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A cyclical feeding model for pathogen transmission and its application to determine vectorial capacity from vector infection rates

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A Cyclical Feeding Model for Pathogen Transmission and Its Application to DetermineVectorial Capacity from Vector Infection RatesAuthor(s): A. J. Saul, P. M. Graves and B. H. KaySource:
Journal of Applied Ecology,
Vol. 27, No. 1 (Apr., 1990), pp. 123-133Published by:
British Ecological Society
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Journal f Applied cology 1990), 27, 123-133
A CYCLICAL FEEDING MODEL FOR PATHOGEN TRANSMISSION AND ITS APPLICATION TO DETERMINE VECTORIAL CAPACITY FROM VECTOR INFECTION RATES
BY A. J. SAUL*, P. M. GRAVES
AND
B. H. KAY Queensland nstitute f Medical Research, ramston errace, risbane, Queensland 006, Australia SUMMARY
(1) A model of disease ransmission as formulated hich ssumed hat ector eeding behaviour s cyclical rather han continuous. This model led to equations which were analogous to those developed by Macdonald (1957) for non-cyclical eeding. (2) Equations were derived from he cyclic feeding model to evaluate parameters describing isease transmission rom ntomological ata or from nfection ates n the vector. (3) Where ector nfection ates an be measured, he model predicts hat stimates f vectorial apacity, ransmission arameters e.g. the probability hat mosquito will be infected) nd vector urvival made from hese ates would be, n principle, ess subject o uncertainties n factors uch as the vector's urvival function, he presence f mixed populations of vectors, nd changing vector numbers nd transmission ates, than methods ased upon entomological arameters lone.
INTRODUCTION Many of the urrently sed models f the ransmission f disease by vectors re based, t least n part, n the work f Macdonald 1957), who derived set of equations o describe the transmission f malaria by mosquitoes. Most subsequent models, e.g. Dietz, Molineaux & Thomas (1974) have differed n their reatment f superinfection nd immunity n the vertebrate ost but rely n equations derived y Macdonald to describe the transmission f nfection n the vector. Garrett-Jones 1974) separated hese vector- related actors f Macdonald's equations nto factor efined s the vectorial apacity f the vector opulation. his was defined y Garrett-Jones s the number f new nfections produced y the vector er ase per day'. However,
oth his mathematical ormulation f vectorial apacity n terms f Macdonald's model, .e.
c = ma p /(- log p) (1)
(where m is the ratio of mosquitoes to man, a is the man-biting abit frequency f feeding human blood index), is the daily mosquito urvival ate nd n s the ength f the xtrinsic ncubation eriod), nd his nd subsequent uthors' se of vectorial apacity has been as a measure f the maximal otential mplification ate of the pathogen y the
*Correspondence uthor.
123
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124 Cyclic athogen ransmission odel vector, .e. the ratio f nfectious ites elivered y the vector o proportion f bites y the vector n an infectious ost per day where he probability f superinfection f the vector is zero. It is in this ense that vectorial apacity s used
n this paper. In Macdonald's formulation, t was implicitly ssumed that a is constant, .e. the probability f a mosquito eeding n a human n any time eriod does not depend on the
immediate istory f the mosquito. This s not true, ince mosquito which has taken full blood meal will not feed again for ome time. As shown n this paper, this s an unnecessary onstraint, s is another ssumption equired n Macdonald's derivation, that he daily probability f urvival emains onstant. he alternative odel derived ere assumes yclical eeding, nd n ts implest orm, constant robability f urviving rom one feed o the next. Although his model has been used n publications y Birley nd co- workers Birley 984), by Kay, Saul & McCullagh 1987) and by Saul (1987), ts detailed
derivation as not been published. As the equations of this model are critical or the interpretation f Graves et al. (1990) who estimate vectorial apacity from vector
infection ates, ts derivation s reported n full n this paper.
