A dynamic model of some malaria-transmitting anopheline mosquitoes of the Afrotropical region. I. Model description and sensitivity analysis
© Lunde et al.; licensee BioMed Central Ltd. 2013
Received: 21 September 2012
Accepted: 7 January 2013
Published: 23 January 2013
Most of the current biophysical models designed to address the large-scale distribution of malaria assume that transmission of the disease is independent of the vector involved. Another common assumption in these type of model is that the mortality rate of mosquitoes is constant over their life span and that their dispersion is negligible. Mosquito models are important in the prediction of malaria and hence there is a need for a realistic representation of the vectors involved.
We construct a biophysical model including two competing species, Anopheles gambiae s.s. and Anopheles arabiensis. Sensitivity analysis highlight the importance of relative humidity and mosquito size, the initial conditions and dispersion, and a rarely used parameter, the probability of finding blood. We also show that the assumption of exponential mortality of adult mosquitoes does not match the observed data, and suggest that an age dimension can overcome this problem.
This study highlights some of the assumptions commonly used when constructing mosquito-malaria models and presents a realistic model of An. gambiae s.s. and An. arabiensis and their interaction. This new mosquito model, OMaWa, can improve our understanding of the dynamics of these vectors, which in turn can be used to understand the dynamics of malaria.
KeywordsAnopheles gambiae complex Model Malaria
This is the first of two papers describing a dynamic model (Open Malaria Warning; OMaWa) of Anopheles arabiensis and Anopheles gambiae s.s. Our aims in this article are 1) to formulate recent research on the Anopheles gambiae complex in a mathematical framework, and 2) to show how the new formulations influence the dynamics of malaria and mosquito populations.
In this paper, we describe a model of the dynamics of the two species and then show how parameters can influence the success of the two species, and how temperature, humidity and mosquito size can influence malaria transmission.
Climate and malaria
Most of the 149-274 million cases and 537,000-907,000 deaths from malaria occur in sub-Saharan Africa [1, 2]. Climate has been one of the main drivers of this disease , governing the spatial extent and year-to-year variations. The pathway from climate to malaria goes through the parasite and the mosquito. Although it is well established  how parasite development is influenced by temperature , the vector’s response to weather and climate is more complex. Mosquito density depends not only on temperature but also on the abundance of breeding sites (rainfall and evaporation) , desiccation (humidity) , and competition between mosquitoes . In the past 20 years, a shift in the distribution of An. arabiensis and An. gambiae s.s. has been observed in Kenya , showing that the species composition is not static over time. In the context of climate change , variability in vector populations is a factor that has not been considered so far.
Malaria and mosquito models
At the turn of the 20th century the work of several researchers, including Battista Grassi and Ronald Ross, resulted in the discovery that mosquitoes of the Anopheles genus transmit malaria [11, 12]. Over the next 20 years, Ross, and later Lotka and Waite, developed mathematical models that became central in malaria control [13–19]. In the 1950s, George MacDonald refined these models and showed that DDT could be used to interrupt malaria transmission . Since then, several modelers have followed in the footprints of Ross, Lotka, and MacDonald [21–30]. Some have designed models to show how temperature alone influences malaria transmission , while others have focused on the theoretical effect of bed nets , multiple interventions  or climate change [34–36]. There is also a growing number of models that address the dynamics of immunity within individuals  and in communities [21, 38].
In 2011, The malERA Consultative Group on Modeling  provided a review of the current state of mathematical models and pointed to the importance of good mosquito models for assessing the impact of climate change on malaria.
Many traditional models rely on a threshold principle. The idea has been to find thresholds for longevity, number of bites or days to recovery that must be reduced to interrupt the transmission. With increased computational power it is now possible to make more complex models and hence explore a wider range for the dynamics of malaria and mosquito survival. By integrating the knowledge from simpler models into a complex system, it is possible to test if the assumptions are true over a wider geographical range. In addition, these complex models can make quantitative predictions about strategies for control .
Model summary and motivation
A model is mental copy that describes one possible representation of a system. We present an alternative formulation of the dynamics of An. gambiae s.s. and An. arabiensis. The model is a system of ordinary differential equations (ODEs) with three compartments: eggs, first to fourth instar larvae, and pupae; an age-structured formulation of adult mosquitoes; and size prediction for adult mosquitoes (measured as wing length in mm). This can be considered the skeleton of the model. As demonstrated later, the model structure can be simplified when mosquito size can be neglected or when we assume no births. The model can be run with a spatial structure in which we include or exclude mosquito dispersion, or as an idealized model in which the model is evaluated at a single point.
The ODEs parametrize daily mortality rates, which are size-dependent for adult mosquitoes; development rates in the aquatic stages; biting rates; fecundity; the probability of finding a blood meal; and mortality related to flushing of eggs, larva and pupa out of oviposition sites. These parametrization schemes are driven by air temperature, relative humidity, relative soil moisture, water temperature, and runoff. As already mentioned, the model can be applied in a spatial domain. In this case, temperature and other environmental data are taken from a regional climate model, the Weather Research and Forecasting Model (WRF) . In the examples shown later, we run the model at a resolution of approximately 50 km and a temporal resolution of 5-20 years in steps of 3 h. In addition to weather data, human  and cattle [43, 44] densities are introduced to estimate the probability of feeding.
At this spatial resolution, the model should potentially be able to define larger foci of mosquito productivity, while the ability to identify hotspots will be limited . However, 50 km is the standard for regional climate models addressing long-term changes in climate . In addition, the true accuracy of historical cattle and human population density estimates for Africa in general is not likely to be greater than 50 km.
