STAGE-STRUCTURED POPULATION DYNAMICS OF AEDES AEGYPTI

Transcription

1 Internatonal Conference Mathematcal and Computatonal Bology 211 Internatonal Journal of Modern Physcs: Conference Seres Vol. 9 (212) World Scentfc Publshng Company DOI: /S STAGE-STRUCTURED POPULATION DYNAMICS OF AEDES AEGYPTI Int. J. Mod. Phys. Conf. Ser : Downloaded from by on 4/16/18. For personal use only. NURAINI YUSOFF Faculty of Computer and Mathematcal Scences, Unverst Teknolog MARA, 445 Shah Alam, Selangor, Malaysa nuran@tmsk.utm.edu.my HARUN BUDIN Faculty of Computer and Mathematcal Scences, Unverst Teknolog MARA, 445 Shah Alam, Selangor, Malaysa harun@tmsk.utm.edu.my SALEMAH ISMAIL Faculty of Computer and Mathematcal Scences, Unverst Teknolog MARA, 445 Shah Alam, Selangor, Malaysa salemah@tmsk.utm.edu.my Aedes aegypt s the man vector n the transmsson of dengue fever, a vector-borne dsease affectng world populaton lvng n tropcal and sub-tropcal countres. Better understandng of the dynamcs of ts populaton growth wll help n the efforts of controllng the spread of ths dsease. In lookng at the populaton dynamcs of Aedes aegypt, ths paper explored the stagestructured modelng of the populaton growth of the mosquto usng the matrx populaton model. The lfe cycle of the mosquto was dvded nto fve stages: eggs, larvae, pupae, adult1 and adult2. Developmental rates were obtaned for the average Malaysan temperature and these were used n constructng the transton matrx for the matrx model. The model, whch was based only on temperature, projected that the populaton of Aedes aegypt wll blow up wth tme, whch s not realstc. For further work, other factors need to be taken nto account to obtan a more realstc result. Keywords: Aedes aegypt; stage-structured; matrx populaton model. 1. Introducton Dengue fever (DF) s a vector-borne nfecton that has become a major publc health concern worldwde. Its symptoms of hgh-fever, severe headache, muscle and jont pans, and rashes may develop nto deadly complcatons known as the dengue haemorrhagc 364

2 Stage-Structured Populaton Dynamcs of Aedes Aegypt 365 Int. J. Mod. Phys. Conf. Ser : Downloaded from by on 4/16/18. For personal use only. fever (DHF). World Health Organzaton (WHO) estmates that, worldwde, there may be, about, 5 mllon dengue nfectons every year 1. Aedes aegypt s a domestc mosquto and has been dentfed as the prmary vector for dengue and yellow fever. The classfcaton of ts genome sequence s dscussed n detal n Ref. 2. Ths speces can be found n the tropcal and subtropcal regons, hence the places where DF and DHF cases are wdespread. WHO reported that, out of more than 1 countres that have ndcated the presence of DF and DHF, those n Southeast Asa and Western Pacfc are most serously affected 1. One of the effectve ways n controllng the spread of DF and DHF s to destroy the breedng stes of Aedes aegypt. In addton, understandng the effect of clmate change n the growth of mosquto populaton wll certanly be an advantage. The lfe cycle of mosquto s modeled usng the system of frst order dfferental equatons and the matrx model. The number of the stages used n the lfe cycle of mosquto determnes the number of dfferental equatons n the system, or the dmenson of the transton matrx n the matrx model. There are four man stages n the lfe cycle of mosquto; egg, larva, pupa and adult. The frst three stages need water as a medum whle the last one needs ar. In lterature, varous researchers used dfferent sets of stages, dependng on the problems needed to be solved. In Ref. 3, the authors dvded the lfe cycle of mosquto nto aquatc and wnged stages. Another classfcaton used was by takng two egg stages, mmature stage and mature stage, one larval stage and two adult stages, before and after the frst egg layng 4. Next extensve study on the populaton growth of mosquto was done by dvdng the stages nto egg, larva, pupa, adult1 and adult2 5. Another verson s the one proposed n Ref. 6, where the adult stage s dvded nto three categores, mmature, gestatng and reproducng adults, resultng n a system of sx equatons. In studyng the populaton dynamcs of mosqutoes, varous models had been used such as system of ordnary dfferental equatons 5, system of partal dfferental equatons 3 and the matrx model 4,6. Models represented as a system of dfferental equatons are solved usng numercal methods whle the soluton to the matrx model s obtaned through matrx multplcaton. However, obtanng the transton matrx s a bg task, dependng on the classfcaton of stages/age classes of the speces, determnaton of the fertlty and survval rates, and other factors nvolved n the lfe-cycle. In ths paper, we wll construct a matrx model by usng the classfcaton of Aedes aegypt used n Ref. 5. A system of dfference equatons wll be formed based on the transton stages of the lfe-cycle, whch then wll be transformed nto a matrx model. In the next secton we wll dscuss all related data and theores n constructng the model. In Secton 3, the matrx model obtaned based on Malaysan average temperature wll be presented and dscussed. Fnally, n the fnal secton, the concluson of the research and further work wll be gven.

