Theoretical Population Biology 83 (2013) 82–94

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Theoretical Population Biology

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Stage and age structured Aedes vexans and Culex pipiens(Diptera: Culicidae) climate-dependent matrix population modelŽeljka Lončarić a, Branimir K. Hackenberger b,∗a BioQuant, Našička 4, 31000 Osijek, Croatiab Department of Biology, Josip Juraj Strossmayer University in Osijek, Trg Ljudevita Gaja 6, 31000 Osijek, Croatia

a r t i c l e i n f o

Article history:Received 15 July 2011Available online 24 August 2012

Keywords:MosquitoesVariable carrying capacityClimate dependencyPopulation modelTransient dynamics

a b s t r a c t

Aedes vexans and Culex pipiens mosquitoes are potential vectors of many arbovirial diseases. Due tothe ongoing climate changes and reappearance of some zoonoses that were considered eradicated,there is a growing concern about potential disease outbreaks. Therefore, the prediction of increasedadult population abundances becomes an essential tool for the appropriate implementation of mosquitocontrol strategies. In order to describe the population dynamics of A. vexans and C. pipiens mosquitoesin temperate climate regions, a 3-year period (2008–2010) climate-dependent model was constructed.The models represent a combination of mathematical modeling and computer simulations, and includetemperature, rainfall, photoperiods, and the flooding dynamics of A. vexans breeding sites. Both modelsare structured according to the developmental stages, and by individuals’ ‘‘age’’ (i.e., time spent ineach developmental stage), as we wanted to enable a time delay between the appearances of differentdevelopmental stages of mosquitoes. The time delay length is temperature dependent, with temperaturebeing the most important factor influencing morphogenesis rates in immatures and gonotrophic cycledurations in adult mosquitoes. To determine which developmental stages are the most sensitive and arethose at which control measures should be aimed, transient elasticities were calculated. The analysisshowed that both mosquito species reacted to perturbation of the same matrix elements; however, inthe C. pipiens model, the stage with greatest proportional sensitivity (i.e., elasticity) during most of thethree-year reproduction season contained adults, while in the A. vexans model it contained larvae. Themodels were validated by comparing 7-day model outputs with data on human bait collection (HBC)obtained from the Public Health Institute of Osijek-Baranja, with both model outputs showing validcompatibility with field data over the three-year period. The proposed models can easily be modifiedto describe population dynamics of other mosquito species in different geographical areas, as well as forassessing the efficiency and optimization of different mosquito control strategies.

© 2012 Elsevier Inc. All rights reserved.

1. Introduction

During the past few years, the geographical distribution ofarthropod-borne zoonoses has expanded considerably (Chevalieret al., 2004). Climate changes can cause the emergence and re-emergence of vector-borne diseases by changing their geographi-cal distribution aswell as their dynamics (Confalonieri et al., 2007).Several studies predict that diseases such as malaria, dengue andWest Nile viruswill have increased transmission intensity and thattheir spatial distribution will expand in correlation with climatechanges (Hales et al., 2002;Martens et al., 1995; Ogden et al., 2008;Hongoh et al., 2012). In August 2007, the first indigenous trans-mission of chikungunya in Europe was reported from a rural area

∗ Corresponding author.E-mail addresses: hack@biologija.unios.hr, hackenbergerb@gmail.com

(B. K. Hackenberger).

0040-5809/$ – see front matter© 2012 Elsevier Inc. All rights reserved.doi:10.1016/j.tpb.2012.08.002

in Emilia-Romagna, Italy (Townson and Nathan, 2008). Consider-ing the ongoing climatological changes and reappearance of somezoonoses that are considered to be eradicated (Kallio-Kokko et al.,2005; Akritidis et al., 2010; Polley, 2005), there is a growing con-cern about potential disease outbreaks. Therefore, in order to pre-dict increased mosquito abundances and implement appropriatecontrol strategies it is very important to knowwhich environmen-tal parameters govern the population dynamics of mosquitoes.

In particular, temperature is one of the most important en-vironmental factors influencing insect physiology and behavior(Ratte, 1985), and mosquitoes like all poikilotherms are highly de-pendent on the ambient temperature for successful development(Ahumada et al., 2004). As all mosquitoes have aquatic larval andpupal stages and thus require water for breeding and develop-ment, heavy rainfall was correlatedwith increasedmosquito abun-dances and in some areas with subsequent disease outbreaks byseveral authors (Hu et al., 2006; Kelly-Hope et al., 2002; Lindsayet al., 1993). Foreseeable annual changes in environmental factors

Ž. Lončarić, B. K. Hackenberger / Theoretical Population Biology 83 (2013) 82–94 83

such as photoperiod and temperature are used by mosquitoes ascues to initiate or cease their activity, because seasonal activitypatterns change with latitude and differ among mosquito species(Knight et al., 2003). Diapause as an adaptation for hibernationis common in mosquitoes from northern and temperate latitudes(Vinogradova, 2007), so, in the temperate belt, a seasonal cycle ofAedes and Culex species involves its development and reproductionduring the spring–summer period and a reproductive diapauseduring the autumn–winter time (Mori and Wada, 1978; Thomsonet al., 1982; Vinogradova, 2007).

There are severalways to predict peaks inmosquito populationsand risks of disease outbreaks. The most common one is bycorrelating mosquito abundance data with various environmentalvariables such as temperature, precipitation, rainfall, and tidalevents (Reisen et al., 2008; Yang et al., 2009), orwith climatologicaland hydrological model outputs (Thomson et al., 2006; Shamanet al., 2002). All these models have a relatively good correlationwith mosquito abundance data and are able to predict an increasein mosquito populations or an increase in disease outbreak risks;however, these models can be constructed only for areas in whichsystematicmonitoringwith available large quantities of data aboutmosquito abundance has been carried out.

