A Set of Discrete Formulae for the Performance of a Tsetse Population During Aerial Spraying
aa r X i v : . [ q - b i o . O T ] O c t A Set of Discrete Formulae for the Performance of aTsetse Population During Aerial Spraying
S. J. Childs
Department of Mathematics and Applied Mathematics, University of the Free State,P.O. Box 339, Bloemfontein, 9300, South Africa.Tel: +27 51 4013386 Email: [email protected]
Acta Tropica, 125: 202–213, 2013
Abstract
A set of discrete formulae that calculates the hypothetical impact of aerial spraying on atsetse population is derived and the work is thought to be novel. Both the original popula-tion and the subsequent generations which survive the aerial spraying, may ultimately bethought of as deriving from two, distinct sources. These origins are, however, neither dis-tinct, nor relevant by the third generation. It is for this reason that the female populationis considered to be composed of the following four categories for the purposes of deriva-tion: Original flies which existed as such at the commencement of spraying; original pupaewhich existed as such at the commencement of spraying; the immediate descendants of boththe aforementioned categories, during spraying; third and higher generation descendants.In theory, the latter category is a recurrence relation. In practice, the third generation’spupal stage has hardly come into existence, even by the end of a completed operation. Im-plicit in the formulae is the assumption of one, temperature-dependent mortality rate forthe entire pupal stage, a second for the period between eclosion and ovulation and yet athird for the entire, adult life-span. Gravid female resistance to the insecticide is assumedto be inconsequential. A further assumption of the formulae is that at least one male isalways available (degree of sterility variable).
Keywords: Tsetse;
Glossina ; aerial spraying; trypanosomiasis; nagana; sleeping sickness.
The
Glossina genus is the vector of trypanasomiasis in Africa. There are about thirty threespecies and subspecies of tsetse fly, whereas about half as many trypanosomes of the salivarian1
Childs S. J. clade are thought to exist (Gooding and Krafsur, 2004; Stevens and Brisse, 2004). Thirty sixAfrican countries are still afflicted by human, African trypanosomiasis (HAT), although naganais still of veterinary and economic importance in others e.g. South Africa (Anonymous, 2012).The most common causes of nagana in livestock are
Trypanosoma congolense and
T. vivax , inthat order of priority. Neither pathogen has ever been known to infect an human host, althoughboth domestic and wild animals serve as the reservoir for the human afflictions,
T. brucei gam-biense and
T. brucei rhodesiense . T. gambiense is associated with chronic disease in WestAfrica, while
T. rhodesiense is associated with acute disease in East Africa. Although theadvance of HAT is spectacularly rapid in the case of
T. rhodesiense , T. gambiense can be dan-gerously insidious, the symptoms often only becoming manifest once it is too late to treat (95%of all HAT cases are attributed to
T. gambiense according to Anonymous, 2012). Not enough isknown about the vector competence of the various tsetse species, as was recently illustrated bythe findings of Motloang et al. (2009). The fusca and palpalis groups are largely confined toWest Africa, while the morsitans group is largely confined to the Eastern side of the continent,a few exceptions to this rule occurring in both the fusca and morsitans groups. Three membersof the morsitans group, namely
Glossina morsitans , Glossina pallidipes and
Glossina austeni could be considered to be mainly problematic in the Southern and East African theatres, whilethe problem assumes a far greater diversity around Lake Victoria and to the west of it. Mem-bers of the palpalis group are notorious vectors in the West African theatre (
Glossina fuscipesfuscipes , in particular, having been implicated by the focus of numerous epidemics).Trypanosomiasis is regarded by many African countries to be largely of veterinary and eco-nomic importance, in modern times. This has certainly not always been the case and Leak(1999) provides a grim reminder that in the opening years of the twentieth century, around200 000 people died of trypanosomiasis in the provinces of Buganda and Busoga alone andthat these provinces eventually had to be evacuated. F`evre et al. (2004) put the figure closerto around two thirds of the lake-shore population, for a slightly longer period and the epidemicreached similar proportions in the Congo river basin (Anonymous, 2012). Today HAT has allbut vanished, largely as a result of the all-out war waged against tsetse during the twentiethcentury. So great has been the success that, in 2010, only 7139 new cases were reported, thebiggest contributor being the Democratic Republic of the Congo (Anonymous, 2012).This is no small achievement and such success has not come without a price. Du Toit (1954)put the cost of
G. pallidipes eradication from KwaZulu-Natal, in the first half of the twenti-eth century, at well in excess of £
100 000. Properly planned aerial spraying has proved to bethe most effective means of tsetse control and it is with the prohibitive costs in mind that thisresearch attempts to make operations as efficient and successful as possible. The modern opera-tion conventionally utilizes a relatively harmless pyrethroid such as endosulfan or deltamethrin(Allsopp, 1984). An aerosol of insecticide is discharged from a formation of aircraft, flying atlow altitude (less than 100 metres a.g.l.) and guided by G.P.S. Adult flies are extremely sus-ceptible to the insecticide and kill rates very close to 100% can be anticipated under favourableconditions.The main challenge to controlling tsetse by aerial spraying is that the pupal stage is largelyprotected from insecticides. Repeat spray cycles therefore need to be scheduled to kill new . . Aerial Spraying of Tsetse Fly
Glossina genus are entirelytemperature dependent and are therefore readily predictable. By knowing the mean temperaturespray cycles can be scheduled two days short of the time to the production of the first larva; thetwo-day safety margin being designed to ensure that there is no variance in time-to-first-larvato levels below the length of the spray cycle.The strategy explored in Childs (2011) was one in which the repeated spray cycles are con-tinued until two sprays subsequent to the eclosion of the last, pre-spray-larviposited, femalepupae. None of the observations in that work are, however, valid in the event that the operationis terminated one, or more, sprays short and there could be many reasons for pursuing such astrategy in the modern scenario. Costs, environmental considerations and an area-wide, inte-grated approach to pest management which contemplates the use of the sterile insect technique(Barclay and Vreysen, 2010), are only a few of the reasons why a curtailed operation has in-creasingly been entertained as a ‘knock-down’, rather than as an agent of eradication, in recenttimes. A recent shift in interest from savannah, to the more inaccessible, riverine and forestedhabitats, in combination with a better understanding of odour-baited targets, pour-ons and dips(Childs, 2010 and Esterhuizen et al. 2006), has led to these alternative counter measures re-cently having been assigned a far more significant role in control and eradication, than in thepast. In the event that spray cycles are not continued for the full duration, the formula derivedin Childs (2011) is not appropriate. A more comprehensive set of formulae is required, onewhich, for example, also accounts for other categories of pupae, such as the immediate, pupaldescendents of pre-spray-existing flies, as well as actual flies themselves.The effect of temperature on aerial spraying, through the reproductive cycle and general pop-ulation dynamics of the tsetse fly, can easily be taken into account. The same cannot, how-ever, be said for the effect of temperature on spray efficacy, it being a property unique to eachand every environment and the conditions prevailing at the time. Very high kill rates usually(though not always) come about as a result of the sinking air associated with cooler weather.It favours the settling of insecticidal droplets. The inherent toxicity of deltamethrin and manyother pyrethroids also decreases with temperature, contrary to the toxicity of most insecticides.The effects of anabatic winds, the protection afforded by the forest canopy and multifariousother variables, are just as relevant to spray efficacy. No account is taken of the mechanism ingravid females, whereby lipophilic toxins are excreted, sacrificing larvae in utero for survival,either. The effect of temperature and age on spray efficacy is therefore not modelled and itis, instead, a variable in the formulation. Spray efficacy is usually measured in the field, withhindsight, rather than predicted. Three kill rates of around 99%, 99.9% and 99.99% respec-tively are entertained in this work. They should be thought of as being broadly associated withthe warmer, intermediate and cooler parts of the low-temperature range respectively. It is inthis way that the hypothetical impact of aerial spraying on tsetse fly populations is formulated.The formulae derived in this work are largely a predictive tool. They provide a convenientmeans of calculating theoretical levels of control in the aerial spraying of tsetse, by way of
Childs S. J. spreadsheets and simple algorithms, in which the outcome is based on mean temperature andspray efficacy. They also provide a convenient means of making ‘back-of-an-envelope’ esti-mates based on first order terms. The formulae provide a means to calculating the outcome atmean temperature. The data presented in Hargrove (1990), for example, suggest that the tem-perature in tsetse environments often varies little. The restriction to mean temperature is notproblematic from a point of view of prediction, since one can usually only forecast mean tem-perature. An algorithm is the next logical step, brought about by the introduction of variabletemperature.
