Modeling the role of environmental variables on the population dynamics of the malaria vector Anopheles gambiae sensu stricto
© Parham et al.; licensee BioMed Central Ltd. 2012
Received: 4 April 2012
Accepted: 31 July 2012
Published: 9 August 2012
The impact of weather and climate on malaria transmission has attracted considerable attention in recent years, yet uncertainties around future disease trends under climate change remain. Mathematical models provide powerful tools for addressing such questions and understanding the implications for interventions and eradication strategies, but these require realistic modeling of the vector population dynamics and its response to environmental variables.
Published and unpublished field and experimental data are used to develop new formulations for modeling the relationships between key aspects of vector ecology and environmental variables. These relationships are integrated within a validated deterministic model of Anopheles gambiae s.s. population dynamics to provide a valuable tool for understanding vector response to biotic and abiotic variables.
A novel, parsimonious framework for assessing the effects of rainfall, cloudiness, wind speed, desiccation, temperature, relative humidity and density-dependence on vector abundance is developed, allowing ease of construction, analysis, and integration into malaria transmission models. Model validation shows good agreement with longitudinal vector abundance data from Tanzania, suggesting that recent malaria reductions in certain areas of Africa could be due to changing environmental conditions affecting vector populations.
Mathematical models provide a powerful, explanatory means of understanding the role of environmental variables on mosquito populations and hence for predicting future malaria transmission under global change. The framework developed provides a valuable advance in this respect, but also highlights key research gaps that need to be resolved if we are to better understand future malaria risk in vulnerable communities.
Among the potential effects of climate change on human health, the impact on infectious diseases has attracted increasing attention in recent years. Vector-borne diseases (VBDs) are likely to be particularly vulnerable given the poikilothermic nature of vector survival and development, as well as the effects of temperature on pathogen development. Although the link between climatic variables and transmission has attracted interest for VBDs such as dengue and schistosomiasis, the combined global mortality of these diseases is less than 7% of that due to malaria, and this, combined with the significant effects of climatic variables on multiple stages of the transmission cycle, has led to malaria remaining an important focus of ongoing debate regarding climate change and VBDs[3, 4].
In the context of better understanding the role of weather and climate on transmission, two modeling approaches are possible. Statistical models use empirical relationships between climatic variables and past (or current) disease incidence (or prevalence) to predict future disease trends[5, 6]. Mechanistic models, on the other hand, adopt a process-based approach, incorporating known biological, epidemiological and entomological relationships affecting vector and pathogen vital rates and formulating mathematically how these combine[7–9]. Both types of model have important roles to play in improving our understanding of climate-driven transmission changes, but the focus here is on exploiting the explanatory power of the latter.
A vital component in developing reliable VBD transmission models is establishing a realistic model of the vector population dynamics, yet only a few studies have explicitly modeled and parameterized the impact of climatic drivers on vector vital rates[8, 10–12]. While these studies have greatly improved our understanding of the relative importance of temperature, rainfall and relative humidity (RH) on vector populations, they also highlight the need to develop a comprehensive mathematical framework for analysing how a range of environmental factors, arising at different spatial scales, combine at the level of breeding sites to affect stage-specific vector abundance in malaria-affected regions.
This work aims to provide such a framework by formulating and parameterizing environment-vector relationships through surveying and modeling relevant experimental and field data, and incorporating these relationships within a low-dimensional, deterministic mathematical framework. Model simplicity permits ease of integration into malaria transmission models and the model is calibrated and validated against longitudinal Anopheles gambiae abundance data from Tanzania. The model also highlights where further experimental and modeling work is required to improve parameterization, in addition to developing a framework readily generalized to different Anopheles species and other disease vectors.
where and M is the projection matrix, the high-dimensional nature of M increases by an order of magnitude as temperature measurements become more precise. The dependence of development on other factors (such as RH for adults) also increases the complexity of M, as well as making an implicit assumption about the linearity of development with temperature that is often violated. Thus, a low-dimensional approach is instead adopted here, providing a simple, structurally-parsimonious, deterministic model that more transparently illustrates the basic structure that may be built upon in future model development, is considerably easier to construct, analyse and interpret, and may be readily appended to malaria transmission models.
where F 4 is the average number of eggs laid per day per female adult, P i is the proportion of vectors surviving and remaining in stage i in t to t + 1, and G i the proportion surviving and progressing from stage i in t to t + 1. To calculate P i and G i , the expressions from are used, namely and(for all values of i), where 0 ≤ p i ≤ 1 is given by (2) and d i > 1 the average duration spent in stage i. To parameterize the model, the literature is reviewed to source relevant data, as well as using previously unpublished data, to develop, where appropriate, functional forms for F 4 , d i and the components of p i in (2). The resultant population model (3) is then calibrated and validated against vector abundance data from.
