A Reduce and Replace Strategy for Suppressing Vector-Borne Diseases: Insights from a Deterministic Model

Genetic approaches for controlling disease vectors have aimed either to reduce wild-type populations or to replace wild-type populations with insects that cannot transmit pathogens. Here, we propose a Reduce and Replace (R&R) strategy in which released insects have both female-killing and anti-pathogen genes. We develop a mathematical model to numerically explore release strategies involving an R&R strain of the dengue vector Aedes aegypti. We show that repeated R&R releases may lead to a temporary decrease in mosquito population density and, in the absence of fitness costs associated with the anti-pathogen gene, a long-term decrease in competent vector population density. We find that R&R releases more rapidly reduce the transient and long-term competent vector densities than female-killing releases alone. We show that releases including R&R females lead to greater reduction in competent vector density than male-only releases. The magnitude of reduction in total and competent vectors depends upon the release ratio, release duration, and whether females are included in releases. Even when the anti-pathogen allele has a fitness cost, R&R releases lead to greater reduction in competent vectors than female-killing releases during the release period; however, continued releases are needed to maintain low density of competent vectors long-term. We discuss the results of the model as motivation for more detailed studies of R&R strategies.

In the main text, we describe the development of a system of ordinary differential equations that we use to simulate population dynamics and population genetics of Aedes aegypti following the introduction of an R&R strain into a wild-type population. Analysis of this model is limited by the model complexity; however, we are able to obtain equilibrium population density of the wild-type population in the absence of transgenic releases.
In a completely wild-type population, the systeṁ for i = 1...9, where i = 9 represents the wild-type genotype, reduces to J 9 = λF 9 − µ J J 9 − α β−1 J β 9 − νJ 9 F 9 = 1 2 νJ 9 − µ F F 9 M 9 = 1 2 νJ 9 − µ M M 9 . (2) Here,Ṁ 9 is decoupled from the system, so we can analyze the reduced systeṁ (3) This system has a trivial equilibrium at (J (1) 9 ) = (0, 0), and one non-trivial equilibrium at 9 . (4) We rearrange the expression for J (2) 9 by noting that where Here, 1 µ f is the average lifespan of adult females, λ 2 is the rate of production of female offspring, and ν µ J +ν is the fraction of juveniles that survive to emerge as adults. Thus, R 0 is the basic reproductive number of the population. We rewrite (4) in terms of R 0 .
In order for population to have a positive equilibrium (i.e., J 9 > 0), R 0 > 1. Thus, we analyze the stability of the equilibrium only for the case when R 0 > 1.
To verify the stability of the equilibrium in (7), we first find the Jacobian of system (3).
We then evaluate the Jacobian at the equilibrium in (7).
We now study the eigenvalues of J by studying the determinant and trace of J . The equilibrium point (J (2) 9 , F (2) 9 ) is stable when Tr(J ) < 0 and det(J ) > 0 (i.e., both eigenvalues of J must be negative). First, we calculate Tr(J ).
Rearranging the terms, we get In order for det(J ) > 0, So we have that the equilibrium (J * 9 , F * 9 ) is stable when β > 1.

Equilibrium Values for Model Runs
Here, we list the values of the equilibrium density of juveniles, adult males, and adult females that are used for model runs in the main text. Note that the release size of R&R individuals is always defined as a function of the equilibrium wild-type male population density so that release rates are always relative to the population density. This allows for a general study of R&R releases in an Ae. aegypti population. While changes in α will result in changes in the density of the population, the qualitative results for relative density are the same.