The female tsetse fly mates only once in her life with the chance η that she is successfullyinseminated ( η is usually taken to be unity). She also produces only one larva at a time. Thetime between female eclosion and the production of the first larva is known as time-to-first-larva, τ . Thereafter she produces pupae at a shorter interlarval period, τ . The effect oftemperature on the first and subsequent interlarval periods has been estimated in the field,using G. pallidipes . The predicted mean time taken from female eclosion to the production ofthe first pupa is obtained using Jackson’s (Anonymous, 1955) temperature-dependent formula, τ i = 1 k + k ( T − i = 1 , , in which k = 0 . and k = 0 . (Hargrove, 1994 and 1995). The subsequent interlarvalperiods are predicted using k = 0 . and k = 0 . (Hargrove, 1994 and 1995). Theinterlarval periods are therefore entirely temperature dependent and readily predictable. Useof this formula needs, however, to be tempered by a knowledge of the large standard deviationpresented in Hargrove (1994 and 1995), as well as the fact that larviposition usually takes placein the late afternoon, for G. morsitans (Potts, 1933, reported by Jackson, 1949, and Brady,1972), or afternoon shade in the case of
G. palpalis (Jackson, 1949, and Buxton, 1955). Thereexists an ever present risk in interpreting the output to have a precision any better than thedaily cohort and a discrete model may be more appropriate than a continuous one under thesecircumstances.What is the relevance of the above formula? Since the pupae present in the ground are unaf-fected by insecticide, the idea is to schedule follow-up operations shortly before the first fliesto eclode, after spraying, themselves mature and become reproductive. Subsequent sprays areconsequently scheduled two days short of the time to the first larva. This length of the sprayinterval is denoted σ in the formulae to follow. For temperatures of ◦ C and below, bothJackson’s curve and the data reported in Hargrove (2004) suggest that spraying two days be-fore the time to first larva (the one predicted using the Hargrove, 1994 and 1995, coefficients)is sufficient to ensure that none of the recently eclosed female flies ever give birth prior to beingsprayed. This observation is supported by the success of operations such as those of Kgori et.al (2006). Caution may, however, need to be exercised in the case of G. austeni , in that bothperiods could be shorter than the above formula predicts. This suspicion is based on the small . . Aerial Spraying of Tsetse Fly
G. austeni . For
G. brevipalpis one suspects longer periods based on diammetrically oppositearguments. The only relevance to aerial spraying in this latter case is economic, inefficiencybeing the only expected consequence.Pupae that are successfully larviposited remain in the ground for a period of time. The dura-tion of the period between larviposition and the emergence of the first imago is known as thepuparial duration and is denoted τ in this work. The puparial duration is also a function oftemperature and may be predicted using the formula τ = 1 + e a + bT k (Phelps and Burrows, 1969). For females, k = 0 . ± . , a = 5 . ± . and b = − . ± . (Hargrove, 2004). The fact that pupae usually eclode in the evening (Vale et. al. 1976)again begs the question of over-interpreting precision. There exists an ever present risk ininterpreting the output to have a precision greater than the daily cohort and a discrete model isagain indicated as being more appropriate than a continuous one.What is the relevance of the above formula? A cautious strategy advocates that spray cyclesshould be repeated until after the last pre-spray-larviposited pupae eclode and it is safer tocontinue until at least two sprays after their eclosion due to variation in the environment. If,under such circumstances, s denotes the total number of sprays, the total duration of the entirespraying operation is s − cycles. Again, caution needs to be exercised in that Parker (2008)reports that G. brevipalpis takes a little longer than the above formula predicts, whereas thepuparial durations of all other species are thought to lie within 10% of the value predicted. Forthe same conditions which produce a
G. morsitans puparial duration of 30 days,
G. brevipalpis has a puparial duration of 35 days. This has important implications for the aerial spraying of
G.brevipalpis . The shortest puparial duration is that of
G. austeni . G. austeni ’s puparial durationwas 28 days under the aforementioned conditions. These observations are noteworthy giventhe South African context of a sympatric,
G. brevipalpis - G. austeni population.Other aspects of tsetse population dynamics are also largely temperature-dependent(Hargrove, 2004), although soil-humidity can play an as, or more, important role in early mor-tality, depending on the species (Childs, 2009). While the effects of both temperature andhumidity on pupal mortality are known to be important, they vary profoundly according to theexact stage of development and are cumulative, rather than instantaneous. One might thereforesurmise that the age-dependence which characterises post-pupal mortality (observed by Har-grove, 1990 and 1993) is largely a consequence of pupal history. Fortunately, variables suchas soil-humidity and vegetation index have little to do with metabolic rate, hence the timingof spray cycles, and worst-case values might therefore be used. Alternatively, they can be re-garded to vary (and therefore be relevant) only in the medium to long term. In many regions,the level of humidity and temperature are sometimes linked. Pupae are therefore taken to dieoff at some temperature-dependent, daily rate, δ , and those flies which subsequently emergehave a probability γ of being female. Some comfort can be taken from the knowledge that theeffects of natural mortalities are very small in comparison to those due to aerial spraying. They Childs S. J. have little bearing on the overriding trends and, to a certain extent, this knowledge permits aprimitive approach. The question of pupal mortality can also be substantially avoided throughthe use of a steady-state eclosion rate, β . Hargrove (2004) suggests adult mortality to be pre-dictable, almost entirely temperature-dependent, and a knowledge of post-eclosion mortalitiesinfers the eclosion rate and vice versa, assuming the population to be in equilibrium. It is withthis wisdom in mind that the derivation will commence.During the first few hours subsequent to eclosion, the young, teneral fly’s exoskeleton is softand pliable, its fluid and fat reserves are at their lowest and a first blood meal is imperative forits survival. It is at this time that the insect is at its most vulnerable and it is also at this timethat its behaviour is least risk averse (Vale, 1974). Post-pupal survival can be defined as e − δ per day for the period between female eclosion and ovulation. Thereafter the female tsetse fly’schances of survival are higher and can be defined as e − δ per day.The accumulated mortality described above can be modelled linearly as δ ( t, T ) = δ tδ ( t − τ ) + δ τ δ [ t − ( τ − τ ) − τ ] + δ ( τ − τ ) + δ τ for t < τ τ ≤ t < τ − τ + τ t ≥ τ − τ + τ , where t denotes age, for the present. For the purposes of later brevity, it is convenient to definea second cumulative mortality, one which commences at eclosion. If t denotes the time elapsedsince eclosion, then δ ∗ ( t, T ) = (cid:26) δ tδ [ t − ( τ − τ )] + δ ( τ − τ ) for t < τ − τ t ≥ τ − τ , is the aforementioned mortality desired. Some actual values of the various mortalitites, theirassociated temperatures and the justification for their selection can be found in Childs (2011).The spray-survival rate will, in contrast, be assumed to be independent of age, whereas, in ac-tual fact, a mechanism in gravid females exists whereby lipophilic toxins are excreted, sacrific-ing larvae in utero for the mother fly’s own survival. The older the fly, the more developed thismechanism is usually found to be. The dependence of spray efficacy on age has been ignoredfor two reasons. Firstly, one might reason that a simple trade-off exists between a fly livingand a larva dying and further pregnancies should similarly terminate in spontaneous abortion.Secondly, the spray-survival rate is a small number. Whatever the exact value of φ may be forthese older flies, the value of φ s , or similar, should ensure that such cohorts are decimated bythe end of the operation. Of course, ignoring gravid female resistance to the insecticide mayresult in slightly altered eclosion rates and the use of inappropriate natural mortalities. Somecomfort can be taken from the knowledge that the effects of natural mortalities are very smallin comparison to those due to aerial spraying. They have little bearing on the outcome. The emphasis in the derivation is on the female population, since the male tsetse fly’s role inreproduction is relatively insignificant. At least one male is always assumed to be available and . . Aerial Spraying of Tsetse Fly original pupae.higher generationThird andDaughters of Daughters offemale female Original Original flies. pupae.daughters.original flies.
