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The Effect of Temperature on
Anopheles
Mosquito
Population Dynamics and the Potential for Malaria
Transmission
Lindsay M. Beck-Johnson
1,2
*, William A. Nelson
3
, Krijn P. Paaijmans
1,4
, Andrew F. Read
1,2,4,5
,
Matthew B. Thomas
1,4
, Ottar N. Bjørnstad
1,2,4,5
1Center for Infectious Disease Dynamics, The Pennsylvania State University, University Park, Pennsylvania, United States of America, 2Department of Biology, The
Pennsylvania State University, University Park, Pennsylvania, United States of America, 3Department of Biology, Queen’s University, Kingston, Ontario, Canada,
4Department of Entomology, The Pennsylvania State University, University Park, Pennsylvania, United States of America, 5Fogarty International Center, National
Institutes of Health, Bethesda, Maryland, United states of America
Abstract
The parasites that cause malaria depend on Anopheles mosquitoes for transmission; because of this, mosquito population
dynamics are a key determinant of malaria risk. Development and survival rates of both the Anopheles mosquitoes and the
Plasmodium parasites that cause malaria depend on temperature, making this a potential driver of mosquito population
dynamics and malaria transmission. We developed a temperature-dependent, stage-structured delayed differential equation
model to better understand how climate determines risk. Including the full mosquito life cycle in the model reveals that the
mosquito population abundance is more sensitive to temperature than previously thought because it is strongly influenced
by the dynamics of the juvenile mosquito stages whose vital rates are also temperature-dependent. Additionally, the model
predicts a peak in abundance of mosquitoes old enough to vector malaria at more accurate temperatures than previous
models. Our results point to the importance of incorporating detailed vector biology into models for predicting the risk for
vector borne diseases.
Citation: Beck-Johnson LM, Nelson WA, Paaijmans KP, Read AF, Thomas MB, et al. (2013) The Effect of Temperature on Anopheles Mosquito Population Dynamics
and the Potential for Malaria Transmission. PLoS ONE 8(11): e79276. doi:10.1371/journal.pone.0079276
Editor: Fabio T. M. Costa, State University of Campinas, Brazil
Received June 1, 2013; Accepted September 19, 2013; Published November 14, 2013
Copyright: ß2013 Beck-Johnson et al. This is an open-access article distributed under the terms of the Creative Commons Attribution License, which permits
unrestricted use, distribution, and reproduction in any medium, provided the original author and source are credited.
Funding: This research was supported by United States Department of Agriculture National Research Initiative 2006-35302-17149 to ONB, www.usda.gov, and by
National Science Foundation-EID program grant EF-0914384 to MBT and AFR, www.nsf.gov. The funders had no role in study design, data collection and analysis,
decision to publish or preparation of the manuscript.
Competing Interests: The authors have declared that no competing interests exist.
* E-mail: lmb404@psu.edu
Introduction
Mosquitoes are very efficient vectors of human diseases and are
responsible for transmitting some of the most devastating diseases
today. For many of these diseases, the age structure and
abundance of female adult mosquitoes are key in determining
the ability of a mosquito population to vector the disease
effectively; malaria is one such disease. Malaria is the most
prevalent human vector borne disease, with one half of the world
population living in areas where there is risk of infection [1].
Despite the widespread transmission it is still difficult to predict
future malaria intensity, particularly in the face of climate change.
Because the parasites that cause malaria are so strongly tied to
mosquitoes for transmission, malaria incidence will change as the
climate changes; however, it is still unclear and a matter of debate
how the change(s) in transmission will occur [2–10].
Mathematical models of malaria transmission have a long
history dating back a century [11]. The classic Ross-MacDonald
model has been particularly influential and assumptions made in
the model have, in various forms, been included in the majority of
malaria models that followed [12–14]. The focus of Ross-
MacDonald and many subsequent models is the human popula-
tion, assuming that there is a constant adult mosquito population
capable of transmitting parasites. The mosquito lifecycle is
generally ignored because eggs, larvae and pupae are not involved
in the transmission cycle. This is a useful simplification of the
system but unfortunately the results of these models do not predict
malaria intensity in most endemic regions [14]. There have been
exceptions to this generalization, with some models focusing on
the mosquito population, and/or the influence of environmental
drivers, such as temperature and rainfall [8,15–20]. Of these
models, the ones that explicitly include temperature predict a peak
in abundance of vectors at temperatures that are higher than those
observed to occur in conjunction with malaria transmission in the
field [54]. We propose that a disconnect exists between classic
model predictions and observed epidemiology that is caused by
mosquito population dynamics that depend on ambient environ-
mental conditions and are strongly influenced by juvenile stage
dynamics.
Malaria is caused by Plasmodium spp. protozoan parasites.
Female Anopheles mosquitoes pick up Plasmodium parasites in a
blood meal taken from an infectious person; blood is required in
order to develop eggs. The parasites then go though several
developmental stages before they migrate to the mosquito salivary
glands. Once in the salivary glands the parasites can be
transmitted to a susceptible human host when the mosquito takes
PLOS ONE | www.plosone.org 1 November 2013 | Volume 8 | Issue 11 | e79276
another blood meal [21]. The time spent developing in the
mosquito is known as the extrinsic incubation period (EIP) and its
duration is determined by temperature [7,12,22].
Mosquitoes have four main life stages: egg, larva, pupa and
adult. The three juvenile stages, egg, larva and pupa, are aquatic.
Typically 1 to 10% of the eggs that are laid emerge as adult
mosquitoes [23–29]. The larval stage is the longest of the three
juvenile stages and is the only one that feeds. Previous studies
indicate that larvae experience the majority of the effects of
density-dependence [30,31]. Density-dependence is thought to
manifest in several different ways depending on species. Increased
larval mortality and decreased developmental speed are two of the
most commonly measured density-dependent effects on juveniles.