The mosquito model described here is designed to capture the spatial distribution and the time-dependent density of An. gambiae s.s. and An. arabiensis. If the model can capture the current distribution and density of the two species and how they are related to malaria, a future version of this model, including infections, could be used to explore the long-term impact of current interventions under a changing climate. To have confidence that the model has these abilities, several aspects not considered here should be evaluated (papers under preparation). In addition, if malaria modelers move towards the ensemble thinking widely adopted in the climate community, this model could be one representation of historical and future changes for malaria. The aim of such an ensemble would be to deal with uncertainties in the system. Ultimately, the goal would be to produce policy-relevant information including uncertainty.
We have chosen to represent the non-exponential mortality of An. gambiae s.s. and An. arabiensis as observed in laboratory settings , semi-field conditions , and in the field . A common assumption is that in the field, mortality rates are constant with age because of predation . To date, few studies have confirmed this, while there is field-based evidence of age-dependent Ae. aegypti mortality , which has implications for malaria transmission . In the model, we also describe how mosquito size changes over the season. This might seem to be an overcomplication of the model. The motivation, however, is that we have observed substantial improvements for arid regions such the Sahel when we included mosquito size prediction. Fouet et al. reported that mosquito size is an important adaptation strategy in arid environments .
We do not claim that the additional complexity adds any value. Stating this before the model has been fully evaluated and compared to simpler models would be dangerous. The model is thus one possible way of describing the dynamics of An. gambiae s.s. and An. arabiensis. It is under continuous development, and we expect to add and alter components as new data become available.
To highlight some of the components that contribute to the dynamics of An. gambiae s.s. and An. arabiensis in the model, five sensitivity experiments focus on the effect of temperature, relative humidity and mosquito size on malaria transmission. We also show how An. gambiae s.s. and An. arabiensis respond to changes in the probability of finding blood, carrying capacity, initial conditions, and dispersion.
Material and methods: model description
Summary of the model
As mentioned above, the model comprises a system of ODEs for eggs, first to fourth instar larvae, and pupae; an age-structured formulation for adult mosquitoes; and size prediction for adult mosquitoes (measured as wing length in mm). The first limitation in the aquatic stage is the availability of ovipositing sites, which is parametrized in terms of relative soil moisture and the potential for puddle formation in a specific location. Once ovipositing sites have been formed, adult female mosquitoes are allowed to deposit eggs until the site is full, defined as the biomass relative to the carrying capacity for the location. To account for density-dependent mortality, first instar larvae can be preyed on by fourth instar larvae , and an extra density-dependent mortality term is added to account for prey-independent mortality . The numbers of eggs, larvae and pupae are reduced when the precipitation rate exceeds the infiltration rate. The larval density in the aquatic habitat influences the size of adult mosquitoes . We account for this by predicting mosquito size at emergence as a function of larval density. In addition to temperature and relative humidity , mosquito size influences the daily adult survival probability [7, 51, 54, 55] (, Aedes aegypti). We therefore describe an adult survival model that takes temperature, relative humidity and mosquito size into consideration. In addition, adult mortality and fecundity can increase if there are no or few sources of blood. This follows the idea that a mosquito living in an environment where much energy has to be used to find blood will do this at the cost of survival.
We adopt these general ideas for two species, An. gambiae s.s. and An. arabiensis. It should be noted that we have less confidence in the model for the An. gambiae s.s. M form, since aestivation (as documented by Lehmann et al.  and Adamou et al. ) is not included. In addition, there are some indications that the M form breeds in larger pools  and hence the puddle parametrization might have limited validity for this form.
In addition to time, the model can include two (three, since space is two-dimensional) additional dimensions, namely age and space. The space dimension allows dispersion of mosquitoes, meaning that (re)establishment through migration to areas that were previously free of An. gambiae s.l. is possible. The gradual invasion of Brazil by An. arabiensis in the 1930s  is one example of dispersion.
Near surfacetemperature (2 m)
25, 26, 30, 36
Potential number ofnew eggs
Number of mosquitoes ineach age group
Daily probability of gettinga blood meal
14, 16, 18
0-10 cm soil temperature
βN,L(T water )
Natural mortiality rate,eggs, larva, and pupa
14, 1, 2, 3, 4, 5, 6
τ g a m b
An. gambiae s.s. develop- ment rate, aquatic stages
An. arabiensis develop-ment rate, aquatic stages
An. gambiae s.l. development rate, eggs
An. gambiae s.l. development rate, instar 1-4
 2, 3, 4, 5
An. gambiae s.l. development rate, pupa
Aquatic development rate modification An. arabiensis
f g a m b
Aquatic development rate modification An. gambiae s.s.
