How malaria models relate temperature to malaria transmission
 Torleif Markussen Lunde^{1, 2, 4}Email author,
 Mohamed Nabie Bayoh^{3} and
 Bernt Lindtjørn^{2}
DOI: 10.1186/17563305620
© Lunde et al; licensee BioMed Central Ltd. 2013
Received: 5 October 2012
Accepted: 15 January 2013
Published: 18 January 2013
Abstract
Background
It is well known that temperature has a major influence on the transmission of malaria parasites to their hosts. However, mathematical models do not always agree about the way in which temperature affects malaria transmission.
Methods
In this study, we compared six temperature dependent mortality models for the malaria vector Anopheles gambiae sensu stricto. The evaluation is based on a comparison between the models, and observations from semifield and laboratory settings.
Results
Our results show how different mortality calculations can influence the predicted dynamics of malaria transmission.
Conclusions
With global warming a reality, the projected changes in malaria transmission will depend on which mortality model is used to make such predictions.
Keywords
Anopheles gambiae sensu stricto Climate Temperature Mathematical modelBackground
Since the 1950s, nearsurface global temperatures have increased by about 0.50.6°C[1], and it is likely that temperatures will continue to increase over the next century [2]. Model predictions, reported widely in climate policy debates, project that a warmer climate could increase malaria caused by the parasites Plasmodium falciparum and P. vivax in parts of Africa [3]. Malaria is transmitted by mosquitoes of the Anopheles genus, with Anopheles gambiae s.s., An. arabiensis and An. funestus being the dominant vector species in Africa [4, 5].
These projections rely on knowledge about how the malaria parasite and anopheline vectors respond to changes in temperature. While a lot is known [6] about how parasite development is influenced by temperature [7], the same cannot be said for mosquitoes. In addition to temperature, humidity [8, 9], breeding site formation [10], and competition between mosquitoes [11, 12] are important factors controlling the number of vectors at any time.
Climate predictions about humidity and precipitation are more uncertain than temperature projections. Therefore, it is of interest to see if a consensus exists between different malaria models about how temperature alone influences malaria transmission. In the past, studies have suggested that the optimal temperature for malaria transmission is between 30 and 33°C[13–15].
Here, we compare six mortality models (Martens 1, Martens 2, BayohErmert, BayohParham, BayohMordecai and BayohLunde) to reference data (control) for Anopheles gambiae s.s., and show how these models can alter the expected consequences of higher temperatures. The main purpose of the study is to show if there are any discrepancies between the models, with consequences for the ability of projecting the impact of temperature changes on malaria transmission.
We have focused on models that have been designed to be used on a whole continent scale, rather than those that focus on local malaria transmission [10, 16, 17].
Methods
Survival models
Six different parametrization schemes have been developed to describe the mortality rates for adult An. gambiae s.s.. These schemes are important for estimating the temperature at which malaria transmission is most efficient. The models can also be used as tools to describe the dynamics of malaria transmission. In all of the equations presented in this paper, temperature, T and T_{ air }are in °C.
Martens 1
Martens 2
BayohErmert
In 2001, Bayoh carried out an experiment where the survival of An. gambiae s.s. under different temperatures (5 to 40 in 5°C steps) and relative humidities (RHs) (40 to 100 in 20% steps) was investigated [24]. This study formed the basis for three new parametrization schemes. In the naming of these models, we have included Bayoh, who conducted the laboratory study, followed by the author who derived the survival curves.
In 2011, Ermert et al.[18] formulated an expression for Anopheles survival probability; however, RH was not included in this model. In the text hereafter, we name this model BayohErmert. This model is a fifth order polynomial.
BayohParham
where β_{0}=0.00113·R H^{2}−0.158·RH−6.61, β_{1}=−2.32·10^{−4}·R H^{2} + 0.0515·RH + 1.06, and β_{2}=4·10^{−6}·R H^{2}−1.09·10^{−3}·RH−0.0255.
BayohMordecai
BayohLunde
From the same data [24], Lunde et al.[27], derived an agedependent mortality model that is dependent on temperature, RH, and mosquito size. This model assumes nonexponential mortality as observed in laboratory settings [24], semifield conditions [28], and in the field [29]. In the subsequent text we call this model BayohLunde. The four other models use the daily survival probability as the measure, and assume that the daily survival probability is independent of mosquito age. The present model calculates a survival curve (ϖ) with respect to mosquito age. Like the BayohParham model, we have also varied the mosquito mortality rates according to temperature and RH.
Because mosquito size is also known to influence mortality [8, 9, 30, 31], we applied a simple linear correction term to account for this. In this model, the effect of size is minor compared with temperature and relative humidity. The survival curve, ϖ, is dependent on a shape and scale parameter in a similar manner as for the probability density functions. The scale of the survival function is dependent on temperature, RH, and mosquito size, while the scale parameter is fixed in this paper.
Biting rate and extrinsic incubation period
The equations used for the biting rate, G(T), and the inverse of the extrinsic incubation period (EIP, pf) are described in Lunde et al. [27]. For convenience, these equations and their explanations are provided in Additional file 1. The extrinsic incubation period was derived using data from MacDonald [7], while the biting rate is a mixture of the degree day model by Hoshen and Morse [32], and a model by Lunde et al.[27]. Since our main interest in this research was to examine how mosquito mortality is related to temperature in models, we used the same equation for the gonotrophic cycle for all of the mortality models. If we had used different temperaturedependent gonotrophic cycle estimates for the five models, we would not have been able to investigate the effect of the mortality curves alone.
Malaria transmission
where H_{ i }is the fraction of infectious humans, which was set to 0.01. G(T) is the biting rate, and pf is the rate at which sporozoites develop in the mosquitoes. The model is initialized with S=1000, E=I=0 and integrated for 150 days with a time step of 0.5. As the equations show, there are no births in the population, and the fraction of infectious humans is held constant during the course of the integration. This setup ensures that any confounding factors are minimized, and that the results can be attributed to the mortality model alone.
Because the Lunde et al.[27] (BayohLunde) mortality model also includes an age dimension, the differential equations must be written taking this into account. Note that the model also can be used in equation 8 if we allow β to vary with time.
We separate susceptible (S), infected (E) and infectious (I), and the subscript denotes the age group. In total there are 25 differential equations, but where the equations are similar, the subscript n has been used to indicate the age group.
Formulating the equation this way means we can estimate mosquito mortality for a specific age group. We have assumed that mosquito biting behaviour is independent of mosquito age; this formulation is, therefore, comparable to the framework used for the exponential mortality models.
Age groups for mosquitoes (m) in this model are m_{1}=[0,1], m_{2}=(2,4], m_{3}=(5,8], m_{4}=(9,13], m_{5}=(14,19], m_{6}=(20,26], m_{7}=(27,34], m_{8}=(35,43], m_{9}=(44,∞] days, and coefficients a_{ n }, where n=1,2,…,9, are 1.000, 0.500, 0.333, 0.250, 0.200, 0.167, 0.143, 0.125, 0.067. The rationale behind these age groups is that as mosquitoes become older, there is a greater tendency of exponential mortality compared to younger mosquitoes.
This model has initial conditions S_{1}=1000, and all other 0.
A note on the use of ODEs and rate calculations can be found in Additional file 4.
Validation data
To validate the models, we used the most extensive data set available on mosquito survival [24] under different temperatures (5 to 40 by 5°C) and RHs (40 to 100 by 20%) [24]; it is the same data that the BayohErmert, BayohParham and BayohLunde models were derived from. These data describe the fraction of live mosquitoes (f_{ a }) at time t, which allows us to validate the models over a range of temperatures. Because three of the models used the Bayoh and Lindsay data to develop the survival curves, this comparison is unrealistic for Martens models.
Hence, to account for this we have used three independent data sets to validate the fraction of infectious mosquitoes and the mosquito survival curves.
Skill scores
Control  AIC Control  Scholte  AF  BL mortality model  SK mortality model  

