Bjerknes Centre for Climate Research, University of Bergen, Norway

Centre for International Health, University of Bergen, Norway

KEMRI/CDC Research and Public Health Collaboration, Kisumu, Kenya

Bjerknes Centre for Climate Research, Uni Research, Norway

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

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.

Background

Since the 1950s, near-surface global temperatures have increased by about 0.5-0.6°

These projections rely on knowledge about how the malaria parasite and anopheline vectors respond to changes in temperature. While a lot is known

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°

Here, we compare six mortality models (Martens 1, Martens 2, Bayoh-Ermert, Bayoh-Parham, Bayoh-Mordecai and Bayoh-Lunde) to reference data (control) for

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

Methods

Survival models

Six different parametrization schemes have been developed to describe the mortality rates for adult _{
air
}are in °

Martens 1

The first model, which is called Martens scheme 1 in Ermert

Martens 2

In 1997 Martens

Bayoh-Ermert

In 2001, Bayoh carried out an experiment where the survival of

In 2011, Ermert

Overall, this model has higher survival probabilities at all of the set temperatures compared with the models created by Martens.

Bayoh-Parham

In 2012, Parham _{0}
_{1}
_{2}).

where _{0}=0.00113·^{2}−0.158·_{1}=−2.32·10^{−4}·^{2} + 0.0515·_{2}=4·10^{−6}·^{2}−1.09·10^{−3}·

For all models reporting survival probability, we can rewrite

Bayoh-Mordecai

Recently, Mordecai

Bayoh-Lunde

From the same data

Because mosquito size is also known to influence mortality

The mortality rate, _{
n
}(

**Details of the Bayoh-Lunde model, mosquito biting rate, and parasite extrinsic incubation period.**

Click here for file

**This file shows how ****
ζ
**

Click here for file

**Survival curves for all of the models investigated in this study plotted at different temperatures and relative humidities.** The figure on page two shows the legend as well as an example of non-exponential mortality.

Click here for file

Biting rate and extrinsic incubation period

The equations used for the biting rate,

Malaria transmission

We set up a system of ordinary differential equations (ODEs) to investigate how malaria parasites are transmitted to mosquitoes. Four of the mortality models (equations 1, 2, 3, and 4) are used in a simple compartment model that includes susceptible (

where _{
i
}is the fraction of infectious humans, which was set to 0.01.

Because the Lunde

We separate susceptible (

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.

The number of infectious mosquitoes is the sum of _{
n
}, where

Age groups for mosquitoes (_{1}=[0,1], _{2}=(2,4], _{3}=(5,8], _{4}=(9,13], _{5}=(14,19], _{6}=(20,26], _{7}=(27,34], _{8}=(35,43], _{9}=(44,_{
n
}, where

This model has initial conditions _{1}=1000, and all other 0.

A note on the use of ODEs and rate calculations can be found in Additional file

**A note on the use of ordinary differential equations, age structure (with an example), and rate calculations.**

Click here for file

Validation data

To validate the models, we used the most extensive data set available on mosquito survival _{
a
}) at time

Hence, to account for this we have used three independent data sets to validate the fraction of infectious mosquitoes and the mosquito survival curves.

Scholte

**Control**

** AIC Control**

**Scholte**

**AF**

**BL mortality model**

**SK mortality model**

Skill scores as defined in equation 11. “Control” represents the validation of infectious mosquitoes using the data from Bayoh and Lindsay

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

Bayoh-Ermert

0.27

30 (1, 58)

0.16

0.43

0.79

0.56

Bayoh-Parham

0.16

26 (-11, 55)

0.05

0.31

0.79

0.59

Bayoh-Lunde

0.90

-111 (-148, -81)

0.83

0.94

0.90

0.81

Bayoh-Mordecai

0.62

-53 (-82, -29)

0.58

0.70

0.57

0.49

Using the data from Bayoh and Lindsay, Afrane

Because some of the schemes do not include RH, we have displayed the mean number of infectious mosquitoes,

Validation statistics

Skill scores (

where _{0}=1 is the reference correlation coefficient, and _{
f
}/_{
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.

Another important aspect is determining at which temperatures transmission is most efficient. If mosquitoes have a peak of infectiousness at, for example, 20°

For the transmission process we also report Akaike information criterion (AIC) _{
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, non-correlated errors. Median AIC, with 95% confidence intervals are reported in Table

Results

Figure

The percentage of infectious mosquitoes over time and temperature

**The percentage of infectious mosquitoes over time and temperature.**

While Martens 1 has the most efficient transmission at 20.4°C, Martens 2 and Bayoh-Ermert show the transmission efficiency peaking at 26.8 and 27.5°

Integral of infectious mosquitoes over temperature

**Integral of infectious mosquitoes over temperature.** Models: Bayoh-Ermert (blue solid line), Martens 1 (black solid line), Martens 2 (blue dashed line), Martens 3 (grey solid line), Bayoh-Parham (red solid line), Lunde (black dashed line), and the mean value of the five models (green thick solid line). Black dots indicate the results for the control, and vertical lines show the temperature at which the maximum can be found (equation 12).

The numerical solution of the Bayoh-Ermert 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°

To evaluate the skill of the models, with emphasis on spatial patterns and variance, we investigated the skill score that was defined in equation 11. The standard deviation, root mean square and correlation coefficient are summarized in a Taylor diagram (Figure

Taylor diagram

**Taylor diagram.** The model names are indicated next to the symbols. The x and y axes represent the standard deviations, the curved grey lines are the root mean square, while the dashed lines represent the Pearson correlation coefficient.

When validating the transmission process using the data from Bayoh and Lindsay (Table

The relatively simple Martens 2 model ranked third among the models. We re-calibrated

The newly calibrated Martens 2 model (hereafter called Martens 3), can be seen in Figure

To investigate how sensitive the results of the Mordecai _{0} model (equation 2 in _{0} according to temperature (with _{0}ranges from 10 (Martens 1) to 206 (Bayoh-Parham).

_{0}as a function of temperature calculated using equation 2 in Mordecai

_{0 }**as a function of temperature calculated using equation 2 in Mordecai **** [****], but with different mortality models.** Blue dots represent optimal temperatures using the methods in this paper, and red crosses is the optimal temperature using the methods from Mordecai

**This paper**

**
R
**

**Relative**

**
et al.
**

**difference %**

Optimal temperature calculated using the methods in this paper, and by using methodology in Mordecai

Martens 1

20.4

23.0

11.98

Martens 2

26.8

27.0

0.74

Martens 3

24.7

26.0

5.13

Bayoh-Ermert

27.5

27.2

1.10

Bayoh-Parham

26.3

26.9

2.26

Bayoh-Lunde

25.2

Bayoh-Mordecai

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 Bayoh-Ermert models suggest that areas of the world where temperatures approach 27°

Table

The Martens 1 model has been used in several studies

It is likely that regions with temperatures below 18°

Most countries in Sub-Saharan Africa have annual mean temperatures between 20 and 28°

Abbreviations

BL: Bayoh and Lindsay; EIP: Extrinsic incubation period; ODEs: Ordinary differential equations.

Competing interests

The authors declare that they have no competing interests.

Authors’ contributions

The work presented here was carried out in collaboration between all of the authors. BL, MNB and TML defined the research theme. MNB provided the data for the control. TML designed the methods and experiments, did the model runs, analysed the data, interpreted the results, and wrote the paper. All authors read and approved the final version of the manuscript.

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.