Number of larvae
Mortality rate modification
f g a m b
Mortality rate modification
scaling factor for winddispersion
F r m
Number of eggs
Induced mortalityin aquatic and adult stages
1, 2, 3, 4, 5, 6,7, 8
S M r
Dimensionless time varying water constant, or rate at which ovipositing sites are found
Number of 1 st instar larva
Number of 2 nd instar larva
Number of 3 rd instar larva
Number of 4 th instar larva
Number of pupa
Predation constant.Currently set to 0
F g o n o t
part of gonotrophic cycle formulation
Adult mortality related to feeding
Number of humans
Dimension in age grid
Size of newly emerged mosquitoes
Size of mosquitoes in age group n
Prediction of larva size
a s p p
b s p p
Potential river length in km
Equally spaced riverdataset resolution indegrees
Earth radius inkm (6371.22)
latitude in radians
Diurnal modification fortransport of mosquitoes
Human blood index
Size dependent mortality
Natural mortality of adultmosquitoes
32, 7, 8
Survival curve for adultmosquitoes
Shape parameter for adult survival
Sub-function forequation 33
Probability of finding cattle
Probability of findinghumans
Differential equations for the aquatic compartment
The aquatic compartment consists of six stages: eggs (E), four larval stages (L1,L2,L3,L4), and pupae (P). Transitions between the different compartments can be expressed in terms of delayed equations. To simplify the solution and avoid numerical instabilities, we approximate the model as ODEs . Lunde et al. reported on the errors introduced by this approximation .
where represents potential new eggs from each age group, G(T) is either constant or dependent on temperature T, S M r is a function of the relative soil moisture and the potential puddle formation area, K is the maximum larval biomass a grid cell can hold, βN,E(T) is natural mortality rate for eggs [Eqs. (16) and (18)], βI,Eis the induced mortality rate for eggs (not specified) and τ E is the inverse of development time from eggs to first instar larvae.
The term 1-B L /K is used as a scaling factor to modify the growth rate. When the population is low compared to the breeding sites available, its growth is high. As the population grows, there is more competition for food, predators become more abundant, and the growth slows. In the egg compartment this represents the idea that the mosquitoes will lay fewer eggs when breeding sites are already occupied .
First instar larvae (L1) are added as eggs develop into larvae. Additional mortality is added in the transition stage in relation to how much biomass there already is in a given location . This approximation of increased (density-dependent) mortality arises because of competition and predators; if a puddle already is full, the number of eggs developing to first instar larvae is reduced, whereas if a puddle is empty (1-B L /K = 1), no extra mortality occurs. Similar terms could have been added to the second, third and fourth instar larvae, but we assume that earlier life stages will be affected more by density-dependent competition and predation.
Shoukry looked at how fourth instar larvae of An. pharoensis prey on first instar larvae during a 24-h experiment . Using these data, we add additional mortality for first instar larvae according to the density of fourth over first instar larvae. The constant C pred is tunable to both limit the predation on L1 and make it more specific to species in the future. At most temperatures, this constant does not influence the density of mosquitoes (Additional file 1).
where β is the daily mortality rate, with the first subscript denoting natural (N) or induced (I) mortality and the second subscript denoting the aquatic stage. The subscript for the development rate, τ, corresponds to the aquatic stage. The parametrization schemes and data sources used to estimate the rate at which eggs are laid (G(T) and ε), mortality (β) and the development rate (τ) are discussed later.
Differential equations for adult mosquitoes
The life history and mortality rate vary over the lifespan of a mosquito population. We formulated a model to account for this variation. Adult mosquitoes are denoted by m n , where n indicates the age group; n = 1 is the youngest group and n = 9 refers to the oldest mosquitoes. The age groups in the model are m1 =[0,1], m2 = (2,4], m3 = (5,8], m4 = (9,13], m5 = (14,19], m6 = (20,26], m7 = (27,34], m8 = (35,43] and m9 = (44,∞] days, with ageing coefficients a n of 1.000, 0.500, 0.333, 0.250, 0.200, 0.167, 0.143, 0.125 and 0.067 for , respectively. Mosquito ageing is represented by Ψ n , where n denotes the age group. Ageing is time-invariant and is thus not related to the number of gonotrophic cycles.
Although there is no ageing from age group 9, the term Ψ9 is included to limit the concentration of old mosquitoes. This is a user-specified variable and in the model results shown here we set this to for An. arabiensis and An. gambiae s.s.; this value should be set to ensure that mosquito populations can survive during dry periods [66, 67], but still hinder accumulation of old mosquitoes. This can be particularly useful if the mortality model described later is replaced with a model in which mortality is independent of age.
When m is written with subscripts ı and ȷ in addition to n, this denotes inclusion of mosquitoes from neighboring areas. For example, subscript ı-1 indicates that mosquitoes to the west of the point of interest are interacting with the point of interest. The formulation presented here includes movement of mosquitoes, and where appropriate we denote mosquitoes by mn,ı,ȷ.
Again, β denotes mortality, with the first subscript denoting natural (N) or induced (I) mortality and the second subscript denoting the age group (m n ) of the mosquitoes. Φ represents the mosquito flux (transport) and subscripts ı and ȷ define which boundaries are evaluated. This is discussed in the section “Movement of mosquitoes”.
The number of adult mosquitoes of a specific age in a grid point is controlled by new mosquitoes from mn-1, as well as the flux to and from the point of interest , natural mortality , induced mortality , ageing to mn+1, and mortality due to lack of food (P(B)). Parametrization schemes related to mortality are discussed later.
Differential equations predicting mosquito size
Mosquito size (m size ) is important for the efficiency of mosquito multiplication. There are also some indications that increased body size is a strategy for survival in arid environments . In general, high larval density leads to a smaller body size as adults, and vice verse . Where only one species is competing for a resource, such as in a small puddle, mosquito size, and hence the number of eggs laid by each mosquito, will be of less importance. If two species are competing for the same resource (e.g. An. arabiensis and An. gambiae s.s.), the trade off between development time and size can be important in competition for breeding sites. An. gambiae s.s. generally develop faster than An. arabiensis, but end up with a smaller body size. An. arabiensis spends more time in the aquatic stages and develops larger bodies, and can thus produce more eggs. Since our model includes competition between those species, we describe mosquito size as a function of competition for breeding sites. In theory this should improve our ability to separate geographical and seasonal distributions of An. arabiensis and An. gambiae s.s.
