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

The first model, which is called Martens scheme 1 in Ermert et al.[18], and described by Martens et al.[19–21], is derived from three points, and shows the relationship between daily survival probability (p) and temperature (T). This is a second order polynomial, and is, mathematically, the simplest of the models.

p ( T ) = − 0 . 0016 · T 2 + 0 . 054 · T + 0 . 45 (1)

Martens 2

In 1997 Martens [21] described a new temperature-dependent function of daily survival probability. This model has been used in several studies [13, 14, 22, 23]. In the subsequent text this model is named Martens 2. Numerically, this is a more complex model than Martens 1, and it increases the daily survival probability at higher temperatures.

p ( T ) = e − 1 − 4 . 4 + 1 . 31 · T − . 03 · T 2 (2)

Bayoh-Ermert

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 Bayoh-Ermert. This model is a fifth order polynomial.

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

p ( T ) = − 2 . 123 · 1 0 − 7 · T 5 + 1 . 951 · 1 0 − 5 · T 4 − 6 . 394 · 1 0 − 4 · T 3 + 8 . 217 · − 3 · T 2 − 1 . 865 · 1 0 − 2 · T + 7 . 238 · 1 0 − 1 (3)

Bayoh-Parham

In 2012, Parham et al.[25] (designated Bayoh-Parham in subsequent text) included the effects of relative humidity and parametrized survival probability using the expression shown below. This model shares many of the same characteristics as the Bayoh-Ermert model. The mathematical formulation is similar to the Martens 2 model, but constants are replaced by three terms related to RH (β 0 β 1 β 2 ).

p ( T , RH ) = e − T 2 · β 2 + T · β 1 + β 0 − 1 (4)

where β 0 =0.00113·R H2−0.158·RH−6.61, β 1 =−2.32·10−4·R H2 + 0.0515·RH + 1.06, and β 2 =4·10−6·R H2−1.09·10−3·RH−0.0255.

For all models reporting survival probability, we can rewrite p to mortality rates, β according to:

β = − ln ( p ) (5)

Bayoh-Mordecai

Recently, Mordecai et al.[26] re-calibrated the Martens 1 model by fitting an exponential survival function to a subset of the data from Bayoh and Lindsay [24]. They used the survival data from the first day of the experiment and one day before the fraction alive was 0.01. Six data points were used for each temperature.

p ( T ) = − 0 . 000828 · T 2 + 0 . 0367 · T + 0 . 522 (6)

Bayoh-Lunde

From the same data [24], Lunde et al.[27], derived an age-dependent mortality model that is dependent on temperature, RH, and mosquito size. This model assumes non-exponential mortality as observed in laboratory settings [24], semi-field conditions [28], and in the field [29]. In the subsequent text we call this model Bayoh-Lunde. 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 Bayoh-Parham 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.

The mortality rate, β n (T,RH,size) (equation 7) is fully described in Additional file 1, with illustrations in Additional files 2, and 3.

β n ( T , RH , size ) = ln ϖ N , m t 2 ϖ N , m t 1 Δt (7)

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 temperature-dependent gonotrophic cycle estimates for the five models, we would not have been able to investigate the effect of the mortality curves alone.

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 (S), infected (E) and infectious mosquitoes (I) (equation 8):

dS dt = − ( β + G ( T ) · H i ) · S dE dt = ( G ( T ) · H i ) · S − ( β + pf ) · E dI dt = pf · E − β · I (8)

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 set-up 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] (Bayoh-Lunde) 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.

The number of infectious mosquitoes is the sum of I n , where n=2,…,9.

d S 1 dt = − ( β 1 + a 1 ) · S 1 d S n dt = a n − 1 · S 1 − ( β n + a n + G ( T ) · H i ) · S n n = 2 , 3 , .. , 9 d S n dt = G ( T ) · H i · S 2 − ( β 2 + a 2 + pf ) · E 2 d E n dt = G ( T ) · H i · S n + a n − 1 · E n − 1 − ( β n + a n + pf ) · E n n = 3 , 4 , .. , 9 d I 2 dt = pf · E 2 − ( β 2 + a 2 ) · I 2 d I n dt = pf · E n + a 2 · I n − 1 − ( β n + a n ) · I n n = 3 , 4 , .. , 9 (9)

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 Bayoh-Ermert, Bayoh-Parham and Bayoh-Lunde 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.

Scholte et al. (Figure two in [33]) published a similar data set, but this was based on a temperature of 27±1°C and a RH of 80±5%, whereas Afrane et al. (Figure two in [28]) used mean temperatures of 21.5 to 25.0 and RHs of 40-80%. Use of these data sets will allow us to complement the validation to determine if the patterns of malaria transmission are consistent with that of the control (Table 1). In addition to the data from Scholte et al.[33], we also found the following data set, which is suitable for validation of the survival curves but not the transmission process itself, because the data does not show the survival curve until all of the mosquitoes are dead [Kikankie, Master’s thesis (Figures three to eight, chapter 3, 25°C, 80% RH) [34]]. These results are also shown in Table 1. The additional validation only gives information about the model quality between 21, and 27°C; however, it serves as an independent model evaluation to determine if the results are consistent and independent of the data set used to validate the models.

Table 1 Skill scores Full size table

Using the data from Bayoh and Lindsay, Afrane et al. or Scholte et al.[33], we can calculate the fraction of mosquitoes that would become infectious at time t, using equation 8. We replace β with the time-dependent β(t), which is a time varying mortality rate. This approach was used for the data from [24] and [33].

β ( t ) = − ln f a t + 1 2 f a t − 1 2 (10)

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

Skill scores (S) are calculated following Taylor [35]:

S s = 4 · ( 1 + r ) 4 ( σ ̂ f + 1 / σ ̂ f ) 2 · ( 1 + r 0 ) 4 (11)

where r is the Pearson correlation coefficient, r 0 =1 is the reference correlation coefficient, and σ ̂ 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.

Another important aspect is determining at which temperatures transmission is most efficient. If mosquitoes have a peak of infectiousness at, for example, 20°C in one model, temperatures above this will lead to a smaller fraction of mosquitoes becoming infectious. A different model might set this peak at 27°C, so that at temperatures from 20-27°C, the fraction of infectious mosquitoes will increase, followed by a decrease at higher temperatures. Isolating the point at which the mosquitoes are the most efficient vectors for malaria parasites is important for assessing the potential impacts of climate change. To show the differences between the models, we report the temperature where the maximum efficiency for producing infectious mosquitoes was observed. This can be done by maximizing equation 12.

arg max T ∈ [ 10 , 40 ] ∫ t = 0 ∞ Idt (12)