Over the last two decades different types of model havebeen developed to describe the population dynamics of variousmosquito species (Focks et al., 1993; Fouque and Baumgartner,1996; Shone et al., 2006; Shaman et al., 2006; Erickson et al., 2010),two of them being matrix models (Schaeffer et al., 2008; Ahumadaet al., 2004). Age-structured matrix models were developedindependently by Bernardelli (1941), Lewis (1942), and Leslie(1945, 1948). These models classified organisms into discreteage classes and incorporated age-specific vital rates such assurvival probability and fecundity of each age class from one ageclass to the next. Lefkovitch (1965) developed a stage-structuredmodel for species whose development is better described usingcharacteristics such as organism size, weight, and physiological,morphological, or developmental state. In our work, stage andage structured models were combined for modeling the dynamicsof the two most common mosquito species in the city of Osijek(45°33

′

4′′

N, 18°41′

38′′

E, Croatia), A. vexans and C. pipiens. C.pipiens mosquitoes are cosmopolitan species that can be foundin almost all known urban and suburban temperate and tropicalregions. They are known to be very potent disease vectors forseveral diseases such as the West Nile and St. Louis encephalitisviruses, avianmalaria, and filarial worms (Bogh et al., 1998; Reisenet al., 1992; Turell et al., 2002). A. vexans vector competence isusually considered as negligible, but several studies have shownthat this mosquito species could be involved in vector-diseasetransmissions (Yildirim et al., 2011; Goddard et al., 2002a,b;Molaeiand Andreadis, 2006).

The goal of this research design was to construct a stageand age structured model that could enable time delay in thepresence of different developmental stages of C. pipiens and A.vexans mosquitoes. This research design also implemented amechanistic model construction approach using biological andecological characteristics ofmosquitoeswith factors that influenceand limit their growth anddevelopment in temperate geographicalareas. The models were constructed based on available literatureand monitoring data on C. pipiens and A. vexans mosquitoes andwere used for a simulation ofmosquito population dynamics in thecity of Osijek during 2008–2010. Model outputs were comparedwith data on human bait collection (HBC), and general correlationexisted in both models. To determine which developmental stagesare the most sensitive and are those at which control measuresshould be aimed, transient elasticities were calculated for eachmosquito species.

Fig. 1. Environmental parameters used in the model construction. Mean dailytemperature (°C) and number of rainy days in Osijek during the three-year period(2008–2010). In the model the photoperiod at latitude 45 °N was used. Danube’swater levels (cm) from measurement station Batina in the three-year period(2008–2010).

2. Methods

A discrete-time, stage and age structured matrix model, withone-day projection interval, was constructed in order to simulatethe population dynamics of C. pipiens and A. vexans mosquitoes.The model is constructed for a three-year period (2008–2010),and only females are modeled. In this section, the mosquito life-cycle and the specific biological characteristics of the modeledspecies are described. Then, three-dimensional projection matrixconstruction, specific influences that are modeled for eachspecies, and the general structure of the models is described.Environmental parameters used in the model construction areshown in Fig. 1.

2.1. Study area

The city of Osijek is located in the continental part of Croatia,and, according to Koppen’s classification, it belongs to the climatetype Cfwbx. This is a moderately warm rainy climate withno dry periods and with precipitation uniformly distributedthroughout the year. The mean temperature of the coldest month(January) usually does not drop below −0.4 °C, while the meantemperature of the warmest month (July) usually does not exceed21.4 °C. Minimum temperatures during the winter can sometimesbe below −25 °C, and, during the summer, the maximumtemperatures can sometimes exceed 40 °C. Annual precipitationamounts to 900 mm, with a maximum in June, and a secondarymaximum in September, while the annual relative air humidityranges between 77% and 92%. However, during the past ten years,strong and sudden temperature and rain variations have becomequite usual. Located 10 km northeast of the city of Osijek is amarshland area and Kopački Rit Nature Park. The basic ecologicalfeature of Kopački Rit is given by its flooding dynamics; thelandscape of the whole region depends on the flood intensity.The parts of the marshland change size, form, and functiondepending on the quantity of risen water originating mostlyfrom the Danube, and to a smaller extent from the River Drava.

84 Ž. Lončarić, B. K. Hackenberger / Theoretical Population Biology 83 (2013) 82–94

This area represents an ideal breeding site for different speciesof floodwater mosquitoes, among which A. vexans is the mostabundant, being 75.59% of mosquito fauna. C. pipiens mosquitoesare also continuously present in the city area throughout the year(Merdić et al., 2010).

2.2. Mosquito biology

Common characteristics: Mosquitoes are holometabolous insects,which means that they undergo complete metamorphosis. Theirlife cycle consists of four different developmental stages: embryo(egg), larva, pupa, and imago or adult. A few days after oviposition,larvae hatch from the eggs and start their development in thewater. Several days later, depending on environmental conditionsand food abundance, larvae turn into pupae which do notfeed anymore. After approximately two days, adult mosquitoesemerge from the pupae. Shortly after emergence, females arefertilized, with mating occurring only once in their lifetime.The female usually passes through several gonotrophic cycles,whose number depends on numerous environmental factors. Eachgonotrophic cycle consists of host seeking, blood feeding, andoviposition. Temperature is one of the most important abioticfactors affecting the complete mosquito life cycle—development,growth, and survival of immature mosquitoes (Clements, 1992),and blood digestion rates, ovary development, and gonotrophiccycle duration in adult females (Eldridge, 1965; Madder et al.,1983).C. pipiens biological characteristics. The females lay their eggs onlyupon standing water, and the eggs are not drying resistant. In cooltemperate areas, C. pipiens hibernate as nulliparous, inseminatedfemales that enter a facultative reproductive diapause (Mitchell,1983). The adult diapause in females is induced by shorter daylength and the low temperature experienced during larval andpupal development (Spielman, 2001).A. vexans biological characteristics. A. vexans is a floodwatermosquito and it is known that fluctuations in the abundance ofthe larvae of the genus Aedes are influenced by the flood regime oftheir breeding sites (Maciá et al., 1995). Females of this mosquitospecies lay their eggs in moist substrates without standing wa-ter, with eggs usually being resistant to desiccation and hatchingwhen flooded. If the environmental conditions are unfavorable, theeggs are dormant and can hatch 5–7 years later (Kliewer, 1961).This species overwinters as eggs. The egg diapause is an adapta-tion to the seasonality of climatic conditions, with a winter eggdiapause being typical for mosquito species occurring in temper-ate zones. The photoperiod and temperature are themain environ-mental factors responsible for the induction of an egg diapause inmosquitoes, and the photoperiod is the major diapause-inducingstimulus (Vinogradova, 2007).

