A theoretical model of the dynamics of an Anopheles gambiae population under challenge with eggs giving rise to sterile males*

Bull. Org. mond. Sante )1969, 40, 205-212Bull. Wld Hith Org. A Theoretical Model of the Dynamics of an Anopheles gambiae Population under Challenge with Eggs Giving Rise to Sterile Males* C. B. CUELLAR 1 An explanation is given of the probable effects ofseeding a breeding place ofAnopheles gambiae with hybrid eggs that produce almost exclusively sterile males. For the calcula- tions, the dynamics of an A. gambiae population in a single breeding place have been simulated in a computer programme. General considerations and implications are discussed with reference to practical applica- tions, and the conclusion is reached that it will be, feasible to eradicate the species from any breeding place only if the number offactory-produced eggs distributed daily bears a known, optimal relationship to the numbers deposited daily on the site. The period of treatment, if the first-day ratio of normal: factory-produced eggs is 1: 1, was estimated to be in the region of9 weeks. Several runs of the computer have been condensed in a graph which might be used in the early evaluation ofprogress in the field. One of the pressing concerns of applied biology today is the development of resistance by insects to chemicals originally thought to be the means to eradicate or control insect pests. The difficulty in controlling insects lies primarily in their great reproductive potential. In the case of anophelines, Anopheles gambiae was eradicated from north-eastern Brazil (Soper, 1943) and from Upper Egypt (Shousha, 1948), but it has not been possible to achieve eradication in places where this species-complex is indigenous. However, in Cyprus, Mauritius and other places, indigenous anopheline populations of other species have been eradicated. A new approach was convincingly demonstrated by the eradication of Callitroga hominivorax, first from the island of Cura9ao (Baumhover et al., 1955), and then from Florida-a brilliant success which had the effect of stimulating research into the practical uses of artificially induced sterility in other insects (Cole, La Brecque & Burdon, 1961; Potts, 1958; Davis et al., 1959). However, the * This work was made possible by a grant from the Tropical Diseases Research Fund of the London School of Hygiene and Tropical Medicine. 1 Epidemiologist, Ministry of Health, Mexico City, Mexico. Present address: Research Assistant, Ross Institute of Tropical Hygiene, London School of Hygiene and Tropical Medicine, University of London, England. weakness produced by the sterilizing procedure is one of the limitations of this method. This handicap is not apparent in hybrid males produced by crossing species within the A. gambiae complex (Davidson, 1969). Moreover, by crossing certain of these species it has proved possible to obtain a population composed almost entirely of infertile males. Furthermore, it has been shown in the laboratory that these males can compete success- fully with normal males in mating with the available females. This finding followed important discov- eries made by Davidson (1964) while studying the genetics of the A. gambiae complex. It is not yet known whether the infertile males will prove to be equally competitive upon release in the field, but cage experiments have shown an unmistakable in- crease in the mating ability of the sterile hybrids, compared with that of non-hybrid males. The ultimate value of a method of this kind must depend upon the results of large-scale trials. How- ever, modem computational techniques permit the simulation of field trials, from which estimates