How important is vertical transmission in mosquitoes for the persistence of dengue?Insights from a mathematical model
Ben Adams a,b,⁎, Michael Boots c
a Department of Mathematical Sciences, University of Bath, Bath, BA2 7AY, UKb Department of Biology, Kyushu University, Fukuoka, 812-8581, Japanc Department of Animal and Plant Sciences, University of Sheffield, Sheffield, S10 2TN, UK
⁎ Corresponding author. Department of MathematicaBath, BA2 7AY, UK. Fax: +44 1225386492.
Article history:Received 24 August 2009Revised 5 January 2010Accepted 13 January 2010
Keywords:DengueVertical transmissionAedesMosquitoPersistenceMathematical model
In many regions dengue incidence fluctuates seasonally with few if any infections reported in unfavourableperiods. It has been hypothesized that vertical transmission within the mosquito population allows the virusto persist at these times. A review of the literature shows that vertical infection efficiencies are 1–4%. Using amathematical model we argue that at these infection rates vertical transmission is not an important factorfor long term virus persistence. In endemic situations, increases in reproductive number, half-life andpersistence times of the disease only become significant when vertical infection efficiency exceeds 20–30%.In epidemic situations vertical infection accelerates the course of the outbreak and may actually reducepersistence time. These results stem from the fact that the mosquito life-cycle is relatively rapid andvertically acquired infections are multiplicatively diluted with every generation. When the efficiency ofvertical infection is as low as reported from empirical studies, the virus is rapidly lost unless there is regularamplification in the human population. Processes such as asymptomatic human dengue cases are thereforemore likely to be important in persistence than transmission within the vector population. The empiricaldata are not, however, unequivocal and we identify several areas of research that would further clarify therole of vertical transmission in the epidemiology of dengue.
One of the many challenges of dengue epidemiology is to under-stand how the virus can remain apparently endemic, in the sense thathuman infections are detected regularly, even if there are long periodsof extremely low, or zero, incidence. Although dengue virus occursthroughout tropical and subtropical regions of the world, intra-annualpatterns of human infections vary geographically (Rodhain and Rosen,1997; Rogers et al., 2006). Apparent incidence may be roughlyconstant throughout the year, fluctuate seasonally but never fall tozero, fluctuate seasonally and fall to zero for several months or becharacterized by sporadic outbreaks several years apart. It has longbeen hypothesized that vertical transmission (from mother tooffspring) in the Aedes mosquito population may allow the virus topersist during periods unfavourable for transmission to humans(Angel and Joshi, 2008; Guo et al., 2007; Hull et al., 1984; Joshi et al.,2002; Mitchell and Miller, 1990; Pherez, 2007; Rosen et al., 1983).There is no doubt that vertical transmission does occur, but itsimportance for the persistence of dengue is less clear. Here we useinsights from a mathematical model to argue that vertical infection at
rates so far reported from laboratory experiments are only likely tomake a small contribution to virus persistence.
Over the last three decades Bangkok and some other large urbancentres have seen year round incidence of dengue haemorrhagicfever, which reflects dengue incidence, with a pronounced seasonalpeak in the wet season from June to October (Chareonsook et al.,1999; Nisalak et al., 2003). Prior to this the annual oscillation was stillevident but, due to an overall lower prevalence, very few cases werereported in the cooler, drier months of December and January.Seasonal patterns in dengue prevalence have also been reported forregions of Thailand, Brazil, Puerto Rico and Trinidad (Barbazan et al.,2002; Chadee et al., 2007; Keating, 2001; Siqueira et al., 2005) andmay be a consequence of both mosquito and virus biology. Duringcold or dry periods poor availability of breeding sites or highmortalitymay cause the mosquito population density to fall (Alto and Juliano,2001; Scott et al., 2000). Additionally, dengue virus must incubate inthe mosquito gut to generate sufficient levels for onward transmis-sion. The incubation period is similar to the mosquito life expectancy.But it becomes longer at lower temperatures, introducing seasonalvariations in transmission efficiency (Watts et al., 1987). This maylead to a seasonal variation in incidence in the human population evenif the mosquito population size is stable.
Over the last few decades, it has been demonstrated that verticaltransmission in the insect vector is a component in the epidemiology
Table 1Laboratory vertical infection (ν) and, where available, vertical transmission (vt) andfilial infection (fi) rates for Aedes mosquitoes. All studies found significant variationbetween individual mosquitoes and for different combinations of dengue andmosquitostrains. Values shown here are the highest average rates of all the virus–mosquito strainpairings tested. Most strain pairings resulted in much lower rates.
Species vt (%) fi (%) ν (%) Source
Ae aegypti – – <0.0025 (Rosen et al., 1983)Ae aegypti 3.0 0.13 0.039 (Bosio et al., 1992)Ae aegypti (F1) 2.8 (Joshi et al., 2002)Ae aegypti (F3) 13.0 (Joshi et al., 2002)Ae albopictus – – 1.3 (Rosen et al., 1983)Ae albopictus 41.2 2.9 1.2 (Bosio et al., 1992)Ae albopictus (F1) 0.4 (Shroyer, 1990)Ae albopictus (F2) 4.2 (Shroyer, 1990)Ae scutellaris 57.0 6.7 3.8 (Freier and Rosen, 1987)
2 B. Adams, M. Boots / Epidemics 2 (2010) 1–10
of many pathogens. Examples include La Crosse virus (Miller et al.,1978; Tesh and Gubler, 1975), St Louis Encephalitis virus (Nayar et al.,1986), West Nile virus (Baqar et al., 1993) and Yellow fever virus(Diallo et al., 2000). Vertical transmission of dengue virus has beendemonstrated in the lab in Ae aegypti, Ae albopictus and Ae scutellaris(Bosio et al., 1992; Freier and Rosen, 1987; Joshi et al., 2002; Mitchelland Miller, 1990; Rosen et al., 1983; Shroyer, 1990). In the field,vertical transmission can be inferred by detecting the virus in larvaeor non-biting male mosquitoes. At least one study has failed to detectsuch infections (Watts et al., 1985). Numerous other studies haveprovided clear evidence of vertical transmission of dengue in wild Aeaegypti and Ae albopictus populations (Angel and Joshi, 2008; Hullet al., 1984; Joshi et al., 2006; Kow et al., 2001). It may thereforeappear reasonable to hypothesize that an independent transmissionloop within the mosquito population can maintain the virus throughperiods when human infections are absent. In support of this amathematical model has previously been used to show that thereproductive potential of the virus is indeed enhanced by verticaltransmission. The relationship was found to be non-linear andparticularly sensitive when the efficiency of vertical infection wasclose to 100%. The prevalence of infection was also increased byvertical transmission. The accumulation of immunity meant that thischange was much less pronounced in the human than the mosquitopopulation (Esteva and Vargas, 2000). This model provides importantinsights but does not address the question of viral persistence,particularly when there are seasonal fluctuations in the mosquitopopulation size or the efficiency of vertical infection is relatively low,as found in laboratory studies.