3 366 N. Yusoff, H. Budn and S. Ismal 2. Materals and Methods Ths secton of the paper wll dscuss the related theores and bologcal parameters requred n constructng a stage-structured model of the populaton dynamcs of Aedes aegypt. As mentoned n the prevous secton, the stage-structured model used n ths paper s the matrx populaton model Lfe-cycle and bologcal parameters of Aedes aegypt Int. J. Mod. Phys. Conf. Ser : Downloaded from by on 4/16/18. For personal use only. There are four dstnct stages n the lfe cycle of any mosquto, whch are egg, larva, pupa and adult. The frst three stages are n the water whle the adult stage has ar as ts medum. However, n ths paper, the adult stage s further dvded nto 2: adult1 (before egg-layng) and adult2 (after the frst egg-layng). Fg.1 represents the lfe cycle of Aedes aegypt that we wll refer to n ths paper. w 1 Adult 2 w 1 w 2 Adult 1 w 9 Eggs w 1 w 8 Pupae Fg. 1. Lfe cycle of Aedes aegypt Table 1 gves the explanaton of the transton stages denoted n Fg. 1. It s adapted from a smlar table gven n Ref. 5. The transton stages represent the developmental events takng place n the lfe cycle. Table 1. Transton stages of the lfe cycle of Aedes aegypt Larva Transton stage Symbol Event w 1 Ovr1 Gonotropc cycle coeffcent for adult females n stage 1 w 2 Ovr2 Gonotropc cycle coeffcent for adult females n stage 2 w 3 me Mortalty of eggs w 4 elr Hatchng rate w 5 ml Mortalty of larvae w 6 lpr Pupaton rate w 7 mp Mortalty of pupae w 8 par Pupae nto adults development coeffcent w 9, w 1 ma Mortalty of adults w 3 w 4 w 7 w 6 w 5 We wll use the symbols E, L, P, A1, and A2 to represent the number of populaton n the stages egg, larva, pupa, adult1 and adult2, respectvely. In the modelng process, besdes the above transton stages, we wll also have the parameter egn, the average number of