Results and discussion
Modelling breeding site hydrodynamics
Here, Δ is the slope of the vapour pressure curve (kPa°C-1) (which depends on T A ), R n the daily net radiation transferred to the breeding site (MJm-2 day-1) (which, for a given location and day number, depends on the daily cloud fraction CF (through its relationship with the number of sunshine hours per day), dew-point temperature T DP (°C), minimum daily temperature T min (°C) and maximum daily temperature T max (°C)), G the soil heat flux (MJm-2 day-1), γ the psychrometric constant (kPa°C-1) (constant for a given site), U 2 the wind speed at 2 m (ms-1), e s the saturation vapour pressure (kPa) (dependent on T min and T max ), and e a the actual vapour pressure (kPa) (dependent on T DP ). The climatic variables R t , T A , T DP and CF are readily available from the ECMWF ERA-40 re-analysis dataset, while U 2 may be approximated from U 10 (the wind speed at 10 m, available from ERA-40) using the conversion U 2 = 0.748U 10 . The outgoing heat conduction between the water body and surrounding soil G is typically negligible compared to R n  and, as in, is neglected here.
Key model variables, parameters, and climatic variables
n i (t)
The number of An. gambiae s.s. in stage i on day t (where i = 1, 2, 3, and 4 corresponds to eggs, larvae, pupae, and adults respectively)
The daily survival probability of stage i
The average duration spent in stage i (days)
The volume of the breeding site on day t (ml)
Evaporation from the breeding site on day t (mm)
The number of consecutive days without water in the breeding site (days)
Daily mean air temperature (°C)
Daily mean water temperature in the breeding site (°C)
Total rainfall on day t (mm)
Dew-point temperature (°C)
Relative humidity (%) (can be calculated from knowledge of T A and T DP )
Minimum daily temperature (°C)
Maximum daily temperature (°C)
Wind speed at 2 m (ms-1)
Environmental influences on immature development
The prolonged absence of water also affects immature longevity; anopheline egg survival in desiccating conditions is two to three weeks, while An. gambiae s.l. eggs are viable for up to 12 days without water. To model the decrease in egg viability in dry habitats, the findings of are used, which demonstrate that the duration of exposure to desiccating conditions is a better measure of egg viability than soil moisture content. If p i (D) is the daily survival probability of stage i given D days without water, the functional form p i (D) = 2exp(−ω i D)/(1 + exp(−ω i D)) (i = 1, 2, 3) is fitted, where ω i quantifies the sensitivity of stage i to desiccation and the functional form ensures that survival is near unity when D is small and approaches zero as desiccation increases. Least-squares estimation using field populations under medium-moisture conditions gives ω i = 0.405days−1 (R2 > 0.99). Survival of larvae and pupae may be similarly parameterized using, which demonstrates that L4 larvae survive significantly better than L1, L2 and L3 instars in such conditions – weighting by the average duration in each instar stage gives ω2 = 0.855days−1 (R2 = 0.97). In the absence of data on pupal survival, pupae are assumed to demonstrate a similar response to L4 larvae, whereupon using gives ω2 = 0.602days−1 (R2 = 0.94) (Figure2b).
Average duration d i ( T W ) of immature stage i at water temperature T W (from Bayoh and Lindsay (unpublished data))
d 1 (T W )
d 2 (T W )
− d1(T W )
d 3 (T W )
Of the juvenile stages, larval survival demonstrates the strongest dependence on temperature and the effect of competition between An. gambiae s.s. and Anopheles arabiensis on temperature-dependent survival has been examined. The relationship between survival, development and water temperature, and age-dependent mortality, for An. gambiae s.s. is considered in. Larval duration is parameterized as a function of T W in, but this is An. gambiae s.l., rather than An. gambiae s.s. Moreover, this parameterization is based only on temperatures between 23.0 and 32.8°C and extrapolating to temperature extremes gives inconsistent results with experimental findings in (such as development times around 30 days at 18°C in the former compared to 15 days in the latter). While provides a literature survey of larval development times as a function of T W , eight of the twelve data points for An. gambiae s.s. are calculated from on the assumption of eggs and pupae developing within one day, which is inconsistent with experimental data in the latter. The revised coefficients from Bayoh and Lindsay (unpublished data) are therefore used to determine d 2 (T W ).