Martens 1  0.01  76 (56, 96)  0.00  0.03  0.36  0.25 
Martens 2  0.38  9 (14, 30)  0.55  0.37  0.54  0.45 
Martens 3  0.65  38 (75, 9)  0.53  0.77  0.65  0.52 
BayohErmert  0.27  30 (1, 58)  0.16  0.43  0.79  0.56 
BayohParham  0.16  26 (11, 55)  0.05  0.31  0.79  0.59 
BayohLunde  0.90  111 (148, 81)  0.83  0.94  0.90  0.81 
BayohMordecai  0.62  53 (82, 29)  0.58  0.70  0.57  0.49 
β(t) is linearly interpolated at times with no data. The reference data from Bayoh and Lindsay [24] are hereafter designated as the control data in the subsequent text, whereas data from Scholte et al.[33] is called Scholte in Table 1. Table 1 also shows the skill scores of the mortality model alone (for the figures in Additional file 3).
Because some of the schemes do not include RH, we have displayed the mean number of infectious mosquitoes, I, for schemes that do include it. For the validation statistics, RH has been included. However, for schemes where the RH has not been taken into account, single realization at all humidities has been employed.
Validation statistics
where r is the Pearson correlation coefficient, r_{0}=1 is the reference correlation coefficient, and ${\widehat{\sigma}}_{f}$ is the variance of the control over the standard deviation of the model (σ_{ f }/σ_{ r }). This skill score will increase as a correlation increases, as well as increasing as the variance of the model approaches the variance of the model.
The Taylor diagram used to visualize the skill score takes into account the correlation (curved axis), ability to represent the variance (x and y axis), and the root mean square.
For the transmission process we also report Akaike information criterion (AIC) [36] from a generalized linear model with normal distribution. Since the observations are not independent, and residuals do not follow a normal distribution, we sample 100 values from the simulations 1000 times. We set the probability of sampling y_{i,j} equal to normalized (sum = 1) fraction of infected mosquitoes of the control. This method allow us to generate a model with normally distributed, noncorrelated errors. Median AIC, with 95% confidence intervals are reported in Table 1.
Results
The numerical solution of the BayohErmert mortality model also reveals that it has problems related to enhanced mosquito longevity at all of the selected temperatures; this effect was especially pronounced around 20°C. We also found that the BayohParham model has issues with prolonged mosquito survival.
When validating the transmission process using the data from Bayoh and Lindsay (Table 1, column 1), the majority of the penalty for the Martens 1 and 2 models was due to the low variance, indicating that the mortality is set too high compared with the reference. Further analysis found that the BayohErmert model correlated poorly with the reference, and the variance, ${\widehat{\sigma}}_{f}$, was too high. The BayohParham model also suffered from low correlation, as well as too high variance. Overall, the BayohLunde model has the highest skill score, followed by the BayohMordecai model. The patterns are consistently independent of the data used to validate the models with respect to the malaria transmission process. Validation of the survival curves alone, and their relationship with the transmission process, is discussed in the next section.
The newly calibrated Martens 2 model (hereafter called Martens 3), can be seen in Figure 2; the skill scores are reported in Table 1.
Optimal temperature for malaria transmission
This paper  R_{0} from Mordecai  Relative  