can be obtained of the scale of intervention needed to achieve success. Thus, it is possible to reduce the risks of a scheme failing because of theoretical weaknesses in its planning. The basic computer programme used (see Fig. 1) allows for great flexibility in the variation of the 2289 -205- 3 C. B. CUELLAR parameters bearing on the reproductive ability of the mosquito population in question. The para- meters are as follows. (1) The probability of daily survival of an adult mosquito (p) (Macdonald, 1952; Beklemishev, Detinova & Polovodova, 1959). (2) The average number of eggs in a clutch or batch (b) (Detinova & Gillies, 1964). (3) The length of the aquatic cycle in days. (4) The probability of survival of an egg to successful imago emergence (M). (5) The oviposition rhythm, as expressed by the average times between emergence and the first oviposition and between successive ovipositions (Gillies & Wilkes, 1965). Although there is some doubt about the details of the mating behaviour of A. gambiae, it appeared reasonable to assume that the females are fertilized only once and usually within the first 24 hours of adult life. Different possible rhythms of oviposition behaviour were dealt with by successive runs of the computer, trying different combinations each time, and by selecting for more detailed analysis the rhythms thought to be common and those favourable for the reproductive ability of the insect. The sum to infinity of the ovipositions can be estimated on the basis of the value ofp (see Annex 1) which, multiplied by the average clutch, b, will give the average number of eggs laid per female (Table 1). However, in an attempt to be more realistic, the programme sets an assumed limit, varied from 30 to 60 days, to the life of the female mosquito. These values are based on the maximum number of follicular relics observed in females captured in the field and on knowledge of the average length of the gonotrophic cycle, allowing for the more prolonged first one (Gillies & Wilkes, 1965). Several factors must influence the densities of A. gambiae populations in different areas (Holstein, 1954), but there must clearly be a limit to the mosquito breeding capabilities of any locality. Its output will normally tend to approach that limit and to remain there as long as there are only minor changes in the size of the breeding places or other important factors, for instance, the quantity of larval food supply. If we consider a typical breeding place in the middle of the mosquito season, the salient facts are that the site is producing mosquitos at the maximum possible rate and that, concurrently, it is being saturated with eggs. If, in these conditions, upwards of 100 eggs per female are produced, then 98% or more of the eggs will not survive to a successful imago emergence. This means that a pair of mos- quitos of one generation will give rise to an average of 2 adult mosquitos in the next. This value repre- sents a probability of survival through the aquatic stages of 0.02, at most-a very low value indeed, which contrasts with the much higher values pre- valent in insectaries where the larvae thrive in optimum conditions of space, food, etc.; here survivals of 0.9 have been recorded. These considerations lead one to conclude that variation of the probability of survival from egg to successful imago emergence (given the symbol M) constitutes a powerful mechanism for the natural regulation of the adult densities of practically all mosquito species. The actual variation of M in the field is presently unknown. Therefore, in the com- puter programme, M is allowed to vary in a self- regulating manner from the lowest value which will