# American Institute of Mathematical Sciences

November  2018, 23(9): 3837-3853. doi: 10.3934/dcdsb.2018113

## The impact of releasing sterile mosquitoes on malaria transmission

 1 School of Mathematics and Statistics, Central China Normal University, Wuhan 430079, China 2 School of Mathematics and Statistics, South-Central University for Nationalities, Wuhan 430074, China 3 Department of Mathematical Science, University of Alabama in Huntsville, Huntsville AL 35899, USA

* Corresponding author: Cuihong Yang

Received  March 2017 Revised  October 2017 Published  November 2018 Early access  April 2018

The sterile mosquitoes technique in which sterile mosquitoes are released to reduce or eradicate the wild mosquito population has been used in preventing the malaria transmission. To study the impact of releasing sterile mosquitoes on the malaria transmission, we first formulate a simple SEIR (susceptible-exposed-infected-recovered) malaria transmission model as our baseline model, derive a formula for the reproductive number of infection, and determine the existence of endemic equilibria. We then include sterile mosquitoes in the baseline model and consider the case of constant releases of sterile mosquitoes. We examine how the releases affect the reproductive numbers and endemic equilibria for the model with interactive mosquitoes and investigate the impact of releasing sterile mosquitoes on the malaria transmission.

Citation: Hongyan Yin, Cuihong Yang, Xin'an Zhang, Jia Li. The impact of releasing sterile mosquitoes on malaria transmission. Discrete & Continuous Dynamical Systems - B, 2018, 23 (9) : 3837-3853. doi: 10.3934/dcdsb.2018113
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##### References:
With the parameters given in (25), the threshold values are $\bar b = 5.3960$ and $b_c = 8.0033$. By using $b$ as an independent variable, the horizontal axis is for $b$ and the vertical axis is for $R_0^c$. The curve in this figure represents the reproductive number $R_0^c(b)$ for $0\le b \le b_c$. The reproductive number $R_0^c(0) = R_0 = 1.1284 >1$ at $b = 0$. At $b = \bar b$, the curve for $R_0^c(b)$ crosses the horizontal line $R_0^c = 1$ so that $R_0^c(b) < 1$ for $\bar b < b \le b_c$
With the parameters given in (25), the threshold values are $\bar b = 5.3960$ and $b_c = 8.0033$, respectively. The curve on the left figure is for $\lambda_h(b)$ at the endemic equilibrium for each $b$. The upper and lower curves are for $I_h(b)$ and $I_v(b)$, respectively, at the endemic equilibrium for each $b$ as well in the right figure. Clearly, $\lambda_h(b)$, $I_h(b)$, and $I_v(b)$ all become negative for $b > \bar b$ which implies that no endemic equilibrium exists for $b \ge \bar b$ although positive $N_{vb}^\pm(b)$ exist for $\bar b < b < b_c$
With the parameters given in (25), the reproductive number for system (1) and (6) is $R_{0} = 1.1284>1$ and hence the infection spreads when there are no sterile mosquitoes released as shown in the left figure. After the sterile mosquitoes are introduced, for $b = 6>\bar b = 5.3960$, the reproduction number becomes $R_{0}^{c} = 0.9773 < 1$ and hence the infection goes extinct as shown in the right figure.
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