Here we employ a suite of mathematical models, includingversions with seasonal variation in the oviposition rate and mosquitodiapause, to examine how the reproductive number and extinctionprobability of the virus are related to vertical infection efficiency. Weshow that, in endemic situations, vertical transmission enhances viralreproduction and persistence. However, it only has a strong impact ifit is much more efficient than so far reported from laboratoryexperiments. Furthermore, in epidemic situations vertical transmis-sion may actually reduce virus persistence. These results lead us toconclude that, if vertical infection is important in dengue ecology, theefficiency in nature must be substantially greater than that found inthe lab.
Observed vertical infection rates
We briefly review observations of vertical transmission made inthe laboratory over the last 25 years. To avoid confusion we need toclarify some definitions. In the empirical literature the ‘verticaltransmission rate’ (vt) is generally defined as the proportion ofinfected females that infect at least one of their progeny. The ‘filialinfection rate’ (fi) is defined as the proportion of the progeny of aninfected females that are infected given that vertical transmissionoccurred. In the theoretical literature the ‘vertical transmission rate’ issometimes defined as the average number of infected progeny perinfected female i.e. the empirical ‘vertical transmission rate’ multi-plied by the ‘filial infection rate’ (vt× fi). Here, we use the definitions inthe empirical literature and call vt× fi the ‘vertical infection rate’ (ν).Table 1 summarizes results from five laboratory studies. In some casesthe experimental methodology involved examining pools of eggsfrom pools of infected mosquitoes to estimate the minimum verticalinfection rate. In other cases the methodology involved isolatingindividual females and their eggs to give exact figures for the verticaltransmission, filial infection and vertical infection rates. All studiesfound considerable variation between individual mosquitoes anddifferent pairs of dengue and mosquito strains.
Table 1 shows the highest average rates in the population selectedfrom all the virus–mosquito strain pairings tested in a given ex-periment. Most strain pairings resulted in much lower rates than
these. The earliest study found minimum vertical infection rates of1.3% in Ae albopictus and vanishingly small rates for Ae aegypti. Twosubsequent studies broadly agreed with this. In recent years, it hasbecome common to apply polymerase chain reaction (PCR) techni-ques to amplify small samples of DNA and then confirm the identity ofthe virus, and serotype, by electrophoresis (Morita, 1996). Such PCRbased studies have found vertical infection rates as high as 2.5% infield caught Ae aegypti. In artificial selection experiments, where thefirst (F1) and subsequent (Fn) generations were composed only ofmosquitoes reared from infected eggs from the previous generation,the vertical infection rate was found to increase from 0.4% to around3% in Ae albopictus and from 2.5% to 13% in Ae aegypti. However, thelargest average vertical infection rates found in the absence ofartificial selection, and assuming that the PCR primers used were notdetecting other flaviviruses in addition to dengue, have been of theorder 1% for Ae albopictus, 3% for Ae aegypti and 4% for Ae scutellaris.
Mathematical model
Previous studies have incorporated vertical infection into a host–vector–pathogen framework to show that it can increase thereproductive potential of dengue (Esteva and Vargas, 2000) andmay aid the persistence of Ross River virus (Glass, 2005). Verticalinfection has also been incorporated into a model for dengue withseasonal fluctuations in the hatching rate, although its role was notexplicitly examined (Coutinho et al., 2006). In common with these,and most host–vector epidemiological models (Anderson and May,1991) the core of our model is formed by dividing the humanpopulation into susceptible (Sh), infected/infectious (Ih) and recov-ered/immune (Rh) classes. The population of adult femalemosquitoesis divided into susceptible (Sv), infected but not infectious, (Ev) andinfectious (Iv) classes. Males are ignored as they play no part in thetransmission dynamics. The residence time in the mosquito isexpected to be a key determinant of viral persistence. So we introduceadditional classes representing the pre-adult mosquito population,effectively eggs and larvae, in either susceptible (Se) or infected (Ie)states.
A flow diagram of the model system is shown in Fig. 1. Briefly, allhuman classes decrease due to deaths at rate (μh). The susceptible classis increased by births at the same rate so the total population sizeremains constant. We assume the biting rate is frequency dependent(Wonham et al., 2006). So the rate at which infected mosquitoestransmit the virus to susceptible people is given by βIv(Sh/Nh), whereNh=Sh+ Ih+Rh is the total human population and β is the transmis-sible biting rate. Infected people recover at rate γ and gain completelifelong immunity. The pre-adult mosquito class increases at rate bvwhere bv represents the oviposition rate of eggs that will eventuallymature into adults. This term implicitly incorporatesmortality in the pre-
Fig. 1. Flow diagram of host–vector–pathogen model with vertical infection. Sh, Ih, Rh —susceptible, infected, recovered people. Se, Ie — susceptible, infected pre-adultmosquitoes, Sv, Ev, Iv — susceptible, incubating, infectious adult mosquitoes.
Table 2Parameter values used throughout this paper. Nh and Nv were chosen to be as large ascomputationally practical. The population sizes relative to one another are generally themost pertinent measures, but the absolute sizes are important in the stochastic model.The rate parameters correspond to a human life expectancy of 60 years and duration ofinfection of 7 days, a mosquito life expectancy of 8 days, incubation period of 8 days,maturation period of 15 days. The oviposition rate is, when constant, equal to the deathrate, ensuring the total population size remains constant. Baseline values for the adultmosquito population size and biting rates in the model without diapause are 3.84×105
and 0.33 respectively, corresponding to a biting frequency of once every 3 days. Thebiting rate was chosen to give a basic reproductive number of 1.2 in the absence ofvertical transmission. This value is lower than most estimates for dengue butappropriate for our assumption of a region where persistence is borderline. In themodel with diapause the population size, and the magnitude of the seasonal oscillationare determined by the baseline oviposition rate bv=4.5×105. In some cases we wishedto adjust the vector population size but did not want to change the basic reproductiverate so also adjusted the biting rate to compensate.