allow survival of the species to a possible maximum of 0.9. This last value corresponds to optimal laboratory conditions unlikely to be found in the field. Very probably M reaches its highest value in nature at the beginning of the mosquito season, since there are then many new breeding places and relatively little competition for survival, and con- sequently the losses during the aquatic cycle would be minimal. High values ofMmay explain thesudden increases in adult densities often observed in A. gam- biae populations (Holstein, 1954; Symes, 1930). A. gambiae might be eradicated from an isolated area, using eggs which will give rise to sterile males, by swamping every breeding place daily with very large numbers of such eggs, thus reducing to a low level the probability that a virgin female will mate with a fertile male, even in the generation which begins its aquatic cycle on the first day of the scheme. The number of days needed will be governed by the longevity of the longest-lived female emerging from an egg deposited 1 day before the beginning of the scheme, plus an interval of 1 aquatic cycle to allow for the development of the last batch of fertile eggs laid. Although this method of attack would cut down the number of days required to achieve eradication, it has the disadvantage of requiring enormous numbers of " factory-produced" eggs, which might be difficult to produce and distribute. On the other hand, insufficient treatments cannot possibly achieve eradication. It would be best, therefore, to seek a 206 FIG. 1 THE BASIC COMPUTER PROGRAMME OBT RSADY ANY OP SmBL DRSD CB TO STORB TS DAILYR OR OEPOSITIXOS, AC/D OIE TO TNDZ T IXITIAL VALW OP NRO BAD \ TNIC DAILY OUTEW 0P Ttffl IDIllO P>CS TO 1B 8I1ZUITZD \ T AWRAOB R 0P S IX A BAtH\ , TS *ItOIZBILITY 0P DAILY 81JIt=vAL OP A11 ADBIt tlOS*lYITO THS hXI P088I tf 0P M (the proWsbility or *ursttgl to *ser6enee of any *a) \ T} lmR OP P^cEOltY sao8 PD TO 1B DBPOrRPBD DAILY \ TRS NUI=R OP DAY8 PROWI BBROSlICX TO 1PIRW OTIPOSITION \ SHS IIUIWR 0P AQUATIC IRVJkLJ TO YL^P8X 1sP01E SB \ lBaIlilIIllO OF TR1£ATh11S (sinimun 6 weeks if the feleJ lite 30 dAys) \ THIC tWIlilJ# I^NOTIi OP LIN 0P A \ THS XllmR 011' DAY8 I=NStil SUCCSSS OVIt08ITION8 \ THS NR OP DAYS OP TSt AQUATIC Il"'SRtffi CO"PIS TllB PMCTIOW 0P ROINO msus WICH 8IW TO THZ PIRW OPOSNIOW CO"X T PMCTIOW 0P M=S ICH S ^0" 0 bSNIOX TO TB Grr TlA "R OP W^TlAY'S ROI"O ASUCCB38Y M H - IU S CO"P-B TB ACTUAL R OPSM CHIXO THIC DAY OP TBIR PI OENSNIOW AM STOM z IW THAT DAY 'S SPSCIAL CZLL no PI!21) O1XT ROW "AW tIZ ARIB ACTUALLY B8ROINO TODATFI1ND Of HOW h - TU AM ROW"ANY I - IU "A=S OTWO TODAY BJiSZD ON TS PMCBDINO PIO, COMPUN TNS "tILZY PAXORw APIIfD OW HOW MN - IU zS IIACTIIID*9! S STERILE MALES FOR CHALLENGE OF MOSQUITO POPULATION: A THEORETICAL MODEL 207 TABLE 1 AVERAGE NUMBER OF OVIPOSITIONS AND EGGS LAID PER FEMALE, WHERE THE AVERAGE BATCH IS 150 EGGSa Oviposition frequencies and Number of weefrtsubsequent ovi-where i st ovi- positions ov- positions per position at the emergingoccurs following female on day: intervals (days) 2 3 4 5 2 3 4 5 2 3 4 2 2 3 2 3 4 2 3 4 5 2 2 3 2 3 4 2 3 4 5 2 2 3 2 3 4 2 3 4 5 0.563 0.338 0.276 0.203 0.165 0.149 0.122 0.099 0.089 0.084 0.961 0.673 0.522 0.471 0.365 0.316 0.330 0.256 0.221 0.202 1.286 0.964 0.730 0.723 0.547 0.463 0.542 0.410 0.347 0.311 Number of eggs laid per emerging female (E) 84 51 41 30 25 22 18 15 13 13 144 101 78 71 55 47 50 38 33 30 193 145 110 108 82 69 81 62 52 47 Probability of an adult female surviving throughout I day (p) 0.8 0.85 0.9 0.95 Oviposition frequencies and where first subsequent ov- ovi-povitio positionsposition at the occurs: followingoda: intervals (days) 