Because larval conditions determine adult characteristics, density
can play through into the adult stage by changing the number of
emerging adults and the size, fecundity and survival of adults
[30,32–34].
The EIP is often relatively long compared to the life expectancy
of mosquitoes. For this reason, the age structure of a given adult
mosquito population is a major determinant of that population’s
vectorial capacity (the ability of the population to transmit the
parasite). The common assumption is that only around 10% of the
adult population survives to the epidemiologically relevant age
[7,35]. It is unknown, however, what changes occur in the
proportion surviving in response to changes in juvenile population
makeup or temperature conditions, on which they are dependent.
Even a small shift in the adult age structure can have big
consequences in terms of the disease burden.
Both Anopheles and Plasmodium are sensitive to temperature.
Because mosquitoes are ectotherms, each life stage is dependent
on temperature in the developmental and mortality rates. The
blood meal-egg laying cycle, known as the gonotrophic cycle, in
adult females is also dependent on temperature. Interestingly, the
temperature-dependencies are not the same among the stages,
leading to nonlinearities in population responses to temperature
[2,33,36–38]. Additionally, the optimum temperature for parasite
growth does not necessarily correspond to the vector optimum.
The effects of temperature on mosquito life history and parasite
development have been acknowledged for many years; however,
these are rarely included in models used to predict malaria
transmission.
We developed a model that begins to take into account the
complex, nonlinear temperature relationships present throughout
the life cycle, as well as intra-stage competition among larvae. The
framework draws on a rich body of previous theory that has been
developed for modeling stage-structured invertebrate populations
[39–43]. The model is comprised of a set of temperature-
dependent delayed differential equations (DDE). Temperature is
included in all the developmental delays, egg-laying and mortality
rates. Using the model, we ask how temperature affects adult
mosquito age structure and population densities and thus the
potential for disease transmission. The combination of nonlinear
temperature-dependencies and within stage density-dependence
lead to non-intuitive dynamics that emphasize the potential
importance of including vector dynamics in future malaria models.
Additionally, the predicted age structure of the adult population
points to a greater influence of temperature and juvenile stages
than previously thought. Furthermore, our model predicts
estimates of the peak temperatures for malaria transmission at
temperatures that are more in line with the observed biology than
when the classic assumption of a static vector population is used.
The ability to predict a peak in potentially infectious mosquitoes
that lines up more closely with observed malaria incidence is an
important development because it allows for a better understand-
ing of the population drivers and dynamics.
Materials and Methods
Model
The framework of the stage-structured, temperature-dependent
delayed differential equation (DDE) model reflects details of the
mosquito lifecycle (Figure 1). The stage structuring corresponds to
the four main life stages in the mosquito life cycle (egg, larva, pupa,
and adult) and it allows us to incorporate stage specific life history
rates and processes. The stage durations, given by the delays (or
lags), are temperature-dependent and allow for biologically
realistic developmental times. The temperature-dependence in
the delays also allows for the stage duration to change with
changing temperatures. We assume that juvenile and adult
mosquitoes experience the same temperature. Egg-laying rate,
and mortality in all stages, are temperature-dependent, with the
larval stage experiencing extra mortality because of the effects of
density-dependence. By limiting the effects of density-dependence
to larval mortality, the delays present in the model become
dependent on temperature alone. The four state equations
corresponding to egg (E(t)), larva (L(t)), pupa (P(t)) and adult
(A(t)) are as follows:
dE(t)
dt ~RE(t){RL(t){dE(t)E(t)ð1Þ
dL(t)
dt ~RL(t){RP(t){dL(t)L(t)ð2Þ
dP(t)
dt ~RP(t){RA(t){dP(t)P(t)ð3Þ
dA(t)
dt ~RA(t){dA(t)A(t)ð4Þ
where Ri(t)(i~E,L,P,orA) is the recruitment or the flux of
individuals into or out of a state and di(t)represents the per capita,
stage-specific mortality rate. The recruitment into a stage is
dependent on the recruitment into the previous stage according to,
RE(t)~b(t)A(t)ð5Þ
RL(t)~RE(t{tE(t))SE(t)hE(t)
hEt{tE(t)ðÞ ð6Þ
RP(t)~RL(t{tL(t))SL(t)hL(t)
hLt{tL(t)ðÞ ð7Þ
RA(t)~RP(t{tP(t))SP(t)hP(t)
hPt{tP(t)ðÞ ð8Þ
where, ti(t)is the duration of stage iat time t. The egg-laying rate
(the number of eggs per female per day) is given by b(t)and the
temperature-dependent, stage specific development rate is hi(t).
The length of the delay, ti(t), is determined by the temperature-
Temperature and Vectorial Capacity in Mosquitoes
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dependent development rate hi(t)of stage iat time t. The
implication of temperature-dependent time delays in a model is
that when temperature changes, delays become variable, which
makes analysis more difficult. To alleviate this we re-scaled the
model to a physiological time scale so that the delays become fixed
[41,43]. The details of this transformation are presented in the
(Text S1 and Text S2). The ratio at the far right hand side of
equations 6–8 corrects for any changes in the speed of
development within a stage that occur because of any temperature
changes during the stage and allows for time varying delays. When
temperature is held constant, as in this application of the ratio, the
ratio is one and does not impact the recruitment [42]. For the full
derivation of this correction, see Nisbet and Gurney [42]. The
model works well for fluctuating temperatures and can be driven
with stylized or realistic temperature drivers; as a demonstration of
this we have included the predicted adult abundance trajectories
for a 10uC seasonal fluctuation (Figure S1). However, the
systematic exploration of temperature variability is beyond the
scope of the current study (see Beck-Johnson et al. in prep).
Si(t)represents the survival through stage iand expands as
follows,
SE(t)~exp {ðt
t{tE(t)
dE(j)dj
!