Figure 1: The tsetse population is deconstructed into logically natural categories for the pur-poses of formulation.Both the original population and the subsequent generations which survive the spraying, maybe thought of as ultimately deriving from two distinct sources (refer to Fig. 1). These originsare, however, neither distinct, nor relevant by the third generation. To understand why this isso, cognizance should be taken of the fact that the number of flies in a given cohort dependson the number of mothers which survived long enough to successfully larviposit, not just onthe cohort’s own chances of survival since larviposition. There are two distinctly differentancestral origins for the second generation, since mothers existed as either a pupa, or a fly atthe commencement of spraying. The same is not true for third generation cohorts since theonly generalisation that can be made is that all mothers simply eclosed sometime, subsequentto one puparial duration into the operation, and happened to larviposit on the same day.
Childs S. J. symbol unit descriptionN ~ original, steady-state, equilibrium number of females η - probability of insemination β flies ~ − day − eclosion rate γ ~ flies − sex ratio δ day − puparial mortality δ day − post-puparial, pre-ovulatory mortality δ day − adult mortality τ days puparial duration τ days time between eclosion and first larva τ days interlarval period σ days length of a spray cycle s sprays total number of sprays φ - probability of surviving a single spray ˘ t days time to eclosion since first spray E pre-spray (˘ t ) flies time- ˘ t -ecloding cohort which existed aspupae at the commencement of spraying E a (˘ t ) flies time- ˘ t -ecloding cohort, larviposited by original,adult females during spraying (second generation) E ps (˘ t ) flies time- ˘ t -ecloding cohort, larviposited by original, femalepupae which existed as such at the commencement ofspraying (second generation) E is (˘ t ) flies time- ˘ t -ecloding cohort, immediately descended frominter-spray-deposited, female pupae (third generationand higher)Table 1: Symbols used and the quantities they denote. . . Aerial Spraying of Tsetse Fly remaining pupaeand flies.Formulae fororiginal flies andoriginal pupae.Daughters ofSchematicdiagramsremaining pupaeand flies.Formulae for remaining pupaeand flies.Formulae forhigher generationdaughters.Third andSchematicdiagramremaining flies.Formula forfemale flies.Original remaining fliesFormula forpupae.female Original τ ?(s−1) < σ Final tally offemale fliesand pupae.Flow of Variablesand SequenceSequence OnlyFlow of Variables Only NoYes
Figure 2: Flow chart of both the strategy for obtaining the complete set of formulae and thecalculation itself.The observation of two initially distinct origins is, in some sense, an artefact of having seperate,‘start-up’ pupal and fly populations, something which was rendered possible by the assumption0
Childs S. J. of an equilibrium. Yet the existance of that equilibrium prior to spraying is no artefact.For the reason that there are originally two distinct sources, it is expedient to deconstruct thesurviving population into the following, categories, in the derivation:1. Original, female flies which existed as such at the commencement of spraying.2. Original, female pupae which existed as such at the commencement of spraying.3. Daughters larviposited after the commencement of spraying, including:(a) Daughters of 1 above.(b) Daughters of 2 above.(c) Third generation and higher daughters of this self-same category, 3 above.Fig. 2 summarises the strategy for both formulation and calculation.
The actual flies themselves, as distinct from pupae, which survive the last spray are usually ofno real consequence to the outcome of spraying (Childs, 2011). This is not necessarily the casein instances in which the operation has been curtailed, or kill rates are low. The state of theadult fly population during spraying is, nonetheless, what ultimately determines the size of theremnant population at the end of spraying.
How many of the original flies survive spraying? If a fly survives one spray cycle with proba-bility φ , then the probability that it survives s consecutive sprays is φ s , assuming the probabilityof survival for each spray is identical. The fly must also survive the normal hazards of life forthe ( s − σ days from the first through to the last spray. The maximum number of femalesfrom the original population which survive to the conclusion of spraying, is therefore N e − δ ( s − σ φ s , (1)in which N is the original, steady-state, equilibrium number of females prior to spraying and δ is the worst-case-scenario, adult mortality rate.A simplification made in this formula is that no age distribution profile has been assumed forthe natural mortality. It should, however, be pointed out that, while the original proportion offemales which have not yet ovulated is significant, the duration of the time preceding ovulationis insignificant when compared to the length of the spraying operation itself. The average . . Aerial Spraying of Tsetse Fly φ , is a smallnumber. The chances of any of the original flies surviving several spray cycles are usuallypractically zero. How many flies initially eclosed from such pupae? Assuming a population which was in equi-librium at some mean temperature prior to the commencement of spraying, the daily numberof flies ecloding from pre-spray-deposited pupae, is a constant E pre-spray = βN, in which β is the steady-state eclosion rate previously described. Such flies continue to eclodefor a period of one puparial duration subsequent to the commencement of spraying.How many spray cycles will a given cohort be subjected to? The total number of spray cyclesthat a fly will be subjected to is determined by its day of eclosion, ˘ t . The time, during theoperation, that it spent above ground is the length of the operation less the time before eclosion,that is σ ( s − − ˘ t . The total number of insecticidal spraying cycles the fly will be subjectedto is one more than the number of times a complete spray cycle fits into the period spent aboveground. More succinctly, floor (cid:26) σ ( s − − ˘ tσ (cid:27) + 1 , where floor { . } is the greatest integer function and ˘ t is the time from the first spray cycle toeclosion. The spray-survival rate, φ , must be applied this many times, so that the fraction offlies which survives the entire operation is φ floor n σ ( s − − ˘ tσ o +1 . What of natural mortality? The flies die off naturally at some age-dependent mortality, δ ∗ ( t − ˘ t, T ) .What is the total number of flies of such origins remaining at the end of spraying? Collectingthe above three observations, the number of female flies surviving at some later time, t , is γ τ ( T ) X ˘ t =1 E pre-spray e − δ ∗ ( t − ˘ t,T ) φ floor n σ ( s − − ˘ tσ o +1 . At the completion of the operation, the total time elapsed is ( s − σ , and taking cognisance ofthe fact that the number of flies ecloding from pre-spray-deposited pupae must be constant for2 Childs S. J. a population which was in equilibrium at some mean temperature, prior to the commencementof spraying, yields γβN τ ( T ) X ˘ t =1 e − δ ∗ (( s − σ − ˘ t,T ) φ floor n σ ( s − − ˘ tσ o +1 . (2)What if the length of the operation is less than one puparial duration? In the event that theoperation is curtailed to such an extent that the cycles are terminated before τ , then not allthe original pupae have the opportunity to eclode as flies and the remaining fraction contributeto the pupal population, still in the ground at the end of spraying. Under these circumstances,the above summation is truncated so that the upper limit, τ ( T ) is replaced with σ ( s − . Afurther, extraordinary, pupal contribution must then also be added to the tally of pupae, stillpresent in the ground at the end of spraying. The last of the category “original pupae” eclode the moment one puparial duration since thecommencement of spraying