2 2 3 2 3 4 2 3 4 5 2 3 4 5 2 3 4 5 2 3 4 5 2 3 4 5 2 2 3 2 3 4 2 3 4 5 2 3 2 3 4 2 3 4 5 2 2 3 2 3 4 2 3 4 5 Number of Number of ovi- eggs laidpositions per per emerging emerging emaerEnfemale f 1.778 1.422 1.049 1.138 0.839 0.694 0.910 0.672 0.555 0.487 2.604 2.213 1.591 1.881 !1.353 1.092 1.599 1.150 0.928 0.792 4.263 3.837 2.690 3.454 2.421 1.908 3.108 2.179 1.717 1.442 9.256 8.794 6.013 8.354 5.712 4.391 7.936 5.426 4.171 3.421 267 213 157 171 126 104 137 101 83 73 391 332 239 282 203 164 240 173 139 120 639 578 404 518 363 286 466 327 258 216 1 388 1 319 902 1 253 857 659 1 190 814 626 513 a Different values of p are considered and the maximum length of life is taken to be a function of p only. Complete fertilization is assumed (over-all fertility factor = 1.0). Probability of an adult female surviving throughout 1 day (p) 0.6 0.7 0.75 - C. B. CUELLAR compromise with the aim of reducing the required supply of factory-produced eggs as much as possible without jeopardizing the success of the attack or increasing unduly the period during which the sites have to be treated. Since uniformity in breeding places is rarely found in nature, treatments have to be adjusted. The obvious adjustment should be made by relating the number of factory-produced eggs to the number of normal eggs deposited daily in a breeding place. This might be achieved directly or indirectly in the following ways: (1) By estimation of the number of larvae of a particular instar, allowing for previous mortality and non-hatching of eggs; or (2) By a count of pupae, correcting for emergence losses, adult mortality, efficiency of mating, and oviposition rhythm. The direct method is self-explanatory, but the second requires a simple formula described in Annex 2, which is based on Table 1. With a base-line of the number of natural eggs deposited daily in a breeding place it was possible to understand more easily the results of successive runs of the computer, changing the number of factory-produced eggs distributed daily in the simulated breeding place (see Table 2). As expected, the minimum period theoretically required to clear a breeding place (defining " clearance" as the absence of any surviving female able to lay normal eggs) was found to be roughly equivalent to the maximum length of life of a female plus 2 aquatic intervals. A reduction in the number of factory- produced eggs deposited daily increased the period required to clear the place, and the increase was found to be directly related to the ratio normal: factory-produced eggs deposited daily, as measured for the first day of the scheme. When that ratio was greater than 5: 1 the breeding place could not be cleared, even if the daily treatment with factory eggs was prolonged indefinitely. The most favour- able ratio was found to be about 1 : 1, since the num- ber of days required to attain clearance is near to the absolute minimum and at the same time the number of eggs required is not excessive (see Fig. 2). FIG. 2 EXPECTED DECLINE IN OUTPUT OF FEMALES FROM AN ANOPHELINE BREEDING PLACE TREATED WITH DIFFERENT FIRST-DAY RATIOS OF NATURAL TO FACTORY-PRODUCED EGGS,a WHERE THE AQUATIC INTERVAL IS 7 DAYS a Ratios are indicated for each curve. 