ð9Þ
SL(t)~exp {ðt
t{tL(t)
dL(j)dj
!
ð10Þ
SP(t)~exp {ðt
t{tP(t)
dP(j)dj
!
ð11Þ
SA(t)~exp {ðt
t{tA(t)
dA(j)dj
!
:ð12Þ
The stage-specific per capita mortality rate equations (di(t)) are
given by,
dE(t)~cEm3exp T(t){m4
m5
2
!
ð13Þ
dL(t)~cLm3exp T(t){m4
m5
2
zsL(t)
!
ð14Þ
dP(t)~cPm3exp T(t){m4
m5
2
!
ð15Þ
dA(t)~m0exp T(t){m1
m2
4
!
ð16Þ
where mj(j~0,1,2,3,4,or5) is a scalar and ci(i~E,L,orP) is the
proportion of the juvenile life cycle that the eggs, larvae and pupae
take up respectively (see Text S1, for derivation). The extra
mortality experienced by the larvae because of density-depen-
dence is denoted by s. We assume that the larvae are the only
stage to experience density-dependent mortality. The Gaussian
(and squared Gaussian) functional forms for the temperature-
dependence were chosen to fit empirical patterns (see below).
Figure 1. Diagram of the model setup. Each stage experiences temperature-dependent, stage-specific mortality, di(i~E,L,Por A). Recruitment
into a stage iat time tis given by Riand is also dependent on temperature. Density-dependent mortality is only experienced in the larval stage, s.
doi:10.1371/journal.pone.0079276.g001
Temperature and Vectorial Capacity in Mosquitoes
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Parameterization and Data
All parameter values used in the model are based on data from
laboratory studies. The functional relationships used in the model
were fit to the data using nonlinear least squares optimization. The
data that we used to parameterize temperature-dependent
mortality rates in the juvenile mosquito stages come from two
studies by Bayoh and Lindsay [37,38]. They monitored mortality
at a range of constant temperatures using An. gambiae mosquitoes,
the main African malaria vector (Figure S2). The data suggest a
Gaussian dependence of mortality as expressed in equations 13–
15. The adult temperature-dependent mortality data also come
from a laboratory study on An. gambiae [36], in which mortality
rates are followed across a range of temperatures at several
humidities (Figure S2). For the purposes of our model, we used the
adult mortality data at 60% and 80% humidity, which are both
acceptable humidity levels for An. gambiae mosquitoes. The
functional form that fits adult mortality best is similar to the
juvenile stage except that it is raised to the fourth power instead of
squared, (equation 16). Parameter values are given in Table 1.
The data used to parameterize temperature-dependent devel-
opmental rates came from multiple laboratory studies on An.
gambiae sensu lato complied by Depinay et al. [17]. The adult
gonotrophic cycle rate, or egg development rate, was parameter-
ized with data from An. pseudopunctipennis across different constant
temperatures [44] (Figure S2). We fit a power function of the form
hi(t)~aiT(t)bð17Þ
where hi(t)is the development rate through stage i(i=E,L,P,or
A). Further information on hi(t)is presented in the (Text S1).
Temperature is represented by T(t)and aiand bare parameters
empirically derived.
The data for parameterizing density-dependent larval mortality
came from a laboratory study on four species of mosquito
conducted at 27uC; we used data from An. stephensi (Figure S3)
[31]. Mosquitoes within the Anopheles genus appear to respond to
density in different ways, through a mixture of increased larval
mortality, slowed larval development and feedbacks on adult size,
fecundity and survival [30,31,33,34]. In this study we assume
increased larval density to increase per capita larval mortality; An.
stephensi appears to have a relatively strong response in larval
mortality to increased density when compared with the response in
An. gambiae [31]. The data about density-dependence in Anopheles is
not very comprehensive and therefore it is difficult to draw
conclusions about the type of functional response of the population
to increasing density. To that end, we tried several functional
forms of density-dependence in our model, including exponential,
linear, quadratic and log-linear. The exponential form was chosen
because it is the best fit to the data and results in mosquito
population abundances peaking in the mid-20uC range and larval
populations not growing larger than 2000 larvae per liter; these
latter two results are more in line with what is known about the
biology of these mosquitoes than the results from any of the other
functional forms of density-dependence. The exponential form is
incorporated in equation 14 (see Text S3, Figures S4, S5, S6, S7,
S8, S9, S10, and Tables S1, S2 for results assuming linear form of
density-dependence). There is evidence that there may be
interactions between temperature and density [45]; however, this
relationship has yet to be fully described and therefore can not be
incorporated into the model. In our model, we make the
simplifying assumption that temperature-dependent mortality is
the baseline mortality rate and that the mortality in the larval stage
resulting from density-dependence is additive.
The temperature-dependent relationship of the length of
parasitic EIP differs from that of the adult mosquito age structure,
making it more difficult to predict the temperature at which we
would expect to see a peak in mosquitoes that survive to the
epidemiologically relevant age. Assuming for simplicity that all
mosquitoes become infected as they enter the adult stage, we used
both the classic Detinova EIP prediction curve [2,46] and the
curve recently proposed by Paaijmans et al. [7], hereafter in this
paper referred to as the Paaijmans curve, to predict the number of
mosquitoes that potentially survive to infectiousness. In this study,
we are using the EIP predictions based on P. falciparum
development, because this is the most virulent of the human
parasites and the most prevalent in Africa. The Detinova curve
was proposed in 1962 and is based on a study of Plasmodium
development within An. maculipennis mosquitoes, a vector of
malaria found in Russia. This curve takes the form of a Blunck
hyperbola, predicting extremely long development at cool
temperatures and fast development at warm temperatures [46].