Macdonald 1957) derived series f quations which would llow estimates f ome of
the parameters n his model based upon the sporozoite rates n captured mosquitoes. These techniques ere sed by Davidson & Draper 1953) but have not ubsequently een widely sed, n part through he difficulty f capturing nd determining he nfectious state of sufficient umbers. There re at least three easons for re-examining he use of vector nfection ates s a means of determining he mportant arameters f the ransmission ycle. First, with he
development f sensitive ucleic cid and antibody robes for etecting athogens n the vector, t has become easier to make reliable measurements n sufficient umbers f
infected ectors. For example, the development f monoclonal antibodies directed against malarial porozoites acilitates he creening f arge numbers f mosquitoes o determine porozoite ates nd the species of Plasmodium resent Collins et al. 1985). Secondly, his paper demonstrates hat the use of vector nfection
ates nstead of vector urvival an lead, in principle, o better stimates f transmission here here s heterogeneity n the vector population, n the environment, r age-dependent ector survival. Whether uch estimates an be achieved n practice will depend upon the availability f suitable ampling rocedures. Thirdly, here has been a new emphasis on control measures aimed at reducing transmission o and from he vector y host nti-gamete ntibodies locking ransmission of malaria to mosquitoes nd host anti-sporozoite ntibodies locking ransmission f malaria from he mosquitoes Miller et al. 1986) and by selection f resistant ectors (Collins et al. 1986). This emphasis n vector abits nd life xpectancy s not surprising since he ontrol measures which cted s the mpetus or many f the xisting modelling studies were directed owards reducing vector ongevity. While estimates f vector longevity an be made from measurements f both vector nfection ates and age structures, stimates f factors uch as transmission locking mmunity nd vector resistance annot be obtained rom tudy f the ife ycle f vectors nd require tudy f their nfection ates. In the companion paper Graves et al. 1990), the practical se of some aspects of this model is illustrated hrough he analysis of data obtained by measuring he malarial vector nfection ates n anopheline mosquitoes ollected from villages near Madang, Papua New Guinea.
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A. J.
SAUL,
P. M.
GRAVES AND
B. H.
KAY
125
BASIC TRANSMISSION MODEL Assumptions nd terms The model presented n this paper describes n quantitative erms one type of transmission f pathogens y vectors. his model, which s applicable o a wide range f diseases, including rboviruses uch as Japanese encephalitis nd parasites such as malaria, s described n qualitative erms elow. The vector ecomes nfected y feeding n an infectious ost. Not all hosts which re
infected will necessarily e infectious, .g. people with malaria will only be potentially infectious f hey ave circulating, ature ametocytes. n this model we use the ymbol to denote he proportion f feeds y the vector opulation n hosts which re potentially infectious. Not all feeds by the vector n hosts which re potentially nfectious ill result n the vector becoming nfected. here are factors which relate to the
vector, he host, the pathogen and the environment hich nfluence his probability. he probability f infection ill depend on the density f pathogens n the blood meal and the size of the meal; transmission-blocking ntibodies n the blood meal may kill the pathogen n the vector ut; different ndividuals n a vector opulation nd different ector pecies may have nnate esistance o infection. he symbol k will be used to signify he probability that vector eeding n an infectious ost will become nfected. Where
host factors ffect k e.g. the presence f ntibodies n the blood meal), he distinction etween n infectious host with low k and a non-infectious ost s a matter f definition nd experimentally equivalent. denotes he probability f feeding n a host nd becoming nfected K= xk) in situations where he two possibilities re equivalent. Most vectors end o feed n a cyclic manner nd having ed will not usually eed gain for ome time. n this model, he probability hat vector eeds n a host hen eturns o feed gain s emphasized. his probability s designated f, he probability f surviving feeding ycle, he ength f time rom ne feed o the next. Other workers ave equated this o the probability f surviving gonotrophic ycle the time rom ne egg-laying o the next) ut disease