2.3. Matrix dimension determination

The mosquito population of both species is divided into threeimmature stages: eggs, larva, and pupae; and six adult stages:one nulliparous stage in which females do not reproduce, andfive parous stages or five gonotrophic cycles in which femalesreproduce (Fig. 2). It is well known that temperature changeshave a significant effect on the duration of the immature stages(de Meillon et al., 1967); therefore the developmental stages arefurther divided. The dimensions of each projection submatrixfor developmental stages were determined as the maximal timerequired to complete all stages at low mean daily temperatures,according to data of Kamura (1959). The egg stage is thereforefurther divided into durations of 20 days, as this is supposed tobe the maximum time needed for eggs to develop into larvae.

Fig. 2. Overview of the mosquito life-cycle as used in the models. The mosquitolife-cycle is divided into nine stages, and each stage is further divided into daysof duration according to the maximum time required to complete developmentin each stage at low mean daily temperatures. Egg stage (E) is divided into a20-day duration, the larval (L) stage into a 26-day duration, the pupal (P) stageinto a 5-day duration, and the nulliparous (N) stage and all five gonotrophic cycles(GCs) into 8-day durations. The life-cycle graph shows possible transitions betweendevelopmental stages.

Larval stages are not separated to instars, but this stagewas furtherdivided into a 26-day duration. The pupal stagewas further dividedinto 5 days as this is the maximum time needed for development.In C. pipiens mosquitoes, the average duration of the gonotrophiccycle is 5.54 ± 1.73 days (Faraj et al., 2006), so the nulliparouscycle and all five gonotrophic cycles were further divided into8-day durations. The projection matrix had dimensions 99 × 99,and distinguished individuals according to their stages and timespent in each stage. The matrix dimensions were determinedaccording to literature data available, and the same matrixdimensionswere used for bothmosquito species. As themodelwasconstructed for a three-year period (2008–2010), 1096 projectionmatrices were constructed for every day in a year, and thenthe matrices were combined to one three-dimensional projectionmatrix (TDPM), with dimensions 99 × 99 × 1096.

2.4. C. pipiens model

2.4.1. Temperature-dependent transition probabilitiesTransition probabilities were calculated based on the data of

stage durations at various temperatures for C. pipiens mosquitoes(Kamura, 1959). First, the data were fitted and the functions werechosen based on correlation coefficients. Using those functions, wecould calculate stage duration (SD) values for each developmentalstage at different mean daily temperatures (T ). The functionsused for modeling the stage durations at various mean dailytemperatures, their general form, and function parameters arelisted in Table 1.

We assumed that the calculated values of stage duration atmean daily temperature in time (t) represent the time at which50% of the population in each of stage develop to the next stage;and therefore that value can be used as an inflection point of a two-parameter logistic function that we used for modeling transitionprobabilities:

Pi(t) =1

1 + exp(−b(ln(TS(t)) − ln(SD(t)))), (1)

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Table 1Best-fit functions and function parameters used for calculations of stage durations at various mean daily temperatures (T ). The same functions were used for both mosquitospecies. Two different parameterizations (1, 2) of the Weibull model exist within the drc package (dose-response curve) under an R software environment, and they do notyield the same fitted curve for a given dataset (Seber and Wild, 1989).

Stage Function General form Function parameter values Correlation (r)b c d e

Egg Four-parameter Weibull (1) SDE(t) = c + (d − c) exp(− exp(b(ln(T (t)) − ln(e)))) 2.978 1.916 8.343 15.485 0.9989Larva Four-parameter Weibull (1) SDL(t) = c + (d − c) exp(− exp(b(ln(T (t)) − ln(e)))) 9.872 11.932 26.251 20.533 0.9999Pupa Four-parameter logistic SDP (t) = 1/1 + exp(b(ln(T (t) − ln(e)))) 43.246 2.999 5.000 18.999 0.9999Adult Four-parameter Weibull (2) SDA(t) = c + (d− c)(1− exp(− exp(b(ln(T (t) − ln(e)))))) −5.561 2.585 25.398 13.802 0.9843

Fig. 3. Probabilities of moving from the larval to the pupal stage, based on thetime spent in the larval stage and the mean daily temperature. It is not possiblefor larvae to move to the pupal stage before day 7, as this is the minimal timeneeded for morphogenesis to be completed. The transition probability increaseswith increasing mean daily temperature and increasing time spent in the larvalstage.

where Pi(t) is the probability of moving from stage i to stage i+ 1,TS(t) is time spent in the developmental stage (days) and b isslope of the two-parameter logistic function. We assumed thatthe slope (b) is equal in functions describing stage durations andtransition probabilities. The probability of moving from stage i tostage i + 1 thus depends on the time spent in stage i and themean daily temperature (Fig. 3). As the temperature increases,the transition probability increases, while the development timedecreases. So, during one projection interval, an individual cancontinue development through the same stage (becoming one dayolder), or can ‘‘jump’’ directly to the next developmental stage,which depends on the mean daily temperature.

The dependence of the developmental rates on ambient tem-perature introduces complications in the population models(Rueda et al., 1990), as the transition probabilities describingdevelopment (e.g., probability of egg becoming larva) are heldconstant. However, in our model, the transition probabilities aretemperature dependent, and the projection interval is one day, sothe elements of the projection matrix are changing daily.

2.4.2. Density dependenceIn 1948, Leslie introduced a density-dependent model to illus-

trate limited population growth. Several authors have reportedthat the density dependence in the mosquito populations occursduring early larval stages (Gilpin and McClelland, 1979; Service,1985; Juliano, 2007), and that the density-dependent competi-tion among larvae is an important factor regulating the growthof mosquito populations (Agnew et al., 2000). For those reasons,the population carrying capacity (K ) was chosen to be in the larvalstage.

q(t) =K + (λ(t) − 1)

K

nLAR(t), (2)

where q(t) is Leslie’s density-dependent factor, λ(t) is thedominant eigenvalue of the projection matrix in time t , nLAR(t) isthe sumof all larvae present in time t , andK is the carrying capacityfor larvae.

