Incorporating the effects of humidity in a mechanistic model of Anopheles gambiae mosquito population dynamics in the Sahel region of Africa

Background Low levels of relative humidity are known to decrease the lifespan of mosquitoes. However, most current models of malaria transmission do not account for the effects of relative humidity on mosquito survival. In the Sahel, where relative humidity drops to levels <20% for several months of the year, we expect relative humidity to play a significant role in shaping the seasonal profile of mosquito populations. Here, we present a new formulation for Anopheles gambiae sensu lato (s.l.) mosquito survival as a function of temperature and relative humidity and investigate the effect of humidity on simulated mosquito populations. Methods Using existing observations on relationships between temperature, relative humidity and mosquito longevity, we developed a new equation for mosquito survival as a function of temperature and relative humidity. We collected simultaneous field observations on temperature, wind, relative humidity, and anopheline mosquito populations for two villages from the Sahel region of Africa, which are presented in this paper. We apply this equation to the environmental data and conduct numerical simulations of mosquito populations using the Hydrology, Entomology and Malaria Transmission Simulator (HYDREMATS). Results Relative humidity drops to levels that are uncomfortable for mosquitoes at the end of the rainy season. In one village, Banizoumbou, water pools dried up and interrupted mosquito breeding shortly after the end of the rainy season. In this case, relative humidity had little effect on the mosquito population. However, in the other village, Zindarou, the relatively shallow water table led to water pools that persisted several months beyond the end of the rainy season. In this case, the decrease in mosquito survival due to relative humidity improved the model’s ability to reproduce the seasonal pattern of observed mosquito abundance. Conclusions We proposed a new equation to describe Anopheles gambiae s.l. mosquito survival as a function of temperature and relative humidity. We demonstrated that relative humidity can play a significant role in mosquito population and malaria transmission dynamics. Future modeling work should account for these effects of relative humidity.

where u and v are the flow velocities in the x and y directions, respectively, h is the water depth, P is precipitation, I is infiltration, and ET is evapotranspiration [1].
The momentum equations for the x and y directions are: (3) where g is the gravitational acceleration, and S fx and S fy are the friction slopes in the x and y directions, respectively. For the diffusion wave approximation, we neglect the first three terms which represent inertial effects. We make the replacement H = h + z for water level above a datum.  (6) where n is the Manning's roughness coefficient which determines resistance to overland flow. The y direction velocity is formulated similarly. Following Lal [11], we reformulate equation 6   Equations 1, 4, 5, 7 and 8 are then solved using the alternating-direction implicit (ADI) method, as described in Lal [11].
Topography at very high resolution is a critical parameter for overland flow simulation and prediction of pool formation. Topography determines the cell-to-cell bed slope, which is then used to determine intercell flow potentials [11]. The model uses a digital elevation model (DEM) which was derived from a combination of a ground topographic survey and Envisat synthetic aperture radar data [12]. In addition to topography, Manning's n in equation 6 strongly controls the timing and volume of hydrographs entering topographic depressions. This roughness parameter depends on the vegetation cover and soil type at the grid cell, and influences overland flow velocities [1].

LAND SURFACE MODEL
The model presented borrows heavily from the land surface scheme LSX of Pollard and Thomson [13].
The model simulates six soil layers and two vegetation layers for a detailed representation of hydrologic processes in the vertical column. LSX simulates momentum, energy, and water fluxes between the vegetation layers, soil, and the atmosphere. Vegetation type and soil type strongly influence soil moisture profile simulation, and spatially variable soil and vegetation properties are used to assign roughness in the runoff routing model.
Vertical soil layer thicknesses are assigned to allow simulation of a low-permeability structural crust commonly observed at the land surface in bare soil and sparsely vegetated areas of the Sahel [14].
Precipitation at each grid cell is partitioned between runoff and infiltration, based on hortonian runoff processes. The resulting infiltration flux is redistributed in the unsaturated zone with a Richard's equation solver, with soil hydraulic parameters assigned for each layer and grid cell. The Richards equation governs vertical water movement through the unsaturated zone, for which the model uses an implicit solver. The Richards equation is presented in equation 9: Groundwater representation is similar to the lumped aquifer model of Yeh and Eltahir [15]. The regional unconfined aquifer is represented using a lumped model in which groundwater table fluctuations are simulated. The depth to the water table varies from cell to cell and is a function of topography. This addition to the model allows areas of groundwater penetration of the surface and the resulting extended pool persistence to be predicted [1].

MODEL INPUTS
Necessary model inputs come from a variety of sources. The climate data for model forcing can come from meteorological stations in the field, and/or from regional climate model simulations. Meteorologic variable inputs for the hydrology model are temperature and humidity, wind speed and direction, incoming solar radiation, and precipitation. These six variables can be assumed spatially invariant over the model domain, or can be represented as distributed rasters, based on either multiple measurements or assumptions, to account for the existence of mosquito microhabitats.

MODEL INPUTS
Air temperature, water temperature, humidity, wind speed, wind direction, and distributed water depths are the primary inputs for the entomology model. Water depth and temperature for each grid cell are predicted by the hydrology model, and the remaining four can be either field measured or supplied by climate models [1].