Symbol Meaning Value
Nh Human population size 106
μh Human death rate 4.57×10−5
γ Human recovery rate, per day 1/7Nv Adult mosquito population size 9.5×104–106
μv Adult mosquito death rate, per day 1/8bv Total oviposition rate (constant), per day μvNv
ω Pre-adult mosquito maturation rate, per day 1/15ε Virus incubation rate in mosquito, per day 1/8β Transmissible biting rate, per day 0.2–0.67ν Vertical infection rate, proportion 0–1δ Magnitude of seasonal fluctuation in bv 0–1d Duration of favourable season, days 270p Diapause survival rate, proportion 0–0.5
3B. Adams, M. Boots / Epidemics 2 (2010) 1–10
adult stage. bv may be constant or fluctuate seasonally bSv(t)=(1+δsin(2πt/360))bv. Here δ controls the magnitude of the fluctuations and, forcomputational convenience, a year has 360 days. If δ is less than 1,oviposition continues throughout the year, albeit at a lower rate duringunfavourable seasons. Aproportionν(Iv/Sv+Ev+Iv) of eggs are verticallyinfected, and he remainder susceptible. Pre-adults mature into theequivalent adult class at rate ω. Since the virus has been presentthroughoutdevelopment,weassumenoadditional incubation is requiredbefore vertically infected mosquitoes become infectious. All adultmosquito classes decrease due to death at rate μv. Susceptible adultsbecome infected at rate βSv(Ih/Nh). We assume that the underlyingtransmission probability between human and mosquito is the same inboth directions. To our knowledge, there are no data from which tounequivocally justify any alternative. Infected mosquitoes becomeinfectious at rate ε and never recover. There are 10 parameters in thebaseline model. The key parameter that will be varied is the verticalinfection rate (ν). Other parameter values are kept fixed at reasonablevalues for dengue, as detailed in Table 2. The system is described by eightordinary differential equations:
dSedt
= bv 1−υIv
Sv + Ev + Iv
� �� �−ωSe
dIedt
= bvυIv
Sv + Ev + Iv
� �−ωIe
dSvdt
= ωSe−βIhNh
Sv−μvSv
dEvdt
= βIhNh
Sv− ε + μvð ÞEv
dIvdt
= εEv + ωIe−μvIv
dShdt
= μhNh−βShNh
Iv−μhSh
dIhdt
= βShNh
Iv− γ + μhð ÞIh
dRh
dt= γIh−μhRh
ð1Þ
Vertical infection and the reproductive number
The basic reproductive number R0 is defined as the expected num-ber of secondary infections resulting directly from a single infectedindividual in an otherwise naïve population. In host–vector systems R0is sometimes defined to be the number of host infections resulting, viaimplicit stepping stones of infected vectors, froma single infected host.However, as lucidly discussed by other authors (Roberts andHeesterbeek, 2003), thinking of R0 strictly as the reproductive rate ofthe pathogen should not make a distinction between human andmosquito infections. When strictly defined as the reproductive rate ofthe pathogen R0 is often conveniently found using a next generation
method (Diekmann and Heesterbeek, 2000; van den Driessche andWatmough, 2002, 2008). As detailed in the Appendix A, for ourmodel,the basic reproductive number is given by:
Note that Λ1/2 is the basic reproductive number in the absence ofvertical infection. The previouslymentioned concept of a reproductivenumber that represents the number of infected hosts resulting from asingle infected host has been formalised as the ‘type’ reproductivenumber T1
h, where the type is the host (Roberts and Heesterbeek,2003; Heesterbeek and Roberts, 2007). An expression for T1h can alsobe found by matrix methods. Another type reproductive number T1v isassociated with the vector population. In the absence of verticaltransmission the host and vector type reproductive numbers are bothequal to the square of the basic reproductive number. For our modelthis is Λ . In the presence of vertical infection R0
2 gives the expectednumber of secondary infections after one average, complete (host–vector–host or vector–host–vector) transmission cycle but does notcorrespond to a specific population type. For the host population, oneinfected host leads to some secondary host infections in the nexthost–vector–host transmission cycle. However, further secondaryhost infections may also occur after any number of vector–vectortransmission cycles. The type reproductive number accounts for thisadditional independent cycle to give the total expected number ofsecondary host infections:
Th1 =
Λ1−ν
ð3Þ
The vector type reproductive number is simply T1v=Λ+ν because
all secondary infections resulting from a single infected vector mustoccur in the next vector–host–vector transmission cycle. SeeAppendix A for technical details on the construction of both typereproductive numbers.
Setting R02=1, T1h=1 or T1v=1 gives the same parameter thresholdsfor an epidemic to occur but the behaviour away from unity may be
4 B. Adams, M. Boots / Epidemics 2 (2010) 1–10
quite different. As Fig. 2a shows vertical infection leads to an effectivelylinear increase in the basic reproductive number and vector typereproductive number but amore rapid, and accelerating, increase in thehost type reproductive number. In a single transmission cycle a singlegeneration ofmosquitoes is infected vertically. Therefore, more efficientvertical infection results in proportional increases inR0
2 and T1v. However,T1h is enhanced by additional host infections from vertical infected
mosquitoes in future generations. In each generation, the number ofhost infections is proportional to the number of infected mosquitoes,which decreases proportionally to the vertical infection efficiency. If theefficiency is low, only thefirst one or two generationsmake a significantcontribution. As the efficiency increases, many more mosquito genera-tions have an impact. The divergence between T1
h and R02 gives an idea of
the degree to which viral persistence is facilitated by verticaltransmission. All the reproductive numbers are virtually indistinguish-able for vertical infection rates up to 10%. Substantial divergence onlybegins when the rate is around 25%.
One of the main goals of public health strategies is to force thereproductive number of a pathogen below 1 and eradicate it. Formosquito-borne pathogens such as dengue, transmission may bereduced by controlling the mosquito population. We define theeradication effort to be the percentage reduction in the mosquitopopulation size required to prevent endemic transmission of the virus.In the context of our model, eradication thus requires reducing thebasic reproductive number, or equivalently either of the typereproductive numbers, below the threshold of 1. So, using the hosttype reproductive number, the eradication effort is 100(1–1/T1h)=100(Λ+ν–1)/Λ . Therefore, the efficiency of vertical infection islinearly related to the eradication effort, as shown in Fig. 2b, andvertical infection at q% can be responsible for at most q% of therequired eradication effort. If the type reproductive number in theabsence of vertical transmission (Λ) is not close to 1 vertical infectionwill have much less impact than this. Vertical infection rates of lessthan 10%, therefore, are unlikely to have much impact on the controleffort required to reduce the mosquito population to a level thatcannot sustain endemic dengue circulation.
Vertical infection and the persistence of dengue
The potential for sustained endemic transmission expressed in thereproductive number is not necessarily the same as the potential forlong term circulation. Persistence may be affected by stochasticfluctuations or periods when themosquito–human transmission cycle
Fig. 2. a: Basic reproductive number for a complete transmission cycle R02 (grey), host typ
depending on percentage efficiency of vertical infection 100ν. b: Percentage of mosquito poperadicate the virus, depending on efficiency of vertical infection. In both panels Nv=3.84×
is less active due to low population densities, high levels of humanimmunity, a long viral incubation period or some other factor. To getsome insight into viral persistence we will consider the results ofstochastic model simulations. First, however, we analyse a simplifiedmodel. We focus on the role of vertical infection by assuming thatsome proportions of the adult and pre-adult mosquito populations areinfected but there is no transmission betweenmosquitoes and people,or vice versa. Then the virus only persists by vertical infection, whichcan be described by a pair of differential equations for the number ofinfectious mosquitoes in the pre-adult and adult classes:
dIedt
= bvυIvNv
−ωIe = μvυIv−ωIe
dIvdt
= ωIe−μvIv
ð4Þ
Unless vertical infection is 100% efficient the inevitable outcome inthe absence of mosquito–human transmission is viral extinction. Butthis takes time. The reduced system can be solved subject to the initialcondition of Ie0 infected pre-adults and Iv
0 infected adults. Part of thesolution decays very rapidly and can be discarded to give a goodapproximate solution for the total number of infections at time t, I(t)=Ie(t)+Iv(t):
It follows that the approximate time until the number of infectedindividuals is reduced by 50% is:
t2 =2
λ−μv−ωlog
I0e + I0v� �
λ
I0e λ + μv + ωð Þ + I0v λ−μv + 2μvν + ωð Þ
0@
1A
24
35ð6Þ
Note that the half-life does not normally depend on the initialcondition. Ie0 and Iv
0 appear here as a result of the approximation buttheir impact is minimal. The approximate time until the number ofinfected individuals is less than 1 is:
te =2
λ−μv−ωlog
2λI0e λ + μv + ωð Þ + I0v λ−μv + 2μvν + ωð Þ
� �� �ð7Þ
e reproductive number T1h (black) and vector type reproductive number T1
v (dashed)ulation that must be removed to reduce the host type reproductive number below 1 and105, β=0.33, bv is constant and all other parameters are as in Table 2.