The Detinova equation is the temperature-dependent parasite
development function most used in mechanistic malaria models to
date (e.g., [4,20]). The Paaijmans curve is based on the
temperature-development function proposed by Briere et al.
[47], which also leads to long development times at cool
temperatures but also a slowing and eventual cessation of
development beyond the optimum temperature for development.
This curve is based on parasite development data from several
Table 1. Parameter Values.
Parameter Value Description Reference
b1.726 fit exponent in development
rate function
[17]
a
I
2.87e24 fit scalar in egg to adult
development rate function
[17]
r0.156 egg-laying rate scalar E
0
a
G
m
0
8.86e22 fit scalar in adult mortality
rate function
[36]
m
1
21.211 fit scalar in adult mortality
rate function
[36]
m
2
14.852 fit scalar in adult mortality
rate function
[36]
m
3
2.00e22 fit scalar in juvenile
mortality rate function
[37,38]
m
4
23.00 fit scalar in juvenile
mortality rate function
[37,38]
m
5
6.50 fit scalar in juvenile
mortality rate function
[37,38]
s
exp
1.33e23 fit scalar in density-
dependent mortality
exponential function
(Larvae/Liter)
[31]
c
E
6 estimate for the proportion
of time spent in egg stage
[17]
c
L
3
2
estimate of the proportion
of time spent in larval stage
[17]
c
P
6 estimate for the proportion
of time spent in pupal stage
[17]
a
G
1.04e23 fit scalar in gonotrophic
cycle rate function
[44]
E
0
150 number of eggs laid per
cycle by a single female
(observed range 50–300)
[58–60]
doi:10.1371/journal.pone.0079276.t001
Temperature and Vectorial Capacity in Mosquitoes
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different Anopheles species [7] and takes the upper thermal limit of
P. falciparum parasites into account. This type of approach has been
widely used to explore the effect of temperature on a wide range of
ecological and evolutionary questions (e.g., [48–50]). These curves
differ most substantially at temperatures greater than 26–27uC,
which is the warmer end of the parasite development range.
We also compared the predictions our model makes about the
potentially infectious age class with predictions made using the
‘‘classic’’ model assumptions of a static, temperature-independent
adult mosquito population. For the classic model predictions, we
used the same assumptions about the potentially infectious age
group as presented above except that to predict survival to that
age, we used a constant adult mortality of 10% per day and
constant recruitment into the adult stage with no temperature-
dependence in either. Additionally, in order to make the
predictions comparable, we took the maximum adult recruitment
predicted by our model and used that as the recruitment
abundance across all temperatures. This assumption means that,
given the abundance of adult mosquitoes chosen, the prediction
curve will move up or down but the shape will remain the same.
To compare our model predictions with observations of
transmission intensity, we compared the number of potentially
infectious mosquitoes predicted by our model and those using
assumptions of a constant adult mosquito population to observed
entomological inoculation rates (EIR) from 14 African countries
complied by Mordecai et al. [54]. The EIR is the rate of infectious
bites on people and is determined by the following functional
relationship:
EIR~masð18Þ
where mis the number of mosquitoes per host, ais the daily rate of
mosquito biting and sis the proportion of the mosquitoes which
have the Plasmodium parasites in their salivary glands. Mordecai et
al. [54] took EIR observations and matched them with mean
transmission-season temperature for each location in the data set,
so we can use these to compare our temperature-dependent
predictions with estimates of transmission across a range of
temperatures.
We ran simulations on the fully parameterized model from 16 to
40uC at one-degree increments, giving 25 different constant
temperature runs. This range of temperatures encompasses the
temperatures that are relevant for malaria transmission, with 16uC
being the lower developmental limit of the malaria parasite P.
falciparum and 40uC being the thermal death point of mosquitoes
[2]. The equilibrium at each temperature was examined to see if
the mosquito population was predicted to crash or persist through
time. The age structure of the adult population was determined
using both the recruitment into the adult stage at equilibrium for
each temperature and the adult survival. This was combined with
the Detinova and Paaijmans EIP prediction curves to determine
the abundance of potentially infectious mosquitoes at each
temperature. We ran a local sensitivity analysis looking at the
effect of changing each of the 12 parameters on adult recruitment,
adult and larval equilibrium abundance and the potentially
infectious adult populations using both the Paaijmans and
Detinova curves. The sensitivity of the model outputs was
calculated using
sensitivity~Oj(1:05pi){Oj(0:95pi)
0:1Oj(pi)ð19Þ
where, Ojis the model output j(j~adult, or larval abundance or
adult recruitment or potentially infectious adult abundance
calculated using the Paaijmans or Detinova curves), and piis the
parameter i(i~r,cE,cL,cP,aI,s,m0,m1,m2,m3,m4,orm5). The
sensitivity is the percent change in the model output in response to
a percentage change in the parameter [51].
Results
The model predicts that mosquito populations will persist (i.e.
have a population size greater than 1) from 17 to 33uC, which is in
line with the experimental data used to parameterize the model
[37,38]. From 17 to 19uC and from 27 to 33uC, mosquito
abundance dynamics were predicted to be stable. Between 20 and
26uC, the equilibrium point lost stability and the dynamics
followed small amplitude cycles, which are consistent with
oscillations seen in other DDE systems [52]. Across the
temperature range where populations persisted, the larval
equilibrium abundance was 20 to 50 fold the adult abundance
(Figure 2). The model predicts that the adult population will be 2.1
to 4.7% the size of the larval population, which is consistent with
empirical estimates of larval survival that range between 1 to 10%
[23–29]. In addition to numerical analysis, we also derived the
equilibria analytically. The non-trivial equilibria found by
analytical analysis of the model matched the numerical solutions
at those temperatures where populations converged on the
equilibrium point. Because the model is deterministic with a
single attractor, the results are not dependent on the initial
conditions.