208 SrERILE MALES FOR CHALLENGE OF MOSQUITO POPULATION: A THEORETICAL MODEL TABLE 2 SIMULATION OF THE DYNAMICS OF A BREEDING PLACE TREATED WITH FACTORY-PRODUCED EGGS Input values Daily output of breeding place (maximum) 10 000 Proportion of adult females surviving daily 0.8 Days to first oviposition 4 Subsequent oviposition intervals (days) 2 Maximum length of life of a female (days) 30 Average number of eggs in a batch or clutch 150 Maximum value of M (proportion surviving from egg to adult) 0.9 Days from egg to emergence (aquatic cycle) 7 Total daily Fertile Mated with Sterile Ratio of Week beedroutput of Generation's females fertile males naturalyWeek ngplce value of M emerging male emerging fatrbreedingp ac ~~~(per day) III eggs Pre-treatment - 10 000 0.01174 5 000 5 000 -_ - 10 000 0.01174 5 000 5 000 - (a) Treatment begins; 1 630 000 factory-produced eggs deposited daily I 10000 0.00403 1 716 I 355 6568 0.52 2 10000 0.00455 1 289 191 7422 0.35 3 10 000 0.00564 406 17 9 188 0.09 4 10 000 0.00592 175 3 9 650 0.04 5 10 000 0.00609 35 0 9 930 0.01 6 10 000 0.00612 8 0 9 983 0.01 7 10 000 0.00613 1 0 9 998 0.01 Breeding place cleared (b) Treatment begins; 1 165 450 factory-produced eggs deposited daily 1 10 000 0.00496 2 111 564 5 778 0.73 2 10000 0.00573 1 660 330 6680 0.50 3 10 000 0.00745 656 46 8 688 0.15 4 10000 0.00800 336 12 9329 0.07 5 10 000 0.00843 85 1 9 830 0.02 6 10 000 0.00854 25 0 9 949 0.01 7 10 000 0.00857 4 0 9 992 0.01 8 10 000 0.00858 1 0 9 998 0.01 Breeding place cleared (c) Treatment begins; 815 000 factory eggs deposited daily 1 10000 0.00600 2555 I 877 4890 1.05 2 10 000 0.00708 2 117 568 5 766 0.73 3 10000 0.00963 1 076 130 7847 0.27 4 10 000 0.01064 662 47 8 676 0.15 5 10 000 0.01171 230 5 9 540 0.05 6 10 000 0.01205 90 1 9 819 0.02 7 10 000 0.01222 19 0 9 961 0.01 8 10 000 0.01226 5 0 9 990 0.01 Breeding place cleared 209 C. B. CUELLAR TABLE 2 (continued) Total daily Fertile Mated with Sterile Ratio of Week output of veealueof'M females fertile males facuator breeding place evalue of M merging male emerging factory(per day) II eg (d) Treatment begins; 500 000 factory eggs deposited daily I 10 000 0.00740 3150 1 449 3 699 1.70 2 10 000 0.00882 2 794 1 084 4 411 1.27 3 10 000 0.01233 1 916 454 6 167 0.62 4 10 000 0.01409 1 478 256 7 044 0.42 5 10 000 0.01666 836 76 8 328 0.20 6 10 000 0.01800 501 26 8 999 0.11 7 10 000 0.01922 194 4 9 612 0.04 8 10 000 0.01968 80 1 9 841 0.02 9 10 000 0.01993 18 0 9 963 0.01 10 10 000 0.01998 5 0 9 990 0.01 Breeding place cleared (e) Treatment begins; 250 000 factory eggs deposited daily 1 10 000 0.00908 3 865 2 436 2 269 3.41 2 10 000 0.01059 3 676 2 137 2 648 2.78 3 10 000 0.01406 3 242 1 555 3 516 1.84 4 10 000 0.01582 3 023 1 310 3 954 1.53 5 10 000 0.01867 2 666 969 4 668 1.14 6 10000 0.02061 2 423 775 5 154 0.94 7 10 000 0.02345 2 069 540 5 862 0.71 8 10 000 0.02558 1 803 396 6 395 0.56 9 10 000 0.02858 1 427 238 7 145 0.40 10 10000 0.03084 1 145 148 7711 0.30 11 10 000 0.03378 777 65 8 846 0.18 12 10 000 0.03576 531 30 8 939 0.12 13 10 000 0.03784 270 7 9 461 0.06 14 10 000 0.03889 139 2 9 722 0.03 15 10 000 0.03964 45 0 9 910 0.01 16 10 000 0.03987 16 0 9 969 0.01 Breeding place cleared (f) Treatment begins; 100 000 factory eggs deposited daily 1 10 000 0.01051 4 475 3 624 1 051 8.52 2 10 000 0.01153 4 423 3 509 1 153 7.67 3 10 000 0.01348 4 326 3 299 1 348 6.42 4 10 000 0.01415 4 292 3 228 1 415 6.07 5 10 000 0 01493 4 253 3 148 1 493 5.70 6 10000 0.01529 4235 3112 1 529 5.54 7 10 000 0.01565 4 218 3 076 1 565 5.39 8 10 000 0.01582 4 209 3 059 1 582 5.32 9 10 000 0.01599 4 200 3 042 1 599 5.25 10 10 000 0.01608 4 196 3 034 1 608 5.22 11 10 000 0.01616 4 192 3 026 1 616 5.19 12 10 000 0.01620 4 190 3 021 1 620 5.17 13 10000 0.01624 4188 3018 1 624 5.16 14 10 000 0.01626 4 187 3 016 1 626 5.15 15 10000 0.01628 4 186 3 014 1 628 5.14 16 10 000 0.01629 4185 3 013 1 629 5.14 17 10 000 0.01630 4 185 3 013 1 629 5.14 Clearance of breeding place cannot be achieved with this treatment 210 STERILE MALES FOR CHALLENGE OF MOSQUITO POPULATION: A THEORETICAL MODEL With regard to the practical possibilities of this method, consideration