has elapsed. All the flies ecloding thereafter are of an inter-spray-larviposited origin. If, however, spraying is curtailed to the extent that its duration is less thanone puparial duration, then none of this latter category ever eclode. Under such circumstancesthey exist solely as pupae, still in the ground at the end of spraying.Otherwise, the survival of flies ecloding from inter-spray pupae can be deduced by similarreasoning to the aforementioned case, one difference being that the number of emergent fliesis no longer constant over time (the ecloding population no longer being in equilibrium, orconstant). Contributions to the time- ˘ t -ecloding cohort arise as a result of pupae which werelarviposited ˘ t − τ days before. The number of such flies at the conclusion of spraying is γ σ ( s − X ˘ t = τ ( T )+1 E (˘ t ) e − δ ∗ (( s − σ − ˘ t,T ) φ floor n σ ( s − − ˘ tσ o +1 . (3)The pupae were deposited by the previous, two survival categories and the inter-spray pupae,themselves. That is, E (˘ t ) = E a (˘ t ) + E ps (˘ t ) + E is (˘ t ) , in which the time- ˘ t -ecloding cohorts are defined in Table 1, according to their ancestral origins.What of the ‘knock-down’ approach to the aerial spraying of tsetse? For instances in whichthe duration of an operation has been curtailed to the length of one puparial duration, or less,there is clearly no such contribution to flies, only pupae. Under such circumstances this secondand higher generation category of flies may be completely disregarded. They need only beconsidered from the point of view of a pupal population. . . Aerial Spraying of Tsetse Fly E a (˘ t ) Contribution to E (˘ t ) This is the contribution attributed to larviposition by original, adult females, those which ex-isted as such prior to the commencement of spraying and which larviposit during the operation.By far the largest mass of the pupae larviposited by original adults are larviposited during thefirst spray cycle, between the first and second sprays. Their eclosion commences immediatelyafter the last of the pre-spray-larviposited pupae have emerged. They and a varying propor-tion of the pupae larviposited during the second cycle, eclode during the aerial spraying, fora completed operation. The majority of them are exposed to the last, or last two, sprays, forsuch a completed operation. Terminating the operation one spray short allows all the pupaelarviposited in the second cycle and a varying proportion of those larviposited during the firstcycle, never to be sprayed, in theory.How many original mothers larviposit on a given day during the spraying? If there were N females prior to spraying, to assume that all have already ovulated is a wost-case scenario,therefore a safe assumption. Inseminated females, all ηN of them, are expected to deposit onepupa every τ days; that is, the larviposition of ηNτ pupae every day. τ fliespupae Ε a t η N Figure 3: Schematic diagram of second generation flies eclosed from pupae that werelarviposited during spraying by original, pre-spray-existing adults.How many of these potential mothers survive until a given day into the spraying operation?The proportion of these adult mothers which survive naturally as long as ˘ t − τ into sprayingis e − δ (˘ t − τ ) and they are, in turn, subjected to floor (cid:26) ˘ t − τ − σ (cid:27) + 1 sprays (by contemplating Figure 3 and assuming larviposition is successfully accomplishedshortly before spraying on the day in question). If a fly survives one spraying cycle withprobability φ , then the probability that it survives the above number of cycles is φ floor n ˘ t − τ − σ o +1 , Childs S. J. always assuming the probability of survival for each cycle is identical.How many of their daughters, in turn, survive to eclode? Taking natural mortality into account,the proportion of their pupae which survive to eclode is e − δ τ .Hence, the final expression E a (˘ t ) = η Nτ e − δ (˘ t − τ ) − δ τ φ floor n ˘ t − τ − σ o +1 H (˘ t − τ ) , (4)in which H is the version of the Heaviside step function with H (0) = 0 . One prerequisitefor such E a contributions to a second generation of flies is a restriction on the eclosion of thecohorts, ˘ t > τ (again, by contemplating Figure 3). Otherwise they need only be consideredfrom the point of view of a pupal population. E ps (˘ t ) Contribution to E (˘ t ) This is the contribution attributed to larviposition by mothers which existed as pupae at thecommencement of spraying. Many such pupae eclode subsequent to the last spray, even fora completed operation. Under normal circumstances, this category may be thought of as theproblem category. How many such mothers come into existence on a given day during theoperation? The number of potential mothers, ecloding daily (for a limited period), from pre-spray-deposited pupae that will subsequently be inseminated, is γηE pre-spray = γηβN, in which β is the steady-state, maximum possible, eclosion rate previously described, N is theoriginal, steady-state, equilibrium number of females prior to spraying, γ is the probability ofbeing female and η is the probability of insemination. Mothers of this category cease eclodingone puparial duration into the operation.What is the subsequent mortality of these mothers? These pre-spray-larviposited mothers suffera daily natural mortality of δ ∗ ( τ + iτ , T ) and, by contemplating Figure 4, are subjected to atotal of floor (cid:26) ˘ t − τ − σ (cid:27) − floor (cid:26) ˘ t − τ − τ − iτ − σ (cid:27) sprays, this being the difference between the total number of sprays to larviposition and thetotal number of sprays up to the day before the mother’s eclosion. If a fly survives one sprayingcycle with probability φ , then the probability that it survives the above number of cycles is φ floor n ˘ t − τ − σ o − floor n ˘ t − τ − τ − iτ − σ o , always assuming the probability of survival for each cycle is identical and that larviposition willbe successfully accomplished before spraying on relevant days. The survival of such mothersis therefore readilly quantifiable in terms of the above. . . Aerial Spraying of Tsetse Fly τ τ + i τ fliespupae Ε ps Ε pre−spray η t γ Figure 4: Schematic diagram of second generation flies ecloding from pupae that werelarviposited by flies from original pupae that existed as such at the commencement of spraying.What are the temporal restrictions on the eclosion of the second generation cohorts these moth-ers produce? By contemplating Figure 4, the first requirement for second-generation descentfrom such mothers, is a restriction on the cohorts to ˘ t > τ + τ . The mothers would otherwisehave had to have eclosed prior to the first spray, a fact which would exclude them from thecategory presently under consideration, altogether. Secondly, only for a limited period of time(one puparial duration) do mothers which originate from pre-spray-deposited pupae continueto emerge from the ground. That is, if all τ s are integer cohorts, ≤ ˘ t − τ − τ − iτ ≤ τ i = 0 , , . . . , yielding a restriction on i , i ≤ floor (cid:26) τ (˘ t − τ − τ − (cid:27) , and completing those on the time of eclosion, τ + τ + iτ < ˘ t ≤ τ + τ + iτ . Lastly, only an e − δ τ fraction of the pupae survive to eclode. Collecting all of the aboveinformation E ps (˘ t ) = γηβN floor n τ (˘ t − τ − τ − o X i =0 (cid:20) e − δ ∗ ( τ + iτ ,T ) − δ τ φ floor n ˘ t − τ − σ o − floor n ˘ t − τ − τ − iτ − σ o (cid:2) − H (˘ t − τ − τ − iτ ) (cid:3) H (˘ t − τ − τ − iτ ) i , (5)in which H is the version of the Heaviside step function with H (0) = 0 . Notice that thelast Heaviside factor becomes a precaution once i is greater than zero, since it is derived fromthe same inequality used for the restriction on i . Clearly there is no E ps contribution to fliesfor instances in which the duration of the operation has been curtailed to, or below, the timebetween parturition and the production of the first larva, although pupae of this category willcertainly exist.6 Childs S. J.