Since the size of An. arabiensis and An. gambiae s.s. stabilizes after approximately 4 days  and ovoposition does not start before this, it is not necessary to differentiate the maximum and minimum size depending on age to mimic changes in the number of eggs per mosquito with age. However, this may be required if mortality based on desiccation [7, 69] is used. Although mosquito size at a given time can be approximated using finite differences, we develop a different approach that is more efficient in terms of computational time in our model framework. Mosquito size for the first age group depends on larval size. Since the pupation time is short, this assumption is justified, although it might introduce minor errors. In a future version of the model, we plan to predict larval size dynamically. The limitations set on mosquito size (described in “Parametrization schemes in the aquatic stages”) in this model might lead to An. arabiensis that are slightly too small compared size in the field study of Ye-Ebiyo et al. , but the size is in line with studies by Huestis et al.  and Kirby et al. . Kirby et al. also noted that mixed populations of An. arabiensis and An. gambiae s.s. had a negative effect on mosquito size at some temperatures. This mechanism is not included in the current work. However, the most important aspect of modelling of mosquito size is to capture seasonal and spatial variations.
For size prediction we use the symbol , where n is the age group as described above.
The constants a s p p and b s p p are 0.45 and 0.12 for An. arabiensis and 0.383 and 0.147 for An. gambiae s.s., respectively .
Therefore, the size of newly emerged mosquitoes () depends on the number of newly emerged pupae and the relative density of larva at the breeding site.
Therefore, the size in age groups 2-9 only depends on the number of mosquitoes surviving from one age group to the next (mn-1) and the size of mosquitoes in the younger age group ().
To drive a dynamic malaria model it is necessary to have boundary conditions that are consistent over time and space. Temperature, relative humidity, and rainfall data from weather stations are point measures. Hence, they might not be representative of larger areas over shorter time scales. This is especially true in areas with varying topography or where convective rainfall is dominant [73–75]. Despite the limitations of rainfall stations, they can provide a robust estimate of large-scale events. By pooling data from several stations, the error for a single station is reduced and the data can provide a good estimate for dry and wet years, for example. Hence, weather stations are useful tools for validating climate models.
The problems of point measurements are described later, and represent one of the reasons why OMaWa is tightly linked to a climate model. As shown in sensitivity experiments, the model can also be run with constant forcing (e.g. temperature) or with data from weather stations.
Where we present results for Africa as a whole, OMaWa is driven by data from WRF 3.3.1. This realization (TC50), described in part two of this paper, has a tropical channel set-up in which set-up, the domain consists of boundaries above and below a certain latitude and no side boundaries. The model was run at 50-km resolution from January 1, 1989 to January 1, 2009. At the northern (45°N) and southern (-45°N) boundaries the model was driven by Era Interim. The Kain Frisch cumulus parametrization scheme was used [76, 77]. This experiment was not designed to reproduce observed year-to-year weather variability, but to assess the mean mosquito density and distribution. The driving experiment is described in the section on model validation.
Climate and weather models
Currently, our best guess of (future) climate at multidecadal time scales comes from general circulation models (GCMs). These models are designed to close the energy budget of the Earth and include an interactive representation of the atmosphere, ocean, land, and sea ice. A set of scenarios with different emissions describes how sensitive the climate is to atmospheric constituents (greenhouse gasses) . While climate is the average weather over time and space, weather can change over minutes, hours, days and seasons. The same equations used to predict climate are used to predict weather. However, weather forecasts are more dependent on current observations of the atmosphere. Hence, weather predictions are initial value problems, whereas climate simulations are rather boundary value problems.
Both climate and weather models are mostly structured on a grid, with coordinates from west to east (x), north to south (y) and bottom to top (z). In the grid, one square (or polygon) represents the weather within that square. While climate models often have a horizontal resolution of more than 10000 k m2, operational weather models such as the European Centre for Medium-Range Weather Forecast (ECMWF) model are run at approximately 160 k m2. If the state of the atmosphere is observed correctly, higher resolution can lead to better local skill in predicting the weather. A hybrid between a weather model and a climate model is a limited-area model (LAM), which relies on initial and boundary conditions from a weather or climate model. Given these conditions (weather), the LAM can be run at a higher resolution over a limited area, which potentially improves the spatial accuracy of the coarse model . The WRF model is a widely used LAM .
In tropical regions, most rainfall comes from convective clouds. This type of rainfall is generally intense and of short duration. The geographical extent of such rainfall episodes may be limited. Therefore, rainfall measurements in regions where convective rainfall is dominant should be handled with care [74, 75, 80, 81], especially when extrapolating station data to areas with no data. While station data are accurate at a specific point, climate models and satellite estimates give a more general description of the weather within a certain area; Chen and Knutson reviewed how models compare to observations at varying scales . Since future climate is projected using climate models and considering the limitations of weather stations, construction of a mosquito/malaria model around a LAM is a good choice. The LAM will have higher resolution than most climate models, with higher-resolution orography, coastlines, and land use, but will still give a general description of the weather within a certain area.
Parametrization schemes in the aquatic stages
To relate a variable such as mortality to the physical environment, we need simplified equations that describe this relationship. An equation in which temperature influences mortality only states that there is a relationship between the two, but does not explain why temperature modifies mortality. In this paper we use parametrization schemes to represent the influence of the environment on mosquitoes. This section describes the aquatic parametrization schemes used, excluding water availability, which is discussed later.
where m n is the number of mosquitoes in age group n. Note that this limits the number of eggs laid by a single mosquito per gonotrophic cycle to approximately 184, which is somewhat less than the number observed by Yaro et al. , but in line with that reported by Howard et al. .