The cycles displayed by the system from 20 to 26 C result from
over-compensatory density-dependence in the larval stage. At
these temperatures, the temperature-dependent mortality, which
we assume is the baseline mortality, is at its lowest point in both
juvenile and adult mosquitoes. Additionally, juvenile development
rate is slow to moderate through the range of temperatures in
question; this, in combination with low mortality, leads to a large
larval population. Density-dependent mortality then becomes very
strong and causes the observed over-compensatory crash. The
cycles are of such a small amplitude that it is unclear whether it
would be discernible in the face of variability in most natural time
series of mosquitoes. The cycle period is determined by the length
of the juvenile delay (egg to adult maturation time) and
corresponds to approximately twice the length of that delay
(Table S3). The cycle period is consistent with the dynamics of
many insect populations that experience larval competition [52].
Because the adults are the epidemiologically important subset of
the mosquito population, we explored the changes in this stage
across temperatures and in response to juvenile stage dynamics
through the adult recruitment. The juvenile stage mortality rate
data showed greater variability to temperature and stayed low over
a smaller temperature range than the adult mortality rates. The
data suggested that adult mosquito mortality rates did not change
much across the temperature range we are interested in, except at
the extremely warm temperatures (Figure S2). Daily adult survival
is therefore predicted by the model to be high for all temperatures
explored except those at the high end of the range. Interestingly,
we found adult abundance to be more sensitive to temperature
than one would predict based on the adult survivorship alone,
having a more defined peak and a sharp decrease in abundance at
cooler temperatures (Figure 2). The juvenile stage temperature
sensitivities impact recruitment into the adult stage, which made
the abundance of adults more temperature-dependent.
We explored the effects of temperature and juvenile stage
dynamics on adult age structure because it is an important
determinant of population vectorial capacity. The combination of
Temperature and Vectorial Capacity in Mosquitoes
PLOS ONE | www.plosone.org 5 November 2013 | Volume 8 | Issue 11 | e79276
temperature-dependencies and intra-stage density-dependence in
the larval stage impacted recruitment into the adult stage, and
therefore the age structure, in nonlinear ways (Figure 3). The
number of mosquitoes emerging as adults was determined by the
egg-laying rate and the juvenile stage dynamics. However, because
there was no further feedback from the juvenile stages once
recruits were in the adult stage, the survival of newly emerged
adults was determined by the adult temperature-dependent
mortality alone. The model predicted the largest abundance of
long-lived mosquitoes to be across the 20–30uC temperature
range, with the most noticeable drops in longevity at the extremely
warm and cool temperatures where recruitment was low. This
corresponds with observations of decreased longevity at temper-
atures above 32uC and in the East Africa highlands were
temperatures are cool [2,53].
We found that the adult population old enough to potentially be
capable of transmitting malaria was strongly influenced by juvenile
stage dynamics through adult recruitment and more strongly
temperature-dependent than previously predicted, regardless of
the EIP prediction curve used. The predictions about the
potentially infectious populations differed based on the EIP
prediction curve used, most noticeably at the warmer end of the
temperature range. With both the Detinova and the Paaijmans
curves (Figure 4), the model predicted a peak in the abundance of
mosquitoes potentially able to transmit parasites at cooler
temperatures than when using the classic assumptions of a
temperature-independent adult population (Figure 5).
The shift to peaks at cooler temperatures is important
biologically for this system, because malaria transmission peaks
at temperatures in the mid-20s rather than in the 30s [2,54,55].
The Detinova curve predicts that parasite development speed
continued to increase until it was curtailed by the imposed lethal
temperature of 40uC; thus the observed drop in abundance at high
temperatures was the result of the population dynamics in the
model. Our model, in combination with the Detinova curve,
predicted that the abundance of potentially infectious mosquitoes
will start to decrease above 30uC. The combination of our model
predictions and the Paaijmans curve resulted in a peak in the
abundance of epidemiologically relevant mosquitoes at the slightly
cooler temperature of 28uC. This observed two-degree drop in the
peak occurs because the Paaijmans curve predicts that parasite
development will begin to slow above 30uC and eventually halt at
35uC. It should be noted that the Paaijmans curve, because of its
predicted 30uC peak, does as well alone in predicting a biologically
realistic peak as the Detinova Curve does in combination with our
mosquito model. However, the most biologically realistic peak
found in this study is the combination of the Paaijmans curve and
the predictions of our model. This can be seen when the EIR data
points are compared to the predicted results (Figure 5). The
transmission intensity is generally higher at cooler temperatures
than are predicted by the curves without mosquito dynamics and it
drops off rapidly at higher temperatures.
We ran a local sensitivity analysis on the twelve parameters in
our model at each of the 25 temperatures from 16 to 40uC. This
allowed us to determine which model parameters were most
sensitive to change but also whether the sensitivity of a given
parameter changed across the temperature range. The outputs
used to gauge model sensitivity to changes in parameters were
larval and adult abundance, recruitment into the adult stage and
the abundance of mosquitoes old enough to potentially transmit
malaria using both the Detinova and Paaijmans EIP prediction
curves. For larval abundance the parameters that were the most
Figure 2. Larval and adult equilibrium abundances. (A) The larval equilibrium abundances across temperatures with exponential density-
dependence. (B) The adult equilibrium abundances (solid line, left axis) and daily survival (dashed line, right axis) across temperatures. The gray points
and bars in both panels are the stable and cyclic abundances, respectively. The solid line connecting the points is the average abundance across
temperature. Notice that the y-axes have different scales.
doi:10.1371/journal.pone.0079276.g002
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sensitive to change were the three juvenile mortality parameters
(m3,m4, and m5), density-dependent mortality (sexp) and juvenile
development rate (aI). These five parameters displayed high
sensitivity across the majority of the temperature range. The
parameters that showed the greatest sensitivity for adult abun-
dance were larval density-dependence (sexp), juvenile development
rate (aI) and the proportion of time spent in the larval stage (cL).