Modifications for a Continuous Model
What if a continuous rather than discrete model were to be entertained? What if the τ s had notbeen rounded off to integer cohorts? What if they involved fractions of a day, instead? The i would, nonetheless, still be integers in such a model although, by analogous reasoning to thatabove, < ˘ t − τ − τ − iτ < τ i = 0 , , . . . , This would lead to a modification of the upper bound in the above summation, one based on max { i } < τ (˘ t − τ − τ ) , as well as the replacement of (cid:2) − H (˘ t − τ − τ − iτ ) (cid:3) with H ( − ˘ t + 2 τ + τ + iτ ) . The new switch differs in that it turns off when the argument zero, instead of immediatelyabove it. So far as the number of sprays is concerned, ‘the-moment-before’ replaces the ‘the-day-before’ of the discrete case, so that the relevant factor becomes φ floor n ˘ t − τ σ o − floor n ˘ t − τ − τ − iτ σ o . E is (˘ t ) Contribution to E (˘ t ) This is the contribution attributed to female flies descended from the mothers which were them-selves larviposited during spraying. They are the immediate descendants of the E a category,the E ps category, or this very same E is category itself. The first prerequisite for such third, orgreater, generation contributions is that ˘ t > τ + τ (by contemplating Figure 5). τ τ τ + i τ fliespupae η Ε Ε is t γ Figure 5: Schematic diagram of flies emerging from inter-spray pupae that are descended frominter-spray pupae themselves (third generation and higher). . . Aerial Spraying of Tsetse Fly τ + 1 , (just whenthe pre-spray pupae, have ceased to eclode). That is, if the various τ s are integer cohorts, τ + 1 ≤ ˘ t − τ − τ − iτ i = 0 , , ... , yielding a restriction on i , i ≤ floor (cid:26) τ (˘ t − τ − τ − (cid:27) . The probability that such mothers survive the relevant number of spray cycles is formulatedin the same way as in the previous case; as is the natural mortality. The pupal mortality ofthe mothers and the mortality of the grandmothers is already taken care of by the E a and E ps categories. Taking cognizance of the fact that the emergent population is not constant over timeunder such circumstances, E is (˘ t ) = γη floor n τ (˘ t − τ − τ − o X i =0 h E (˘ t − τ − τ − iτ ) e − δ ∗ ( τ + iτ ,T ) − δ τ φ floor n ˘ t − τ − σ o − floor n ˘ t − τ − τ − iτ − σ o H (˘ t − τ − τ − iτ ) (cid:21) . (6)Notice, once again, that the Heaviside factor becomes a precaution once i is greater than zero,since it is derived from the same inequality used for the restriction on i . There is no E is contribution to flies for instances in which the duration of the operation is equal to, or below,the length of two puparial durations and the time to the first larva. In fact, normal circumstancesmake it difficult to imagine the category E is as ever having eclosed by the end of spraying,therefore as having any relevance to the total fly tally at all. E is may usually be neglected inthe fly calculation. Neither is there any E is contribution to pupae for instances in which theduration of the spray operation is equal to, or below, the length of time between parturitionand the production of the first larva. E may, in practice and under normal circumstances, beassumed to have only two contributions, E a and E ps .From this point on the origins of the inter-spray pupae are no longer relevant. Generationshigher than the third are accounted for through recursion, in theory. In practice, the relativedurations of the spraying operation, the puparial stage and the time between eclosion and theproduction of the first larva are such that it is difficult to imagine a scenario involving a fourthgeneration, consequently any recurrence relation at all. Modifications for a Continuous Model
What if a continuous rather than discrete model were to be entertained? What if the τ s had notbeen rounded off to integer cohorts? What if they involve fractions of a day, instead? The i would still be an integer in such a model, however, by analogous reasoning to that above, τ < ˘ t − τ − τ − iτ i = 0 , , ... , Childs S. J. leading to a replacement of the upper bound in the above summation, one based on max { i } < τ (˘ t − τ − τ ) . So far as the number of sprays is concerned, ‘the-moment-before’ replaces the ‘the-day-before’of the discrete case, so that the relevant factor becomes φ floor n ˘ t − τ σ o − floor n ˘ t − τ − τ − iτ σ o , as in the previous case. The total number of female pupae, which are still in the ground at the end of spraying and which will survive to eclode , is the γ fraction of flies destined to begin ecloding as a series ofcohorts immediately subsequent to the last spray. That is, starting at σ ( s −
1) + 1 , and endingwith σ ( s −
1) + τ , in the discrete case. Contributions to this pupal population arise as a resultof female pupae larviposited after the commencement of spraying, in a completed operation.They may be categorized as:1. Daughters of original, female flies.2. Daughters of original, female pupae.3. Third generation and higher daughters of females which were larviposited after the com-mencement of the spraying.A fourth contribution, γβN τ ( T ) X ˘ t = σ ( s − e − δ ∗ (( s − σ − ˘ t,T ) φ floor n σ ( s − − ˘ tσ o +1 H ( τ − ( s − σ ) , that due to the pressence of original pupae, must also be taken into account in an operationwhich has been curtailed to the extent that its duration is less than one puparial duration.Otherwise, the total number of such female pupae remaining in the ground at the end of spray-ing and which will survive to eclode , is γ σ ( s − τ ( T ) X ˘ t = σ ( s − (cid:2) E a (˘ t ) + E ps (˘ t ) + E is (˘ t ) (cid:3) . (7)As it transpires, one of the above categories is far and away more important than any of theothers in a completed operation. The pre-eminent category is the second one above, the pupae . . Aerial Spraying of Tsetse Fly E ps eclosion after spraying. The implications of thisdiscovery are that, under certain conditions, one formula can be adapted to provide a goodestimate of the outcome of aerial spraying. This fact is revealed when considering that thereis only one O ( φ ) contribution and this observation is further corroborated by the algorithm ofChilds (2011). A compositional analysis of the origins of female pupae, still in the ground,reveals that summation of the second term in the above summation formula is a good indicatorof the entire outcome of spraying, given a kill rate of 99.9%, or better. It accounts for well over90% of the pupal population at a kill rate of 99%. At ◦ C , four sprays which define three spray cycles, of length 14 days each, are required. Theaerial spraying scenario at this temperature is slightly simplified and lends itself favourably tomanual calculation for two reasons. The first is that the time to the second last spray is, forall practical purposes, exactly one puparial duration. All the pupae deposited during the firstspray cycle therefore eclode during the last spray cycle and the pupae still in the ground atthe end of the operation were deposited during the second and third spray cycles. The secondreason is that the spray operation ends early from a metabolic point of view, meaning that athird generation never exists during spraying, as is so often the case. For the aforementionedreasons any problems with the formulae should be relatively easy to detect. Surviving Flies
Although the number of spray cycles is relatively small, the number of original adults whichsurvive is still insignificant, it being of O ( φ ) . Eq. 1 can accordingly be dismissed as neg-ligeable. This is usually the case in a completed operation. Those pre-spray-deposited pupaewhich eclode for the duration of the second cycle must survive only the last two sprays, insteadof three, and Eq. 2 therefore becomes γβN X ˘ t =15 e − δ ∗ (42 − ˘ t, φ floor { − ˘ t } +1 + O ( φ ) . The only categories left to consider are the second and third generations, calculated accordingto Eq. 3. Both the E is and E ps terms can be dismissed as irrelevant to the fly population, sincethe length of the operation is shorter than the time from parturition to the production of the firstlarva. Relevant second generation flies are therefore all descended from the original, pre-spray-existing flies, those which survived the first spray. This E a contribution is also a significant, O ( φ ) contribution, since the pupae were deposited during the first spray cycle and eclode forthe duration of the last spray cycle. Eq. 3 therefore becomes γ X ˘ t =29 η N e − . t − − . · φ floor { ˘ t − − } +1 · · e − δ ∗ (42 − ˘ t, φ floor { − ˘ t } +1 + O ( φ ) . Childs S. J.