Estimation of water temperature
Using the 0-10-cm soil temperature (T soil ) from the NOAH land surface model [91–94] to approximate the mean water temperature (T water ) in larval habitats, we assume that evaporative cooling and heat fluxes at the water boundaries are negligible. Hence, the water temperature is equal to the top soil temperature. Paaijmans et al. showed that the 5-cm soil temperature represents the water temperature in small ponds reasonably well . Therefore, the model will have limited validity in areas where larger puddles are the main breeding sites. There is also a chance that diurnal fluctuations will be slightly over- or underestimated. When a grid cell covers several k m2, this effect should be negligible, although we do not have data to support this. We hope to improve the prediction of water temperature in the future, either by modelling this explicitly or using a parametrized version based on data from Huang et al. .
Parametrization of mortality
We used two approaches to calculate mortality in the aquatic stages. In the simpler approach, we assume that mortality and development time in the aquatic stages are independent of the species. We also assume that the relationship between the mortality rate and temperature is the same for eggs, instars and pupae. In this method we do not consider competition effects as described by Paaijmans et al. . This type of parametrization is suitable when the model is used for one species only (e.g. if the model represents an area where only one of the two species is present).
Species-independent mortality (BLL)
Constants for equation 14 and 33
Species-dependent mortality (KBLL)
f gamb and f arab are the ratio of An. gambiae s.s. to An. arabiensis larvae and An. arabiensis to An. gambiae s.s. larvae, respectively. At each time step, L size is estimated as a function of B L and K. As the density increases, there will be more competition and hence less food for each larva, which leads to smaller larvae.
Parametrization of the development rate
for An. arabiensis. f arab and f gamb is the fraction of An. arabiensis and An. gambiae s.s., respectively.
Parametrization of breeding sites
The formation of puddles can be described as a balance of runoff, infiltration, evaporation, and rainfall entering the puddle. The formulation of an idealized puddle can be found in Additional file 2.
Modelling of every single breeding site requires high enough resolution to resolve the puddle. In practice this is not possible and the problem has to be simplified.
Mushinzimana et al. described typical breeding sites in a Kenyan highland area . Most of the puddles were located at less than 100 m from rivers, which means we can assume that semi-permanent puddles will mostly form in the proximity of rivers and lakes. They also found that the number of breeding sites was close to threefold higher in the rainy season compared to the dry season, and grouped breeding sites by surface area.
If we assume that breeding mainly occurs in the vicinity of potential rivers and lakes, the availability of breeding sites can be expressed as a function of potential river length and soil saturation. At high resolution this might not always be true , but since the model is designed to be applied to coarser grids, we believe the assumption is as reasonable as or more reasonable than the common assumption that puddle formation is only dependent on rainfall . The newest version of the NOAH land surface model in WRF 3.4 also includes groundwater and dynamic vegetation, and future versions might change the way in which puddles are parametrized. In OMaWa we introduce a simple parametrization scheme to represent breeding sites.
The Hydrological Data and Maps based on SHuttle Elevation Derivatives at Multiple Scales (HydroSHEDS) 15s river data set from the US Geological Survey (USGS)  was used to derive the total potential river length within a grid cell. Since the algorithm used to develop this data set describes where water would collect if it were available within the catchment, it also represents a general description of the potential for water aggregation within an area. However, the validity might decrease on moving to finer scales .
where Ξ is the equally spaced river data-set resolution in degrees, where Δ lon = Δ lat, ER is the radius of the Earth (6371.22 km) and φ is latitude in radians.
where is the maximum larval biomass per km of river (2400 mg, estimated from data collected by Munga et al. ) and S M r is the relative soil moisture content (fraction).
In the current implementation we do not distinguish between fast- and slow-flowing rivers. It should be noted that this way of approximating breeding sites has limited validity in areas with irrigation or around rivers where breeding sites could form as rivers recede [66, 67, 102]. Some special cases, such as along the River Nile in Sudan, where breeding sites form as a result of rainfall hundreds of kilometers away, will not be captured at all .
Parametrization of the gonotrophic cycle
where T air is the air temperature (°C), D d is degree days, and T c is the critical temperature from Hoshen and Morse , with D d = 37, and T c = 7.7.
Parametrization of the age-dependent mortality of adult mosquitoes
Our new survival curves are based on unpublished data from Bayoh and Lindsay . The validity ranges from 5 to 40°C by 5°C and 40-100% by 20% relative humidity. We name the scheme BLLad (Bayoh-Lindsay-Lunde adult mortality). The data set and the curves are valid for An. gambiae s.s. The lowest agreement between the model and the data is at 40% relative humidity and 40°C. While the data suggest that all An. gambiae s.s. would be dead after approximately 2 days, the survival curve would result in no mosquitoes after approximately 4 days at 40% relative humidity and 40°C. To correct for this error, we include data from Kirby and Lindsay , who described the responses of An. gambiae s.s. and An. arabiensis to high temperatures. By assuming that maximum survival is 480 min for An. gambiae s.s. and 1440 min for An. arabiensis at temperatures greater than 40°C, we can set the mortality rate to 3day-1 and 1day-1, independent of age group. However, there are uncertainties at relative humidity below 40%. The lack of studies in this range is a limitation of this survival model, and could make the model less accurate for An. gambiae s.l. in some regions. The basic principle of these survival curves is that mortality will be low in the first few days after emergence. In addition, mosquitoes that survive up to a certain age have a higher survival probability (depending on T air and relative humidity). In Figure 5, survival at 60% relative humidity and 0, 10, 20, 30, and 40°C is plotted.