Additionally, adult abundance was sensitive to changes in one of
the juvenile mortality parameters (m4), with the other two (m3, and
m5) becoming more important at the extreme edges of the
temperature range. The responses of adult recruitment and of the
potentially infectious mosquito population were quite similar;
across most of the temperature range the parameter that resulted
in greater sensitivity was the proportion of time spent in the larval
stage (cL). Additionally, across shorter ranges of temperatures
juvenile development rate (aI), larval density-dependence (sexp),
one of the adult mortality parameters (m0), and one of the juvenile
parameters (m4), were important (Figure 6, Figures S11, S12, S13,
and TablesS4, S5).
Discussion
The population dynamics and adult age structure of Anopheles
mosquitoes are important when determining a given population’s
ability to transmit malaria. Ambient temperature conditions affect
both mosquito life history processes and the Plasmodium EIP. The
sensitivities to temperature change between the mosquito juvenile
stages and the adult as well as between life history traits such as
development and mortality. Additionally, malaria parasites have a
temperature-dependent development curve that does not match
up with the mosquito temperature curves. All these factors pull the
system in different directions at certain temperatures, making the
population response hard to predict. By running simulations of our
DDE model across a broad temperature range (16–40uC) we were
able to explore population responses to changes in temperature.
Our results indicate that non-linear temperature sensitivities
throughout the mosquito life cycle have a large impact on the
adult population dynamics and therefore on a population’s ability
to vector malaria effectively. Additionally, our results suggest that
juvenile stage dynamics influence adult stage structure dramati-
cally.
Juvenile mosquitoes are not infected by Plasmodium parasites and
live in an entirely different habitat from the adults, and so juvenile
mosquitoes are frequently left out of malaria transmission models.
It is well known that conditions experienced by juvenile
mosquitoes determine adult characteristics, only some of which
have been included in our model [30,32–34]. Our results
demonstrate that the effects of temperature on juvenile stages
are important in determining the age structure of the adult
population. Here we made the simplifying assumption that water
and air temperatures are the same. However, it has been been
shown that water temperatures in the pools that are preferred by
An. gambiae mosquitoes are warmer than air temperatures in
western Kenya [56]. When data are available across a range of
environments, the relationship between air and water temperature
can be easily incorporated into the model.
Adult mortality is less sensitive across much of the temperature
range in question than the juvenile mortality rate; in fact, across
much of the range, mortality in adults is almost constant (Figure
S2). If adults were independent of the juvenile stages we would
expect to see a very broad flat curve of equilibrium adult
abundance. Instead we see a curve that resembles the shape of the
larval equilibrium abundance curve and has a more defined peak
and rapid declines on both edges of the temperature range
(Figure 2). Additionally, our results show that larval density-
dependence has a significant regulatory impact on mosquito
populations and can lead to low amplitude overcompensation
cycles.
Larval density-dependence is an important regulatory process in
the model. It is also one of the most sensitive parameters across
most of the temperature range for adult recruitment, adult and
larval abundance and the abundance of potentially infectious
mosquitoes. From the literature, it appears that different species of
Anopheles respond to increases in density in different ways. For
example, An. stephensi, an important vector in Southeast Asia,
shows an increase in daily mortality rate with increased density in
the larval stage [31]. In contrast, An. gambiae, an important vector
in Africa, does not appear to respond strongly to density through
morality rate, but does show a increase in the developmental
period [30,31]. In the model, we assumed that the influence of
density is manifested in daily mortality rates and, because of this,
used data from An. stephensi (Figure S3). This assumption allows for
the delays in the model to be determined solely by temperature,
making the system more tractable. It also provides a starting point
Figure 3. Age-specific adult abundance and adult recruitment
across temperature. (A) The age-specific abundance of a single
cohort of adult mosquitoes for each temperature. High abundance is in
dark blue decreasing to zero in white. (B) Recruitment (the mean
abundance of new recruits) into the adult stage over the temperature
range.
doi:10.1371/journal.pone.0079276.g003
Figure 4. Extrinsic incubation period curves. Temperature-
dependent extrinsic incubation period in days; the solid line is the
Detinova prediction curve and the dashed line is the Paaijmans curve.
doi:10.1371/journal.pone.0079276.g004
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to explore the ways in which the type of density-dependence
influences population dynamics.
We also assumed that density-dependence in mortality takes an
exponential form. This assumption was based on the fit to the data
and on the fact that this form of density-dependence resulted in
population abundances and peaks that most closely resemble the
biology of the system. Past experimental work has shown a
relationship between key life history traits and larval density
[30,31,33,34]; however, much of those data are in a format that
could not be used to parameterize the model. The data needed to
parameterize this model are daily mortality rates for a single
Anopheles species. The scarcity of data is a great hinderance to
understanding Anopheles mosquito population dynamics. This is
particularly true if we want to use models to predict the potential
impact of a mosquito control program, as changing densities of
larval populations could have unexpected ramifications in the
adult population. The effects of density-dependence on Anopheles
mosquito populations merit further exploration both theoretically
and empirically, as our sensitivity analysis reveals that it is one of
the critical parameters.
In addition to density-dependence, other key parameters such as
juvenile mortality and development were important for all the
population metrics we looked at in our sensitivity analysis. It is
interesting to note that all of the parameters that appear to be
important to the population across temperatures have been found
to be affected by density-dependence [30,31,33,34]. Juvenile
mortality and development have been fairly well studied across a
constant temperature range in optimized food and density
conditions; however, much more data is needed in order to
understand how these parameters respond to density and to
temperature and density together. The scarcity of field data on
density-dependence has recently been addressed by Muriu et al.