2.4.3. FecundityThe fecundity of C. pipiens females changes with the season

and with the female’s age through each subsequent gonotrophiccycle. For modeling procedures, the seasonal changes in fecundityexperimental data of Sichinava (1978) were best fitted using thefour-parameter Brain–Cousen function:

Fec(t) = 81.13 +−57.52 + 1.67ODY

1 + exp(ln(ODY ) − ln(216.71)), (3)

where ODY is the Ordinal Day of Year.The fecundity also changes with every subsequent gonotrophic

cycle. Using experimental data (Rouband, 1944), egg raft ratioswere calculated with respect to the first gonotrophic cycle.During the projection interval, the fecundity in each gonotrophiccycle was multiplied by its associated ratio. As females do notreproduce continuously in time, the fecundities are distributed at8-day intervals in the first four gonotrophic cycles, with femalesovipositing at the beginning of each, and at the end of the lastgonotrophic cycle. The general form of the fecundity vector is

F =160F · · · 0 2

68F · · · 0 376F · · · 0 4

84F · · · 0 0 · · ·599F

, (4)

where abF is the fecundity (average number of eggs) laid by a female

b days old, and in the ath gonotrophic cycle.

2.4.4. Rainfall dependenceFrequent rainfalls increase the abundance of habitats available

for the mosquito females to oviposit their eggs. If rainfall is absentfor a long period of time, the females have nowhere to deposit theireggs; so in the model, we assumed that the fecundity is rainfalldependent. Tomodel the influence of rainfall on fecundity,we usedthe cumulative number of rainy days in an 8-day period:

rf (t) =

t−8t

rd · rpf , (5)

where rf (t) is the rain factor which incorporates the cumulativenumber of rainy days (rd) during the past 8 days. The rain potentialfactor (rpf ) accounts for the strength of the rainfall influenceon the Culex mosquito population. The rain factor decreases asthe number of rainy days in the 8-day interval decreases and,accordingly, the fecundity decreases:

Fec(t) = Fec(t)rf10

+ rrf , (6)

where rrf is the rest-rain fecundity, which denotes the minimumnumber of eggs laid by one female in extremely unfavorableenvironmental conditions. This parameter was introduced to themodel because, although there might not be any rain, even for along period of time, there are always places where some femaleswill be able to deposit their eggs (i.e. fecundity is never 0).

86 Ž. Lončarić, B. K. Hackenberger / Theoretical Population Biology 83 (2013) 82–94

Fig. 4. Modeling changes of carrying capacity in the A. vexans model. With increase of the Danube’s water level, the flooded area increases and thus the carrying capacityfor Aedes larvae increases. Flooding starts when the Danube’s water level is above 200 cm.

2.4.5. Diapause inductionThe C. pipiens model includes an overwinter period for adult

females during unfavorable environmental conditions (i.e., thetemperature and photoperiod are below threshold values).Temperature diapause induction

The threshold temperature, below which no development oc-curs, is 8 °C (Farghal et al., 1987), so this temperature was usedas the threshold value in the model (trshTEMP). In the model, whenthe mean daily temperature is below the threshold value, the val-ues on the main diagonals of the adult projection submatrices areset to 1, and all other values in the projection matrix are set to 0(e.g., all females remain in their stages, and all other stages die).During those unfavorable temperature conditions, females do notoviposit, so the fecundity is also set to 0.Photoperiod diapause induction

Adult females enter a diapause in response to a shorter daylength. As photoperiods shorter than 12 h induce a reproductivediapause in all females (Spielman and Wong, 1973), we used thatvalue as the threshold value (trshPHP) at which all females entera diapause. In the model, when the photoperiod is below thethreshold value, the values on the main diagonals of the adultprojection submatrices are set to 1, and all other values in theprojectionmatrix are set to 0 (e.g., all females remain in their stagesand all other stages die, as no other stages are present during theoverwinter period). The fecundity is also set to 0.

2.5. A. vexans model

In the A. vexansmodel, the same transition probabilities and thesame fecundity changes were used as in the C. pipiensmodel.

2.5.1. Flooding dynamics of A. vexans breeding sitesFlooding dynamics has several important influences on Aedes

mosquitoes. An increase in the surface of the flooded area enhancesegg hatching and subsequent larval survival which in effect leadsto increased adult mosquito abundances. In the A. vexans model,we modeled the variable carrying capacity that is dependent onthe Danube’s water level. If we assume a conic shape of theperiodically flooded areas (i.e.,mosquito breeding sites), then,withevery unit of increase in the Danube’s water level, the surface ofthe flooded area increases exponentially. For those reasons, thecarrying capacity coefficient was modeled using an exponentialfunction:Kf (t) = 1 + 0.4391 exp(0.005WLD(t)), (7)where Kf (t) is the carrying capacity coefficient at time t , andWLD(t) is the Danube’s water level (cm) at time t . The

parameters of the function (7) were obtained empirically basedon environmental experiments andmonitoring programs (Merdić,2002). The Danube’s flooding threshold is 200 cm, so an increaseof the Danube’s water level above that value causes an increaseof the flooded area, and the carrying capacity (K ) for A. vexanslarvae increases (Fig. 4). The carrying capacity for larvae at eachtime interval (t) is calculated usingK(t) = K0Kf (t), (8)where K0 is the carrying capacity when there is no flood (i.e., the

Danube’s water level is ≤200 cm), and Kf is the carrying capacitycoefficient at time t .

The density-dependent factor, which is set only for larvae, isthereforemodified accordingly to account for the variable carryingcapacity:

q(t) =K(t) + (λ(t) − 1)

K(t)

nLAR(t). (9)

The second important characteristic of A. vexansmosquitoes is theresistance of the eggs to desiccation and freezing. Eggs can hatchafter several years of estivation; so, during one season, eggs from aprevious season are hatching too, given the right environmentalconditions. To model this influence we also used the Danube’swater level. With the increase of water level, the flooded areaincreases, and thus more of the previously laid eggs hatch. Thisinfluence is modeled using a second-degree polynomial function.The parameters of the function are obtained empirically based onmonitoring programs and by model calibration.

