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 mosquito-borne illnesses. Diseases such as chikungunya, dengue fever, malaria, and Zika virus can be life-threatening, 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.
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 self-sustaining Wolbachia infection to a wild mosquito population. Their two-sex 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 mosquito-borne 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 single-sex 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 multi-stage system of nine ODEs that integrate both genders, pregnant and non-pregnant 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 two-sex model differentiates between non-pregnant (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.”
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 (egg-laying rate) that accompanies an outbreak and limits its permanence. “There are three coexisting equilibria for the proposed model: a stable zero-infection equilibrium, an unstable intermediate-infection endemic, and a stable high-infection 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 zero-infection 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 real-life mitigation strategies for employment prior to the release of Wolbachia-infected 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 pre-release 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 pre-release 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.”
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 Wolbachia-infected 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 multi-season 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 mosquito-borne diseases.”
Acknowledgments: This research was partially supported by the NSF-MPS/NIH-NIGMS award NSF-1563531.
Source article: Qu, Z., Xue, L., & Hyman, J.M. (2018). Modeling the Transmission of Wolbachia in Mosquitoes for Controlling Mosquito-Borne Diseases. SIAM Journal on Applied Mathematics, 78(2), 826-852.
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 vector-borne diseases. Mac Hyman is a past-president of SIAM and an applied mathematician at Tulane University specializing in modeling the spread of infectious diseases.