must be given to the capacity of the " egg factory ", the extent of the area to be treated, the length of the dry season and the number and output of the breeding places. Estimation of the maximum possible value of M in the field would be of importance in mosquito eradication schemes relying mainly on insecticides or sterile adult releases, a subject elaborated in another paper (Cuellar, 1969). However, if the breeding places are treated with eggs, that deter- mination becomes irrelevant since the increased over-all number of eggs deposited daily (natural plus factory-produced) will increase the larval density and may thus be expected to reduce the probability of survival of each individual (M). It is important to remember that the results obtained by computer techniques cannot possibly be better than the observations on which they are based. The work of the field entomologist is now as important as ever but when he is faced with all the factors influencing vector dynamics, mathema- tical models may help to reveal those factors that are the most sensitive, and what may be ex- pected to happen when one or more factors are modified. ACKNOWLEDGEMENTS I am most grateful to the late Professor G. Macdonald, a brilliant scientist and an irreplaceable friend, to Dr G. Davidson, Dr G. K. Matthew and Dr G. F. Mason, Ross Institute, University of London, and to Dr M. T. Gillies, University of Sussex, for their valuable advice and stimulating discussions. Thanks are also given to the Institute of Computer Science, University of London, for the use of the ATLAS computer, and to Miss J. Maddox for her untiring secretarial assistance. RItSUME MODtLE THtORIQUE DE LA DYNAMIQUE D'UNE POPULATION D'ANOPHELES GAMBIAE PERTURBtE PAR L'INTRODUCTION D'CEUFS PRODUISANT DES MALES STERILES Toute tentative de representer par un modele math& matiques les parametres biologiques caracteristiques d'un vecteur doit tenir compte de la capacit6 de reproduction de l'espece etudiee. L'importance de ce facteur apparait souvent lorsque les mesures de lutte modifient la dyna- mique et la stabilit6 d'une population d'insectes. Anopheles gambiae, complexe groupant les principales especes vectrices du paludisme en Afrique tropicale, est difficilement accessible a l'eradication par les methodes classiques dans son habitat naturel. La nouvelle methode de lutte g6n6tique decrite par Davidson (1969) permet cependant d'entrevoir une solution du probleme. Elle prevoit la production de masse et l'introduction dans le milieu d'aeufs hybrides donnant des males steriles les- quels, en laboratoire tout au moins, font preuve d'une aptitude sexuelle superieure a celle des males normaux. II reste 6videmment a fixer le nombre d'ceufs qui doivent etre ainsi produits et introduits dans une population nor- male, d'importance et de caracteristiques donnees, pour parvenir a son elimination. La technique proposee par l'auteur a cet effet comporte deux stades.On identifie d'abord les principaux facteurs qui agissent sur la dynamique de la reproduction et on les incorpore au modele. Leur importance relative est ensuite evalu6e en modifiant successivement la valeur de chacun d'entre eux et en notant les resultats. Les para- metres suivants ont ete retenus: probabilit6 de survie quo- tidienne d'un moustique adulte; nombre moyen d'ceufs par ponte; duree, en jours, du cycle aquatique; probabi- lite de survie de l'aeuf jusqu'a l'eclosion imaginale (M); rythme de ponte, expriln6 par les intervalles moyens entre l'eclosion imaginale et la premiere ponte ainsi qu'entre les pontes successives. Differentes combinaisons de ces parametres ont ete dprouvees face a l'introduction d'un nombre croissant