IC/96/118

United Nations Educational Scientific and Cultural Organizationand

International Atomic Energy Agency

INTERNATIONAL CENTRE FOR THEORETICAL PHYSICS

MODELLING THE POPULATION DYNAMICSOF SIMULIUM DAMNOSUM USING A STAGE CLASSIFIED

MATRIX MODEL WITH APPLICATION TO DATA

F.W.O. Saporu1

International Centre for Theoretical Physics, Trieste, Italy.

ABSTRACT

A piecewise time invariant stage classified matrix model is proposed for modelling the

population dynamics of female blackflies (Simulium damnosum). A piecewise homoge-

neous transition matrix is adopted in order to reflect the response of the flies to seasonal

changes in the environment. The lower level parameters of the projection matrix are ob-

tained by fitting a negative binomial model to the data giving the age distribution of the

flies. The model is compatible with the data. The rhythm in the dynamical behaviour of

the fly population in response to seasonal environmental changes is identified. Eigenvalue

sensitivity of the estimates of the matrix coefficients to the asymptotic growth rate is

examined. The results and its epidemiological consequences are discussed.

MIRAMARE - TRIESTE

July 1996

Permanent address: Department of Mathematics and Statistics, University of Maid-uguri, Maiduguri, Borno State, Nigeria.

1. INTRODUCTIONSimulium darnnosum, the blackfly,is the vector responsible for thetransmission of Onchocerciasis(river blindness) infection in man.Their habitat is usually associated with flowing streams orrivers. The life cycle [4] is described as follows. The eggs arelaid in fast flowing water. They hatch into filter feeding larvaewhich attach themselves to the bottom. An adult female Simuliumemerges from its pupae, floats to the water surface, mates andperforms its maiden flight in search of a bloodmeal from ahuman(the definitive host). The bloodmeal is necessary forovarian development. On getting a bloodmeal it goes back to theriver to oviposit, after which it emerges again for anotherbloodmeal. This process repeats itself until the fly eventuallydies. Eggs layed at each oviposition go through the larvae, andpupae stages, after which they emerge from water as adult fliesfor their maiden flight. It is thought that blackflies which donot obtain bloodmeal during the maiden flight die[2].

At each bloodmeal time a blackfly can either take an "infected"bloodmeal, if she bites an individual with the disease, or an"uninfected" bloodmeal. An "infected" bloodmeal is so calledbecause it contains the microfilariae of Onchocerca volvulus.After about six days the microf ilariae develop in the fly intoinfective larvae. The fly is capable of transmitting infectivelarvae into man through the bite wound of a blood meal. It is theinfective larvae received by a susceptible human, that developinto mature worms and produce the symptoms of Onchocerciasis.

Because blackflies harbour Onchocerca volvulus, its populationdynamics is of epidemiological significance. The interest of thispaper is to study this dynamics: highlighting the salientfeatures of its epidemiological consequences.

The following terms used in the text are defined below.A female blackfly is NULLIPAROUS if she has not taken its firstbloodmeal; otherwise she is PAROUS.The LENGTH OF A GONOTROPHIC CYCLE is the time interval betweentv/o consecutive bloodmeals.An "INFECTED" BLACKFLY is that which has taken a bloodmealcontaining microfilariae.An "INFECTIVE" FLY is an "infected" fly which has themicrofilariae in her fully developed into infective larvae and iscapable of transmitting such to a host through the bite wound fora bloodmeal.A female blackfly is said to have SINGLE INFECTION if sheharbours only one brood of parasites(microfilariae or infectivelarvae) resulting from the same "infected" bloodmeal.A female blackfly is said to have DOUBLE INFECTION if she takesan "infected" bloodmeal at two different bloodmeal times, whichare days apart; that is, containing two broods of parasites.

The following demographic indices used in the text are defined

2

below.The NET REPRODUCTIVE RATE, RQ, is the expected number ofoffspring by which a newborn individual will be replaced at theend of its life.

The GENERATION TIMES are quantified in terms of(i) the time T required for the population to increase by a

factor of Rc,(ii) the mean age, ja-,, of the parents of the offspring produced,

by a cohort,

(iii)the mean age, A, of the parents of the offspring producedby a population at stable age distribution and

(iv) t2G is the time required for the contribution of thedominant eigen value "k- to become 20 times as great asthat of X2.

The following terms for the population projection matrix used inthe text are defined below.

(i) X- denotes the DOMINANT EIGEN VALUE of the projectionmatrix, M, and its associated RIGHT and LEFT EIGEN VECTORSare denoted by W, and V: respectively.

(ii)Estimate for (a)A-, gives the ASYMPTOTIC GROWTH RATE forthe population, (b)W1, gives the STABLE STAGE DISTRIBUTIONand (c)V-, gives the REPRODUCTIVE VALUES for the respectivestages.

Henceforth in the text, a female blackfly will be referred to asfly.

2.DATAThe data for analysis are part of • the entomological datacollected by Duke in Bolo, a Cameroun rainforest village andreported in two separate papers[2,3]. Here Simulium damnosum isthe principal vector responsible for the transmission ofOnchocerciasis infection to man and the infectivity of the fly isperennial.