5B. Adams, M. Boots / Epidemics 2 (2010) 1–10
The half-life, as shown in Fig. 3a, increases with vertical infectionefficiency. The rate of increase is accelerating and so efficient verticalinfection extends the half-life considerably. Inefficient verticalinfection has little impact, with a rate of 10% only increasing thehalf-life from 11.4 to 12.3 days. The time to extinction can be ex-tended considerably by vertical infection, as shown in Fig. 3b. Onceagain, however, the effect of inefficient vertical transmission isamplified with each generation. Even with an extremely large initialinoculum of 105 infectious mosquitoes in each of the pre-adult andadult classes, 10% vertical infection only extends the time to ex-tinction by 31 days.
The relationship between vertical infection and viral extinctionwhen the mosquito–human transmission cycle is included is nowassessed by stochastic simulation. The structure of the original model(Fig. 1) is not modified. However, populations are rendered discretelywhen small, and the solution iterated by a stochastic algorithm(Gillespie, 1977). More details are given in the caption of Fig. 4. Wefirst consider the probability of extinction when the virus is endemic.In this case, the total mosquito population size is constant throughoutthe year. A large proportion of the human population is immune toinfection and there is low level virus circulation prone to extinction bydemographic stochasticity. In the absence of vertical infection thebasic reproductive number is 1.2 and the average endemic persistencetime is around 2.5 years. As Fig. 4a shows, when circulation is endemicand transmission is not seasonal, vertical infection rates of around 5%extend persistence by a matter of months. Rates of 10% may extendpersistence by 2 years but it is only when efficiency exceeds 20–30%that there is a powerful effect and viral persistence moves to thedecadal timescale.
We next consider the probability of extinction following anepidemic outbreak. If the virus is rarely present then the level ofimmunity in the human population will be much lower. So, when thevirus does enter the population it will cause a large epidemic. Weassume that there is no immunity in the human population. Then, inthe absence of vertical transmission the average persistence time isaround 5 years. The end of the epidemic is largely determined by theaccumulation of herd immunity. Demographic stochasticity has lessdirect influence. Chance is, however, important at the beginning of theepidemic. It determines the initial acceleration, which is proportionalto the duration of the epidemic, and whether there will be a large-scale outbreak at all. Around 60% of simulated outbreaks are small andover within a year. All other outbreaks continue for at least 4 years. AsFig. 4b shows, vertical infection has a counterintuitive impact. Itreduces the average duration of epidemics that last more than 1 year.
Fig. 3. a: Time taken for the number of infections in the mosquito population to fall by hmosquitoes and people. Solid line is from approximate analytic expression (6), dashed line itransmission, taken for the number of infections in mosquito population to fall to 1, effectiveladult mosquitoes were Ie
0= Iv0=104 and other parameter values were as in Table 2. Results
Fig. 4c illustrates why. Vertical infection increases the reproductivenumber and this enhances the initial, exponential, growth of theepidemic. When the initial growth rate is higher, the epidemic peak islarger but earlier, and the outbreak is over more quickly.
Finally we consider a time-dependent oviposition rate so that themosquito population size fluctuates seasonally but is never zero. If thevirus is endemic, infection incidence varies throughout the year but ina regular, non-explosive manner due to immunity in the humanpopulation. As Fig. 4d shows, the relationship between vertical in-fection rate and persistence time has the familiar exponential form,indicating that single digit vertical infection rates have only a modestimpact on persistence. When the mosquito population size fluctuatesseasonally, an epidemic may occur still following the introduction ofthe virus into a human population with no immunity. In this casevertical infection generally reduces persistence times, as Fig. 5ashows, unless the magnitude of seasonal fluctuations is very large.Highly efficient vertical infection can lead to a small upturn in theaverage persistence time (not shown) that is not observed in the non-seasonal model. Fig. 5b shows that the seasonal fluctuations lead tomore complex epidemic curves. Nevertheless, as in themodel withoutseasonal fluctuations, vertical transmission accelerates epidemics,making them more severe but shorter.
A slightly different result is seen if the seasonal fluctuations arevery large. Generally, increased vertical transmission reduces persis-tence, as before. However, as shown in Fig. 5c, very low levels ofvertical transmission may increase persistence slightly. The epidemiccurves in Fig. 5d indicate that, although vertical transmission reducespersistence by accelerating the epidemic, it may also improvepersistence by maintaining a larger pool of infections during theunfavourable season. When seasonal variation is very strong, soextinction is highly probable between seasonal peaks, and verticaltransmission is weak the net result is a slight increase in persistence.
Diapause in the mosquito life-cycle
In regions with large environmental fluctuations there may beperiods of the year when no adult mosquitoes are present. Themosquito population may then persist as eggs in a diapausal state.This pattern can be incorporated into the model in a straightforwardway by modifying the oviposition rate. In place of the constant andsinusoidal forms used previously, we now define bSv(t)=sin(πt/d)bvwhen 0≤ t≤d and 0 otherwise. So the oviposition rate variessmoothly from 0 to bv and back again over a favourable season of ddays, and is zero for the rest of the year. At the end of the favourable
alf depending on vertical infection efficiency when there is no transmission betweens an exact numerically determined solution. b: Additional time, compared to no verticaly indicating viral extinction. In both panels the initial numbers of infected pre-adult andare similar for initial infection numbers of 105 and 106.