4 Stage-Structured Populaton Dynamcs of Aedes Aegypt 367 eggs lad durng every ovposton. Eq. (1) gves the system of dfference equaton that represents the dynamcs of the populaton of every stage of the mosquto. ( t + 1 ) = egn( ovr1 A1( t) + ovr2 A2( t) ) me E( t) elr E( t) ( t +1 ) = elr E( t) ml L( t) lpr L( t) ( t +1 ) = lpr L( t) mp P( t) par P( t) 1 ( t + = par P( t) ma A1( t) ovr1 A ( t) 2 ( t 1) = ovr1 A1( t) ma A ( t) E L P A 1 A + 2. (1) Int. J. Mod. Phys. Conf. Ser : Downloaded from by on 4/16/18. For personal use only. The bologcal parameters of Aedes Aegypt can be dvded nto two, temperature dependent and non-temperature dependent. Parameters that do not depend on temperature are egn, number of eggs lad durng every ovposton (63 eggs/ovposton), me, egg mortalty (.11/day) and ma, adult death (.91/day) 5. Mortalty of larva (ml) and pupa (mp) depend on temperature 5 and are gven by Eq. (2), where T s temperature n degree Kelvn. ( ( T 278) / 2 735) mp = ml = exp.. (2) Parameters that depend on temperature are the developmental rates, namely the egg hatchng (elr), pupaton (lpr), adult emergence (par) and gonotrophc cycles (ovr1 and ovr1). The computaton of the developmental rates orgnated from the thermodynamc model dscussed n Ref. 7. It was later on smplfed n Ref. 8 and gven as Eq. (3). R D ( ) ( ) ( ο ) ( )( ο ) ο T T = RD 298 K ( )( ) 298 K exp H A R K 1 T. (3) 1+ exp H R 1 T 1 T In the formula, R s the unversal gas constant, whle H A and H H are thermodynamcs enthalpes characterstc of organsm. Ther values are gven n Table 2. The term T 1/ 2 represents the temperature when half of the enzyme s actvated wth T measured n degree Kelvn. Table 2. Coeffcents for the enzymatc model of maturaton (Eq. (3)) 5 Develop. cycle ( T ) ο R D ( K ) H H T 1 / 2 R D 298 H A Eggs hatchng elr Larval develop. lpr Pupal develop. par Gonotrophc cycle (A1) ovr Gonotrophc cycle (A2) ovr H 1 2 It can be shown that the developmental rates wll ncrease wth temperature. Ths ndcates that each stage of the mosquto develop much faster durng the summer, or, almost all year round n Malaysan temperature. The parameters for the selected temperature wll be computed and they contrbute to the computaton of the entres for the transton matrx.

5 368 N. Yusoff, H. Budn and S. Ismal 2.2. Matrx populaton model Int. J. Mod. Phys. Conf. Ser : Downloaded from by on 4/16/18. For personal use only. Matrx model was ntated by P.H. Lesle n , whch enables analyss and projecton of female populaton beng done accordng to age-classes. In ths model, the female populaton s dvded nto n age classes wth the class nterval h, and ther brth and death patterns are requred. The projecton nterval has to be done for every h years (or whchever tme unt used) and the transton matrx s known as the Lesle matrx. However, there are speces n our bologcal systems whose lfe-cycles are not welldefned by age, but rather, by stage or sze. For example, nsect s lfe-cycle s dvded nto the stages of egg, larva, pupa and adult, whle sze s more sutable n the classfcaton of plants. Moreover, the duraton of each stage s dfferent. The transton matrx constructed from such lfe cycle s known as the Lefkovtch matrx 1. Let A be the transton matrx and ( k X ) s the populaton at any tme k. A projecton of populaton at tme k wll be a matrx multplcaton between matrx A and populaton at tme k-1, as gven n Eq. (4). X ( k ) = AX ( k 1 ). (4) An example of a fve-stage Lefkovtch s gven n Eq. (5). P1 G1 A = F P G F 3 P G 3 3 F 4 P G 4 4 F5 P 5 Element P denotes the probablty of survvng and stayng n stage, G s the probablty of survvng and growng from stage to stage ( + 1), and F represents the fertlty of stage. One bg advantage of the Leftkovtch model s that t allows dfferent duraton n each stage and projecton can be done by days, weeks or whatever unt used. If s s the survval rate for stage and d s the duraton n stage, then we can compute P and G as follows: (5) G d ( s ). s 1 = 1 d s (6) Snce the total populaton of stage that survve equals to the proporton that remans n stage plus the proporton that has grown to stage ( + 1), or, s = P + G, then P s = d 1 ( 1 s ). 1 s d (7)