The experiment involved catching blackflies that came to feed onhuman baits. Fourteen catching sessions were used and these wereheld at intervals over the period from March 1966 to February1967. Each session comprised of six days in the week excludingSundays. A total of 164,863 wild flies were caught of which35, 558 were dissected. The probable age (in bloodineal times anddays ) of each dissected fly was determined based on the stagesof development of the microfilariae in it. Some simplifyingassumptions were necessarily made in order to extend the agedistribution of the infected flies up to the sixth blood mealtime, as follows:

To allow for statistical analysis, it was found necessary tomodify the groupings of the flies. Towards this end, it was firstnoted that it is the age distribution of the total fliesdissected that is of interest and not that for the total numberof flies/man/day as presented in Duke[3]. This was easilyobtained using some of the assumptions in Duke [3] . Also at eachbioodmeal time, the "infected" flies were grouped into only twoage classes of approximately three days apart(interval betweenbloodmeals) based on the information of the age distributionprovided. The "infected" flies were further classified intosingle/double infection. The age classification for the"uninfected" fly population was understandably not done byDuke[3]; there is no relevant information for doing this. Howeverthey were distributed over the blood meal times using the sameratio of distribution for "infected" flies on 2nd, 3rd, 4th and5th blood meal round. It must be mentioned that the distributionso obtained for the "uninfected" flies may be very crude. Table 1presents the age distribution of the flies in the modified formdescribed above.

Also of interest for analysis is the stage distribution of theflies. These are presented in Duke[2]. The stages reported arenulliparous, "uninfected" parous, "infected" parous fly withdeveloping infection, and "infected" parous fly with single anddouble infection respectively.

Duke highlighted the seasonal fluctuations in the data byplotting over each month, the calculated values for each of threemeasures of transmission index. These are (i) the total number offlies per man per day, (ii) the total number of infective larvaeper man per day and (iii) the geometric mean number of infectivelarvae per infective fly. These show similar trends as depictedby the plot for the total number of flies per man per day shownin Fig. 1. The trends clearly suggest three seasons; A, March toMay , B, October to february and C, June to September, as used byDuke. The values for the total number of flies per man per dayand the range of the mean hourly temperature shown on the graphare indices which also show clearly the differences between theseseasons. Consequently the grouping of the data by these seasonsis retained in this study.

3. THE MODELSA. Stage classified matrix population model

The type of model constructed was largely dictated by the natureof available data {Table 1) . The fly population biting man hasthree distinct stages; nulliparous, "uninfected" parous and"infected" parous, suggesting that a stage classified matrixmodel is feasible. The "infected" parous stage is furtherclassified as flies with developing single infection, infectiveflies with single and double infection respectively. The intervalof the projection matrix is chosen to coincide with the interval

between consecutive blood meals, which is assumed to be threedays as suggested in Duke[2]. This allows the sampling times ofthe flies to coincide with bloodmeal times.

The stage of developing infection to that of becoming aninfective fly requires about six days, that is two gonotrophiccycles. Pseudo-stages are therefore introduced between these twostages so that the projection interval and the assumed intervalfor moving from one stage in the life cycle to the next will becompletely synchronised in all the stages considered. The stagesso chosen are such that a fly with a single developing infection,at the next time epoch, can either become (i)a fly with a doubledeveloping infection, if she took a previously "infected"bloodmeal or (ii)a fly with with a single developing infectionbut aged one gonotrophic cycle more, if she took a previously"uninfected" bloodmeal. This will, in effect, create thenecessary time delay for the development of microfilariae in thefly into infective larvae. Thus all the stages used in themodelling are labelled as follws:

(i) Nulliparous(ii) "Uninfected" parous(iii)a "Infected" parous with developing single infection(iii)b "Infected" parous with developing double infection(iii)c Aged "infected" parous with developing single infection(iv) "Infective" parous with single infection and(v) "Infective" parous with double infection.

These stages roughly correspond to that recorded in the datashown in Duke[2] and hence a model can be fitted on the availabledata.

As this is a matrix projection model, only the female flies areconsidered and the following assumptions are made about • themodel.

(i) All births are nulliparous and contributions to birth cancome from all stages except the first.

(ii) The fate of a nulliparous fly is either to take abloodmeal or die.

(iii) An "infective" fly does not loose all its infectivelarvae from one brood, at any given bloodmeal time,

{iv} The response of the fly population to environmentalvariability is seasonal.

The life cycle graph for this model is given in Fig. 2.

The population projection matrix arising from the life cyclegraph is of the form

M = 0000

F2

^30,2

0000

00

00

00000

00004,3c

5 3c

0000

PiC5.4

000000

where F.A (i = 2 , 3 , 4, 5) is the fertility coefficient for stagei, defined as, the number of nulliparous flies at time t + 1 perstage i fly at time t, P., is the chance of surviving andremaining in stage i and GHi is the chance of surviving andgrowing out of stage i to stage j . The matrix coefficients areexpressed as functions of lower level parameters, a._, 01 and y^_with

Pj =i=2 and 4

3.1

where 9, is the chance of surviving stage -\, y,L is the chanceof growing out from stage i to stage j and a. ,"is the chance oftaking a bloodrneal in stage i.