If we assume that differences in adult mortality for An. gambiae s.s. and An. arabiensis can be explained by differences in body size, these BLLad curves can be used for both species. We explore this mortality model in .
AL adult mortality
Parametrization of air temperature
Hence, T air can be partly or fully replaced by T indoor , depending on the proportion of mosquitoes indoors.
It should be noted that we still do not include temperatures in resting places described by Holstein, such as holes in rocks and cracks in soil, covered pigsties, rabbit hutches, hen coops and dry wells , and by de Meillon (, under stones).
Approximation of mosquito movement
The role of diffusion and advection in vector borne diseases have been explored in several papers [102, 121–127]. Considering the gradual invasion of Brazil in the 1930s by An. arabiensisit can be argued that movement of mosquitoes is important over decades. Here we include the active and passive transport of mosquitoes as fluxes across grid boundaries. Passive transport is movement of mosquitoes caused by wind, while active transport is movement due to flying. On shorter time scales the role of such movement will be limited. However, on long time scales it is necessary to allow mosquitoes to travel to allow them to establish in new locations.
Transport of mosquitoes is defined by fluxes (s-1) at the grid boundaries. In the model we allow fluxes from the eight neighboring grid points. A special case is implemented when a neighbouring cell is water. In this case, fluxes to water are reduced to 0.1% of the original flux to avoid large losses of mosquitoes along the coastline. Given strong winds from land to the ocean, such an assumption could lead to accumulation of mosquitoes along the coast. Conversely, allowing free movement to the ocean could lead to undesired loss of mosquitoes.
Since the movement of mosquitoes has a high computational cost, the spatial fluxes do not change the size calculations. This will introduce some minor errors when the movement of mosquitoes is low compared to their density, with larger errors if many mosquitoes are moved relative to their density. When a cell free of mosquitoes is colonized, the size is set to 3.05 mm.
The possible flight range of anophelines varies with food availability . We do not include vegetation types in the model and hence it is hard to justify differences in flight performance based on, for example, land use. The dispersion coefficient describes how far mosquitoes can move in a day. We assume that the dispersion coefficient D is constant, independent of geographical location. For An. gambiae s.s. and An. arabiensis, real flight performance outside the laboratory of only a few hundred meters per day (approx. 300-700 m) has been reported [102, 129, 130]. In this experiment we subjectively chose D=30m day-1 independent of age group. Anophelinae also travel with humans , which adds to the transport equation and makes the dispersion coefficient uncertain. Gillies noted that wind direction mostly has a minor effect on dispersal , while de Meillon  and Adams  reported distances of 2-4.5 miles (3-7 km) in the direction of the prevailing wind. Thus, it cannot be ruled out that wind plays a role on longer time scales. Hence, we express movement caused by wind as a function of 10-m zonal (u) and meridional (v) wind components (m s-1). This can be understood by considering the following example. For a constant u-wind of 10m s-1 and v-wind set to 0, mosquitoes will be moved a distance related to a scale factor S f , which is equal to the distance travelled at 20m s-1 to the east. For example, with S f = 750m day-1, the eastward distance traveled will be m in 1 day, but since each mosquito is not modelled individually, it would be more natural to describe this as a fraction moving a certain distance. Different wind directions and speeds will result in other distances/fractions and directions. D and S f are unknown tunable constants.
In the presence of wind, we obtain additional transport as a function of zonal and meridional wind components.
Mortality related to feeding
where ρ human and ρ bovine is the probability of finding a human and bovine source, respectively. ρ humans is defined as the human population density per km2 multiplied by 0.1 (since a smaller area on a human is accessible) and ρ bovine is defined as the bovine density per km2, each with a user-defined threshold at which the density is so low that P(B) is virtually zero. Since P(B) is a conceptual parameter, it can be tuned.
Since blood meals, besides sugar meals, are important for the mobility  and survival of female anophelines , the success of a species is likely to be linked to the presence of the preferred host. The dominant blood source for An. arabiensis is bovine and human blood, while it is human blood for An. gambiae s.s. . In reality there are strong indications that the human blood index is a dynamic quantity rather than a constant [139–142]. In the current implementation, HBI is a static number and hence there are probably errors related to this term. To find the probability of feeding on humans at each time step, we combine two data sets. Between 2000 and 2010 we use population densities from the Gridded Population of the World (GPW) , and for before 2000 and after 2010 we use growth rates from the Population Division of the Department of Economic, and Social Affairs of the United Nations Secretariat . Since there are no projections of cattle densities, this quantity is time-invariant and based on Food and Agriculture Organization (FAO) 2005 estimates . We are currently working to include time-varying cattle densities.
Results and discussion
Sensitivity experiments are useful in understanding which parameters are important for the success of An. arabiensis and An. gambiae s.s. and which are important for malaria transmission. Classical sensitivity analysis investigates the robustness of a study when parameters are estimated from statistical modelling. Our model uses parametrization schemes to represent the influence of the environment on the two species. We show how the model responds to changing temperature, humidity, mosquito size, dispersion and the probability of finding blood. This approach does not allow us to directly measure the robustness of each parametrization scheme, but gives us an insight into which external factors influence the model and where it is of importance to have improved parametrization schemes. We use the term sensitivity experiments for this analysis.