6 Stage-Structured Populaton Dynamcs of Aedes Aegypt 369 For a detaled dervaton of Eq. (6) and Eq. (7), readers may refer to Ref. 11. In the next secton, we wll dscuss on how these nformaton can be obtaned based on the development rates of each stage. The postve unque egenvalue of A, λ, wll have a correspondng egenvector, from whch the lmtng behavor of the age-specfc populaton can be analyzed. In general, f λ >, the populaton wll grow, but f λ <, the populaton wll de out, as tme ncreases 9. Int. J. Mod. Phys. Conf. Ser : Downloaded from by on 4/16/18. For personal use only Constructon of Lefkovtch matrx In the constructon of the Lefkovtch matrx, we wll refer to ths system of dfference equatons, Eq. (1). For example, n larval stage, the number of larval present s equal to the number of larva hatched from eggs that had survved after beng n the egg stage for several days mnus the larva that ded daly, mnus the ones that have developed nto pupa. The element P 2 represents the probablty of survvng and stayng n the larval stage whle G 2 s the probablty of survvng larval stage and developed to be pupae. Note that n formulae gven by Eq. (6) and Eq. (7), we need to know s 2, survval rate n larval stage (1 ml), and d 2, duraton the mosquto s n ths stage. In order to compute long wll each stage lasts, we wll let CDt be the cumulatve developmental rate such that CD t = n t= R D ( T ) t. (8) Formula gven by Eq. (8) accumulates the developmental rate for t days, based on the temperature of the day. The developments elr, lpr and par are consdered complete when CD t >.95 whle ovr1 occurs on the day when CD t > 1.. Subsequent cycles (ovr2) occur on the day when CDt ncreases an addtonal Ths nformaton enables us to compute how long the mosquto stays n each stage, based on the temperature. For example, for the larval stage, takng the average Malaysan temperature of 27.2 o C, we computed the R D (27.2) for larva to be and the cumulatve value CD t >.95 occurs on day 5. Smlar calculatons are done for each stage. The values F 4 and F 5 are the fertlty rate of stages adult1 and adult2, respectvely. Ths s computed by takng the average number of eggs lad per ovposton, dvded by the duraton the mosquto stay n the respectve stages. Note that only half of the number of eggs are taken snce n the matrx model, only female ones are consdered and we assume that the rato of male:female s 1:1. Values of F 2 and F 3 are zero snce both larva and pupa do not lay egg. 3. Results and Dscusson As mentoned n prevous secton, to estmate the duraton the mosquto stays n that each stage, we requre the developmental rates. The survval rate (1- mortalty rate) and the duraton value or each stage wll be substtuted nto formulae gven by Eq. (6) and

7 37 N. Yusoff, H. Budn and S. Ismal Eq. (7). Table 3 gves the summary of the computatons of the developmental rates for each stage. Int. J. Mod. Phys. Conf. Ser : Downloaded from by on 4/16/18. For personal use only. Table 3. Developmental rates at temperature 27.2 C o Stage R ( 27. C) D 2 Egg.2791 Larva Pupa Adult Adult Snce there s no bg fluctuaton n Malaysan temperature throughout the year, an average temperature s used. The average temperature used s 27.2 o C, computed based on the daly average temperature at Subang meteorology staton from January 26 untl Jun 21, as provded by the Malaysan Meteorology Department. As mentoned n the prevous secton, only female populaton s consdered, so only half of the average number of eggs lad wll be taken. It s averaged over duraton before frst ovposton and n between subsequent ovpostons. The age (n days) when the mosquto stays n each stage s based on duraton untl the development s complete. Table 4. Ages, daly survvorshp and number of eggs lad per day for each stage Stage Ages (days) Daly Survvorshp Eggs lad Eggs < Larva Pupa Adult Adult a a The subsequent ovposton occurs every two days for a total of fve tmes 5 Larval daly survvorshp computed n Table 4 s under the assumpton that there s no overcrowdng n the breedng ste. In addton, t s assumed that after the frst ovposton, the subsequent ones wll occur every two days, for a maxmum of fve tmes, f temperature reaches 3 o C 5. Hence, the Lefkovtch matrx for Aedes aegyt at the average temperature of 27.2 o C s as gven n Eq A = (1) The egenvalues obtaned are λ 1 = , λ 2 = I, λ 3 = I, λ 4 = and λ 5 = The largest, real postve egenvalue s λ 1,