The seasonal fluctuations in the infectivity of the flies isreflected in the model by assuming a piecewise time invariantprojection matrix as follows. Let S,_ represent a column for thestage distribution(with stages as defined above) of the flypopulation at time t. The model projection of St over- time isgiven by

M-"SC. 0 <t<tx

So t:<t<t2

MLlS0 t2<t<t2

where M1,M2 and M3 are respectively the projection matrices for

seasons A,B and C defined earlier.

It is of interest to see whether the relaxation of some of theassumptions could produce a better understanding of the evolutionprocess for the fly population. In particular, assumption(ii) wasmodified to be that the fate of the fly at all stages is eitherto take a bloodmeal or die. Thus the parameters of the projectionmatrix M, have a different form in this case given by

G,

1= 2 and 4

i= 1,2,3, .. .,53 . 2

B o t h m o d e l s a r e f i t t e d t o d a t a a n d a r e n a m e d m a t r i x m o d e l I a n dI I r e s p e c t i v e l y .

B . M O D E L S F O R A G E D I S T R I B U T I O N O F F L I E ST h e p a r a m e t e r y^- r e l a t e s t o t h e c h a n c e o f a f l y t a k i n g a n" i n f e c t e d " o r " u n i n f e c t e d " b l o o d m e a l s i n c e g r o w i n g o u t i n t h i ss i t u a t i o n i m p l i e s t a k i n g e i t h e r a n " i n f e c t e d " o r " u n i n f e c t e d "b l o o d m e a l a s t h e c a s e m a y b e . H e n c e o b t a i n i n g e s t i m a t e s f o r t h el o w e r l e v e l p a r a m e t e r s o f t h e p r o j e c t i o n m a t r i x , M , n e c e s s i t a t e sm o d e l l i n g t h e a g e d i s t r i b u t i o n o f t h e f l y p o p u l a t i o n . S u c h m o d e l sa r e c o n s i d e r e d b e l o w .

1 . B I N O M I A L M O D E LT h e n u m b e r o f b l o o d m e a l s a f l y h a s t a k e n r e l a t e t o h e r a g e

s i n c e t h e i n t e r v a l b e t w e e n b l o o d m e a l i s a s s u m e d f i x e d .C o n s e q u e n t l y a b i n o m i a l m o d e l i s f i r s t c o n s i d e r e d , t r e a t i n g t h ee v e n t a f l y t a k e s a n " i n f e c t e d " b l o o d m e a l , a t e a c h b l o o d m e a lt i m e , a s a n i n d e p e d e n t B e r n o u l l i t r i a l w i t h p r o b a b i l i t y o fs u c c e s s [ 3 . L e t t h e r a n d o m v a r i a b l e X d e n o t e t h e n u m b e r o f" i n f e c t e d " b l o o d m e a l s t a k e n b y a f l y a t . t h e n " - h b l o o d m e a l t i m e .T h e n

P ( X = r ) =

Hrv

A(?JPi-p;

(I-p)

1 i\\ " ''

n = 2

r = 0 ,

n>2

r=0,l,2

Truncation becomes necessary because the maximum number of"infected" bloodmeal taken by a fly that was observed is 2.

2.NEGATIVE BINOMIAL MODELThe binomial model is a simple one. There are two

information in the sample that the nature of it does not warrant.These are (i) the waiting time for the n t n bloodmeal and (ii) thefact that the rth(=1,2,...) "infected" bloodmeal was taken at then t h bloodmeal time by the fly; this is because an infected humanwas used as bait. This is useful information which the featuresof a negative binomial model can contain, as such, it isconsidered. The following simplifying assumptions are made inorder to be able to use them in this situation.

(i) The interval between bloodmeals, is the length of agonotrophic cycle, which is assumed to be fixed andapproximately three days.

(ii) The chance of a fly taking an infected bloodmeal at eachbloodmeal time is constant and denoted by fi.

(iii) The chance of survival of each period between consecutivebloodmeals is constant and independent of the stages for"infected" flies and is denoted by 8. A different constantchance of survival is assumed for "uninfected" flies,

(iv) At the nth bloodmeal time, the possible ages of a fly are3(n+y-l), y-0,1;reflecting the waiting time for abloodmeal.

(v) The event of taking a bloodmeal and survival of a fly areindependent.

Let the random variable T n denote the waiting time for the rth"infected" bloodmeal of a fly, to occur at her n t n bloodmealtime. Let S n denote the event surviving n bloodmeal times.

P(Tn=y andy = 0 , i ( 2 , . . .

Again truncation is necessary because only two values of y(=0,l)are observed. Truncating the joint distribution above, aftersimplification, becomes

fn+y-l[ r-i J ( P )

P(Tn=y and S n + y-D = j ^ ^-^r y = 0

[r-lj +which is a Bernoulli distribution for a fixed n. The truncationis done -with respect to the joint distribution in order thatestimates for 9 and [3 can be obtained from the same availabledata.

