Mathematical model of mosquito population dynamics
The dynamics of the wild type adult mosquito population, A, through time, t, are described by a time-delayed differential equation model (which captures the overlapping generations characteristic of Ae. aegypti), with a two-parameter, α and β, density-dependent function . The model, which assumes density-dependent mortality acts on pre-adultlife stages, takes the form
in which T is the mosquito generation time, δ is the per capita daily adult death rate, E is the maximum per capita daily egg production rate, and P is the maximum per capita daily egg production rate corrected for density-independent egg to adult survival. We assume a 1:1 sex ratio.
In the absence of any control, the maximum finite rate of increase (net fecundity after lifetime density-independent mortalities), λ, is
and the equilibrium adult population size, A*, is
In evaluating the performance of the conventional SIT (early-acting lethality) and RIDL (late-acting lethality) systems we assume the population is at pre-control stable equilibrium when SIT control is initiated (i.e. A
0 = A*. The input ratio, I, of released males is defined relative to the pre-control population equilibrium, A*, such that the number of "sterile" males remains constant through time.
Under both release scenarios, females are assumed to select mates proportionately to their relative abundance. Therefore, for the constant number release scenario the proportion of females that mate with wild type males at time t is A
With the early-acting lethal, we assume that the all offspring of sterile males die at early embryogenesis and so do not contribute to density-dependent mortality in pre-adult life stages. The dynamics of the mosquito population under conventional SIT control is given by
In contrast, for late acting lethality, we assume all offspring of "sterile" males survive through the pre-adult life stages and contribute fully to density-dependent mortality, before dying prior to adult emergence. The resulting population dynamics under RIDL control are described by
For any value of I > 0, the population will move away from A* towards a new, control-mediated equilibrium. If the release ratio exceeds a critical threshold, I*, the wild type population will be driven to extinction (i.e. control-mediated equilibrium = 0).
The relative performance of the conventional SIT and RIDL systems was examined by simulating equations4 and 5 over a range of values of I and for different combinations of parameters values for β (0.302 and 1) and P (0.367 and 1.31), representing the range estimated by Dye . For each combination of β and P, the critical release ratio for population eradication was estimated numerically for both early and late lethality approaches. The simulation was conducted using a one-day time step; at each time step the magnitude of A
was divided by A* so as to scale the wild type adult population relative to the pre-control equilibrium. This relative measure of adult population numbers is independent of the magnitude of α and E (for all non-zero values of α and E) and is equivalent to the relative adult female mosquito population (as the egg sex ratio is equal). A simulation approach was adopted because there are no analytical solutions for the critical release ratios, nor the equilibrium population sizes for values of I > 0.
To model the effect of incomplete penetrance of RIDL-induced mortality, the equation describing the dynamics of mosquito population under RIDL control (equation 5) was adjusted to account for the proportion of lethality, L, giving
To model the effect of reduced competitiveness of RIDL larvae, relative to wild type, equation 5 was adjusted to incorporate a competitiveness scaling factor, C, which can take values between 1 (RIDL larvae are fully competitive and contribute equally to density-dependent mortality) and 0 (RIDL larvae contribute nothing to density-dependent mortality giving
C can take values between 1 (RIDL larvae are fully competitive and contribute equally to density-dependent mortality) and 0 (RIDL larvae contribute nothing to density-dependent mortality).
Mosquito transformation and rearing
Aedes aegypti of the Rockefeller strain (obtained from Roger Wood, University of Manchester), were reared in an insectary maintained at 28°C and 75–80% relative humidity with 12-hour light/dark cycle. Mosquitoes were transformed by standard micro-injection methods , using a vector plasmid (e.g. pLA513) concentration of 500 ng/μl coinjected with a 400 ng/μl concentration of piggyBac 'helper plasmid' phsp-pBac  as the source of piggyBac transposase. After injection, eggs were heat shocked at 37°C for 2 hours, then stored for 48 hours at 100% humidity before they were submerged for hatching. Adult injection survivors ('Generation 0' or G0) were back-crossed to wild type: individual G0 males were crossed to 10–15 wild type females and G0 females were combined in pools of 10 with 3 wild type males. G1 eggs were collected and hatched as above (but without heat-shock). Hatched G1 larvae were screened for fluorescence using an Olympus SZX-12 microscope equipped for fluorescence (filters for red fluorescence: excitation 510–550, emission 590LP). Two independent transgenic lines, designated LA513A and LA513B, were recovered from about 200 fertile G0 back crosses, representing a transformation efficiency of approximately 1%. This is lower than published piggyBac-mediated transformation rates for Aedes aegypti of 8–11% [46, 47]. This decrease may reflect the larger size of the LA513 construct, some loss of some transgenics due to the deleterious effect of overexpression of tTAV, and/or variations in experimental technique or environment. DsRed2 fluorescence could be observed in the thorax of all developmental stages of LA513A mosquitoes. For the experiments in Table 1, larvae were reared at 200–250 larvae per liter, with 5–8 pellets of fish food (Omega) every 2 days. Tetracycline (Sigma) was added to the larval water to a final concentration of 30 μg/ml, as appropriate. Eggs were washed carefully after collection to minimize carry-over of tetracycline from one generation to the next. Males and females were separated as pupae to ensure female virginity for all experimental crosses.