et al.  difference %  
Martens 1  20.4  23.0  11.98 
Martens 2  26.8  27.0  0.74 
Martens 3  24.7  26.0  5.13 
BayohErmert  27.5  27.2  1.10 
BayohParham  26.3  26.9  2.26 
BayohLunde  25.2  
BayohMordecai  24.4  25.6  4.80 
Discussion and conclusions
The relationship between sporozoite development and the survival of infectious mosquitoes at different temperatures is poorly understood; therefore, any model projections relating the two should be interpreted with care. The Martens 2 and BayohErmert models suggest that areas of the world where temperatures approach 27°C could experience more malaria. Martens 3, BayohMordecai, and our model (BayohLunde) suggests that transmission is most efficient at around 25°C. The Martens 1 model peaks at 20.4°C, and BayohParham at 26.3°C (Figure 1). None of the models, except BayohLunde, capture all of the characteristics of the reference data, however.
Table 1 also shows the skill score for the mortality model alone. Both the BayohParham and the BayohErmert models have good representations of the survival curves. However, the nature of the exponential mortality curves gives them the choice of rapid mortality giving a reasonable, but underestimated, transmission process (Martens 2), or a good fit to the survival curves, which in turn makes the mosquitoes live too long, resulting in a poor transmission process (BayohParham and BayohErmert). Because the BayohLunde model offers a fair description of the survival curves as well as an age structure in the differential equations, we consider that the transmission process is well described. The Martens 1 and 2, BayohErmert, BayohMordecai and BayohParham models all assume constant mortality rates with age, and would, therefore, not benefit from being solved in an agestructured framework.
The Martens 1 model has been used in several studies [19–21], with the latest appearance by Gething et al. in this journal [39]. Considering the poor skill of the Martens 1 model, the validity, or etiology, of results presented in these papers should be carefully considered.
It is likely that regions with temperatures below 18°C, as is typical for the highland areas of East and Southern Africa, which are too cold for malaria transmission, might experience more malaria if their temperatures increase. However, malaria transmission in the future will be dependent on many other factors such as poverty, housing, access to medical care, host immunity and malaria control measures.
Most countries in SubSaharan Africa have annual mean temperatures between 20 and 28°C. In these areas, linking past and future temperature fluctuations to changes in malaria transmission is challenging. Our data suggest that one way to reduce this uncertainty is to use agestructured mosquito models. These models produce results that agree with the observed data, and nonexponential mosquito mortality has been demonstrated in several studies [33, 40–42], although the true nature of mosquito survival in the field is not fully elucidated. The newly calibrated Martens 2 model described here also produces acceptable results. If simplicity is a goal in itself [43], models that assume exponential mortality will still have utility. To believe in projections of the potential impact of longterm, largescale climate changes, it is crucial that models have an accurate representation of malaria transmission, even at the cost of complexity. For studies of malaria transmission at village level, other approaches might be more suitable [10, 16, 44, 45].
Abbreviations
 BL:

Bayoh and Lindsay
 EIP:

Extrinsic incubation period
 ODEs:

Ordinary differential equations.
Declarations
Acknowledgements
This work was made possible by grants from The Norwegian Programme for Development, Research and Education (NUFU) and the University of Bergen. Our thanks go to Asgeir Sorteberg for commenting on the manuscript, and three anonymous reviewers for their constructive comments, which helped us to improve the manuscript.
Authors’ Affiliations
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