The chance that a fly takes a bloodmeal, a, ("infected" or"uninfected") is a lower level parameter of the projection matrixM defined above. In order to obtain this, a distribution isfitted to the waiting time for obtaining the n t n bloodmeal. Letthe waiting time be denoted by a random variable Bn. A truncatednegative binomial model is also feasible. This gives aftersimplification a joint distribution of the form

fn+y-llI n^l J(i-a) Ye Y

P(Bn=y and Sn+y_!) = — i + n(l-a)B y-0,1.

where 8 is as defined before. The two negative binomial modelsare labelled I and II respectively.

4.1 ESTIMATION:AGE DISTRIBUTION MODELSIt is probably important to distinguish between the required datastructures for fitting the binomial and negative binomial modelsbecause the latter requires extra information not noted in thederivation of Table 1. The flies used for classification byDuke[2] were sampled just before they took their bloodmeal froman infected human bait. Hence they v/ere classified based on thepast history of "infected" bloodmeals; the present was ignored.This is the nature of the data presented in Table 1 and was usedfor fitting the binomial model. The fact that an "infected" humanbait was used in obtaining the sample is exploited by thenegative binomial model thereby introducing differences in thedata form required for estimation in both cases. This differenceis simply met by making r "infected" blodmeals in Table 1 for thebinomial model become r+1 for the negative binomial model whileother figures remain unchanged.

The estimate of the parameter, J3, for the binomial model wasobtained using maximum likelihood estimation procedure,"uninfected" and "infected" parous data were treated separately.Estimate for each of the seasons, A, B, and C were separatelyobtained. A pooled estimate for all seasons was also obtained. Achi-squared test of the goodness of fit of model to data wascarried out. The hypothesis of a common value of p for allseasons was tested using a generalised likelihood ratio testcriterion. This was done in order to ascertain the influence ofseasonality on model parameters. The negative binomial modelswere similarly treated. All results are presented in Table 2.

4.2 DISCUSSION OF RESULTSThere is an overwhelming rej ection of the binomial model in allseasons so the results are not presented.Only those for the

negative binomial model are shown in Table 2. For the"uninfected" parous data, the negative binomial model I wasoverwhelmingly rejected in all seasons except B. This is notsurprising because it is here that the ratio used forclassification of the flies lacked establised scientific basis asearlier mentioned. The assumption of the classification methodmay be wrong a priori and as such the performance of the model inthis circumstance cannot be appropriately judged.

For the infected parous data, v/here there are well establishedcriteria for the classification of the flies, the results showthat a negative binomial model is compatible with data; only inone case, season A, is the model marginally rejected. Even inthis case of rejection about seventy percent contribution to theX2 value came from one observation indicating that that value ishighly suspect.

For season B, where the negative binomial fits for both"uninfected" and "infected" parous data, the hypothesis of acommon value for model parameters is strongly rejected. Thisindicates that the parameters |3 and 9 (which are lower levelparameters for the projection matrix M) are stage dependent; the"uninfected" fly stage tend to have a higher chance of survivaland a lower chance of taking an "infected" bloodmeal than an"infected" fly stage.

Also strongly rejected is the hypothesis of a common value formodel parameters for the "infected" flies in all seasons. Thissuggests seasonal variability in the infectivity of the flypopulation. This seasonal fluctuation of model parameters relatein a different manner to that for the estimated total number offlies per man per day. The trend is a high seasonal total of thenumber of flies per man per day corresponds to high chance ofsurvival and a low chance of taking an "infected" bloodmeal. Thatis, when seasonal influence is unfavourable to transmission, thefly sustains the disease in the population by increasing itsinfectivity(as measured by the chance of taking an "infected"blood meal). This is a similar point noted by Duke[3].

The fit of the second negative binomial model II to data is poor.It is not surprising because the data used here are a combinationof "infected" and "uninfected" parous data. Hence the rejectionof the model could have been infuenced by similar reasonsmentioned earlier for "uninfected" parous data. The results arepresented because the estimate of a are needed, as one of thelower level parameters for the matrix projection model. Thesewill serve as rough estimates.Graphs(observed and fitted) of theage distribution of the fly population are presented in Fig 3.

The parameter p, chance of taking an "infected"bloodmeal(conditional on taking a bloodmeal), of the negativebinomial model I is of epidemiological importance because it

10

relates to the transmission potential of the vector. Theestimated values recorded(Table 2) fluctuate seasonally around1/3. This is a rather low value for a hyper endemic village, suchas Bolo. It corresponds to the low field values for theproportion of parous flies that are "infected" reported inSaporu[7] . The implication of this low value is that the flyfinds it easier,when engorging, to take an "uninfected"bloodmeal, even on an "infected" human host. The reasons forthis, among others, could be atrophy of the skin of an "infected"individual and perhaps hostility of the host to invading flies. Alow value of (3 is good because it acts to reduce transmissionpotential. The fact that at the season v/hen the total number offlies per man per day is highest the vaue for that of p drops toits lowest is indeed very intersting as this suggests a possibleway by which nature controls the disease.