If one very crudely approximates e − δ ∗ (42 − ˘ t, as . , e − . t − − . · as . and e − δ ∗ (42 − ˘ t, as . , in a ‘back-of-an-envelope’ fashion and based on the relevant Childs(2011) mortalities, the sum of the preceding two expressions becomes × . × N (cid:18) . × . × × . × . (cid:19) φ + O ( φ ) , using a β of . . Surviving Pupae
Only Eq. 7 is relevant to the pupal outcome, since the operation is not shorter than one puparialduration. The E is contribution can be dismissed as irrelevant, since the length of the sprayoperation is shorter than τ + τ . In order for pupae to contribute to an E a eclosion subsequentto the completion of the operation, they must have been larviposited in the second or thirdcycles (those which were larviposited in the first cycle have already eclosed by the end ofspraying). This means that their mothers were sprayed at least twice and accordingly theyconstitute an O ( φ ) contribution. There is only one significant, O ( φ ) contribution to the pupalpopulation; that destined to give rise to an E ps eclosion. The pupal outcome can therefore becrudely formulated in terms of Eq. 7 as η γ N β X ˘ t =43 floor { (˘ t − − − } X i =0 h e − δ (28+16+ i , φ floor { ˘ t − − } − floor { ˘ t − − − i − } (cid:2) − H (˘ t − − − i (cid:3) H (˘ t − − − i i + O ( φ ) . This contribution arises as a result of mothers which eclode from original pupae during the firstand second cycles and which subsequently survive a single spray to larviposit in the secondand third cycles. Examination of the above formula reveals significant, O ( φ ) terms only for thecombinations of i and ˘ t represented in × . × N × . " X ˘ t =45 e − δ (44 , + X ˘ t =59 e − δ (44 , + X ˘ t =55 e − δ (44+10 , + X ˘ t =69 e − δ (44+10 , φ + O ( φ ) . Results
Assuming the same mortalites and the same 8 000 000, original, steady-state number of femalesas in Childs (2011), the estimated outcome for surviving, female pupae and flies is as presentedin Table 2. . . Aerial Spraying of Tsetse Fly φ flies log ( flies ) pupae log ( pupae ) ◦ C , based on low order terms.The results only differ from those of the Childs (2011) algorithm insofar as a more cautiouschoice of β has been made. Repeated spray cycles are scheduled at intervals two days short of the time between eclosionand the production of the first larva, σ = 10 .
061 + 0 . T − − , and, in a completed operation, continue until two sprays subsequent to the eclosion of the last,pre-spray-deposited, female pupae. That is, s = ceil (cid:26) e . − . T . σ (cid:27) + 2 , in which ceil { . } is the least integer function.Spray efficacy is found to come at a price due to the greater number of cycles necessitatedby cooler weather. The greater number of cycles is a consequence of a larger ratio of puparialduration to time-to-first-larva at lower temperatures. The prospect of a more expensive sprayingoperation at low temperature, due to a greater, requisite number of spray cycles is, however,one which is never confronted in the real world. In reality, one has to strive towards kill ratesand the only way such rates can be attained is by spraying at as low a temperature as possible(Hargrove, 2009).A refinement of the existing formulae for the puparial duration and the time between eclosionand the production of the first larva might be prudent in the South African context of a sympatric G. brevipalpis - G. austeni , tsetse population.2
Childs S. J.
The complete set of formulae derived for the performance of a tsetse population under condi-tions of aerial spraying is summarised as follows.
Pupae
The following are the contributions to female pupae, still in the ground at the end of spraying, which will survive to eclode . The number of such pupae which are daughters of original adultsis ηγ Nτ σ ( s − τ X ˘ t = σ ( s − e − δ (˘ t − τ ) − δ τ φ floor n ˘ t − τ − σ o +1 H (˘ t − τ ) . (8)The number of pupae which are daughters of original pupae is η γ N β σ ( s − τ X ˘ t = σ ( s − rmfloor n τ (˘ t − τ − τ − o X i =0 (cid:20) e − δ ( τ + τ + iτ ,T ) φ floor n ˘ t − τ − σ o − floor n ˘ t − τ − τ − iτ − σ o (cid:2) − H (˘ t − τ − τ − iτ ) (cid:3) H (˘ t − τ − τ − iτ ) i . (9)The number of pupae which are daughters of inter-spray pupae is ηγ σ ( s − τ X ˘ t = σ ( s − n τ (˘ t − τ − τ − o X i =0 h E (˘ t − τ − τ − iτ ) e − δ ( τ + τ + iτ ,T ) φ floor n ˘ t − τ − σ o − floor n ˘ t − τ − τ − iτ − σ o H (˘ t − τ − τ − iτ ) (cid:21) . (10)If the aerial spraying operation is curtailed to the extent that it is shorter than one puparialduration, then an additional category of pupae must be accounted for. The number of originalpupae, those larviposited before the commencement of spraying, which have not yet eclosedby the end of spraying, is γβN τ ( T ) X ˘ t = σ ( s − e − δ ∗ (( s − σ − ˘ t,T ) φ floor n σ ( s − − ˘ tσ o +1 H ( τ − ( s − σ ) . (11) Flies
The following are the contributions to female flies which survive to the conclusion of spraying.The maximum number of such surviving, female flies from the original population, is
N e − δ ( T )( s − σ φ s . (12) . . Aerial Spraying of Tsetse Fly γβN min { τ ( T ) , σ ( s − } X ˘ t =1 e − δ ∗ (( s − σ − ˘ t,T ) φ floor n σ ( s − − ˘ tσ o +1 . (13)The number of surviving female flies that eclosed from inter-spray-larviposited pupae andwhich will survive until after the last spray is γ σ ( s − X ˘ t = τ ( T )+1 (cid:2) E a (˘ t ) + E ps (˘ t ) + E is (˘ t ) (cid:3) e − δ ∗ (( s − σ − ˘ t,T ) φ floor n σ ( s − − ˘ tσ o +1 , (14)in which the respective time- ˘ t -ecloding cohorts are given by E a (˘ t ) = η Nτ e − δ (˘ t − τ ) − δ τ φ floor n ˘ t − τ − σ o +1 H (˘ t − τ ) ,E ps (˘ t ) = γηβN floor n τ (˘ t − τ − τ − o X i =0 (cid:20) e − δ ( τ + τ + iτ ,T ) φ floor n ˘ t − τ − σ o − floor n ˘ t − τ − τ − iτ − σ o (cid:2) − H (˘ t − τ − τ − iτ ) (cid:3) H (˘ t − τ − τ − iτ ) i ,E is (˘ t ) = γη floor n τ (˘ t − τ − τ − o X i =0 h E (˘ t − τ − τ − iτ ) e − δ ( τ + τ + iτ ,T ) φ floor n ˘ t − τ − σ o − floor n ˘ t − τ − τ − iτ − σ o H (˘ t − τ − τ − iτ ) (cid:21) . Modifications for a Continuous Model
Minor modifications, listed at the ends of Subsections 4.3.2 and 4.3.3, need to be made if thediscrete formulae are to be adapted to the continuous case. The summations over the cohorts P ... ˘ t =1 , P ... ˘ t = τ ( T )+1 and P ... ˘ t = σ ( s − would also be replaced with the integrals R ... , R ...τ ( T ) and R ...σ ( s − , respectively and among other things. The pupae which give rise to the E is eclosion have, at best, hardly come into existence, letalone eclosed, by the end of a completed operation. The E is contribution may therefore usuallybe regarded as irrelevant insofar as flies are concerned, whereas its contribution to the totalpupal tally is usually insignificant, at most. The category E is can be omitted from the Eq. 14fly formula for all reasonable circumstances, that is, unless the operation has been greatly4 Childs S. J. extended. Clearly, there is no E is contribution to the Eq. 14 fly formula for instances in whichthe duration of the operation is τ + τ , or less. There is also no Eq. 10 contribution involved inthe pupal tally for instances in which the duration of the operation is τ + τ , or less. Althoughthis varies from case to case, the Eq. 10 contribution is never significant unless the operation hasbeen greatly extended. The E ps term can be omitted from the Eq. 14 fly formula for instances inwhich the duration of the operation is τ + τ , or less. Take heed, however, that pupae, destinedto give rise to a future E ps eclosion (Eq. 9), are usually the most significant contribution, byfar, in a completed operation. The only circumstances for which these pupae are irrelevant isfor instances in which the duration of the operation involves a single cycle; that is, two spraysonly. The category E a can be omitted from the Eq. 14 fly formula for instances in which theduration of the operation is τ , or less. The Eq. 8 pupae, those destined to give rise to justsuch an E a eclosion, can never be irrelevant, since the larviposition of such pupae commencesimmediately after the first spray. A lengthly spray operation can, however, still render theircontribution insignificant, since such mothers become progressively decimated and the earlierpupal mass will eclode before the operation is complete. Generally, the more curtailed thespraying operation, the fewer contributing categories there are, with one exception: In the eventthat an operation is curtailed to the extent that its duration is less than one puparial duration, aproportion of the original pupae remain in the ground at the end of spraying. This complicatesmatters slightly. Under such circumstances the series of flies, Eq. 2, is truncated as specified inthe formula Eq. 13, the remainder being additional pupae which will survive, as quantified byEq. 11. These same circumstances render the entire Eq. 14 fly formula irrelevant.In a completed operation, by far the most significant category is that calculated in terms of Eq. 9above. These are pupae which are destined to give rise to an E ps eclosion once the operation iscomplete. The