[57], who found that larval density of An. gambiae can impact larval
survival and development rates as well as the size of adults. These
results provide further evidence that understanding the regulatory
processes can increase our understanding of mosquito population
dynamics.
The adult population abundance is more sensitive to temper-
ature than previously assumed, such that the adult population only
persisted at temperatures that were suitable for juvenile mosqui-
toes, despite having high predicted survival across a much broader
range. This does not correspond directly with the vectorial
capacity because the parasitic development rate has a different
temperature relationship. We found that the function used to
describe Plasmodium developmental rate influences the predictions
about a mosquito population’s ability to effectively transmit the
parasite. Both the Detinova and Paaijmans curves predict a rapid
increase in developmental rate over the lower end of the
temperature range. The Paaijmans curve then tapers off and
predicts cessation of development at very high temperatures where
Figure 5. Potential for infectious mosquitoes. The abundance of mosquitoes old enough to be potentially infectious across temperatures. In
both graphs the solid line represents the predictions made using our model in combination with either the Detinova (A) or Paaijmans (B) EIP
prediction curve. The gray points and bars in both panels are the stable and cyclic abundances predicted by our model, respectively. The dashed lines
represent the predictions made using the classic model assumptions of a constant vector population, in combination either the Detinova (A) or
Paaijmans (B) EIP prediction curve. The red points, which correspond to the right y-axis are the observed entomological inoculation rates from 14
countries in Africa, compiled by [54].
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the Detinova curve does not. Using either of these curves with the
predictions from our model gives a more realistic prediction about
temperatures at which we would expect a peak in potentially
infectious mosquitoes [2,54,55]. Our model predicts that the peak
in potentially infectious mosquitoes begins at lower temperatures
than models without vector dynamics (Figure 5). This is important
for understanding the dynamics of mosquito populations and
malaria in the field, and could help to explain some of the mis-
match between previous model predictions and observed malaria
patterns. The prediction from our model that abundance of
potential vectors decreases at high temperatures is a phenomenon
that has been demonstrated in malarious regions [2].
One major advantage of having a model such as this one, which
admittedly does not include all the aspects of mosquito biology, is
that the model is relatively parameter sparse. The inclusion of
other types of environmental drivers, mosquito physiological
responses, or behavior would add considerably to the degree of
difficulty in interpreting the model results. Additionally, many of
those relationships are data scarce and we would therefore be in
danger of over parameterizing relationships that are not well
understood. As it stands, the model represents a starting point and
provides a robust framework that can be built upon. Because there
are only a dozen parameters in the model, we can point to areas
where the sensitivity analysis and/or the data suggest there is good
reason for further empirical exploration. The limited number of
parameters in our model—and our focus on temperature-
dependence—makes it relatively easy to interpret which life
history traits are most important for driving dynamics, in contrast
to the more parameter heavy models (e.g., [17]).
Despite the availability of control systems, such as drug
therapies, insecticides, and bed-nets, malaria continues to be a
major problem for a large part of the world. To date, theoretical
efforts to understand transmission have in large part failed to take
into account vector biology. This is of particular concern when the
impacts of climate and/or climate change on disease risk are
explored. Our results highlight the importance of including
mosquito biology in models of mosquito-borne disease. Incorpo-
rating the juvenile stage dynamics increases our understanding of
potential for transmission because of strong regulatory effects in
the epidemiologically significant adult stage. Furthermore, includ-
ing temperature-dependencies in the entire life cycle has
interesting and non-intuitive impacts on the potential vectorial
capacity of a population. The model framework we have
developed is robust and can be run with a variety of temperature
conditions, including fluctuating temperature regimes (see example
in Figure S1). We can also build upon the model, adding processes
such as malaria infection. The model is relatively parameter
sparse, a considerable bonus for adapting it to different scenarios
quickly and effectively. Because all mosquito vectors share the
same basic lifecycle, the model can also be converted to other
mosquito-borne disease systems, such as Dengue Fever and West
Nile Virus. We propose this model as a useful framework to begin
to interpret mosquito population responses to temperature
sensitivities as well as inter- and intra-stage interactions. Under-
standing the vector population will lead to clearer understanding
of malaria transmission and enhance our ability to predict what
may happen to disease intensity in the future.
Supporting Information
Figure S1 Adult abundance from model simulations
with a 106C seasonal temperature fluctuation. (A) Adult
abundance trajectory over the course of one year, with a mean
temperature of 18uC (B) Adult abundance trajectory over the
course of one year, with a mean temperature of 22uC (C) Adult
abundance trajectory over the course of one year, with a mean
temperature of 26uC (D) Adult abundance trajectory over the
course of one year, with a mean temperature of 30uC. The x-axes
are all time in days over a single year and the y-axes are adult
abundance.
(EPS)
Figure S2 Developmental and mortality data used in the
model parameterization. (A) Juvenile development rate across
temperature. The points are data from [17] and the development
function used in our model is the solid line. (B) Development rate
of the gonotrophic cycle across temperature. The points are data
from [44] and the solid line is the fit function. (C) Temperature-
dependent juvenile mortality rate. The filled circles data are from
[38] and the x’s are data from [37]; the solid line is the fit function.
(D) Temperature-dependent adult mortality rate. The filled circles
are data from 60% humidity and the x’s are data from 80%
humidity; these data were published in [36]. The solid line is the fit
function used in our model.
(EPS)
Figure 6. Sensitivity rank across temperature. The sensitivity of
larval abundance (A), adult abundance (B) and adult recruitment (C) to
changes in the parameters across temperatures ranked from highest to
lowest sensitivity. Red indicates greatest sensitivity to change, followed
by orange, yellow and white. The x-axis is temperature from 17–33uC,
and the y-axis is the parameter.
doi:10.1371/journal.pone.0079276.g006
Temperature and Vectorial Capacity in Mosquitoes
PLOS ONE | www.plosone.org 9 November 2013 | Volume 8 | Issue 11 | e79276
Figure S3 Exponential density-dependence. Exponential
function fit to the larvae mosquito density-dependent daily
mortality rate data. The data points are data from An. stephensi
published in [31].