The chance that an "infected" parous fly survives a gonotrophiccycle is also low: seasonally fluctuating around 30%. Using theestimate for the pooled(over all seasons and all stages) data, incalculation, the chance that a fly will survive and take an"infected" bloodmeal on a third bloodmeal round is about 2%;which makes it an event that is highly unlikely. The low survivalpotential between bloodmeals again acts to naturally reducetransmission potential of the vector.

5.1 ESTIMATION:MATRIX MODELSThe parameters of the matrix models Fi,G1 and Pi are expressed interms of lower level parameters a, 9 and y in equations 3.1 and3.2. The estimates for a and 8 have been derived (Table 2) asexplained above. What remains is y^,the chance of growing outfrom stage i to stage j. This is either the chance of taking an"infected" or "uninfected" blood meal depending on stage i; thatis, P or l-|3,and these estimates are also shown in Table 2. Theselower level parameters are stage dependent as shown earlier. Asthe stage adopted for the matrix model are more than that usedfor the negative binomial model, it is necessary to makesimplifying assumptions so that estimated values of G and P canbe obatained in all cases from Table 2. Towards this end,estimate of P for "uninfected" parous is assumed to be the samefor nulliparous flies and estimates for 6 and p are assumed to bethe same for all stages of "infected" flies. The estimate of 9for nuiliparous flies is derived by calculating the chance ofsurvival of a nulliparous fly over the first three days fromexisting data[8]; .815 was obtained. As the lower levelparameters for all stages can now be calculated, estimates for Giand Fi were obtained by substitution into expressions 3.1 and3.2.

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The estimates for F were obtained by a method of least squaresinvolving minimising the error sum of squares obtained from

e = M.VL - V2

where V1 and V2 are the column vectors for the sample values ofthe stage distribution of the flies, presented in Duke[2], fortwo consecutive periods. Similar method of estimation has beenused by Lefkovitch [5]. Here, F, the fertility coefficient isthe only remaining unknown parameter of M to be estimated. Inorder to simplify the estimation procedure, F is assumed to bethe same for all stages. The fitting was done usingMathematica[6] package on a personal computer and estimates for Fwere obtained separately for each season. Estimates for thedominant eigen values, Xx, and the corresponding eigen vectors,V:and W., were also obtained.

Also of interest are the demographic indices, Rr), T, A, JJ.X, p andt2C (defined in the introduction) for the fly population. Thesewere estimated using life cycle graph method described byCaswell[l]. All results are presented in Tables 3(estimates formatrix coefficients) and 4(estimates for demographic indices).

5.2 DISCUSSION OF RESULTSThe estimated value of X± for matrix model II is consistentlyless than one in all seasons. This suggests a decliningpopulation size at all seasons, which consequently willeventually go to extinction. The trend suggested by this modelis highly unlikely, as such the model is dropped. However theresults(for season A only) for the sensitivity analysis of modelII are tabulated in Table 6, as these will be used fordiscussion; all other results for model II are not presented.

For model I the estimated eigen values(Table 4) are greater thanone for seasons A and C and less than one for season B. That is,during each of the seasons A and C, the fly population size grows{with a higher growth rate in seaon A) . In season B the valuesuggests a decline in the growth of the population size. This ismuch closer to the observed trend in the data and hence the modelcan be said to be capable of explaining the data. In season C itis noted that whereas the fly population size is increasingmonthly, the total number of flies/man/day (Fig.l) is decreasing.This is not entirely surprising because season C is the rainperiod where vector-man contact rate is known to be hampered byrain. Hence the model cannot be said to have failed to explainthe noted observed trend in the transmission index at this point.

The estimated fertility coeffient, F(Table 4), is slightly aboveone in all seasons, with the highest value recorded in season Band lowest in season A. It is interesting to note that the

12

season of the highest fertility coefficient corresponded to thatfor the lowest asymptotic growth rate, A,.,. In all seasons, theresult shows, that the reproductive value rises to a maximum inthe "uninfected" parous stage and drops to a minimum for flieswith multiple infection. Estimated values are highest in seasonB and lowest in season A; again bearing same seasonal trends asresults for the fertility coefficient.

The va.lue of one for Ro is recorded in season A. This implies astable population evolution process. In season C, the estimatedvalue of Ro slightly drops from one. The drop in value inseason B is more pronounced. The estimated values of thegeneration time index, T, show marked seasonal differences. Thisgives an indication that seasonal differences in the length of agonotrophic cycle is highly suspect. The values for other

indices, A and p.1 of the generation time estimated show lesserseasonal variation. They give the mean age of parents ofnulliparous flies as fluctuating seasonally around 2.5 (times thelength of a gonotrophic cycle). Then, the fly is about 7.5 daysold and highly likely to be infected with single developinginfection or infective with single infection.