Fig. 4. a: Efficiency of vertical infection and mean increase in time to viral extinction compared to zero efficiency under endemic circulation. Triangles — Nv=9.5×104, β=0.67,squares — Nv=3.84×105, β=0.33, and circles — Nv=10×106, β=0.26. Initially all variables are close to endemic equilibrium. b: Mean time to viral extinction after an epidemicoutbreak. Nv=3.84×105, β=0.33, initially all individuals naïve except 10 infected people. Grey circles — results of individual trials, black circles — mean of all trials for which theepidemic exceeded one year. c: Examples of the change in epidemic shape, expressed as the number of infections in the human population, with vertical infection efficiencies of 0, 10and 20%. From the deterministic model. d: Mean increase in time to viral extinction, compared to zero-efficiency vertical infection, when circulation is endemic but mosquitooviposition rate bv fluctuates seasonally with magnitude δ. Nv=3.84×105, β=0.33, initially all variables close to the endemic equilibrium. Squares — δ=0.2, circles — δ=0.5, andtriangles — δ=0.8. Panels a,b and d are based on 100 independent trials for each parameter combination. Host variables were always discrete and iterated using a continuous timeMarkov process. Variables for vector classes were rendered discretely and incorporated into the Markov process when less than 100. Otherwise continuous variables and adeterministic Runge–Kutta iteration was applied with a variable time-step determined by the host population Markov process.
6 B. Adams, M. Boots / Epidemics 2 (2010) 1–10
season the dynamics of the adult population continue as before, butwithout reproduction. The dynamics of the pre-adult stage aresuspended and all individuals are assumed to be diapausal. At thebeginning of the next favourable season a proportion p of thediapausal pre-adults are assumed to have survived and reinitiate thecomplete population dynamics. These modifications require the valueof bv to be revised, and two additional parameters: the length of thefavourable season d and the proportion of pre-adults survivingdiapause p. We will focus on the survival parameter. Experimentshave found that, depending on temperature and relative humidity,60–80% of diapausal Ae albopictus eggs survive two to six months ofdormancy in laboratory conditions (Sota and Mogi, 1992). Fieldsurvival is likely to be much lower. Experiments have also found thatvertically acquired dengue infection significantly increases mortalityin non-diapausal Ae aegypti eggs and reduces fecundity of thosefemales that do survive (Joshi et al., 2002). How these laboratorymeasurements relate to the field is an open question, even in terms oforder of magnitude. Here we will consider survival rates of 0 to 50%for both infected and uninfected eggs.Wewill also assume the vertical
infection rate is the same regardless of whether or not eggs arediapausal.
With this transmission framework viral persistence is completelydependent on a combination of vertical infection and diapause sur-vival. Without either of these components an outbreak is inevitablyfollowed by extinction after one season. The impact of both factors issimilar. The key difference is that survival can be controlled byintervention, vertical infection cannot. We consider a situation inwhich there is immunity in the host population commensurate withendemic circulation in the sense of seasonal incidence recurringregularly over many years. As Fig. 6a shows, with inefficient verticalinfection the virus rarely survives more than two seasons, even when50% of the diapausal population survives. The seasonal outbreak isalready winding down by the time diapause begins. The number ofinfected adult mosquitoes is small, the number of vertically infectedpre-adults is smaller still, and there is a low probability that any willsurvive to the following year. Even if they do, a very small initialinoculummeans the outbreak in the next season grows very slowly. Itis then curtailed by the seasonal change, further compounding the
Fig. 5. a: Mean time to viral extinction after an epidemic outbreak when vector birth rate bv shows seasonal variation of intermediate intensity (δ=0.5). Nv=3.84×105, β=0.33,initially all individuals naïve except 10 infected. Grey circles — results of individual trials, black circles — mean of all trials for which circulation continued for at least one year.b: Bottom — deterministic model examples, for δ=0.5, of the change in epidemic shape, expressed as sum of all infected people, pre-adult and adult mosquitoes, with verticalinfection efficiencies of 0 (black), 1 (dashed) and 4% (grey). Top— same curves plotted on logarithmic scale. c: Mean time to viral extinction after an epidemic outbreak when vectorbirth rate bv shows seasonal variation of high intensity (δ=0.8). d: Deterministic examples, for δ=0.8, of the change in epidemic shape vertical infection efficiencies of 0 (black), 1(dashed) and 4% (grey). Initially all variables are at disease free equilibrium. Panels a and b based on 100 independent trials for each parameter combination. Stochastic method asdescribed in the caption of Fig. 4.
7B. Adams, M. Boots / Epidemics 2 (2010) 1–10
impact of vertical infection. There is a clear level of vertical infection,for our parameters around 20–30%, where survival becomes morereliable and persistence times begin to increase rapidly. Theprobability of surviving diapause is also important,with lower survivalrequiring more efficient vertical infection to maintain the virus. AsFig. 6b shows, however, in termsof virus persistence, diapause survivalhas a linear impact and the key factor is vertical infection efficiency.
Discussion
There is no doubt that vertical infection can enhance thepersistence of vector-borne viruses such as dengue by providing atemporary reservoir for the virus. However, the modelling studypresented here suggests that, in the absence of human infections,vertical infection at rates observed in the laboratory has a verymodestimpact on dengue. Under endemic circulation the reproductivenumber, half-life and persistence times predicted by the model allshow the same characteristic exponential relationship with theefficiency of vertical infection: the response is initially very small butstrengthens rapidly when the infection efficiency exceeds 20–30%.
This qualitative form remains the same if the mosquito population isconstant throughout the year, fluctuates seasonally, or undergoesdiapause. There is no reason to believe it would be significantlydifferent in models extended to include other environmental factorsthat retard the mosquito life-cycle such as short periods of eggdesiccation between rainfall events or slower development due to lowtemperatures and resource scarcity. Only epidemic circulation shows adifferent pattern. In this case, vertical infection hastens the growth ofthe epidemic and so actually leads to a reduction in persistence time.
The key question then, is the actual efficiency of vertical infectionin natural populations. If the relatively low efficiencies of 1–4%reported from laboratory experiments are representative then verticalinfection has a no more than a weak influence on the persistence ofdengue. There is, however, tantalising evidence that suggests verticalinfection efficiency in nature may be considerably higher than in thelaboratory. Field studies in Rajasthan, India, found that up to 20% of Aeaegypti and Ae albopictus larvae tested positive for dengue virus(Angel and Joshi, 2008). We cannot infer the vertical infection ratefrom these data since infection prevalence in the adult population isunknown. However, much less than 100% of adult mosquitoes are
Fig. 6. Impact of vertical infection efficiency (a) and probability of surviving diapause (b) on the probability of virus extinction when the mosquito population fluctuates seasonallyand persists in a diapausal state for some of the year. Each point is themean of 100 independent trials, with all variables initially close to the endemic equilibrium. Black lines indicatethe time to extinction in years, up to a maximum of 100. Grey lines indicate the percentage of trials that become extinct within 100 years. The sigmoidal forms to the right of (a) arean artifact of this 100 year maximum, iterating the simulations for longer is expected to lead to an exponential increase in extinction times as ν increases. Nv=2.0×106, β=0.33,bv=4.5×105, d=270 days. a: Solid line, circles— p=0.1, dashed line, triangles— p=0.25, dotted line, and crosses— p=0.5. b: Solid line, circles— ν=0.1, dashed line, triangles—ν=0.2, dotted line, and crosses — ν=0.3.