(EPS)
Figure S4 Linear density-dependence. Linear function fit
to the larvae mosquito density-dependent daily mortality rate data.
The data points are data from An. stephensi published in [31].
(EPS)
Figure S5 Larval and adult equilibrium abundances. (A)
The larval equilibrium abundances across temperatures from the
model with linear density-dependence. (B) The adult equilibrium
abundances across temperature. Notice that the y-axes have
different scales.
(EPS)
Figure S6 Adult recruitment and age-specific adult
abundance across temperature. (A) The age specific
abundance of adult mosquitoes from the model with linear
density-dependence. Time in days is on the x-axis, temperature
(uC) is on the y-axis. High abundance is in dark blue decreasing to
zero in white. (B) The recruitment into the adult stage over the
temperature range, with temperature on the y-axis and recruit-
ment on the x-axis.
(TIF)
Figure S7 Potential for infectious mosquitoes The
abundance of potentially infectious mosquitoes across tempera-
tures from the model with linear density-dependence. In both
graphs the solid line represents the predictions made using our
model and the dashed line represents the predictions made using
the classic model assumptions. The Detinova prediction curve was
used to calculate (A), and (B) was calculated using the Paaijmans
curve.
(EPS)
Figure S8 Sensitivity ranks across temperature from
the model with linear density-dependence. The sensitivity
of larval abundance (A), adult abundance (B), abundance of
potentially infectious mosquitoes using the Detinova (C) and
Paaijmans (D) curves, and adult recruitment (E) to changes in the
parameters across temperatures ranked from highest to lowest
sensitivity. The x-axis is temperature from 17–33uC, and the y-axis
is the parameter. Red indicates greatest sensitivity to change,
followed by orange, yellow and white.
(EPS)
Figure S9 Sensitivity analysis of the across temperature
from the model with linear density-dependence. Sensitiv-
ity of adult equilibrium abundance, solid black line; larval
equilibrium abundance, dashed blue line; and recruitment into
the adult stage, dotted green line; across temperature. Temper-
ature from 16–40uC is on the x-axis and sensitivity is on the y-axis.
(EPS)
Figure S10 Sensitivity analysis of the across tempera-
ture from the model with linear density-dependence.
Sensitivity of adult equilibrium abundance, solid black line; larval
equilibrium abundance, dashed blue line; and recruitment into the
adult stage, dotted green line; across temperature. Temperature
from 16–40uC is on the x-axis and sensitivity is on the y-axis.
(EPS)
Figure S11 Sensitivity rank of the potential for infec-
tious mosquitoes across temperature. The sensitivity of
potentially infectious mosquito population calculated using the
Detinova (a) and Paaijmans (b) curves to changes in the
parameters across temperatures ranked from highest to lowest
sensitivity. The x-axis is temperature from 17–33uC, and the y-axis
is the parameter. Red indicates greatest sensitivity to change,
followed by orange, yellow and white.
(EPS)
Figure S12 Sensitivity analysis of the across tempera-
ture. Sensitivity of adult equilibrium abundance, solid black line;
larval equilibrium abundance, dashed blue line; and recruitment
into the adult stage, dotted green line; across temperature.
Temperature from 16–40uC is on the x-axis and sensitivity is on
the y-axis.
(EPS)
Figure S13 Sensitivity analysis of the across tempera-
ture. Sensitivity of adult equilibrium abundance, solid black line;
larval equilibrium abundance, dashed blue line; and recruitment
into the adult stage, dotted green line; across temperature.
Temperature from 16–40uC is on the x-axis and sensitivity is on
the y-axis.
(EPS)
Table S1 Sensitivity values assuming linear density-
dependence The sensitivity values are the percent change in the
adult and larval equilibrium abundance and adult recruitment in
response to a 5% change in the parameter in the model assuming
linear density-dependence.
(PDF)
Table S2 Potentially infectious abundance sensitivity
values assuming linear density-dependence The sensitivity
values are the percent change in the potentially infectious adult
abundance calculated using the Detinova or the Paaijmans curve
in response to a 5% change in the parameter in the model
assuming linear density-dependence.
(PDF)
Table S3 Periodicity of Fluctuations.
(PDF)
Table S4 Sensitivity values assuming exponential den-
sity-dependence The sensitivity values are the percent change
in the adult and larval equilibrium abundance and adult
recruitment in response to a 5% change in the parameter in the
model assuming exponential density-dependence.
(PDF)
Table S5 Potentially infectious abundance sensitivity
values assuming exponential density-dependence The
sensitivity values are the percent change in the potentially
infectious adult abundance calculated using the Detinova or the
Paaijmans curve in response to a 5% change in the parameter in
the model assuming exponential density-dependence.
(PDF)
Text S1 Model Development and Compression.
(PDF)
Text S2 Model Transformation.
(PDF)
Text S3 Model results when using the linear form of
density-dependence.
(PDF)
Acknowledgments
We thank M. A. Greischar for useful discussions and comments and
members of the RAPIDD program of the Science & Technology
Temperature and Vectorial Capacity in Mosquitoes
PLOS ONE | www.plosone.org 10 November 2013 | Volume 8 | Issue 11 | e79276
Directorate, Department of Homeland Security, and the Fogarty
International Center, National Institutes of Health, for stimulating
discussion.
Author Contributions
Conceived and designed the experiments: LMBJ WAN KPP AFR MBT
ONB. Performed the experiments: LMBJ. Analyzed the data: LMBJ.
Contributed reagents/materials/analysis tools: ONB. Wrote the paper:
LMBJ WAN KPP AFR MBT ONB.
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