The values recorded for the rate of convergence to a stable stagedistribution are particularly low. The highest recorded value,2, is in season A; others are around 1.5. The time reguired forthe contribution of X1 to become 20 times as great as that of X2is also used as a measure of rate of convergence to stable stagedistribution. The lowest value recorded is for season B, 6.7(times the length of a gonotrophic cycle) . This is about 21days, which is less than a month. Thus it is highly likely thatthe fly population settles into a stable stage distributionbefore they approach the next season. In all seasons the stablestage distribution decreases with stage, with the nulliparousflies composing more than 50% of the fly population and theinfective flies about 1%. This, to a large extent corresponds toobserved field values. It is interesting to note that althoughthere are marked seasonal differences in the parameter estimatesobtained, the estimates for the stable stage distribution,particularly the stable proportion of infective flies, are notas pronounced.

6.1 SENSITIVITY ANALYSISIt is of interest to know how sensitive is the asymptotic growthrate X- to changes in the matrix model assumptions. Matrix modelI assumption (iii) (an "infective" fly does not loose all itsinfective larvae from one brood, at any given bloodmeal time,wasfirst investigated. Here X1, was recomputed using the modifiedassumption that an "infective" fly looses all its infectivelarvae from one brood, at any given blood meal time.

13

Of interest is a measure of the proportional change in \±resulting from a proportional change in the matrix coefficientai:. This is given by the elasticity of Xj with respeect to ai,defined [1] by

aX

v i

<W,, v>

where, V i s the vector of reproductive values, W is the stablestage distribution vector and <w,v> is the scalar product of Wand V. The elasticity of X_ with respect to G , Pi and F, formatrix model I were computed for each season using the aboveformula. The scalar product term was ignored in the computationsince the relative sensitivities of X-x to different elements inthe same matrix are being considered. The results are presentedin Table 5.

Also for matrix model I, the sensitivity of X-_ to lower levelparameters a, y and 6 were computed [1] using expressions.

aa±

gf

- y _ _ (I_Y 1 + J y~ f aPi ^ TjiJ + f gG_. t

dXdG,

X 9A Q y

14

The results are presented in Table 6.

It is thought [3] that flies which do not find a blood meal atany blood meal time die. The effect of this assumption on thegrowth rate XL was investigated by computing the sensitivity ofX- to lower level parameters (particularly a) for matrix modelII, which incorporates this assumption. The trend in the resultsare similar for all seasons as such only that for season A ispresented in Table 6.

6.2 DISCUSSION OF RESULTSThe modification in matrix model I assumption (iii) produced onlya marginal difference in the recomputed value of X-. Thisindicates that the growth rate X-_ is not sensitive to suchmodification in model assumption that an "infective" fly loosesall its infective larvae from one brood, at any given blood mealtime.

The results show (Table 5) that the estimate of the contributionof each of the matrix coefficients, Pi, Gi and Fiy for matrixmodel I, to the growth rate X-t is important. The leastcontribution is recorded for P, in all seasons. In seasons A andB, the growth parameter Gi gives the highest contribution,whereas in season C, the fertility coefficient gives the highest.The elasticities of X- with respect to changes in stage specificsurvival probabilities, Pif fertility, F, , and growth, G2, show adecreasing trend with the stage, in all seasons. Substantialcontributions come only from the first three stages (nulliparous,"uninfected" parous and parous with developing single infection);values for other stages are marginal.

Results (Table 6) also show that X1 is sensitive to all lowerlevel parameters, a, 9 and y for model I.. Here again theestimated value show a decreasing trend with stage; in allseasons, only in the first three stages are the estimatedsensitivity values substantial.

For matrix model II, the results show that X_ is sensitive to thelower level parameter a, the chance of taking a blood meal.However substantial contributions to this sensitivity come mainlyfrom the first two stages only. Hence it can be assumed that inthe first two stages, the fate of the fly is either to take ablood meal or die.

7. CONCLUSIONThe population dynamics of female Simulium damnosum is modelledusing a stage classified matrix model. It is necessary tointroduce the assumption of piecewise time invariant projectionmatrix in order to reflect the response of the flies to seasonalenvironmental changes. The assumption that an "infective" fly

15

does not loose all its infective larvae at any one bloodmeal timeis found unwarranted. It is also found that the assumption onthe fate of a nulliparous fly can be modified to equally apply to"uninfected" parous fly; that is, in either the nulliparous orthe "uninfected" parous stages, the fate of the fly is either totake a blood meal or die. Hence at the early stages in the lifeof an adult female blackfly, finding a bloodineal is crucial notonly for ovulation but also for survival.

It appears that the model is reasonably compatible with data. Theresults show that the fly population reaches a stable stagedistribution in any one season before a change to the nextseason. The results also suggest a rhythm.(see Fig.4) in thedynamical behaviour of the fly population which is described asfollows. The flies experience the lowest transmissionindex(total number of flies/man/day) in January. This graduallyincreases to a maximum just before the rains in July andthereafter declines and the cycle continues in the next year.As the transmission index increases so also does the chance ofsurvival of the fly but the fertility coefficient and theinfectivity of the fly (as measured by the chance of taking aninfected blood meal} each drops. The drop in the infectivity ofthe flies is always to such a level that the disease remainsendemic in the human population. This dynamical behaviour israther fascinating as the point where the total number of fliesper man per day reaches its maximum, the chance of a fly taking ablood meal drops to its minimum value. This is suggesting thatnature has its own way of controlling the disease.