NEGG(t) = 0.0264(WLD)2(t) + 9.431WLD(t) − 80.128, (10)

where NEGG(t) is the number of eggs from previous seasons thatare flooded and that can hatch. At each time interval, the numberof ‘‘older’’ eggs is summedwith one-day-old eggs in the populationvector at time t .

2.5.2. Diapause inductionA. vexans mosquitoes overwinter as eggs, and an egg diapause

in mosquitoes can be induced by temperature and photoperioddecrease.Temperature diapause induction

Eggs of this mosquito species are completely dormant attemperatures below 8 °C (Gjullin et al., 1950), so this temperaturewas set as the threshold value (trshTEMP) below which anegg diapause is induced. In the model, when the mean dailytemperature is below the threshold value, the values on the maindiagonal of the egg projection submatrix are set to 1, and all othervalues in the projection matrix are set to 0 (e.g., all eggs remain intheir stages, and all other stages die).Photoperiod diapause induction

As previously stated, the photoperiod is the major diapause-inducing stimulus for the induction of an egg diapause inmosquitoes. A photoperiod of ≤12 h is set as the threshold value(trshPHP) at which all eggs are diapausing. In the model, when thephotoperiod is below the threshold value, the values on the maindiagonal of the egg projection submatrix are set to 1, and all othervalues in the projection matrix are set to 0 (e.g., all eggs remain intheir stage and all other stages die, as no other stages are presentduring the overwinter period). The fecundity is also set to 0.

2.6. Structure of the models

The C. pipiensmodel has the general form

n(t + 1) = TDPM(T , R, Php)q−1n(t), (11)

Ž. Lončarić, B. K. Hackenberger / Theoretical Population Biology 83 (2013) 82–94 87

Fig. 5. General structure of the model. Projection matrices (A) were constructed for every day in a year, for a three-year period (2008–2010), based on the climatologicaldata, biological, and ecological characteristics of the modeled mosquito species. The matrices were then combined into one three-dimensional projection matrix (TDPM).For every projection matrix within the TDPM, the finite rate of the increase (λ) was calculated. n(t) shows the general structure of the population vector.

and the A. vexans model has the form

n(t + 1) = TDPM(T , Php,WLD)q−1n(t), (12)

where t is timemeasured in days and n is a vectorwith the numberof individuals in each stage and ‘‘age’’. Every projection matrixwithin the three-dimensional projection matrix (99 × 99 × 1096)in the C. pipiens model is a nonlinear function of the mean dailytemperature (T ), rainfall (R), and photoperiod (Php). In the A.vexans model, the projection matrices are functions of mean dailytemperature (T ), photoperiod (Php), and the Danube’s water level(WLD). q−1 is the reciprocal of Leslie’s density-dependent factor(based on the number of larvae present at time t). The generalstructure of the three-dimensional projection matrix and of thepopulation vector is shown in Fig. 5. Abbreviations, variables, andparameters used in the models are listed in Table 2.

2.7. Transient model analysis

Themodel analysis becomesmore complicatedwith an increasein projection matrix size and increase in the number of projectionmatrices. Also, as individuals in themodels are separated accordingto their stage and ‘‘age’’, interpretation of the sensitivity andelasticity results would be of little practical value. For thosereasons, before analyzing themodel, the projectionmatrices (A(t))were reduced to size 4×4 to correspond to the four developmentalstages ofmosquitoes: eggs, larvae, pupae, and adults. The elementsof the reduced projection matrices (S(t)) were calculated usingprojection matrices A(t), population vectors n(t), and generationvectors (sum of all individuals in stage at time t) g(t) and g(t + 1).After the projection matrices were reduced in size, the transientsensitivities of n(t + 1) to the elements of S(t) were calculated forevery matrix (Caswell, 2007):

dg(t + 1)dvecTS

= Sdg(t)dvecTS

+ (gT (t) ⊗ I), (13)

where ⊗ denotes Kronecker product, I is the identity matrix, T ismatrix transpose, and the vec operator is used to stack columns

of a matrix into column vector. The result of this calculation is amatrix that contains elements of the vector g(t + 1) sensitive tothe elements of S(t).

The transient elasticities of g(t+1) to the elements of S(t)werecalculated (Caswell, 2007):

diag [g(t + 1)]−1 dg(t + 1)dvecTS

diag [S] , (14)

where diag[x] is a matrix with x on the diagonal and zeros else-where.

The transient population growth rate at time t was calculatedfrom the model outputs:

r(t) = logN(t + 1)N(t)

, (15)

where N is total population size in time t + 1 and t .

2.8. Model validation

To validate the models, we compared the 7-day smoothedmodel outputs with data on human bait collection (HBC) obtainedfrom the Public Health Institute of Osijek-Baranja County. Absolutemosquito population sizes are very difficult, if not impossible, toestimate from the field data, so model outputs are often comparedto field data by looking for a good correlation or agreementrather than absolute numeric agreement (Lord, 2007). Our modelvalidation scaled both field observations and 7-day smoothedmodel outputs, dividing all number of bites with the maximumnumber of bites. TheC. pipiens andA. vexans7-day smoothedmodeloutputs were also divided by the associated maximum outputfrom the model. As the mosquito bite data were not categorizedaccording tomosquito species, we compared those values with thevalues from the C. pipiens and A. vexansmodels and with the sum.

2.9. Weather input data

Mean daily temperature, rainfall (measurement station: Osi-jek), and Danube’s water level (measurement station: Batina) data

88 Ž. Lončarić, B. K. Hackenberger / Theoretical Population Biology 83 (2013) 82–94

Table 2Symbols for abbreviations and parameters used in the model.