To demonstrate some of the capabilities of the model, we set up a series of experiments. Some aspects are best visualized as a one-dimensional model (time and age), while other features are shown using a spatial domain (time, age, and space). For the one-dimensional experiments, the water temperature is set to the air temperature, except for temperature greater than 33°C, for which we set temperature to 33°C. This modification is required since pupae and fourth instar larvae will not develop below 18°C or above 34°C . The results are therefore less robust when temperature is greater than 33°C. Unless otherwise stated, we use size-dependent mortality, correction for indoor temperature, the KBLL method to estimate mortality in the aquatic stages, correction for the development rate in the aquatic stages depending on the ratio of each species, and movement of mosquitoes (in the spatial cases).
Sensitivity to temperature, relative humidity and mosquito size (TempHumSize)
The age-dependent mortality is influenced by temperature, relative humidity and mosquito size [Eq. (32)]. This experiment explores how the dynamics of malaria is sensitive to temperature, relative humidity and mosquito size (measured as mm). We assume that no births occur to isolate the effect of the transmission process, and consequently constant mosquito body size in the course of integration, but include mortality and the biting rate. In this experiment we assume that only one species is present (since the main competition occurs in the aquatic stages). This sensitivity test is designed to observe how the proportion of mosquitoes becomes infected as a function of temperature, relative humidity and mosquito size, given that we start with 1000 newly emerged mosquitoes, with m1 = 1000 and m2-9 = 0 as the initial conditions. In this experiment, 1% of the human population is infectious for Plasmodium falciparum. Mosquitoes are infected with an efficiency of 100%, meaning that biting an infectious human results in gametocyte transmission to the mosquito. In practice, this would be the same as saying that 10% of humans were infectious and gametocyte transmission had an efficiency of 10%. We also neglect the effect of heterogeneous biting. This is the only experiment in which we model the proportion of infectious mosquitoes explicitly. The modified equations describing the transmission process are described in .
where a = 9.5907, b = 0.0051029, c = 0.7349, and d = 17.0325. This expression was derived from the figure in MacDonald page 119  using g3data , and fitted using non-linear least-squares .
The gonotrophic cycle and biting rate are defined in Eq. (26).
These results should be viewed in light of recent findings by Paaijmans et al. that optimal transmission occurs at lower temperatures .
Sensitivity to temperature and carrying capacity (TempCar)
The aim of this sensitivity test was to investigate how carrying capacity and temperature determine the relative proportion of An. arabiensis and An. gambiae s.s. We set the relative humidity to 80% and the probability of getting a blood meal to one. We assumed that the soil was saturated and we varied the temperature between 16 and 38°C (with corrections over 33°C for water temperature) and the carrying capacity between 0.0625 and 125 m g k m-2.
Carrying capacity in the aquatic stages influences larval growth and adult survival. While An. arabiensis invests more time in growth than An. gambiae s.s., the former develops a larger body, and consequently has the potential to oviposit more eggs than the latter. If the two species experience the same mortality rate in the aquatic stages, more An. gambiae s.s. will emerge, but over time An. arabiensis can face this challenge by outnumbering the eggs of An. gambiae s.s. in the habitat. Thus, we are interested in testing how the carrying capacity in the aquatic stages alters the relative proportion of each of the adult species. In this model we only consider the competition between these two species, and hence neglect other competing species .
Sensitivity to temperature and the probability of finding blood (pBlood1D)
This experiment shows how the model responds to changes in the probability of finding a blood meal, which influences the rate at which mosquitoes can oviposit and increases energy consumption if hosts are hard to locate. If, for example, cattle are easier to find compared to humans, An. arabiensis will potentially use less energy per batch of eggs and will also be able to utilize breeding sites at a higher rate than An. gambiae s.s. It is also possible that An. arabiensis uses cattle for navigation . Over time, such differences might lead to dominance by one species. In this experiment, we varied the probability of finding blood, P(B), for An. arabiensis from zero to one, as well as varying the temperature as described for TempCar.
We set the probability of finding blood to one for An. gambiae s.s., independent of the probability of An. arabiensis finding a blood meal. This is a purely theoretical experiment designed to demonstrate a concept. The probability of finding blood is varied between zero and one for An. arabiensis. The scenario in which P(B) = 1 for An. gambiae s.s. and zero for An. arabiensis is not a realistic scenario, but the difference in P(B) is grounded in differences in their feeding behaviour, whereby An. arabiensis can utilize cattle more efficiently than An. gambiae s.s., for example.
Sensitivity to the probability of finding blood in a spatial domain (pBlood2D)
This experiment is similar to pBlood1D, but this time we integrate the model for 5 years over the African domain. The experiment consists of two runs, for which the first has P(B) similar to Figure 8 and the second has P(B)=1 over all land areas for both species. The population density is space-invariant at 400 h u m a n s/k m2 (remember that the number of mosquitoes is limited by the number of hosts). Thus, the only limitation in this experiment is the physical environment (air and water temperatures, relative humidity, wind and run-off), which is updated every 3 h. The initial conditions for the mosquito populations were the same for the two runs.
Even though we have stated that the probability of finding blood P(B) is an expression of the cost of finding a host, it might well be that P(B) also includes a component that describes the environment shaped by cattle and humans. Therefore, it should be noted that it is difficult to distinguish between the true probability of finding blood and the environmental changes caused by the presence of humans or cattle.
It is also worth mentioning that the density of An. gambiae s.l. in South Africa is not very sensitive to the probability of finding a blood meal. Hence, the distribution of An. gambiae s.l. is mainly restricted by climate according to the model.