magnitude of this contribution is easy to see when one considers that many oftheir mothers (original pupae which eclode during spraying) will only be sprayed once (see Fig.4). Since all pupae are, by definition, never sprayed if they are still in the ground at the end ofspraying, the E ps lineage constitutes an O ( φ ) contribution to pupae. In contrast, the mothers ofpupae destined to give rise to an E a eclosion, after spraying, must be sprayed more than once ifthese daughters are still to be pupae by the end of the operation. Otherwise, they would alreadyhave eclosed (see Fig. 3). The Eq. 8 lineage may therefore be regarded as being of O ( φ ) significance. The Eq. 10 lineage (pupae destined to give rise to an E is eclosion, after spraying)also constitutes an O ( φ ) contribution (see Fig. 5). A large number of the pupae destined togive rise to an E ps eclsoion will therefore always be of a lower order than those destined to giverise to an E a or an E is eclosion, in a completed operation. Notice that all lineages which existas flies at the time of the last spray will be of order O ( φ ) , or higher, in a completed operation.This is easy to see when one considers that all these flies must, by definition, be subjected tothe last spray. The length of a complete operation means that their lineage must also have beensprayed at least once during the operation. That the daughters of the original pupae, Eq. 9, area good forecast of the outcome of a completed operation, given a kill rate of 99.9% or better, isfurther corroborated by the algorithm of Childs (2011).Given the high kill rates attainable, it is not surprising that the outcome, for flies (as distinctfrom pupae), is largely determined by the size of the emergent population which was onlysubjected to the last two sprays, in a completed operation. This is why the proportion of . . Aerial Spraying of Tsetse Fly E a eclosion during the last two cycles of thatexample. The actual flies, themselves, which survive the last spray of a completed operationare, however, of no real consequence to the outcome. Under such circumstances, pupae, still inthe ground at the end of spraying, are identified as the main threat to successful control by aerialspraying. The outcome, for kill rates of 99.9%, or higher, was shown in Childs (2011) to bealmost exclusively dependent on the immediate descendants of the original pupae, those pupaewhich were present at the commencement of spraying. Even at kill rates as low as 99% thisEq. 9 category still constitutes around 90% of the surviving female population in a completedoperation while the Eq. 8 contribution accounts for less than 10% of the total pupal population(Childs, 2011).If, however, operations are halted one or more sprays short, these generalisations can not bemade. Not only is the recently-eclosed fly population still significant, there is also a fairly largepupal population descended from the original adults. The contribution of the Eq. 8 categoryand others, becomes significant. The total predominance of Eq. 9 does not exist. Some of thepupae destined to give rise to an E a eclosion become a significant, O ( φ ) contribution, as dosome of the flies that eclosed from original pupae. While flies arising from the E a eclosion,itself, are only of order O ( φ ) , it should also be remembered that they were larviposited duringthe first cycle, when the population of original, adult flies was still strong. The formulae Eq. 8,Eq. 9, Eq. 12, Eq. 13 and part of Eq. 14 all need to be considered if the operation is halted onespray short. The Eq. 9 contribution is always significant and that of Eq. 12 insignificant. Onlythe E a term in Eq. 14 is still relevant under such circumstances. If the last spray falls closeto a full cycle’s length from the one-puparial-duration mark, then the Eq. 13 flies will be lesssignificant, despite the fact that they are an O ( φ ) contribution, and one will find almost a cyclesworth of O ( φ ) , E a eclosion (Eq. 14). If, on the other hand, the last spray falls close to theone-puparial-duration mark, then one will find almost a cycles worth of O ( φ ) , Eq. 13 eclosionand the entire Eq. 14 contribution can be dismissed as insignificant. They will mostly still be inthe ground as Eq. 8 pupae. Of course, a proportion of the alleged pupae might also actually beaging flies, under such circumstances, given the mechanism whereby mature, gravid femalesexcrete lipophilic toxins to sacrifice larvae in utero for their own survival.If the operation is halted two, or more, sprays short, the Eq. 14 fly formula falls away en-tirely while an Eq. 11 contribution comes into existance. Under these circumstances, onlythe formulae Eq. 8, Eq. 9, Eq. 11, Eq. 12 and Eq. 13 are relevant. The significance ofthe Eq. 11 contribution will be proportional to the time between the last spray and the one-puparial-duration mark. Of course, the original, surviving adults (Eq. 12) can almost alwaysbe regarded as insignificant, they being an O ( φ s ) contribution. That is, for all except the mostseverely curtailed operation. It is important to remember that the formulae calculate a theoretical outcome based on thepremise that no practical problems will be encountered in the field. The idea of this work has6
Childs S. J. been to create a simple arithmetic tool which can be used to establish conditions sufficient fora successful operation in the context of a closed tsetse population. Hargrove (2005) quantifiesthe dangers in allowing the smallest of founding populations to survive and re-invasion is anever present threat which will ultimately compromise even the most successful aerial sprayingoperation. A cursory inspection of the Rogers and Robinson (2004) study (based on the Fordand Katondo, 1977, maps) suggests that most tsetse populations cannot be considered closed.The total extent of habitat is a further cause for concern. Even the extant, forest-dwelling,tsetse populations of South Africa cannot be considered closed and extend beyond its borders(Hendrickx, 2007). By far the biggest threat to any aerial spraying operation on mainlandAfrica is re-invasion from adjacent, untreated areas. Closed populations need to be created bytemporary barriers of odour-baited targets such as the one used successfully by Kgori et al.,2006. Childs (2010) and Esterhuizen et al. (2006) comprehensively researched the design ofsuch odour-baited, target barriers for
G. austeni and
G. brevipalpis ; albeit mostly from a pointof view of a control in its own right. In Childs (2011), the same model was re-run with a morestringent,
G. austeni isolation standard than that used for control in Childs, 2010.Quantifying spray efficacy represents a further problem. Temperature doesn’t only effect theaerial spraying of tsetse, through its reproductive cycle and general population dynamics.Cooler weather is preferred for aerial spraying from a point of view of spray efficacy (Hargrove,2009). Very high kill rates usually (though not always) come about as a result of the sinkingair associated with cooler weather. It favours the settling of insecticidal droplets. AlthoughDu Toit (1954) makes mention of the sustained down draught from a slow-moving helicopter,there are obviously distinct disadvantages to such a method of insecticide application. Theinherent toxicity of deltamethrin and many other pyrethroids also decreases with temperature,contrary to the toxicity of most insecticides. The effects of temperature on spray efficacy arenot modelled. For that matter, neither are the effects of anabatic winds, nor the protection af-forded by the forest canopy and multifarious other variables relevant to spray efficacy. Sprayefficacy is usually measured in the field, with hindsight, rather than predicted.It has been assumed that gravid female resistance to the insecticide can be ignored. Although itmay be tempting to consider the model poorer for this lack of detail, it may not be a defficiencyof any consequence, since a simple trade-off exists between a fly living and a larva dying.Further pregnancies during the operation should, similarly, terminate in spontaneous abortion.Another point to bear in mind is that the spray-survival rate is a small fraction. Whateverthe exact value of φ may be for the older flies, the value of φ s , or similar, should ordinarilyensure that they have been decimated. Gravid female resistance is sure to result in the use ofslightly altered eclosion rates and inappropriate natural mortalities, however, some comfort canbe taken from the knowledge that the effects of natural mortalities are very small in comparisonto those due to aerial spraying. They have little bearing on the outcome. . . Aerial Spraying of Tsetse Fly The original conception of this project was entirely John Hargrove’s and it was under his direc-tion that the work originally commenced. The author is indebted to Johan Meyer and NatalieLe Roux of the University of the Free State for hosting this work. Lastly, both the editor andthe anonymous reviewers are thanked for their suggestions and guidance.
