Strategies for Introducing Wolbachia to Reduce Transmission of Mosquito-Borne Diseases

Certain strains of the endosymbiont Wolbachia have the potential to lower the vectorial capacity of mosquito populations and assist in controlling a number of mosquito-borne diseases. An important consideration when introducing Wolbachia-carrying mosquitoes into natural populations is the minimisation of any transient increase in disease risk or biting nuisance. This may be achieved by predominantly releasing male mosquitoes. To explore this, we use a sex-structured model of Wolbachia-mosquito interactions. We first show that Wolbachia spread can be initiated with very few infected females provided the infection frequency in males exceeds a threshold. We then consider realistic introduction scenarios involving the release of batches of infected mosquitoes, incorporating seasonal fluctuations in population size. For a range of assumptions about mosquito population dynamics we find that male-biased releases allow the infection to spread after the introduction of low numbers of females, many fewer than with equal sex-ratio releases. We extend the model to estimate the transmission rate of a mosquito-borne pathogen over the course of Wolbachia establishment. For a range of release strategies we demonstrate that male-biased release of Wolbachia-infected mosquitoes can cause substantial transmission reductions without transiently increasing disease risk. The results show the importance of including mosquito population dynamics in studying Wolbachia spread and that male-biased releases can be an effective and safe way of rapidly establishing the symbiont in mosquito populations.

The mosquito population is divided into classes of juveniles and adults. The juvenile population is divided into classes of eggs, larvae and pupae and the adult population is divided into classes of females and males. The dynamics of larvae and adults are modelled explicitly. Define L(t,l) to be the numbers of larvae at time t which have been in the larval stage for time l. Let the probability a larva survives until time l be  L (t,l) which because larval mortality may be density dependent is a function of time. The total time spent in juvenile development is the sum of the maximum durations of the egg, larval and pupal stages, T O , T L and T P . Egg and pupal mortality are assumed to be density-independent, and total egg and pupal survival are defined as  O and  P .
Define A(t,a) to be the total number of adults of age a at time t. The probability an adult survives until age a is defined as  A (a) which depends on age alone. Let () Lt and () At be the total number of larvae and adults at time t (obtained by integrating over all age classes) and define the female fecundity per unit of time, .
Mosquitoes uninfected and infected with Wolbachia are represented by subscripts U and W. At any one time a proportion p M (t) of male adults are infected by Wolbachia and a fraction s h of the offspring of any uninfected female that mates with an infected male will fail to develop. Mating is assumed to occur at random. Infected females fail to transmit Wolbachia to a fraction  of their offspring (which if fertilised by sperm from uninfected males fail to hatch with probability s h ). In accordance with empirical observations of Wolbachia infection in mosquito populations we will consider that Wolbachia may affect adult mortality, possibly in age-dependent manner, and female fecundity. However for simplicity we assume that Wolbachia infection does not affect the mortality or the development time of eggs, larvae and pupae.
We write separate systems of equations for the dynamics of male and female adults, representing males and females by subscripts M and F. The dynamics of adult females are described by , , (1 ) ( ) ( , ), The dynamics of adult males are given by where the numbers of infected and uninfected larvae are also described by eqns A1a & b. In modelling sex-biased introduction of Wolbachia-infected insects we assume that male and female Wolbachia-infected mosquitoes are introduced into an uninfected population at rates ()

Male infection frequency equilibria for constant male introduction
We solve for the stable and unstable male infection frequency equilibria, * M p , for the limiting case in which the rate of introduction of infected females is negligible, ( ) 0 F It  and the rate of infected male introduction is a constant I M . At the equilibrium, denoted by asterisks, systems (A1-A4) are written as where the average lifespans of adults infected and uninfected with Wolbachia are W  and U  respectively.
Substitute A1a in A1c to give of ε is then given by eqn (A10). Figure S1 shows the equilibrium *   Figure S2 shows how the form of density dependence affects the threshold introduction rate required for Wolbachia to spread for host populations which all have the same equilibrium adult abundance in the absence of Wolbachia. Higher rates of infected male introduction are required when density-dependence is strong. This is because the reduction in densitydependent mortality caused by the introduction of males is greater when density-dependence is stronger, and so less suppression of the (uninfected) adult population occurs. Figure S2. The minimum rate of introduction of infected males required for Wolbachia to spread as a function of the strength of juvenile density dependent mortality. Values of the larval carrying capacity  are chosen so that the equilibrium abundance of adults in the absence of Wolbachia is the same. Other parameters are as in Table 1.