T Mean daily temperature (°C)R RainPhp Photoperiod (hours of daylight)SD(t) Stage duration of different developmental stages at different mean daily temperatures at time t calculated from best-fit functions (Table 1)TS(t) Time spent in developmental stage at time tA(t) Projection matrix in time tS(t) Reduced projection matrixTDPM Three-dimensional projection matrixS(t) Sensitivity matrix in time tODY Ordinal Day of YearPi(t) Two-parameter logistic function used for calculation of transition probabilities at time t for every developmental stageq(t) Leslie’s density-dependent factor at time trf (t) Rain factor at time trd Rainy day (denotes days when it was raining)rpf Rain potential factor (denotes rain influence on mosquito population)rrf Rest-rain fecundity (denotes minimum number of eggs laid by one female in extremely unfavorable environmental conditions (i.e., long periods without rain))WLD Danube’s water level (cm)Kf (t) Carrying capacity coefficient at time t calculated based on the Danube’s water levelK(t) Carrying capacity for A. vexans larvae in time tK0 Carrying capacity for A. vexans larvae when there is no floodK Carrying capacity for C. pipiens larvaeFec(t) Five-parameter Brain–Cousens function that calculates fecundities in time (t)nLAR(t) Number of larvae present in time tn(t) Population vector at time tg(t) Generation vector in time t (obtained by summing all individuals of different ‘‘age’’ that are in the same stage)NEGG(t) Second-degree polynomial function which calculates the number of eggs from the previous season that are flooded and that can hatchtrshTEMP Temperature threshold value (8 °C) below which no development occurstrshPHP Photoperiod threshold value (12 h) below which a diapause is induced

were obtained from the Croatian Meteorological and HydrologicalService in Zagreb, Croatia.

2.10. Computational methods

Computations, simulations, and plotting were performed usingR (version 2.11.1), an open-source language and environment forstatistical computing and graphics (R Development Core Team,2010, Vienna, Austria), an implementation of S-language (Ihakaand Gentleman, 1996). Experimental data were fitted using the drc(dose-response curve) package under an R software environment(Ritz and Streibig, 2005). The elasticitymatriceswere plotted usingthe Plotrix package under an R software environment (Lemon,2006).

3. Results

Simulations were computed for a three-year period (2008–2010) for both mosquito species, with initial parameters for C.pipiens K = 1000, rrf = 5, rpf = 1, N (adults) = 100,and for A. vexans K = 10000, N (egg) = 100 (Fig. 6). In theA. vexans model, the time delay between the first appearancesof the different developmental stages at the beginning of thereproduction season is evident. The number of eggs than canhatch increases during early spring, although no adults arepresent. Those are the eggs laid in previous seasons that canhatch when flooded, if ambient temperatures are suitable. Thepopulation dynamics of the A. vexans adults starts to change15–30 days later, depending on environmental conditions. Thepopulation dynamics of C. pipiens mosquitoes starts to changeduring mid-spring, when the photoperiod is above the thresholdvalue (12 h) and the adult diapause is terminated. Both mosquitopopulations have several peaks during the seasons, with the firstpeak usually occurring in the early to mid spring, followed byseveral others depending on environmental conditions. The adultpopulation of the A. vexans mosquitoes at the beginning ofreproduction seasons in 2008 and 2009 peaked approximately10–15days before the adult C. pipienspopulation,while in 2010 theadult population of C. pipiens peaked first, approximately 10 days

before A. vexans adults. The population dynamics of both modeledspecies during late fall and winter (i.e., during the overwinterseason) is invariable, but the number of overwinter population(A. vexans eggs and C. pipiens adults) between the two reproductionseasons changes.

Daily growth rates (r(t)) were calculated for both mosquitospecies from themodel outputs as log(N(t +1)/N(t)) (Fig. 7). Thegrowth rates show a response of mosquito populations to immedi-ate environmental conditions. A. vexans and C. pipiens mosquitoeshave different growth rate patterns during all three reproductionseasons, as they have different ecological characteristics and theirgrowth and development depends on various different environ-mental factors.

The transient elasticities show a proportional effect on thepopulation structure from the proportional changes in each valueof the projectionmatrix for every day in the three-year period. Bothmosquito population structures (i.e. population stages) reactedto perturbation of the same matrix elements (Fig. 8). Egg stagereacted to perturbation of matrix element S1,1 (probability of eggremaining in the same stage), and matrix element S1,4 (fecundity).The larval stage reacted only to the perturbation of matrix elementS2,2 (probability of larvae remaining in the same stage). Adultsreacted to the perturbation of matrix elements S4,3 (probabilityof pupa moving to the adult stage) and S4,4 (probability of adultsremaining in the stage). Pupae reacted to the perturbation ofmatrix element S3,3 (probability of pupae remaining in the samestage). Values in all other matrix elements were zero at all times,except for matrix elements S2,1 (probability of egg moving tothe larval stage) and S3,2 (probability of larvae moving to thepupal stage), whose values were zero or very close to zero(e.g., less than 10−10). To determine which stage has the greatestproportional sensitivity, i.e., elasticity, results in both mosquitospecies were analyzed by examining howmany days the elasticityvalues for a given stage were smaller or greater than the medianelasticity value during the three-year reproduction seasons. Themedian value was calculated using all elasticity values from threereproduction seasons (i.e., from 1 April to 30 September) for eachmosquito species. In the C. pipiens model, the elasticities of adultsto matrix element S4,4 were above the median value during mostof the reproduction seasons (503 days), the elasticities of larvae

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Fig. 6. 7-day smoothed model outputs: A. vexans eggs (a_egg), larvae (a_lar), pupae (a_pup), and adults (a_ad), and C. pipiens eggs (c_egg), larvae (c_lar), pupae (c_pup), andadults (c_ad).

Fig. 7. Growth rates r(t) of C. pipiens and A. vexansmosquitoes during the three-year reproduction seasons. Periods from 1 April to 30 September of each year (2008–2010)are shown.

90 Ž. Lončarić, B. K. Hackenberger / Theoretical Population Biology 83 (2013) 82–94

Fig. 8. Elasticities of population structure n(t +1) to reduced projectionmatrix elements S in the C. pipiens and A. vexansmodels. Only periods from 1 April to 30 Septemberof each year (2008–2010) are shown.