AQUATIC STAGE SIMULATION
Aquatic stage, or subadult, mosquitoes advance through several stages between eggs and adult mosquitoes. Eggs hatch to become L1, or first stage larvae. They then advance through three more larval stages (instars) as they grow and mature, to finally pupate. Pupae do not feed. They remain in this state for approximately two days before emerging as adult mosquitoes [1].
Simulation of aquatic stage development relies on a compartmental structure model for each grid cell in which the hydrology model assigns a pool. As long as the pool persists in the simulation, the aquatic stage model will continue to advance. In pools predicted by the hydrology model to disappear, any simulated aquatic stages will be killed in the simulation, as is expected of naturally occurring An. gambiae larvae and pupae upon desiccation [10]. An aquatic-stage model structure is embedded within each model grid cell containing water. This model describes the water temperature-dependent stage progression rates of eggs, larvae, pupae, and emerging adults. Only integer abundances are advanced from a previous stage to the next. Progression from eggs to larvae to pupae to adults is calculated using Depinay's temperaturedependent model [16]: where d is the fraction of individuals in a certain stage progressed to the next stage, T k is the temperature (K) over time interval k,  t k is the time step at interval k, and r(T k ) is the temperature-dependent development rate, given by Depinay et al. [16] as a function of water temperature and biochemical parameters specific to each subadult stage. For details of the temperature dependence, the reader is referred to Depinay et al. [16]. For all stages, predation and natural mortality are model parameters [1].
Following Depinay et al. [16], we limit pool biomass using an intraspecific competition coefficient defined as: where w is the sum of total larval biomass in the pool grid cell, and e is the ecological carrying capacity [mg biomass m -2 ]. Ecological carrying capacity is an assigned model parameter and is assumed to be time-invariant [1].
Several other factors influence larvae. Pool water temperatures in excess of 40 degrees result in death of larvae [16]. In addition, we assume that oviposition does not occur in pools deeper than a threshold depth [17]. This is consistent with our own observations that wave action (which generally occurs in deeper, larger, unvegetated pools) seems to deter larvae, either by wave action drowning them or by waves discouraging oviposition. Also, deep water in the center of large pools appears to contain virtually no larvae. In the hydrology simulation, shrinking pools will regularly dry out grid cells at the pool edges as the receding water line causes a retreat of the pool boundaries. As soon as one pool grid cell is predicted to become dry, all subadult mosquitoes are simply moved into the adjacent cells, concentrating larvae and pupae into remaining pool cells [1].

ADULT STAGE SIMULATION
After emergence from the pools, adult mosquitoes are tracked through space and time using an individualbased approach, in contrast to the compartmental structure of the aquatic stage simulation. At each time step in the model, after the aquatic routine has been stepped to simulate newly emerging adult mosquitoes in the simulation, the mosquito matrix is updated. Elapsed times since significant events are updated, and X and Y position and behavior of the mosquito are updated based on radial random walk motion, corrected for wind displacement and including representation of CO 2 plumes as a host-seeking cue, as described below. We assume the following sequence of behavioral events: host-seeking, biting, resting, oviposition, and again host-seeking to repeat the cycle until the mosquito dies. Figure S2  The model incorporates a daily survivability based on daily average temperatures. Above a 42 ˚C daily average temperature threshold, anophelines cannot survive. [18,19]. The survivability dependence on daily average temperature follows the model developed by Martens [19]: (12) where p is the daily survivability probability of each mosquito and T d is the average temperature of the previous 24 hours [1].

EGG DEVELOPMENT AND EXTRINSIC INCUBATION PERIOD
Egg development within the mosquito follows the temperature-dependent model of Depinay et al. [16] as shown in equation 14. For details of the temperature-dependent development rate of eggs within the mosquito, the reader is referred to Depinay et al [16]. Ambient temperature at the mosquito's location regulates this development rate. If the mosquito has finished the full gonotrophic (egg development) cycle, and it encounters a suitable water body, then it deposits a clutch of eggs to add to the subadult mosquitoes of various stages already present in that water body [1].
Once an adult mosquito takes an infectious bloodmeal and becomes infected, the parasite advancement beyond the midgut and into the salivary glands as infectious sporozoites requires 111 degree-days above 18˚ C [20]. Because each individual mosquito tracks degree-days since infection, infective status depends on a simple comparison of this value to 111. If the mosquito reaches this point, it becomes capable of malaria transmission to humans during subsequent blood meals. Once it has become infectious with sporozoites in the salivary glands, all subsequent bites from the mosquito are capable of transmitting malaria [1].

HUMAN AGENTS
The model simulates immobile human agents, representing village inhabitants. Human agents are assumed immobile. While villagers are obviously in reality quite mobile, this may be a reasonable assumption because the mosquitoes actively seek blood-meal hosts at night [21] when villagers are sleeping in their houses. Each model grid cell marked as inhabited (digitized houses from satellite image) contains a finite number of human agents. Unless house survey data allows actual assignment of inhabitant numbers to each house, we assume a constant ten inhabitants per 10 m x 10 m village grid cell.
When a mosquito enters a house to seek a bloodmeal, a certain portion of the inhabitants can be assumed protected by bednets, and if that bednet-protected host is targeted by the mosquito for a bloodmeal it will result in the mosquito's death. If desired, repellent effects can also be included through a simple modification [1].