internal pond calculation, original scheme

OPTION 1:

The net aggregated fractional coverage by temporary pools $w_{pond}$ is derived from a simple water balance model. Ponds are replenished by surface runoff $Q$, while infiltration $I$ (seepage) and evaporation $E$ and pond overflow reduce their water content. An important parameter is the maximum coverage of the temporary ponds $w_{max}$, which described the extent of depressions that could potentially become water filled at the peak of a wet season, which are often small in scale, with total catchments of the gully systems studies in Niger in the HAPEX-SAHEL experiment measuring 0.2 km$^{-2}$. Presently is assumed that the runoff $Q$ that fills the ponds mostly falls within these depressions and thus the run off is set to $Q=w_{max}P$. Thus sub-surface infiltration occurring within the depressions is also considered a water source for temporary pools [#!desconnets:97!#]. Future developments will introduce improved runoff treatment accounting for soil texture and slope.

The source from precipitation is balanced by evaporative, infiltration and overflow losses. In high resolution simulations of surface hydrology in Niger, [#!gianotti:09!#] found overflow losses to be approximately 20% of total losses, more than three times the losses due to evaporation. It should also be noted that overflow losses in field campaigns are difficult to measure and thus are often incorporated in the infiltration, which is calculated as a residual in the water balance calculation. Losses through pond overflow are assumed to increase linearly with pond fraction in VECTRI, achieved by scaling the runoff by a factor $1-\frac{w_{pond}}{w_{max}}$. Once the pond fraction reaches its maximum, all surface runoff overflows and is lost. Infiltration losses vary substantially depending on soil texture and life-scale of the pond in question. Often the infiltration is a highly nonlinear function of water body extent, since silting may significantly reduce infiltration in the lowest part of longer-lived or semi-permanent pools [#!desconnets:97!#]. This results in a fast initial decay after rain events due to high infiltration rates at the pool edges, followed by a slower decay, while temporary, shorter lived water bodies tend to have more uniform infiltration rates. These rates can be very high, exceeding 600 mm day$^{-1}$ [#!martin:03!#]. Presently the VECTRI model simply sets a fixed constant infiltration rate per unit pond area.

Combining these factors, the volume of water $v_{pond}$ in ponds per unit area thus evolves as

$$\frac{dv_{pond}}{dt}=(w_{max}P(1-\frac{w_{pond}}{w_{max}} - w_{pond}(E+I)).$$ (15)

Evaporation is set to 5 mm day$^{-1}$, equivalent to a latent heat flux of 145 Wm$^{-2}$. It is possible to derive evaporation losses from water temperature, and atmospheric wind speed and relative humidity, however, as evaporation is a relative minor loss term relative to infiltration and overflow, a simple fixed evaporation rate suffices. With $I$ set to a reasonable value of 245 mm day$^{-1}$, total losses from infiltration and evaporation are thus 250 mm day$^{-1}$. For closure, the pond fractional coverage needs to be related to the volume. Individual ponds have been modelled previously using a power law approximation [#!hayashi:00!#], which would lead to a relationship $\frac{dw}{dt} \propto w^{-p/2} \frac{dv}{dt}$, where $p$ is the power law exponent. However, [#!hayashi:00!#,#!brooks:02!#] show that $p$ can vary by almost an order of magnitude from one water body to another and depends in particular on the life-time. In the present version model, this factor is presently neglected and the coverage is simply linearly related to volume introducing a tunable factor $K_w$:

$$\frac{dw_{pond}}{dt}=K_w \left ( P(w_{max}-w_{pond}) - w_{pond}(E+I) \right ).$$ (16)

As the pond coverage can change rapidly, eqn. is integrated using a fully implicit solution. An example evolution of the fractional pond coverage for a site near Bobo Dioulasso is given in Fig. 4, which shows that the simple empirical approach mimics the pond evolution modelled by high resolution hydrological models for sites in Niger [#!bomblies:08!#] and ponds modelled in Senegal [#!soti:10!#]. The present empirical formulation is similar to the approaches of [#!alonso:11!#,#!eckhoff:11!#]. It is seen that at the fringes of the rainy season, puddles and small ponds have limited longevity on the order of a few days, implying that they are unsuitable for vector breeding. VECTRI represents the bulk behaviour of ponds, rather than the ultra-high resolution model of [#!gianotti:09!#] which individual models puddles at the 10 m scale and thus can model the lifetime of ponds as a function of their explicit size (see their figure 7). Obviously, the linear relation between rainfall and the growth of potential breeding sites in parameter $K_w$ is a simplification, while the other terms should be related to atmospheric conditions, soil type, vegetation coverage and terrain slope demonstrated to be important for malaria transmission in the Kenyan highlands [#!wanjala:10!#].

Presently, the two unknowns $w_{max}$ and $K_w$ in the framework are set using a simple Monte Carlo suite of station data integrations in a subset of locations to minimize $EIR$ errors compared to field data. An example sensitivity integration is shown in Fig. 5, which is a integration conducted for Bobo Dioulasso using station data to drive the model (see below for experimental set up details). It is seen that transmission intensity increases with $w_{max}$ and $K_{w}$ as expected, since these increase the pond coverage for a given rainrate, while increasing the loss rate acts in the opposite direction. For a given station there are a range of reasonable parameter values, with the present parameter settings chosen using a small number of locations in west Africa. Nevertheless, the physically based framework facilitates future improvement currently underway, which will include direct validation of the revised hydrological model constants using in situ and remotely sensed data.

Finally, it is recalled that the pond dimension limits larvae mortality rates through the availability of breeding sites governed in eqn. . This is an approximation of the net affect of crowding which leads to higher mortality rates, longer development times and smaller adults [#!gimnig:02!#], which in turn have a competative disadvantage [#!takken:98b!#]. The biomass is considered to be distributed equally through all available breeding sites and variability between breeding sites in neglected, supported by [#!munga:06!#] who noted that females avoid ponds that are overcrowded with existing larvae.

In addition to pond dimension the other important parameter of water bodies is the temperature of the water near the surface. The Depinay model [#!depinay:04!#] developed a complex empirical function for water temperature as a function of ambient relative humidity and water body size. As the VECTRI model is applied regionally, specific information about individual water body size can not be included. The temperature in shallow ponds and puddles is homogeneous to a good approximation and is often one or two degrees warmer than the air temperature [#!paaijmans:08b!#,#!paaijmans:08a!#]. VECTRI therefore assumes that the temperature of pools $T_{wat}$ to have a fixed offset relative to the air temperature. The default value adopted is a positive offset of 2 K, however, in hot locations it is likely that vector will preferentially choose shaded breeding locations and a lower or even negative offset may be more appropriate. If accurate gridded weather information for wind and surface radiation were available, this aspect of the model could be potential improved implementing a single energy balance model along the lines of [#!jacobs:08!#,#!paaijmans:08c!#]. While larger permanent water bodies such as lakes and rivers can have complex stratification of the vertical temperature profile, as discussed above, larvae development occurs mostly in the shallow waters and pools that form on the lake/river boundaries and thus the temperature relation for the permanent water fraction is treated in the same way as the temporary ponds.