AC KNOWLEDGEMENTThis work was done during my associate visit to the INTERNATIONALCENTRE FOR THEORETICAL PHYSICS, TRIESTE, ITALY. The visit wassupported by funds from SAREC.

REFERENCES1. Caswell, H. (1989). Matrix Population Models.

Publishers: Sinauer Associates, Inc. Sunderland,Massachusetts. pp 9 5-177.

2. Duke, B. 0. L.(1968a) Studies on factors influencing thetransmission of Onchocerciasis IV:- The biting cycles,infective biting density and transmission potential of'forest' Simulium Damnosum. Annals of Tropical Medicineand Parasitology, 62, 164, pp. 95-106.

3. Duke, B. 0. L. (1968b). Studies on factors influencing thetransmission of Onchocerciasis. V:~ The stages ofOnchocerca volvulus in Wild 'forest' Simulium Damnosum,the fate of the parasites in the fly, and theage-distribution of the biting population. Annals ofTropical Medicine and Parasitology, 62, 164, pp 107-263.

4. Lawrence, B. R. (1976) Simulium - the bane of the tropical

16

World Tropical Doctor, 6, 136-140.Lefkovitch, L.P. (1965). The study of population growth in

Organisms grouped by stages. Biometrics 21: 1-18.Wolfram, S. M. (1993) Mathematical A System for doing

Mathematics by Computer. 2nd Edition Addison-Wesley-Massachusetts.

Saporu, F.W.O. (1992) The Ultimate size of Onchocerciasis"infected" black flies of the Simulidae family. TheStatistician: 41, 65-70.

Saporu, F.W.O. (1993). Analysis of Survival data for SimuliumDamnosum using regression method. Annals of Trop. Med. &Para. Vol. 87(6) 563-569.

17

Table 1 Age distribution of flies by blood meal times

Season Fly Stage(r)

i (0)ii (1)

A iii (2}iv (1)v (2)

i (0)ii (1)

B iii (2)iv (1)v (2)

i (0)C ii (1)

iii (2)iv (1)v (2)

n 1Y 0

7217'

5030*

4454*

20

1524383

1184274

1014201

1

876220

39892

39378

30

909134464511

6418139199

268311552

1

2624114103

190241253

566311

40

11114842

11314732

232100

1

365220

304

rsi

10

00000

50

203110

304210

21100

r number of infected bloodmeal engorgedn bioodmeal timey distribution of flies at ntJ1 bloodmeal times* nuiiiparous fliesi denotes "uninfected" parous stageii denotes stage with single developing infectioniii denotes stage with developing double infectioniv denotes stage infective with single infectionv denotes stage infective with double infection

18

Table 2. Results for fitting negative binomial models to data onage distribution of flies

MODEL I

Season

ABCPooled

"Uninfected" Parous9

.573

.419

.448

.485

P xh>.201.251.233.209

71.50***5. 9 6r,.o.24 ,44***

"Infected" Parous9

.340

.251

.266

.297

P #5, XVi).300.356.347.327

12 .80*3 . 65n.s.4.74:1,3.

12.52*

Pooled

9

.530

.413

.408

.439

P.135.340.258.238

<y2 £A-<2)

41.2

MODEL II

Season

ABC

e.438.360.396

a.562.640.604

* denotes significant at 5% level.** denotes significant at 1% level.*** denotes significant at .1% level.n.s. denotes not significant.# Chi-squared value for generalised likelihood

ratio test.

19

Table 3. Estimated values for elements of projectionmatrix in all seasons for Matrix Model I

M =

M =

M =

03650920000

02551310000

0376115

0000

1.140.458.1150000

1.630.314.105

0000

1.265.344.104

0000

1.14000

.102

.23800

1.63000

.089

.16200

1.26500

.092

.17400

SEASON A

1. 140

000

.340

SEASON B

1.6304

4

000

.251

SEASON C

1.265

0000

.266

1.140

.102

.238

1.630

.089

.162

1.265

.092

.174

1.14(

.238

.102

1.631

. 162

. 089

1.26!