References [1] R. Allsopp. Control of tsetse flies (
Glossina spp) using insecticides: a review and futureprospects.
Bulletin of Entomological Research , 75:1–23, 1984.[2] Anonymous. Notes for field studies of tsetse flies in East Africa. Technical report, EastAfrica High Comission, Nairobi, 1955.[3] Anonymous. Human African trypanosomiasis (sleeping sickness).
World Health Organ-isation , Fact Sheet No. 259, 2012.[4] H. J. Barclay and M. J. B. Vreysen. A dynamic population model for tsetse (Diptera:Glossinidae) area–wide integrated pest management.
Population Ecology , 53(1):89–100,2010.[5] J. Brady. Spontaneous, circadian components of tsetse fly activity.
Journal of InsectPhysiology , 18:471–484, 1972.[6] P. A. Buxton.
The Natural History of Tsetse Flies . London School of Hygiene andTropical Medicine Memoir No. 10. H. K. Lewis & Co. Ltd., 1955.[7] S. J. Childs. A model of pupal water loss in
Glossina . Mathematical Biosciences , 221:77–90, 2009.[8] S. J. Childs. The finite element implementation of a K.P.P. equation for the simulationof tsetse control measures in the vicinity of a game reserve.
Mathematical Biosciences ,227:29–43, 2010.[9] S. J. Childs. Theoretical levels of control as a function of mean temperature and sprayefficacy in the aerial spraying of tsetse fly.
Acta Tropica , 117:171–182, 2011.[10] R. Du Toit. Trypanosomiasis in Zululand and the control of tsetse flies by chemical means.
Onderstepoort Journal of Veterinary Research , 26(3):317–387, 1954.[11] J. Esterhuizen, K. Kappmeier Green, E. M. Nevill and P. Van Den Bossche. Selective useof odour–baited, insecticide–treated targets to control tsetse flies
Glossina austeni and
G.brevipalpis in South Africa.
Medical and Veterinary Entomology , 20:464–469, 2006.8
Childs S. J. [12] E. M. F`evre, P. G. Coleman, S. C. Welburn and I. Maudlin. Reanalzing the 1900–1920sleeping sickness epidemic in Uganda.
Emerging Infectious Diseases , 10(4):567–573,2004.[13] J. Ford and K. M. Katondo. The distribution of tsetse flies in Africa (3 maps).
OAU, Cook,Hammond & Kell, Nairobi , 1977.[14] R. H. Gooding and E. S. Krafsur.
The Trypanosomiases . Editors: I. Maudlin, P. H.Holmes and P. H. Miles. CABI publishing, Oxford, U.K., 2004.[15] J. W. Hargrove. Age–dependent changes in the probabilities of survival and capture ofthe tsetse,
Glossina morsitans morsitans
Westwood.
Insect Science and its Applications ,11(3):323–330, 1990.[16] J. W. Hargrove. Age dependent sampling biases in tsetse flies (
Glossina ). Problems as-sociated with estimating mortality from sample age distributions.
Management of InsectPests: Nuclear and Related Molecular and Genetic Techniques. International Atomic En-ergy Agency, Vienna. , pages 549–556, 1993.[17] J. W. Hargrove. Reproductive rates of tsetse flies in the field in Zimbabwe.
PhysiologicalEntomology , 19:307–318, 1994.[18] J. W. Hargrove. Towards a general rule for estimating the day of pregnancy of field caughttsetse flies.
Physiological Entomology , 20:213–223, 1995.[19] J. W. Hargrove.
The Trypanosomiases . Editors: I. Maudlin, P. H. Holmes and P. H. Miles.CABI publishing, Oxford, U.K., 2004.[20] J. W. Hargrove. Extinction probabilities and times to extinction for populations of tsetseflies
Glossina spp. (Diptera: Glossinidae) subjected to various control measures.
Bulletinof Entomological Research , 95:1–9, 2005.[21] J. W. Hargrove.
By communication . 2009.[22] Guy Hendrickx. Tsetse in Kwazulu Natal – an update –. Technical report, Agriculturaland Veterinary Intelligence Analysis, 2007.[23] C.H.N. Jackson. The biology of tsetse flies.
Biological Reviews , 24:174–199, 1949.[24] P. M. Kgori, S. Modo and S. J. Torr. The use of aerial spraying to eliminate tsetse fromthe Okavango Delta of Botswana.
Acta Tropica , 99:184–199, 2006.[25] S. G. A. Leak.
Tsetse Biology and Ecology . CABI, 1999.[26] M. Y. Motloang, J. Masumu, P. Van Den Bossche, P. A. O. Majiwa and A. A. Latif.Vector competance of field and colony
Glossina austeni and
Glossina brevipalpis fortrypanosome species in KwaZulu–Natal.
Journal of the South African Veterinary Associ-ation , 80(2):126–140, 2009.[27] A. Parker.
By communication . 2008. . . Aerial Spraying of Tsetse Fly
Entomologia Experimentalis et Applicata , 12:33–43,1969.[29] H. Potts. Observations on
Glossina morsitans
Westw., in East Africa.
Bulletin of Ento-mological Research , 24:293–300, 1933.[30] D. J. Rogers and T. P. Robinson.
The Trypanosomiases . Editors: I. Maudlin, P. H. Holmesand P. H. Miles. CABI publishing, Oxford, U.K., 2004.[31] J. R. Stevens and S. Brisse.
The Trypanosomiases . Editors: I. Maudlin, P. H. Holmes andP. H. Miles. CABI publishing, Oxford, U.K., 2004.[32] G. A. Vale. The responses of tsetse flies (
Diptera: Glossinidae ) to mobile and stationarybaits.
Bulletin of Entomological Research , 64:545–588, 1974.[33] G. A. Vale, J. W. Hargrove, A. M. Jordan, P.A. Langley and A. R. Mews. Survivaland behaviour of tsetse flies (
Diptera , Glossinidae ) released in the field: a comparisonbetween wild flies and animal-fed and in vitro -fed laboratory-reared flies.