Sustained Bacterial Outbreak in Mosquitoes Limits Spread of Lifethreatening Diseases
By Lina Sorg
Medical practitioners, such as those at the Centers for Disease Control and Prevention, frequently utilize mathematical models when determining how to best control the spread of mosquitoborne illnesses. Diseases such as chikungunya, dengue fever, malaria, and Zika virus can be lifethreatening, and no effective vaccine currently exists. While most mitigation strategies aim to eliminate popular mosquito breeding sites through the use of insecticides, the accompanying costs, logistical difficulties, and resistance evolution make these treatment methods unsustainable.
Figure 1. Female Aedes aegypti mosquito feeding.
Wolbachia pipientis is a maternallytransmitted bacteria that occurs naturally in over 60 percent of insect species. Certain strains of
Wolbachia inhibit the transmission of diseaseinducing pathogens to humans; this feature gives the microbe potential medical value, and scientists have been studying its effect on mosquitoes for years. Unfortunately, it is not naturally found in
Aedes aegypti mosquitoes, the primary transmitters of mosquitoborne illnesses. If researchers wish to use
Wolbachia to control the spread of these diseases, they must continually reintroduce it to wild mosquito populations. Such repeated introduction is strategically unfeasible.
In an article publishing today in the SIAM Journal on Applied Mathematics, Zhuolin Qu, Ling Xue, and James Mac Hyman use an ordinary differential equation (ODE)based model to calculate the most effective method of introducing a selfsustaining Wolbachia infection to a wild mosquito population. Their twosex model accounts for the aquatic life stage, heterosexual transmission, and multiple pregnant states for female mosquitoes, thus capturing the entire transmission cycle. “If a stable Wolbachia infection can be established in wild mosquitoes, this will reduce the spread of mosquitoborne diseases,” Qu said.
Past researchers have used a number of models to study the presence of Wolbachia in wild A. aegypti mosquito populations, though many did not distinguish between mosquitoes’ varied life stages and the fitness costs that accompany Wolbachia. “Most previous models assumed that there is a fixed male/female ratio, and use this assumption to reduce the equations to a singlesex model,” Qu said. “This is a good approximation for isolated wild mosquito populations, but is violated when releasing only infected male mosquitoes into a wild population. Our ODE model for Wolbachia transmission does not assume that there is a fixed ratio of males to females.”
Qu et al. employ a multistage system of nine ODEs that integrate both genders, pregnant and nonpregnant females, and all life stages. They group the adult mosquito population into seven compartments based on infection state, pregnancy status, and fertility. Because female mosquitoes typically mate only once during their lifetime and store the sperm for multiple egg clutches, a successful twosex model differentiates between nonpregnant (unmated) and pregnant (mated) females.
“The Wolbachia infection is transmitted vertically from an infected female mosquito to her offspring,” Qu said. “A female mosquito usually mates successfully once, and if an uninfected female mosquito mates with an infected male mosquito, very few of her offspring survive. Because our compartmental model includes the heterosexual interaction of mosquitoes and multiple pregnant stages for females, it can be used to analyze the threshold condition required to sustain endemic Wolbachia for both perfect and imperfect maternal transmissions.”
Figure 2. Maternal transmission of Wolbachia in the mosquito population.
To reach the threshold at which enough mosquitoes are contaminated for a Wolbachia infection to persist, a model must surmount the shortened lifespan and decreased fecundity (egglaying rate) that accompanies an outbreak and limits its permanence. “There are three coexisting equilibria for the proposed model: a stable zeroinfection equilibrium, an unstable intermediateinfection endemic, and a stable highinfection endemic equilibrium (or complete infection for perfect maternal transmission),” Qu said. “These three equilibria are characterized by a backward bifurcation, with the unstable equilibrium points being the threshold condition for endemic Wolbachia. If the fraction of infected mosquitoes is below this threshold, then the population returns to a zeroinfection state. If the fraction is above the threshold, then the infection spreads and eventually almost all of the mosquitoes are infected with Wolbachia.”
After characterizing the threshold condition, the authors simulate and compare reallife mitigation strategies for employment prior to the release of Wolbachiainfected mosquitoes. “Because the threshold condition is characterized by a minimal fraction of mosquitoes that are infected, we could reduce the number of infected mosquitoes that must be released by first reducing the population of uninfected mosquitoes,” Qu said. Tested strategies include indoor residual spraying, since A. aegypti tend to breed in or near houses; larval control, which targets water storage and container removal where mosquitoes often breed; sticky gravid traps/ovitraps, which attract and kill uninfected pregnant females as they prepare to lay eggs; and acoustic attraction, which reduces the number of uninfected males.
“Our simulations indicate that the prerelease mitigations that target pregnant females, such as residual spraying and sticky gravid traps, are more helpful than ones that target only males or the aquatic stage, given that prerelease mitigation stops once the release starts,” Qu said. “Removing uninfected pregnant females greatly slows reproduction of the uninfected offspring, and the gap can be filled up mostly with the infected population.”
Figure 3. Schematic of Wolbachia's complex maternal transmission mating
Ultimately, monitoring both fitness cost and maternal transmission rate are key to establishing and sustaining a Wolbachia outbreak among wild mosquitoes. If researchers introduce enough infected mosquitoes to a natural population to surpass the threshold, the population levels out at a stable endemic Wolbachiainfected equilibrium. As a result, infected mosquitoes will be less likely to spread infectious diseases like dengue, chikungunya, and Zika to the humans they encounter.
Qu et al. recognize that they must further hone their model before their findings can truly guide policy efforts. For example, they assume that all parameters are constant for the sake of simplicity, but temperature, humidity, and other seasonal factors realistically vary. Incorporating seasonality in the model would offer a more accurate projection of multiseason success. “We are currently extending this model to include both spatial heterogeneity and temporal variations using partial differential equations that incorporate the diffusion of mosquitoes and seasonal variations,” Qu said. “Hopefully it could offer more practical insights to help guide mitigation efforts for mosquitoborne diseases.”
Acknowledgments: This research was partially supported by the NSFMPS/NIHNIGMS award NSF1563531.
Source article: Qu, Z., Xue, L., & Hyman, J.M. (2018). Modeling the Transmission of Wolbachia in Mosquitoes for Controlling MosquitoBorne Diseases. SIAM Journal on Applied Mathematics, 78(2), 826852.
About the authors: Zhuolin Qu is an applied mathematician researching the modeling of infectious diseases to help public health researchers better understand the spread of disease and optimize mitigation resources. She received her Ph.D. in 2016 and is now a postdoctoral fellow in the Department of Mathematics at Tulane University. Ling Xue is interested in modeling mitigation strategies for infectious diseases, especially vectorborne diseases. Mac Hyman is a pastpresident of SIAM and an applied mathematician at Tulane University specializing in modeling the spread of infectious diseases.

Lina Sorg is the associate editor of SIAM News. 