8 Stage-Structured Populaton Dynamcs of Aedes Aegypt 371 Int. J. Mod. Phys. Conf. Ser : Downloaded from by on 4/16/18. For personal use only. whch ndcates that the populaton wll ncrease by 44.2% after every cycle. Ths does not seem logcal for the mosquto to ncrease so much. Due to ths, we dd not gve any fgures on the change n the number of mosqutoes present at any tme t. However ths result s obtaned based on the assumpton that temperature s always 27.2 o C and overcrowdng of larva s not consdered. A slght change n temperature s assumed not to affect the developmental rates. To mprove ths model, we are workng on a modfed model that wll take nto account the amount of ranfall per day. A clmate dependent model wll be proposed by modfyng the formula of egg hatchng process and larval survval. We wll take nto account that eggs can survve wth mnmal water but floodng s a necessty for hatchng 4,12. We hope that ths modfed model wll gve a better pcture on the populaton dynamcs of Aedes aegypt n Malaysa, generally. 4. Concluson In ths paper, we have shown how we constructed the Lefkovtch matrx for the stagestructured populaton model for Aedes aegypt. Bref theory on the lfe-cycle of Aedes aegypt and ts bologcal parameters as gven n other lteratures, were mentoned. We have also presented the detaled steps n computng each entry n the transton matrx. Based on the average temperature of Malaysa, we constructed the model. Based on the egenvalue of the matrx, we found out ths crude calculaton does not reflect the actual scenaro n Malaysa, or n other parts of the world. If ths were true, then our world wll be swarmed by mllons of mosquto and wll ncrease every few weeks (duraton of the whole cycle). Hence, to obtan a more realstc model, t wll be modfed by takng nto account the amount of ranfall. Ths, of course, wll be done by assumng that the breedng stes are outdoors and able to contan water. We hope that ths paper has gven some basc knowledge n constructng a more complcated and realstc model n future. Acknowledgments We gratefully acknowledge the fnancal support by the Mnstry of Hgher Educaton, under the Fundamental Research Grant Scheme (6-RMI/ST/FRGS 5/3/Fst (2/21)). References 1. WHO Meda Centre, Dengue and dengue haemorrhagc fever. Retreved March 4, 211, from WHO: (March, 29). 2. S. Snkns, Scence, 316 (5832), 1718 (27). 3. K. Tan, H. Koh, A. Md Ismal, and S. Teh, Modellng Mosquto Populaton wth Temperature Effects n Proc. Internatonal Conference on Envronmental Research and Technology, (Penang, Malaysa, 28), p B. Schaeffer, B. Mondet and S. Touzeau, Infecton, Genetcs and Evoluton 8, 422 (28). 5. M. Otero, H. Solar and N. Schwegmann, Bulletn of Mathematcal Bology 68, 1945 (26).

9 372 N. Yusoff, H. Budn and S. Ismal Int. J. Mod. Phys. Conf. Ser : Downloaded from by on 4/16/18. For personal use only. 6. R. A. Erckson, S. M. Presley, L. J. Allen, K. R. Long and S. B. Cox, Ecologcal Modellng 221, 1273 (21). 7. P. J. H. Sharpe and D. W. DeMchele, J. Theoretcal Bology 64, 649 (1977). 8. R. M. Schoofeld, P. J. H. Sharpe and C. E. Magnuson, J. Theoretcal Bology 88, 719 (1981). 9. H. Anton, and C. Rorres, Elementary Lnear Algebra wth Applcatons, 8th edn. (John Wley & Sons, Inc., New York, 2), p H. Caswell, Matrx Populaton Models: Constructon, Analyss and Interpretaton, 2nd edn. (Snauer Assocates, Inc, Sunderland, 21). 11. J. Boardman, D. Hrozenck, M. Kwon, I. Ul-Haq and A. Zeleke, Usng Populaton Models n the Teachng of Egenvalues (Center for Dscrete Mathematcs & Theoretcal Computer Scence, Rutgers Unversty, Pscataway, 27). 12. D. Focks, D. Hale, E. Danels, and G. Mount, J. Medcal Entomology 3 (6), 13 (1993).