

3 World Academy of cience ngineering and Tecnology nternational Journal of Matematical and Computational ciences Vol: o:8 00 nternational cience ndex Matematical and Computational ciences Vol: o:8 00 wasetorg/publication/5 ( )( ) = σ σ ( )( ) q( )( ) = (( )( σ ) σ ) were = r and σ = = = = q Te caracteristic equation of () for te disease free equilibrium state is ( )( ) ( T ) ( a a a 0) = 0 were a =σ ( T) a = (( T) σ T ) a0 = T( q( ) σ ) Te eigenalues are = = = = 5 = Te remaining tree eigenalues can be found from equation a a a0 = 0 From te out-hurwitz stability condition te eigenalues 6 7 and 8 are negatie wen ) a > 0 ) a0 > 0 and ) aa > a0 Te first condition a =σ ( T) is always positie Te second condition a0 = T( q( ) σ ) is positie wen σ > q( ) or q( ) 0 = < ( r)( q) Te last condition aa a0= q( T ) ( T) ( σ )( σ T ) is always positie From our ealuations we found tat all eigenalues ae negatie real parts for 0 < By te local stability teorem [5] te disease free state is local asymptotically stable for 0 < Te caracteristic equation of () for te endemic disease state is ( c c ) ( b b b b ) = 0 were c = ( T) c = ( (q ) ) 0 T T b = ( T) T ( σ ) b = T( (q ) ) ( ) ( ) σ ( T ( )) b = (( T) σ T ) T( q( ) σ ( σ ( ) T( ( σ )( ) ( (q q q σ σ σ (q (q σ ) ) (q (q σ ) ) (q σ ) ) q( ) b = T( q( ) σ ) T(q qσ σ σ qσ σ q q ( ) ( σ ( (q σ (q ) ) q( ( ) )) ( σ ( (q σ (q ) ) q( ( ) ))) Te eigenalues are gien by = = ( T T ( ) ( q ) = = ( T T ( ) ( q ) ince > ( ) ( q ) are always negatie o Te remaining eigenalues are calculated from b b b b = 0 From te out-hurwitz criteria te eigenalues and 8 are negatie wen ) b > 0 ) b > 0 ) b 0 and ) bbb > b bb From te calculations we found tat te aboe conditions satisfied wen 0 > o te endemic disease state is local asymptotically stability for 0 > B umerical olutions n tis section we use te numerical metods to simulate te results of system () Te comparison for te different tresold conditions ( 0 ) of te endemic equilibrium state is presented Te parameters in tis simulation are: = /(65 70) corresponds to te life expectancy of 70 years for uman q = / correspond to te days wic te exposed ector cange to be te infectious ector nternational colarly and cientific esearc & nnoation (8) 00 9 scolarwasetorg/07-689/5

4 World Academy of cience ngineering and Tecnology nternational Journal of Matematical and Computational ciences Vol: o:8 00 r = /0correspond to te recoery of 0 days = /7 correspond to te life expectancy of 7 days for ector [] nternational cience ndex Matematical and Computational ciences Vol: o:8 00 wasetorg/publication/5 a) b) Fig umerical solutions of te system () demonstrate te time series respectiely for = 5000 = 5000 = 0000 = /(65 70) per day = /7 per day r = /0 per day q = / per day = = a) 0 = Te fractions of populations spiral to te endemic state ( ) b) 0 = 057 Te fractions of populations spiral to te endemic state ( ) V DCUO AD COCLUO Te transmission of Cikunkunya in uman and mosquitoes is studied troug te metod of dynamical modeling out- Hurwitz stability condition is used for determining te beaior of te population Te tresold condition ( 0 ) is found to classify te stability property of eac equilibrium state We compare te dynamical beaior of eac indiidual population in fig We can see tat te sorter days of outbreak will occur wen te tresold number is iger nternational colarly and cientific esearc & nnoation (8) 00 9 scolarwasetorg/07-689/5

5 World Academy of cience ngineering and Tecnology nternational Journal of Matematical and Computational ciences Vol: o:8 00 nternational cience ndex Matematical and Computational ciences Vol: o:8 00 wasetorg/publication/5 Furtermore te indiidual ector proportions are sown as arying of te transmission rate from uman to ector in fig Fig Te indiidual ector proportion as arying of te transmission rate of te disease from uman to ector Te parameters are similarly as fig We can see tat wen te transmission rate of te disease from uman to ector is iger te indiidual ector proportions are iger Tis corresponds to te fact tat te transmission rate of te disease sould be reduced to decrease te fraction of te infectious ector Consequently te outbreak of tis disease will be eliminated [] O Dickman and J AP Heesterbeck Matematical epidemiology of infectious disease Wiley eries in Matematical and Computational Biology Wiley: ew York 000 [] H W Hetcote Te matematics of infectious disease AM e ol pp [] M Derouic A Boutayeb and H Twizell A model of dengue feer BioMed ng Online ol 00 [5] L stea and C Vargas Analysis of a dengue disease transmission model Mat Biosci ol 50 pp 998 [6] L stea and C Vargas A model for dengue disease wit ariable uman population Mat Biosci ol 8 pp [7] Bacaer Approximation of te basic reproduction number 0 for ector-borne diseases wit a periodic ector population Bull MatBio ol 69 pp [8] W oss Te ewala epidemic Te irus; isolation patogenic properties and relationsip to te epidemic J Hyg ol 5 pp [9] W McD Hammon A udnick and G ater Viruses associated wit epidemic emorragic feers of te Pilippines and Tailand cience ol pp [0] U Taara and et al Outbreak of cikungunya feer in Tailand and irus detection in field population of ector mosquitoes Aedes aegypti and Aedes albopictus skuse (diptera:culicidae) outeast Asian J Trop Med Public Healt ol 0 pp [] PMoor Fteffens A computer-simulated model of an artropod- borne irus transmission cycle wit special reference to Cikungunya irus Transaction of oyal oiety Tropical Medicineand Hygiene ol 6 pp [] Bacaer Approximation of te basic reproduction number 0 for ector-borne diseases wit a periodic ector population Bulletin of Matematical Biology ol 69 pp [] Y Dumont F Ciroleu and C Domerg On a temporal model for te Cikungunya disease: Modeling teory and numerics Matematical Biosciences ol pp ACKOWLDGM Te work is supported by Faculty of cience King Mongkut s nstitute of Tecnology Ladkrabang Tailand FC [] M Anderson and M May nfectious diseases of umans: Dynamics and Control Oxford uniersity press: Oxford 99 nternational colarly and cientific esearc & nnoation (8) scolarwasetorg/07-689/5