Fig. 9. Frequencies showing the number of days during the three-year reproduction season (552 days total) when elasticity values for a given stage were smaller orgreater than the median elasticity value in the C. pipiens model (C.ele1.1—elasticities of eggs to matrix element S1,1 (i.e., probability of egg remaining in the same stage);C.ele1.4—elasticities of eggs to matrix element S1,4 (i.e., fecundity); C.ell2.2—elasticities of larvae to matrix element S2,2 (i.e., probability of larvae remaining in the samestage); C.elp3.3—elasticities of pupae to matrix element S3,3 (i.e., probability of pupae remaining in the same stage); C.elad4.3—elasticities of adults to matrix element S4,3(i.e., probability of pupa moving to the adult stage); C.elad4.4 — elasticities of adults to matrix element S4,4 (i.e., probability of adults remaining in the stage)) and the A.vexans model (A.ele1.1—elasticities of eggs to matrix element S1,1 (i.e., probability of egg remaining in the same stage); A.ele1.4—elasticities of eggs to matrix element S1,4(i.e., fecundity); A.ell2.2—elasticities of larvae tomatrix element S2,2 (i.e., probability of larvae remaining in the same stage); A.elp3.3—elasticities of pupae tomatrix elementS3,3 (i.e., probability of pupae remaining in the same stage); A.elad4.3—elasticities of adults to matrix element S4,3 (i.e., probability of pupa moving to the adult stage);A.elad4.4—elasticities of adults tomatrix element S4,4 (i.e., probability of adults remaining in the stage)). Themedian elasticity value was calculated using all elasticity valuesfrom a three-year reproduction season (i.e. from 1 April to 30 September 2008–2010) for each mosquito species.

to matrix element S2,2 were above the median value for 498 days,and the elasticities of pupae to matrix element S3,3 were above themedian value for 491 days. In the A. vexansmodel, the elasticities oflarvae to matrix element S2,2 were above the median value duringmost of the reproduction seasons (516 days), the elasticities ofpupae to matrix element S3,3 were above the median value for508 days, and the elasticities of adults to matrix element S4,4 wereabove the median value for 496 days (Fig. 9).

When the model outputs were compared to the field data,a general correlation or agreement between model outputs andfield data existed. The A. vexans model had a more significantagreement with field data in the sense of accurately predictingtiming and maximum adult population values (Fig. 10). Also, the

model accurately predicted the number of population peaks duringseasons. In 2008, themodel’s first population peak underestimatedabundances, but other increased adult abundanceswere accuratelypredicted. The C. pipiens model outputs had less agreement. Themodel outputs during 2008 and 2009 overestimated the adultabundances, but the timing of the peaks showed relatively goodagreementwith field data. The summed values of the C. pipiens andA. vexans model outputs had the best agreement with field data.

4. Discussion

The climate limits the distribution of infectious diseases, andthe weather affects the dynamics and intensity of disease out-

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Fig. 10. A. vexans (a_ad), C. pipiens(c_ad) adult model outputs and summed model outputs (summ) compared to field data on human bait collection (HBC) obtained fromthe Public Health Institute of Osijek-Baranja County. Both model outputs and field observations were scaled by dividing each dataset by the corresponding maximum value.All model outputs are 7-day smoothed.

breaks; hence the prediction of environmental conditions that leadto an increase in mosquito populations is essential in the preven-tion of possible disease outbreaks and maximization of control ef-ficiencies. In temperate areas, during certain periods, populationgrowth is limited and the main factors responsible are the pho-toperiod, low ambient temperatures, and dry seasons. Overwin-tering periods are usually neglected in the other models usinga similar approach in model construction (Shaman et al., 2006;Erickson et al., 2010; Ahumada et al., 2004; Schaeffer et al., 2008),but they can significantly influence the dynamics and abundance ofmosquito populations in subsequent reproduction seasons, espe-cially in species that overwinter as eggs. In both models presentedin this paper all developmental stages of mosquitoes are includedas well as all environmental factors that influence the populationdynamics of the two modeled mosquito species.

Mosquitoes have four developmental stages, and certain timesegments are characterized by the presence of only one or severaldevelopmental stages. To adequately describe the population dy-namics of mosquitoes and other ecologically similar insect species,especially in strong seasonal environments, it is important that themodel includes and achieves time delay, because not all develop-mental stages are present at the same time. Modeling those pop-ulations by using only stage or age structured matrix populationmodels is very difficult. Our research design implemented mod-els that were structured according to the individual’s stage and thelength of each developmental stage (i.e., ‘‘age’’ of individual in eachstage). During one projection interval, every individual can con-tinue its development through the same stage becoming one dayolder, or it can move directly to the next developmental stage. Thetransition probabilities depend on the mean daily temperaturesand the time spent in the developmental stage. The older the indi-vidual gets, the higher the probability of moving to the next devel-opmental stage. So during one or even several projection intervals,depending on environmental conditions, none of the individuals

have tomove to the next developmental stage, which creates a de-lay in the model. The length of those time delays between the de-velopmental stages is temperature dependent. At lowermeandailytemperatures that delay is longer, as morphogenesis of individualswithin the stage is not completed. At high ambient temperatures,this delay is shorter, because morphogenesis at high ambient tem-peratures is completed in a shorter period of time. Although thisapproach requires the use of quite large dimension projection ma-trices, modern computer and mathematical software enable thesecomplicated and demanding calculations.

Periodic changes of various environmental conditions can bethe cause of changes in carrying capacity for some mosquitospecies. Various theoretical frameworks have shown that theaverage total biomass of a population in a periodic environmentcan be greater than or less than the average total biomass inthe associated constant average habitat (Henson and Cushing,1997). The fact that changes in the carrying capacity can have asignificant influence on the population dynamics of some insectshas been proven experimentally. Jillson’s (1980) experimentwith flour beetles (Tribolium castaneum) showed that the totalpopulation numbers in the periodically fluctuating environmentcan be more than twice of those in the constant environment,even though the average flour volume in which the flour beetleswere grown was the same in both cases. For those reasons, itis important to include variations of carrying capacity in themodels constructed for most of the mosquito species. In our work,the A. vexans model includes variable carrying capacity whichis influenced by flooding dynamics (i.e. changes in the Danube’swater level). It is known that fluctuations in the abundance ofthe larvae of the genus Aedes are basically influenced by theflood regime of their breeding sites (Maciá et al., 1995), as theincreased surfacewetness favorsmosquito reproduction and larvalsurvival, which can subsequently lead to an increase in floodand swamp water mosquito abundances (Shaman et al., 2006).