ransmission s related o feeding nd not to egg-laying feeding s not always inked to egg-laying), nd as age determinations re often made from eeding catches, he use of survival hrough feeding ycle rather han through gonotrophic cycle s more ogical. Vectors which become nfected re usually not mmediately nfectious. he pathogen normally ndergoes cycle f development n the vector. his cycle s called the xtrinsic incubation eriod EIP) and n order o transmit isease, vectors must urvive his eriod. P, is used to designate he probability f urviving he EIP and F to designate he number of feeding ycles per EIP. Not only does the vector ave to survive he xtrinsic eriod before t will be nfectious, but he pathogen lso has to survive nd develop. ome vectors ave defence mechanisms which kill the pathogen or prevent t from becoming nfectious, .g. melanization f malarial oocysts Collins et al. 1986), arbovirus modulation Kramer et al. 1981). The symbol is used to denote he probability hat he pathogen will develop n an infected vector o become nfectious. he use of v mplies hat here s a way of measuring nfected vectors efore he vector kills he pathogen. f the only way of determining f vector s infected s by examining mature athogens, hen arlier illing will go undetected nd in this model will ower he value of k. In his model, Macdonald considered he possibility
This content downloaded from 137.219.154.101 on Wed, 16 Apr 2014 20:52:26 PMAll use subject to JSTOR Terms and Conditions
126 Cyclic athogen ransmission odel
TABLE
1. Symbols used in the text Symbol Meaning Reference a Man biting abit Macdonald (1957) b Proportion f mosquitoes with porozoites hat re nfectious Macdonald 1957) c Vectorial apacity Garrett-Jones 1974)
Db
Proportion f vectors nfected n a biting atch This paper Df Proportion f vectors nfected n a resting fed) catch This paper
D'f
Df for non-random atch This paper F Number f feeding ycles per EIP This paper
HBt
Number f bites host-' day-' Reisen & Boreham 1981) IC Individual ectorial apacity This paper k Probability f vector ecoming nfected er nfectious meal This paper K = xk prob of vector ecoming nfected er bite This paper m Number f mosquitoes host-' Macdonald (1957) n Number f days EIP-1 Macdonald (1957) N Number f vectors eeding or he first ime This paper p Daily probability f survival Macdonald (1957) Pe Probability f surviving he EIP This paper Pf Probability f surviving he feeding ycle This paper Q Proportion f feeds n the vertebrate ost This paper Q' Q for non-randomly aught vectors This paper S Proportion f vectors with nfectious athogens This paper; Macdonald 1957) v Probability f pathogen ecoming nfectious n the vector This paper x Proportion f feeds n a host with nfectious athogens Macdonald 1957)
that mosquito may have sporozoites resent n ts alivary lands nd still e incapable of injecting host with sporozoites. He defined his probability s 'b'. In the model presented ere, therefore s a factor f v. Some authors ave extended he definition f b
to encompass ertebrate ost refractoriness. uch effects re explicitly xcluded rom ur definition f v. An infectious ector will only ransmit he disease f t feeds n a suitable host for he pathogen. Q is used to denote his probability. is usually measured y examining he blood meal of vectors aught fter eeding nd s therefore lso called the blood ndex for
that host species. The frequency ith which host s bitten y nfectious ectors the host noculation rate) depends on the rate t which he vectors re biting. his is designated s the host biting ate HBt, he number f vectors eeding er host per 24 hours. In constructing his basic model, everal implifying ssumptions
re made. Many of these re examined ritically ater. The most mportant f these ssumptions re: i) there are homogeneous ector nd host populations; ii) the probability f an individual ector surviving rom ne feed o the next s constant nd therefore ndependent f the age of that ndividual; iii) vectors which become nfectious i.e. those which urvive he EIP) remain nfectious; iv) the population size of the vector, the susceptible host and alternative osts if ny) do not fluctuate ignificantly ver short eriod f ime, his ime being approximately he ength f the EIP plus the ife expectancy f the vector; v) interrupted lood meals re not ignificant; vi) feeding f n already nfected ector n an infectious ost has no effect n the ourse f the nfection n the vector. A complete ist f all symbols sed in the equations of this paper s given n Table 1.
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