8 B. Adams, M. Boots / Epidemics 2 (2010) 1–10
likely to be infected, even during a major epidemic. So the verticalinfection rates required for 20% of larvae to test positive must beextremely high and much greater than anything so far detected in thelaboratory.
Laboratory studies also show that vertical infection is more likelywith certain mosquito and virus strain combinations. Furthermore,artificial selection can increase the efficiency of infection by a factor ofthree within three mosquito generations (Joshi et al., 2002). It is notclear from these experiments whether it is the mosquito, the virus, orboth that are being selected for vertical infection. In naturalpopulations, there is no obvious reason to suspect that such selectionwould operate on the mosquito. However, the virus may be subject tointense selection in regions where it is vulnerable to local extinctiondue to small human populations or large climatic or environmentalfluctuations. It would therefore be valuable to investigate verticalinfection in the laboratory using mosquitoes and viruses recentlycollected from the same location. It would then be interesting tocompare results from locations with epidemiological conditions in thehuman population ranging from year round endemicity to smallannual outbreaks in well defined seasons. The fact that highly efficientvertical infection is not observed for all dengue strains also suggeststhat such adaptations are disadvantageous, probably for some aspectof the mosquito–human transmission cycle. So we would not expectto observe these adaptations in areas where there is year roundinfection in the human population. If vertical infection is important inmaintaining dengue in particular regions, genetic sequences of virusesisolated from those regions should form lineages that are distinct fromthe sequences of viruses in regions with year round circulation.Failure to detect such lineages would suggest that the metapopulationprocess whereby the virus population in regions with unstabletransmission is regularly reseeded by introduction from more stableregions is more important than vertical infection for virus persistence.This would not be surprising given that the dispersal of epidemicdengue strains has been shown to be rapid and widespread(Cummings et al., 2004). French Polynesia could prove to be a usefulcase study. Until 2001 dengue outbreaks were sporadic and brief.However, low level endemic circulation was maintained betweenepidemics in 2001 and 2006 (Descloux et al., 2009). If data becomeavailable at sufficient resolution, a detailed investigation to disentan-gle the factors that facilitated this long term persistence could provideconsiderable insight.
The potential importance of vertical transmission is greatest inregions with large environmental fluctuations where diapause allows
the mosquito population to survive unfavourable periods. It is notablethat regions with Aedes albopictus populations that survive unfavour-able seasons by diapause generally only show very sporadic dengueoutbreaks. Without vertical infection viral extinction is inevitableeach year. The model results presented here suggest that the lowvertical infection efficiency found in laboratory studies is not effectivein maintaining endemic persistence. On a local scale, the number ofinfections in the mosquito population is likely to be small and the sizeof the vertically infected mosquito population is greatly reduced witheach successive generation. At the end of the favourable season theorally transmitted epidemic is declining. The efficiency of verticalinfection is a key determinant of the level of infection in the diapausalpopulation. At the beginning of the favourable season it takes sometime for the epidemic in the human population to gather momentum.During this period vertical infection can provide protection againstchance extinction, but only if it is efficient enough to span severalmosquito generations. It has been shown that dengue can be verticallytransmitted to diapausal eggs of Ae albopictus (Guo et al., 2007).Again, the key question is just how efficient this transmission is.Additional important and unknown factors are the survival rate ofdiapausal eggs in natural populations and how severely this iscompromised by infection. As these empirical data become availablevertical infection may emerge as a stronger candidate for viralpersistence than the re-introduction from a reservoir population withyear round transmission. If this is the case then more quantitativemodels for the diapausal framework can be developed and analysed tofind the most effective intervention procedures to break thetransmission cycle.
We have made a conscious effort to keep our model simple inorder to focus on the impact of vertical transmission. In particular, wehave assumed that there is only one viral serotype and one species ofmosquito. As shown in Table 1, several species of Aedes mosquitotransmit dengue. However, experimental results have not yetprovided systematic differences in either vector competence orvertical infection rates. Developing models that give different rolesto each species in maintaining dengue is therefore not possible. It isalso well known that the dengue virus exists as four distinct serotypesthat often co-circulate. The experiments summarized in Table 1 foundthat vertical infection rates vary considerably both within andbetween serotypes but, to date, there are no clear systematic dif-ferences. Furthermore, while there is an immunological interactionbetween the four dengue serotypes it is not well understood. There isevidence of cross-immunity such that infection with one serotype
9B. Adams, M. Boots / Epidemics 2 (2010) 1–10
protects a person against subsequent infection with another serotype.There may also be cross-enhancement where infection with oneserotype facilitates infection with another serotype by increasingsusceptibility to infection or increasing transmission and/or mortalityfollowing infection. Under different combinations of these assump-tions viral diversity can lead to an impressive range of epidemiologicaldynamics including periodicity, chaos, synchronised and de-synchro-nised multi-annual cycles (Cummings et al., 2005; Recker et al., 2009;Ferguson et al., 1999; Adams et al., 2006; Schwartz et al., 2005). Wehave chosen to omit these factors because the dynamics are verycomplex and various competing hypotheses are still being explored.We wish to maintain a clear focus on the role of vertical infection indengue persistence rather than examining detailed epidemiologicalpatterns. The interaction of vertical infection with the intricateincidence patterns of multiple co-circulating serotypes promises tobe an interesting area for future research. However, we expect thatour key result will continue to hold — vertical transmission at levelsobserved experimentally is unlikely to have a major effect on denguepersistence.
We are interested in viral persistence on human time-scales ofyears to decades, but the life-cycle of Aedesmosquitoes is measured inweeks and vertically acquired infections are multiplicatively dilutedwith each of these generations. To facilitate virus persistence, verticalinfection must pass the virus through many mosquito generationswith little or no amplification in the human population. Only a smallproportion of each mosquito generation becomes infected vertically,leading to an exponential decay in prevalence.When vertical infectionefficiency is low, almost all of the infections are lost within twogenerations. Other explanations include rare, long lived mosquitoesand horizontal transmission of the virus from vertically infected malemosquitoes back to females. However, like vertical infection thesecircumstances appear to be low probability and low impact.Therefore, we suggest that the main determinants of viral persistenceare related to circulation in the human population. Re-introductionfrom external reservoir populations and local low level asymptomaticcirculation are likely to be important factors.