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Because mosquitoes have a rather short life span, and immaturedevelopmental stages can last from several days to at most twoweeks, it is clear that changes in the flooding dynamics of flood-water mosquitoes’ breeding sites can significantly influence adultmosquito abundance. This is even more important for thosemosquito species that overwinter in the egg stage, as those eggscan hatch several years later, given the right conditions (i.e. floodand temperature), and also contribute to increased adult mosquitoabundances. However, in the C. pipiens model, the mosquitocarrying capacity was held constant, as this mosquito has verydifferent survival strategy: it lays its eggs in virtually any receptaclecontaining water that is rich in decomposing organic material; sowe assumed that the carrying capacity for this mosquito species inurban areas remains constant.

Mosquito development, especially in the immature stages, isvery dynamic, so the population response to changes in the en-vironmental conditions is very rapid. Development can be accel-erated or slowed down within several days, so estimating thepopulation growth over the entire cycle (i.e., whole reproductionseason) cannot provide precise information on immediate popu-lation states. As mosquitoes are organisms with vital rates thatcan vary significantly within period of several days, we deter-mined population growth rates r(t) from the model outputs aslog(N(t + 1)/N(t)), as those values give more precise indicationson population states and show their response to immediate en-vironmental conditions. In both mosquito species, the populationgrowth rate is determined mostly by the ambient temperature.However, for the C. pipiens mosquito, the rain pattern and fre-quency are important for reproduction and egg hatching, while forthe A. vexans mosquitoes the important factor is the flooding dy-namics. From the growth rates (Fig. 7)we can see that in 2010 theA.vexans growth rate was zero or slightly negative during all of May,while becoming positive in June. During 2010 the Danube’s wa-ter level was mostly below the flooding threshold (200 cm), whichcaused delayed egg hatching and resulted in the A. vexans peak-ing after the C. pipiensmosquitoes, which was not the case in 2008and 2009.

The matrix sensitivity analysis is usually based on determiningthe sensitivity of the asymptotic growth rate to changes inmatrix elements aij. Such analysis assumes that the distributionof the population’s age, stage, or size remains stable throughtime and that the population grows according to a constant rate(i.e., λ). Taylor (1979) concluded that many (possibly even most)insect species growing in seasonal environments never experiencea stable age distribution, and various empirical evidence alsosuggests that stable population states rarely occurs in nature(Bierzychudek, 1999; Clutton-Brock and Coulson, 2002). Severalauthors point out the importance of transient dynamics inpopulation management (Ezard et al., 2010; Stott et al., 2011),and show that transient dynamics can be very different from theasymptotic dynamics (Koons et al., 2005; Buhnerkempe et al.,2011). Considering the characteristics of mosquito populations,especially in temperate climate regions, it is highly unlikely thatthose populations ever experience stable stage distributions, so,in order to gain proper knowledge on the mosquito populationcharacteristics, a transient model analysis was implemented.Different methods are available today for transient model analysis(Fox and Gurevitch, 2000; Yearsley, 2004; Caswell, 2007).

Our model used large projection matrices to enable variabledurations of different developmental stages, and we also useddifferent projection matrices for every day, which caused theanalysis of themodel to become computationally very demanding,even not possible at all. Also, the interpretation of results would beof little practical value as individuals in themodels were separatedaccording to their stage and ‘‘age’’. For example, if five- or six-day-old larvae were the most sensitive, that would imply that

control measures applied to those individuals would have had thegreatest impact in reducing the total population size. However,those individuals cannot be separated from the rest of the larvaepopulation. Therefore, the projection matrices were reduced tobiologically meaningful forms. The reduced matrices describe thedevelopmental stages of the mosquitoes: eggs, larvae, pupae, andadults. Transient analysis of those matrices can give us biologicallyrelevant results that can be easily interpreted and used forpractical purposes, especially in mosquito control management.Transient analysis showed that both mosquito species reactedto perturbation of same matrix elements; however, not all ofthose perturbations are important at the same time (Fig. 8).The analysis shows that elasticities are constantly changing dueto population intrinsic properties and changing environmentalconditions. If control measures are planned in some period of time,it is important to know which developmental stages the controlmeasures should be aimed at. Control measures implementedin stages with highest proportional sensitivity, i.e., elasticity,will have the best result in reducing the total populationsize. In both mosquito models, larval and adult stages havegreatest proportional sensitivities duringmost of the reproductionseason (Fig. 9), so implementation of control measures at thesedevelopmental stages seems optimal. However, it is important tonote that, as the elasticities of both mosquito species are changingwith different dynamics, controlmeasures applied at a certain timewill have a diverse impact on different mosquito species.

When comparing our models to the field data, the A. vexansmodel showed relatively good compliance with field data, whilethe C. pipiensmodel fitted to field data to a lesser extent. The reasonfor such a difference in model compliances could be due to the factthat the experimental data did not target specifically these twospecies specifically, but all nuisance mosquito species in the cityand its surrounding area. As A. vexans makes up the majority ofmosquito fauna, it is reasonable to assume that most of the humanbait collection data can be attributed to this mosquito species.This is probably the reason for better agreement between the A.vexans model and field data. The C. pipiens mosquito, accordingto available data, makes up only 5–10% of the mosquito faunain the city of Osijek. Also, almost all mosquito control efforts areaimed at this mosquito species. Mosquito control measures canalter the population dynamics, and thus can be the reason forwrong estimates, especially for the C. pipiensmosquito.

Mechanistic models can be valuable tools in predicting in-creasedmosquito abundances and thus possible disease outbreaks.The implementation of these principles during the model de-sign and development, based on the population’s biological andecological characteristics, allows us to predict their dynamics indifferent environmental conditions and closely examine intrinsiccharacteristics of the entire population. This would not be possi-ble if the models were constructed through correlating weatherpattern data with historical mosquito abundance data. The mainadvantage of this model is that it can be easily modified for im-plementation in different geographical areas, or modified in orderto describe the population dynamics of different mosquito species,evenwhen historical data aboutmosquito abundances are unavail-able. The model can also be modified for prediction of the risk ofmosquito-borne diseases.

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