Aside from the work of on the effects of temperatures from 21.2 to 29.5°C on An. gambiae s.l. pupal mortality and, there is little experimental data to parameterize pupal development and survival. The latter, with the corrected values in Bayoh and Lindsay (unpublished data), are therefore used to parameterize d 3 (T W ).
Finally, it is important to note the importance of using water temperature to calculate juvenile survival and development, rather than air temperature. The difference between mean daily water and air temperatures is typically around 3-6°C depending on factors such as breeding site dimensions, microclimate and weather conditions[32, 37]. To account for this, it is assumed thatwhere is assumed to capture all thermodynamic processes taking place at breeding sites leading to a difference between mean water and air temperatures. Lower and upper temperature thresholds for juveniles are taken from.
Predation and density-dependence
for i = 2 and 3. If n 2 (t) + n 3 (t) = 0 or V t = 0, p i (DD) is assumed to be unity. Predation on eggs is assumed to be negligible by comparison and anopheline rice-field survival data from, and is used to provide seven independent datasets to fit b at ΔT = 3°C, 4°C, 5°C and 6°C. For each dataset, air temperature and rainfall data from the nearest meteorological station (using and where missing values are interpolated) are used to calculate the daily survival and development of larvae and pupae due to climatic influences (assuming fixed vector density and assuming no desiccation effects for rice fields) and estimate the additional mortality required to agree with the study data (attributed to p i (DD)). Two approaches are adopted, namely to (a) calculate the number of juveniles remaining after a fixed number of days (determined by the study design), and (b) track the number of cohort larvae and pupae until less than 0.05% of the original population remain. For method (a), where experimental dates are not specified, b is calculated for a range of plausible start dates and the average computed. No significant difference in calculating b using these methods is found and for and 0.88 for, 5°C and 6°C.
Environmental influences on adult development
despite its basis on fitting a three-parameter function to three data points in the range 9-40°C (with the 40°C point inconsistent with). This relationship assumes no adverse effects of RH on mortality, which is unlikely given that RH < 50% leads to significantly reduced survival. Field observations of An. gambiae adults are only approximately consistent with (18), but reflect the relatively high survival at 22-30°C.
Given the absence of age-structure in this model, each gonotrophic cycle is assumed to be of equal duration for all adults and produce the same number of eggs, although studies have shown variation in both.
No direct influences of rainfall on adult survival are assumed (with indirect effects through changes in RH captured by (19)) and adult survival is assumed to be density-independent following and the weak, but statistically significant, relationship between adult density and survivorship in. There is some evidence of predation on adult An. gambiae s.l. at oviposition sites, with the severity potentially depending on the type of site, but there are few quantitative studies in this respect.
Model calibration and validation
To assess performance, the model is calibrated and validated against longitudinal An. gambiae s.l. abundance data from collected in an environment free of vector controls. Data on T A , T DP (for calculation of RH), (low) cloud fraction CF, and the horizontal and vertical components of 10 m wind speed (to calculate U 2 ) are taken from the ERA-40 re-analysis dataset for the rural community in Masaika, Tanzania (5 16' 0'' S, 38 49' 60'' E) (with the nearest ERA-40 point at 5o, 0’ 0” S, 37o 30’ 0” E). Rainfall data from the Maji Depot Tanga Rainfall station (at 5 4' 58'' S, 39 5' 21'' E), approximately 35 km from Masaika, is used when available (see), with missing data taken from. Since the daily values of T min and T max are not available from, we derive empirical relationships between T A and these variables using data from the nearest meteorological station (Tanga at 5o 4’ 48” S, 39o 4’ 12” E), approximately 34 km from the study site, and apply these relationships (T min = 0.724T A + 14.4, with R 2 = 0.53, and T max = 0.728T A + 28.3, with R 2 = 0.61) to ERA-40 data on T A to estimate the associated values of T min and T max .