Appendix A
The basic reproductive number
Following themethod of (van den Driessche andWatmough, 2002,2008) we write the system of differential equations as φ̇= f–νwhere
φ =
IhIeEvIv
2664
3775; f =
βShNh
Iv
bvυIv
Sv + Ev + Iv
� �
βIhNh
Sv
0
266666666664
377777777775; v =
γ + μhð ÞIhωIe
ε + μvð ÞEv−εEv−ωIe + μv + Iv
2664
3775
ðA1Þ
The Jacobian matrices F and V, associated with f and v respectively,at the disease free equilibrium [Sh=Nh, Se=Ne, Sv=Nv, Ih= Ih=Ev=Iv=0] are:
F =
0 0 0 β
0 0 0 νμv
βNv
Nh0 0 0
0 0 0 0
26666664
37777775; V =
γ + μh 0 0 00 ω 0 00 0 ε + μv 00 −ω −ε μv
2664
3775 ðA2Þ
and the next generation matrix is:
K = FV−1 =
0βμv
βεμv ε + μvð Þ
βμv
0 νεν
ε + μvν
βNv
Nh γ + μhð Þ 0 0 0
0 0 0 0
266666666664
377777777775
ðA3Þ
Equivalently, we can construct the next generation matrix directlyby noting that the element (i, j) is the expected number of newinfections of type i produced by an infected individual of type j(Diekmann and Heesterbeek, 2000). We have four types: Ih, Ie, Ev, andIv. An individual of type Ih causes 0 infections of type Ih, 0 infections oftype Ie, β Nv
Nh γ + μhð Þ new infections of type Ev, and 0 infections of type Iv.
Jumping to the final column, an individual of type Iv causesβμv
new
infections of type Ih, ν new infections of type Ie and no new infectionsof any other type. To fill in the remaining two columns, an individualof type Ie becomes an individual of type Iv with probability 1 and thencauses new infections as described above. An individual of type Ev
becomes an individual of type Iv with probabilityε
ε + μh(and
otherwise dies) and then causes new infections as described above.This completes the next generation matrix. The basic reproductivenumber is then the spectral radius of the next generation matrix(Diekmann and Heesterbeek, 2000, van den Driessche and Wat-mough, 2002, 2008) R0=ρ(K).
Following (Roberts and Heesterbeek, 2003), the host typereproductive number, that is the number of new individuals of typeIh resulting from one individual of type Ih, is given by:
Th1 = eThK I− I−Phð ÞKð Þ−1eh ðA4Þ
where eh=[1, 0, 0, 0], I is the 4×4 identity matrix, Ph is a 4×4 matrixwith P11=1 and all other elements 0.
The vector type reproductive, that is the number of new individualsof type Ev resulting from one individual of type Ev is given by:
Tv1 = eTvK I− I−Pvð ÞKð Þ−1ev ðA5Þ
where ev=[0, 0, 1, 0] and Pv is a 4×4 matrix with P33=1 and allother elements 0. Note that the next generation matrix K asconstructed here cannot be used to determine a type reproductivenumber for Iv because new infections can only be individuals of typesIh, Ie and Ev. An equivalent form of the next generation matrix could,however, be constructed so that new infections can be of types Ih, Ieand Iv, but not of type Ev.
References
Adams, B., Holmes, E.C., Zhang, C., Mammen, M.P., Nimmannitya, S., Kalayanarooj, S.,Boots, M., 2006. Cross-protective immunity can account for the alternatingepidemic pattern of dengue virus serotypes circulating in Bangkok. Proc. Natl.Acad. Sci. U. S. A. 103, 14234–14239.
Alto, B.W., Juliano, S.A., 2001. Precipitation and temperature effects on populations ofAedes albopictus (Diptera: Culicidae): implications for range expansion. J. Med.Entomol. 38, 646–656.
Anderson, R.M., May, R.M., 1991. Infectious Diseases of Humans: Dynamics and Control.Oxford University Press, Oxford; New York.
Angel, B., Joshi, V., 2008. Distribution and seasonality of vertically transmitted dengueviruses in Aedesmosquitoes in arid and semi-arid areas of Rajasthan, India. J. VectorBorne Dis. 45, 56–59.
Baqar, S., Hayes, C.G., Murphy, J.R., Watts, D.M., 1993. Vertical transmission ofWest Nilevirus by Culex and Aedes species mosquitos. Am. J. Trop. Med. Hyg. 48, 757–762.
Bosio, C.F., Thomas, R.E., Grimstad, P.R., Rai, K.S., 1992. Variation in the efficiency ofvertical transmission of dengue-1 virus by strains of Aedes albopictus (Diptera:Culicidae). J. Med. Entomol. 29, 985–989.
10 B. Adams, M. Boots / Epidemics 2 (2010) 1–10
Chadee, D.D., Shivnauth, B., Rawlins, S.C., Chen, A.A., 2007. Climate, mosquito indicesand the epidemiology of dengue fever in Trinidad (2002–2004). Ann. Trop. Med.Parasitol. 101, 69–77.
Chareonsook, O., Foy, H.M., Teeraratkul, A., Silarug, N., 1999. Changing epidemiology ofdengue hemorrhagic fever in Thailand. Epidemiol. Infect. 122, 161–166.
Coutinho, F.A., Burattini, M.N., Lopez, L.F., Massad, E., 2006. Threshold conditions for anon-autonomous epidemic system describing the population dynamics of dengue.Bull. Math. Biol. 68, 2263–2282.
Cummings, D.A., Irizarry, R.A., Huang, N.E., Endy, T.P., Nisalak, A., Ungchusak, K., Burke,D.S., 2004. Travelling waves in the occurrence of dengue haemorrhagic fever inThailand. Nature 427, 344–347.
Cummings, D.A.T., Schwartz, I.B., Billings, L., Shaw, L.B., Burke, D.S., 2005. Dynamiceffects of anti body-dependent enhancement on the fitness of viruses. Proc. Natl.Acad. Sci. U. S. A. 102, 15259–15264.
Descloux, E., Cao-Lormeau, V.M., Roche, C., De Lamballerie, X., 2009. Dengue 1 diversityand microevolution, French Polynesia 2001–2006: connection with epidemiologyand clinics. Plos Neglected Trop. Dis. 3.
Diallo, M., Thonnon, J., Fontenille, D., 2000. Vertical transmission of the Yellow fevervirus by Aedes aegypti (Diptera, Culicidae): dynamics of infection in F-1 adultprogeny of orally infected females. Am. J. Trop. Med. Hyg. 62, 151–156.
Diekmann,O., Heesterbeek, J.A.P., 2000.Mathematical epidemiologyof infectiousdiseases:model building, analysis, and interpretation. JohnWiley, Chichester; New York.
Esteva, L., Vargas, C., 2000. Influence of vertical and mechanical transmission on thedynamics of dengue disease. Math. Biosci. 167, 51–64.
Ferguson, N.M., Donnelly, C.A., Anderson, R.M., 1999. Transmission dynamics andepidemiology of dengue: insights from age-stratified sero-prevalence surveys.Philos. Trans. R. Soc. Lond. Ser. B-Biol. Sci. 354, 757–768.
Freier, J.E., Rosen, L., 1987. Vertical transmission of dengue viruses by mosquitoes of theAedes scutellaris group. Am. J. Trop. Med. Hyg. 37, 640–647.
Gillespie, D.T., 1977. Exact stochastic simulation of coupled chemical-reactions. J. Phys.Chem. 81, 2340–2361.
Glass, K., 2005. Ecological mechanisms that promote arbovirus survival: a mathematicalmodel of Ross River virus transmission. Trans. R. Soc. Trop. Med. Hyg. 99, 252–260.