.174

.092

1.140

00

1 . 6 3 0

00

1 . 2 6 5

00

20

Table 4. Estimated values for demographic indices in all seasonsfor Matrix Model I

Season T h w i V-

2 . 014.28 1.0 .076 2.66 2.94

A

i.56 6.74 .90 9 .24 2.35

B

C

1.65 5.95 .97 3.26 2.40

1.140 1.018

2.47 1.630 .989

2 . 581.265 1.009

528345087020009006006

622235107018009003005

556315096017009004004

12.3711.6311.5781.4181.578.893

12.7482.1442.0361.8022.036.607

12.1621.6641.6031.4641.603.798

\l1 denotes reproductive valuesW-L denotes stable stage distribution

21

Table 5. The estimated elasticity of the asymtotic growth rate,X, with respect to matrix coefficients

Season Parameters

A

B

Gi,:,i

Gi_2,i

Pi

F,

Gi,i,i

G:+2.iP,

Gi+i.i

^i + 2, i

Pi

(i)

0.449.0780

0.440.177

0

0

.357

.084

0

(ii)

.386

.064

0

.368

. 387

.0540

.205

.395

.0430

.185

Stages(iii)

.097

.032

.0120

.176

.036

.0170

.120

.021

.0100

(iv)

.022

0

.007

0

.030

0

. 006

0

.020

0.0040

(v)

.0100

.0010

.0150

.0010

.0110

.0010

Cvi}

.007

.00020

.002

.005

.00020

.001

.0050

.0002

.001

(vii)

.0070

0

0

.0080

0

0

.0050

0

0

Total

.529

.545

.098

.370

.621

.530

.201

.206

.556

.421

.099

.186

22

Table 6. The estimated values of the sensitivity of X to lowerlevel parameters

Model

j

II

Season

A

B

C

A

Parameters

aeY

a

eYa

eY

a

eY

i

.955

.661

.965

1.3131.0341 . 581

.936

.6971.043

1.409. 9761.369

ii

0.767.322

0. 611.211

0.653.122

.491

.471

.483

i J

0.133.088

0.209.103

0.150.079

.049

.082

.055

Stageiv

0.023.012

0.028.012

0.023.011

.007

.011

.007

;sV

0.008.003

0.006.001

0.007.002

0.001.001

vi

0.002.002

0.001.001

0.005.003

.001

.002

.001

vii

000

000

000

000

Total

.9551.5941.392

1.3131.8891.909

.9361.53 51.260

1.9571.5431.916

s

LL

2000 -

1000

74.3- 75,1

<

A,B,C, Seasons* Estimate for total

number of flies perman per day

- trend line for totalnumber of flies perman per day

+ Mean hourly temperature

Months

Fig . 1 Seasonal t rends in entomological data for Simulium damnosuin, fromBolo, in Cameroun.

CM

25

EXPECTED C3SERVED

UNINFECTED PABOUS FOESUNINFSCTED PAROUS FLIES

a s * » * - -

PARCU3 FLIES WTTH SINGL£ 1NFECT1CM PAROUS FLIES WITH SINGLE INFECTION

ft*

0.7

0.6

SI ii

-i

I

4 -f

li

PABOUS FLIES WiTH DOUBLE INFECTIONPAROUS FLiES WITH DOUBLE 1MFECTIO.1

Fig. 3a Graphs of observed and expected values from f i t t ingNegative Binomial Model I to data for season A.

26

EXPECTED OBSERVED

UNINFECTED PAROUS FUES UNINFECTED PAROUS FUES

m

PAROUS FUES WiTH SINGLE INFECTION PAROUS FLsES WiTH SINGLE INFECTION

3 43LDO0MEW. TIMES

PAROUS FUES WiTH DOUBLE INFECTIONPAROUS FUES WITH DOU3LE INFECTION

Fig. 3b Graphs of observed and expected valuesfrom fitting Negative Binomial Model I todata .for season B,

27

EXPECTED OBSERVED

UNINFECTED PAROUS FUES

0.75

UNINFECTED PAROUS FUES

ROGKEAL TIMES S.OCKS-iiiM.Ti'.x:

PAROUS FUES WITH SINGLE INFECTION PAROUS FLIES WITH SINGLE INFECTION

BLOODMESLTMS

PAROUS FL1HS WITH DOUBLE INFECTION

3 4

EUOODtiEAL TIMES

PAROUS FUES WITH D0U3LH INFECTION

Fig 3c Graphs of observed and expected values fromfitt ing Negative Binomial Model I to datafor season C,

28

FIRST CYCLE SECOND CYCLE

3001

C BSEASONS

O •• _ ^

I -.63:

. 1.5-tmo 11.27 k '

S 1o

£0.5

Jo • • ! • — - •

A

0.4

C BSEASONS

0 . 3 1 ^ - - - ^

fc 0.25

§ 0.2oaQ 0 . 1 5a.

0.1 —- -

0.05

0

. •• i : 3 I

1.5-r

.11

0.5

C BSEASONS

0.3

i : rC B

SEASONS

SURVIVAL PROBABILITYINFECTIVITY: CHANCE OF TAKING AM "INFECTED1 BLOOD MEAL

0.25 - -

0 . 2 ' - • • - -

0.15 - —

0.1 -

0.05

0A C B

SEASONS

- SURVIVAL PHOSABILTY• • INFECTMTY: CHANCE OF TAKING AN "INFECTED' BLOOD MEAL

SEASONS : A : MARCH - MAY. C : JUNE - SEPTEMBER . B ; OCTOBER-FEBRUARY

Fis.4 Trends in the estimates of liie ^niomologicitl parameters of Simuliuin Damnsum in Qolo, Cameroun