# 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  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
##### References:

show all references

##### 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.
 [1] Timothy C. Reluga, Jan Medlock, Alison Galvani. The discounted reproductive number for epidemiology. Mathematical Biosciences & Engineering, 2009, 6 (2) : 377-393. doi: 10.3934/mbe.2009.6.377 [2] G.A. Ngwa. Modelling the dynamics of endemic malaria in growing populations. Discrete & Continuous Dynamical Systems - B, 2004, 4 (4) : 1173-1202. doi: 10.3934/dcdsb.2004.4.1173 [3] Jia Li. Malaria model with stage-structured mosquitoes. Mathematical Biosciences & Engineering, 2011, 8 (3) : 753-768. doi: 10.3934/mbe.2011.8.753 [4] Liming Cai, Jicai Huang, Xinyu Song, Yuyue Zhang. Bifurcation analysis of a mosquito population model for proportional releasing sterile mosquitoes. Discrete & Continuous Dynamical Systems - B, 2019, 24 (11) : 6279-6295. doi: 10.3934/dcdsb.2019139 [5] Liming Cai, Shangbing Ai, Guihong Fan. Dynamics of delayed mosquitoes populations models with two different strategies of releasing sterile mosquitoes. Mathematical Biosciences & Engineering, 2018, 15 (5) : 1181-1202. doi: 10.3934/mbe.2018054 [6] Ariel Cintrón-Arias, Carlos Castillo-Chávez, Luís M. A. Bettencourt, Alun L. Lloyd, H. T. Banks. The estimation of the effective reproductive number from disease outbreak data. Mathematical Biosciences & Engineering, 2009, 6 (2) : 261-282. doi: 10.3934/mbe.2009.6.261 [7] Jia Li. Modeling of mosquitoes with dominant or recessive Transgenes and Allee effects. Mathematical Biosciences & Engineering, 2010, 7 (1) : 99-121. doi: 10.3934/mbe.2010.7.99 [8] Expeditho Mtisi, Herieth Rwezaura, Jean Michel Tchuenche. A mathematical analysis of malaria and tuberculosis co-dynamics. Discrete & Continuous Dynamical Systems - B, 2009, 12 (4) : 827-864. doi: 10.3934/dcdsb.2009.12.827 [9] Cruz Vargas-De-León. Global analysis of a delayed vector-bias model for malaria transmission with incubation period in mosquitoes. Mathematical Biosciences & Engineering, 2012, 9 (1) : 165-174. doi: 10.3934/mbe.2012.9.165 [10] Kamaldeen Okuneye, Ahmed Abdelrazec, Abba B. Gumel. Mathematical analysis of a weather-driven model for the population ecology of mosquitoes. Mathematical Biosciences & Engineering, 2018, 15 (1) : 57-93. doi: 10.3934/mbe.2018003 [11] Avner Friedman, Wenrui Hao. Mathematical modeling of liver fibrosis. Mathematical Biosciences & Engineering, 2017, 14 (1) : 143-164. doi: 10.3934/mbe.2017010 [12] Zindoga Mukandavire, Abba B. Gumel, Winston Garira, Jean Michel Tchuenche. Mathematical analysis of a model for HIV-malaria co-infection. Mathematical Biosciences & Engineering, 2009, 6 (2) : 333-362. doi: 10.3934/mbe.2009.6.333 [13] Alexander Bobylev, Åsa Windfäll. Kinetic modeling of economic games with large number of participants. Kinetic & Related Models, 2011, 4 (1) : 169-185. doi: 10.3934/krm.2011.4.169 [14] Gang Bao. Mathematical modeling of nonlinear diffracvtive optics. Conference Publications, 1998, 1998 (Special) : 89-99. doi: 10.3934/proc.1998.1998.89 [15] Baba Issa Camara, Houda Mokrani, Evans K. Afenya. Mathematical modeling of glioma therapy using oncolytic viruses. Mathematical Biosciences & Engineering, 2013, 10 (3) : 565-578. doi: 10.3934/mbe.2013.10.565 [16] Karly Jacobsen, Jillian Stupiansky, Sergei S. Pilyugin. Mathematical modeling of citrus groves infected by huanglongbing. Mathematical Biosciences & Engineering, 2013, 10 (3) : 705-728. doi: 10.3934/mbe.2013.10.705 [17] Bashar Ibrahim. Mathematical analysis and modeling of DNA segregation mechanisms. Mathematical Biosciences & Engineering, 2018, 15 (2) : 429-440. doi: 10.3934/mbe.2018019 [18] Natalia L. Komarova. Mathematical modeling of cyclic treatments of chronic myeloid leukemia. Mathematical Biosciences & Engineering, 2011, 8 (2) : 289-306. doi: 10.3934/mbe.2011.8.289 [19] Jeong-Mi Yoon, Volodymyr Hrynkiv, Lisa Morano, Anh Tuan Nguyen, Sara Wilder, Forrest Mitchell. Mathematical modeling of Glassy-winged sharpshooter population. Mathematical Biosciences & Engineering, 2014, 11 (3) : 667-677. doi: 10.3934/mbe.2014.11.667 [20] Evans K. Afenya. Using Mathematical Modeling as a Resource in Clinical Trials. Mathematical Biosciences & Engineering, 2005, 2 (3) : 421-436. doi: 10.3934/mbe.2005.2.421

2018 Impact Factor: 1.008