Guo, X., Zhao, T., Dong, Y., Lu, B., 2007. Survival and replication of dengue-2 virus indiapausing eggs of Aedes albopictus (Diptera: Culicidae). J. Med. Entomol. 44,492–497.
Heesterbeek, J.A.P., Roberts, M.G., 2007. The type-reproduction number T in models forinfectious disease control. Math. Biosci. 206, 3–10.
Hull, B., Tikasingh, E., De Souza, M., Martinez, R., 1984. Natural transovarialtransmission of dengue 4 virus in Aedes aegypti in Trinidad. Am. J. Trop. Med.Hyg. 33, 1248–1250.
Joshi, V., Mourya, D.T., Sharma, R.C., 2002. Persistence of dengue-3 virus throughtransovarial transmission passage in successive generations of Aedes aegyptimosquitoes. Am. J. Trop. Med. Hyg. 67, 158–161.
Joshi, V., Sharma, R.C., Sharma, Y., Adha, S., Sharma, K., Singh, H., Purohit, A., Singhi, M.,2006. Importance of socioeconomic status and tree holes in distribution of Aedesmosquitoes (Diptera: Culicidae) in Jodhpur, Rajasthan, India. J. Med. Entomol. 43,330–336.
Keating, J., 2001. An investigation into the cyclical incidence of dengue fever. Soc. Sci.Med. 53, 1587–1597.
Kow, C.Y., Koon, L.L., Yin, P.F., 2001. Detection of dengue viruses in field caught maleAedes aegypti and Aedes albopictus (Diptera: Culicidae) in Singapore by type-specific PCR. J. Med. Entomol. 38, 475–479.
Miller, B.R., Defoliart, G.R., Yuill, T.M., 1978. Vertical Transmission of La-Crosse virus(California encephalitis group) — transovarial and filial infection-rates in Aedes-triseriatus (Diptera Culicidae). J. Med. Entomol. 14, 437–440.
Mitchell, C.J., Miller, B.R., 1990. Vertical transmission of dengue viruses by strains of Aedesalbopictus recently introduced into Brazil. J. Am. Mosq. Control Assoc. 6, 251–253.
Morita, K., 1996. The identification of dengue viruses using PCR. In: Clapp, J.P. (Ed.),Species Diagnostics Protocols: PCR and Other Nucleic Acid Methods. Humana Press,Totowa, N.J.
Nayar, J.K., Rosen, L., Knight, J.W., 1986. Experimental vertical transmission of SaintLouis encephalitis-virus by Florida mosquitos. Am. J. Trop. Med. Hyg. 35,1296–1301.
Nisalak, A., Endy, T.P., Nimmannitya, S., Kalayanarooj, S., Thisayakorn, U., Scott, R.M.,Burke, D.S., Hoke, C.H., Innis, B.L., Vaughn, D.W., 2003. Serotype-specific denguevirus circulation and dengue disease in Bangkok, Thailand from 1973 to 1999. Am. J.Trop. Med. Hyg. 68, 191–202.
Pherez, F.M., 2007. Factors affecting the emergence and prevalence of vector borneinfections (VBI) and the role of vertical transmission (VT). J. Vector Borne Dis. 44,157–163.
Recker, M., Blyuss, K.B., Simmons, C.P., Hien, T.T., Wills, B., Farrar, J., Gupta, S., 2009.Immunological serotype interactions and their effect on the epidemiologicalpattern of dengue. Proc. R. Soc. B-Biol. Sci. 276, 2541–2548.
Roberts, M.G., Heesterbeek, J.A., 2003. A new method for estimating the effort requiredto control an infectious disease. Proc. Biol. Sci. 270, 1359–1364.
Rodhain, F., Rosen, L., 1997. Mosquito vectors and dengue virus–vector relationships.In: Gubler, D.J., Kuno, G. (Eds.), Dengue and Dengue Hemorrhagic Fever. CABInternational, Wallingford, Oxon, UK; New York.
Rogers, D.J., Wilson, A.J., Hay, S.I., Graham, A.J., 2006. The global distribution of Yellowfever and dengue. Adv. Parasitol. 62, 181–220.
Rosen, L., Shroyer, D.A., Tesh, R.B., Freier, J.E., Lien, J.C., 1983. Transovarial transmissionof dengue viruses by mosquitoes: Aedes albopictus and Aedes aegypti. Am. J. Trop.Med. Hyg. 32, 1108–1119.
Schwartz, M.D., Man, B., Manwell, L., Mundt, M., Varkey, A.B., Williams, E., Linzer, M.,2005. The chaotic office environment: role of patient ethnicity and impact onphysician stress and burnout. J. Gen. Intern. Med. 20, 206–207.
Scott, T.W., Amerasinghe, P.H., Morrison, A.C., Lorenz, L.H., Clark, G.G., Strickman, D.,Kittayapong, P., Edman, J.D., 2000. Longitudinal studies of Aedes aegypti (Diptera:Culicidae) in Thailand and Puerto Rico: blood feeding frequency. J. Med. Entomol.37, 89–101.
Shroyer, D.A., 1990. Vertical maintenance of dengue-1 virus in sequential generationsof Aedes albopictus. J. Am. Mosq. Control Assoc. 6, 312–314.
Sota, T., Mogi, M., 1992. Survival-time and resistance to desiccation of diapause andnondiapause eggs of temperature Aedes (Stegomyia) mosquitos. Entomol. Exp.Appl. 63, 155–161.
Tesh, R.B., Gubler, D.J., 1975. Laboratory studies of transovarial transmission of La-Crosse and other arboviruses by Aedes-albopictus and Culex-fatigans. Am. J. Trop.Med. Hyg. 24, 876–880.
van den Driessche, P., Watmough, J., 2002. Reproduction numbers and sub-thresholdendemic equilibria for compartmental models of disease transmission. Math.Biosci. 180, 29–48.
van den Driessche, P., Watmough, J., 2008. Further notes on the basic reproductionnumber. In: Brauer, F., van den Driessche, P., Wu, J., Allen, L.J.S. (Eds.), MathematicalEpidemiology. Springer, Berlin.
Watts, D.M., Harrison, B.A., Pantuwatana, S., Klein, T.A., Burke, D.S., 1985. Failure todetect natural trans-ovarial transmission of dengue viruses by Aedes-aegypti andAedes-albopictus (Diptera, Culicidae). J. Med. Entomol. 22, 261–265.
Watts, D.M., Burke, D.S., Harrison, B.A., Whitmire, R.E., Nisalak, A., 1987. Effect oftemperature on the vector efficiency of Aedes aegypti for dengue 2 virus. Am. J.Trop. Med. Hyg. 36, 143–152.
Wonham, M.J., Lewis, M.A., Renclawowicz, J., van den Driessche, P., 2006. Transmissionassumptions generate conflicting predictions in host–vector disease models: a casestudy in West Nile virus. Ecol. Lett. 9, 706–725.