For model calibration, the average number of adult An. gambiae s.l. per light trap is fitted to model output after the burn-in period. To account for the difference in scale between data and the model, the scaled fecundity and adult An. gambiae s.s. abundanceare defined and just three parameters fitted over the calibration period – the scale parameters α1 and α2, and ΔT. All other parameters are derived from parameterizations in this paper and local breeding site properties (altitude and latitude). It is assumed that (based on model calibration in) and breeding site dimensions consistent with the characteristics of typical An. gambiae s.s. habitats (in the presence of multiple An. gambiae s.l. species given the collection of multiple Anopheles species in data collection in) reported in, namely A T = 1.79 x 106mm2 and h 0 = 97mm. An initial water volume of 1 litre is assumed (V 0 = 1000ml). Model fit to data is found to be independent of the initial conditions, so 100 mosquitoes are arbitrarily initially assumed to be in each lifecycle stage.
Along with An. arabiensis and Anopheles funestus, An. gambiae s.s. is one of the principal malaria vectors in Africa and understanding its ecology and dynamics is vital in better understanding the associated impact on malaria transmission and the prospects for eradication, as well as the effectiveness of vector controls in different communities and settings. Vector population dynamics are driven by a range of biotic and abiotic factors and clarifying the role of both is key, particularly in the context of how climate change may influence the future spread and distribution of VBDs. Here, a useful framework for understanding how changes in rainfall, temperature, RH, wind speed and cloudiness (both mean values and temporal variability), and density-dependence, at breeding sites may influence vector abundance is presented. By calibrating and validating the model against longitudinal abundance data, this framework is shown to be capable of reproducing the observations in on long-term timescales, suggesting a mechanistic underpinning of mosquito dynamics in terms of environmental variables, an important result given the ongoing debate regarding the link between malaria transmission and climatic changes in Africa[3, 4]. This work also highlights the power of mathematical models in addressing key questions surrounding the role of environmental variables, compared to the multitude of other ecological, epidemiological, socioeconomic and demographic factors, on disease transmission. An important advance of this work is the construction of a modeling framework enabling the linkage of climatic events at large spatial scales to processes at the localized scale of vector breeding sites, enabling assessments of how climatic phenomena at different scales may affect disease transmission in host communities.
Model reliability may be enhanced with improved parameterization and future experimental and modeling research will lead to further understanding of species-specific Anopheles population dynamics and their response to environmental variables. These include (i) improving our understanding of Anopheles oviposition behaviour, (ii) better quantifying the role of rainfall and temperature on egg, larval and pupal survival, as well as the role of heterogeneities, such as body size, that might influence response, (iii) improved modeling of the relationship between air and water temperatures at breeding sites, (iv) improving our understanding of density-dependent effects on juvenile and adult development and survival (including intra-specific competition, inter-specific interactions between species, cannibalistic tendencies, and predation, as well as their dependence on climatic variables), (v) assessing evidence for age-dependent mortality in juveniles and adults, and (vi) better understanding variability in gonotrophic cycles.
New longitudinal vector studies that simultaneously measure changes in environmental variables are also required to improve the validity and reliability of vector models, which will not only further our understanding of dominant factors driving mosquito dynamics, but will also improve our understanding of the implications for VBD transmission. Nonetheless, the approach here not only provides a useful framework for An. gambiae s.s. modeling, but its structure may be readily applied to other Anopheles species with suitable parameterization, as well as other vectors (such as Aedes or Culex). This will ultimately enable a better understanding of the response of a variety of VBDs to environmental change, an important question given the likely influences of weather and climate on many regions of VBD risk over the coming decades.
The authors would like to thank Henri Tsila for providing unpublished data that formed the basis of Figure4. PEP and EM would like to thank the Grantham Institute of Climate Change at Imperial College London for funding this research. EM also acknowledges the Eck Institute for Global Health at the University of Notre Dame for part funding this work. None of the funding bodies mentioned contributed to the design, collection, analysis or interpretation of data, nor the writing of the manuscript or decision to submit. The authors would like to thank the two anonymous reviewers whose comments greatly improved this manuscript.
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