American Institute of Mathematical Sciences

April  2018, 15(2): 523-541. doi: 10.3934/mbe.2018024

Complex wolbachia infection dynamics in mosquitoes with imperfect maternal transmission

 1 College of Mathematics and Information Sciences, Guangzhou University, Guangzhou, Guangdong 510006, China 2 Department of Parasitology of Zhongshan School of Medicine, Sun Yat-sen University, Guangzhou 510080, China 3 Key laboratory for Tropical Diseases Control, (SYSU) Ministry of Education, Guangzhou, Guangdong 510080, China

* Corresponding author: jsyu@gzhu.edu.cn

Received  December 27, 2016 Accepted  May 02, 2017 Published  June 2017

Dengue, malaria, and Zika are dangerous diseases primarily transmitted by Aedes aegypti, Aedes albopictus, and Anopheles stephensi. In the last few years, a new disease control method, besides pesticide spraying to kill mosquitoes, has been developed by releasing mosquitoes carrying bacterium Wolbachia into the natural areas to infect the wild population of mosquitoes and block disease transmission. The bacterium is transmitted by infected mothers and the maternal transmission was assumed to be perfect in virtually all previous models. However, recent experiments on Aedes aegypti and Anopheles stephensi showed that the transmission can be imperfect. In this work, we develop a model to describe how the imperfect maternal transmission affects the dynamics of Wolbachia spread. We establish two useful identities and employ them to find sufficient and necessary conditions under which the system exhibits monomorphic, bistable, and polymorphic dynamics. These analytical results may help find a plausible explanation for the recent observation that the Wolbachia strain wMelPop failed to establish in the natural populations in Australia and Vietnam.

Citation: Bo Zheng, Wenliang Guo, Linchao Hu, Mugen Huang, Jianshe Yu. Complex wolbachia infection dynamics in mosquitoes with imperfect maternal transmission. Mathematical Biosciences & Engineering, 2018, 15 (2) : 523-541. doi: 10.3934/mbe.2018024
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The phase portrait in the degenerate case. (A) When $\beta\mu\leq 2$, the curve $\Gamma_y$ stays in the domain $x+y>\beta\mu/2$ as a decreasing curve connecting $E_1$ and $E_2$. (B) When $\beta\mu>2$, the curve $\Gamma_y$ starts from $E_2$ in the domain $1/2<x+y<\beta\mu/2$, increases first and then intersects the line $x+y=\beta\mu/2$ to enter the domain $x+y>\beta\mu/2$, and remains in this domain as a decreasing curve until reaching $E_1$. In both cases, given the initial value $(x_0, y_0)\in \mathbb{R}_{+0}^2\setminus \Gamma_y$, solution of (4)-(5) tends to the unique intersecting point (the hollow point) of the curve (21) and the nullcline $\Gamma_y$
The separatrices $y=h(x)$ are sandwiched by $y=h_0(x)$ and $y=h_1(x)$
Dynamics complexity induced by imperfect maternal transmission. (A) When $\beta\leq\delta$, small $\mu$ corresponds to bistable case (C3). Increasing $\mu$ leads to (C9) and (C7), and the positive solutions converge to the monomorphic state $E_2$. (B) When $\beta>\delta$, we consider three subclasses. (B1): $\nu_1<\nu_2$. As $\mu$ increases, (C4), (C6), (C2), (C8) and (C7) occur consecutively, and the system transits from the monomorphic state $E_1$ to the polymorphic state $E^*$ and finally the monomorphic state $E_2$. (B2): When $\nu_1=\nu_2$, only (C1), (C2) and (C7) can occur. The system is polymorphic at $E^*$ when $\mu<\nu_1$, and is monomorphic at $E_2$ when $\mu>\nu_1$. (B3): $\nu_1>\nu_2$. As $\mu$ increases, (C4), (C5), (C3), (C9) and (C7) occur one after another, and the system transits from the monomorphic state $E_1$ to the bistable state and finally the monomorphic state $E_2$
The maternal transmission leakage hinders the Wolbachia invasion. (Left) Fix $y_0=0.02$, we plot the ratio of $x_0$ to $y_0$ against the leakage rate $\mu=0.01, 0.05, 0.1, 0.15, 0.2, 0.3, 0.4, 0.45$ and $0.5$ for the leakage case (4)-(5). (Right) The combination of negative effects of imperfect maternal transmission and incomplete CI could make Wolbachia invasion impossible. Again, fix $y_0=0.02$ and $s_h=0.8369$. When the leakage rate is 0.15, the incomplete CI mechanism leads the threshold ratio increase from 1.95 to 7. Worse than that, when the leakage rate is 0.3, the ratio as high as 200 is still insufficient to ensure successful Wolbachia invasion
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