d'aeufs hybrides dans une serie de gites larvaires simules. Les resultats indiquent qu'il est possible d'eliminer une population d'A. gambiae si la proportion des ceufs hy- brides introduits quotidiennement est sup6rieure a un pour cinq. Le rapport optimal est de un ceuf hybride pour chaque ceuf normal: il permet d'obtenir l'eradication du vecteur en neuf semaines environ. Le nombre d'oeufs nor- maux pondus quotidiennement dans un gite donne peut etre calcule directement par numdration des larves ou indirectement apres comptage des nymphes. L'analyse mathematique a fait ressortir l'importance des variations du facteur M pour la regulation naturelle des populations d'A. gambiae. 211 212 C. B. CUELLAR REFERENCES Baumhzver, A. H. et at. (1955) J. econ. Ent., 48, 462 Beklemishev, W. N., D-tinova, T. S. & Polovodova, V. P. (1959) Bull. Wld Hlth Org., 21, 223 Cole, M. M., La Brecque, G. C. & Burden, G. S. (1959) J. econ. Ent., 52, 448 Cuellar, C. B. (1969) Bull. WId Hlth Org., 40, 213 Davidson, G. (1964) Riv. Malar., 43, 167 Davidson, G. (1969) Bull. Wld Hlth Org., 40, 221 Davis, A. N. et at. (1959) J. econ. Ent., 52, 868 Detinova, T. S. & Gillies, M. T. (1964) Bull. Wld Hlth Org., 30, 23 Gillies, M. T. & Wilkes, T. J. (1965) Bull. ent. Res., 56, 237 Holstein, M. T. (1954) Biology of Anopheles gambiae, Geneva (World Health Organization: Monograph Series, No. 9) Macdonald, G. (1952) Trop. Dis. Bull., 49, 569 Macdonald, G. (1957) The epidemiology and control of malaria, London, Oxford University Press Potts, W. H. (1958) Ann. trop. Med. Parasit., 52, 484 Shousha, A. T. (1948) Bull. Wld Hlth Org., 1, 309 Soper, F. L. & Wilson, D. B. (1943) Anopheles gambiae in Brazil, 1930-1940, New York, Rockefeller Founda- tion Symes, C. B. (1930) Kenya E. Afr. med. J., 7, 2 Annex 1 THE AVERAGE NUMBER OF OVIPOSITIONS PER EMERGING FEMALE The diminishing proportions of females with constant daily mortality which survive to sub- sequent ovipositions form a geometric progression: a, ar, ar2, ar3, ar4 ...I where a = first term and r = common ratio of the series. The sum to infinity of a geometric progression is found by the standard formula: a I -r provided that -1 < r <O orO < r <1. The sum of n terms of a geometric progression is found by the standard formula: -a(l-r")Snpetor 0 <1I1-r provided that r < O or O < r < l. Example. What is the average number of oviposi- tions per emerging female? Answer. Both a and r can be calculated from p, the probability of daily survival; thus for p = 0.8 and oviposition frequencies of 4*2 (the first occurs at day 4 and the following every 2 days), a = 0.4096 and r = 0.64. Then, 0.4096 S = - = 1.138.1 - 0.64 But to obtain the average number of ovipositions per emerging female when the maximum possible life-span is 30 days, we have to evaluate n simply by counting the maximum number of ovipositions that can possibly be made by a female in 30 days. Again using the 4*2 example above, n = 14. Therefore, I 0.4096 (1 - 0.6414) = 1.136.1 -0.64 Annex 2 EQUILIBRIUM OF A BREEDING PLACE (INDIRECT METHOD) If N is the number of living pupae found in a breeding place, then N/2 is the number of pupae belonging to the female sex. If c is the length of the pupal stage in days, N/2c is the number of females emerging in 1 day, and Nd/2c is the number of eggs laid daily in that breeding place, d being the value found in the last column of Table 1. Example. How many eggs are being laid daily in a breeding place where 50 pupae are counted,p = 0.8, the oviposition frequencies are 4*2, all females mate successfully with fertile males, the average batch has 150 eggs, and the pupal stage lasts only 1 day? Answer: (50) x (171)/(2) x (1) = 4275 eggs.