Figures 12 and 13 show the distribution and density of An. arabiensis and An. gambiae s.s. under realistic (P0) and space-invariant (P1) P(B) after 6, 12, and 18 months. The integration was started on January 1 and the model was run for 5 years.
Mosquito transport (mosqTran)
The purpose of this experiment was to demonstrate how the initial conditions and competition influence the distribution of An. gambiae s.s. and An. arabiensis. To explore the theoretical dispersion distance and the influence of the initial conditions, we set up a simple experiment. In mosqTran(a) the model was initialized with An. arabiensis at -4.494381°E, 14.0154°N (Sahel), and An. gambiae s.s. at -4.494381°E, 6.502846°N (Cte d’Ivore, Ivory Coast) on January 1, 1989. The second experiment, mosqTran(b), had the same setup, but without An. arabiensis.
The purpose of this demonstration was to show the importance of mosquito movement and how new areas can or cannot be colonized. In a model in which movement is restricted, the vector range would also be restricted by the initial model conditions. For example, if only one point was specified for mosquitoes at the beginning of the integration, only the same point would have mosquitoes after 100 years. With dynamic movement the mosquitoes could colonize new areas if the environmental conditions, or the probability of finding blood, change over time.
We developed a model to predict the presence and abundance of An. arabiensis and An. gambiae s.s. The model is age-structured and includes mosquito dispersal.
Sensitivity tests showed that as well as temperature, relative humidity and mosquito size are important factors in malaria transmission. The result for body size is in line with several studies [7, 51, 54, 55, 88, 154] and thus the model captures some of the aspects related to higher survival among larger individuals. Note that we have not accounted for the higher metabolism in large mosquitoes , which might reduce survival under warm and dry conditions. There are also contrasting results with respect to body size and egg production . It is likely that there is an optimum size that depends on the environment and is a function of temperature and humidity. Currently there are few results to back up this statement. However, Sanford et al. found significant differences in Anopheles gambiae s.s. wing length between Mali and Guinea-Bissau .
We show that relative humidity can be important for malaria transmission. Several models have neglected the role of (relative) humidity [29, 157] and it is true that desiccation might not be a driver of mortality at moderate humidity (>70%?). The main argument for leaving out this parameter is the corresponding reduction in model complexity. As long as rainfall drive the carrying capacity, mosquito numbers will be restricted at lower humidity (no rain), and as a consequence the resulting number of mosquitoes can be limited for the wrong reasons, but with the correct result. For example, Ermert et al.  handle this deficiency by reducing vector survival during dry atmospheric conditions, defined as a function of 10-day accumulated rainfall. More studies on the survival of An. gambiae s.l. in relation to size and relative humidity in the range 5-40% are needed for more confidence in the role of humidity in the survival of An. gambiae s.l.
Assumption of exponential mortality has several advantages (see Figure 5 for examples of models in which exponential mortality is used). The model becomes fast to solve and it is easier to analyse the equations analytically. However, several studies have shown that mortality of An. gambiae s.l. is not exponential, and that inclusion of an age dimension alters the expected outcome of interventions targeted to reduce the vector population . Therefore, we believe that models in which age-dependent mortality is assumed should be further explored. The sensitivity tests also suggest that carrying capacity within a restricted area plays a role in the distribution of An. arabiensis and An. gambiae s.s. The true carrying capacity is hard to estimate on a continental scale and thus relies on qualified guesswork taking into account rainfall, groundwater and soil saturation, for example. Carrying capacity influences not only the relative distribution of the two species but also the total number of mosquitoes. To correctly estimate the biting rate, a correct estimate of carrying capacity is required, and thus more work is needed to parametrize puddle formation. It should also be noted that no current large-scale models can describe the formation of puddles as rivers retreat, as described by Animut et al. .
Experiment pBlood2D showed how the model responds to the parameter P(B), the probability of finding a blood meal. P(B) is important in describing a realistic distribution of An. arabiensis and An. gambiae s.s. Thus, we hypothesize that the large-scale distribution of bovines is key to the success of An. arabiensis. Likewise, large-scale human density favours the presence of An. gambiae s.s.
Finally, experiment mosqTran showed how the initial conditions influence the dispersal of An. gambiae s.s. (and An. arabiensis). The distribution of An. gambiae s.s. changes dramatically with the presence of An. arabiensis, and thus the initial model conditions are highly relevant for correct description of the distribution of the two species. When rainfall is highly seasonal, the first come, first served principle seems to be important for the success of a species in drier conditions. Whether or not this plays a role in the evolution of aestivation in An. gambiae s.s. M form  is a question that should be further investigated.
The strong influence of initial conditions on dispersal of the An. gambiae complex is not irrelevant when assessing the impact of climate change, since vectorial capacity varies between species.
The availability of mosquito models allows researchers to build on and improve our understanding of the role of the An. gambiae complex in malaria transmission. We hope to refine the model as new data on mosquito biology become available, and to incorporate the effects of interventions.
We are grateful to the National Center for Atmospheric Research (NCAR) for making their WRF model available in the public domain. We also thank the Bergen Centre for Computational Science for computational and other resources provided during this study. This work was made possible by grants from The Norwegian Programme for Development, Research and Education (NUFU) and the University of Bergen. We thank Steve Lindsay for providing data on the survival of An. gambiae s.s. under different temperatures and relative humidities. We thank two anonymous reviewers for their constructive comments, which helped us to improve the manuscript.
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