# 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 and Continuous Dynamical Systems - B, 2018, 23 (9) : 3837-3853. doi: 10.3934/dcdsb.2018113
##### References:
 [1] L. Alphey, M. Benedict, R. Bellini, G. G. Clark, D. A. Dame, M. W. Service and S. L. Dobson, Steril-insect methods of mosquito-borne diseases: An analysis, Vector-Borne Zoonotic Dis., 10 (2010), 295-311. [2] R. M. Anderson and R. M. May, Infectious Diseases of Humans, Dynamics and Control, Oxford Univ. Press, Oxford, 1991. [3] H. J. Barclay, Mathematical models for the use of sterile insects, in Sterile Insect Technique. Principles and Practice in Area-Wide Integrated Pest Management, (V. A. Dyck, J. Hendrichs, and A. S. Robinson, Eds.), Springer, Heidelberg, (2005), 147-174. doi: 10.1007/1-4020-4051-2_6. [4] A. C. Bartlett and R. T. Staten, The steril insect release method and other genetic control strategies, in Radcliffe's IPM world Textbook, 1996, Available from: https://ipmworld.umn.edu/bartlett. [5] A. Berman and R. J. Plemmons, Nonnegative Matrices in the Mathematical Sciences, Academic Press, New York, 1979. [6] L. Cai, S. Ai and J. Li, Dynamics of mosquitoes populations with different strategies for models for releasing sterile mosquitoes, SIAM J. Appl. Math., 74 (2014), 1786-1809.  doi: 10.1137/13094102X. [7] CDC, Malaria Facts, 2017, Available from: http://www.cdc.gov/malaria/about/facts.html. [8] Y. Dumont and J. M. Tchuenche, Mathematical studies on the sterile insect technique for the Chikungunya disease and Aedes albopictus, J. Math. Biol., 65 (2012), 809-854.  doi: 10.1007/s00285-011-0477-6. [9] L. Esteva and H. M. Yang, Mathematical model to assess the control of Aedes aegypti mosquitoes by the sterile insect technique, Math. Biosci., 198 (2005), 132-147.  doi: 10.1016/j.mbs.2005.06.004. [10] J. M. Hyman and J. Li, The Reproductive aumber for an HIV model with differential infectivity and staged progression, Linear Algebra Appl., 398 (2005), 101-116.  doi: 10.1016/j.laa.2004.07.017. [11] J. Li, Malaria models with partial immunity in humans, Math. Biol. Eng., 5 (2008), 789-801.  doi: 10.3934/mbe.2008.5.789. [12] J. Li, Malaria model with stage-structured mosquitoes, Math. Biol. Eng., 8 (2011), 753-768.  doi: 10.3934/mbe.2011.8.753. [13] J. Li, Modeling of transgenic mosquitoes and impact on malaria transmission, J. Biol. Dynam., 5 (2011), 474-494.  doi: 10.1080/17513758.2010.523122. [14] J. Li and Z. Yuan, Modeling releases of sterile mosquitoes with different strategies, J. Biol. Dynam., 9 (2015), 1-14.  doi: 10.1080/17513758.2014.977971. [15] J. Li, New revised simple models for interactive wild and sterile mosquito populations and their dynamics, J. Biol. Dynam., 11 (2017), 316-333.  doi: 10.1080/17513758.2016.1216613. [16] G. A. Ngwa, Modelling the dynamics of endemic malaria in growing populations, Discrete Contin. Dyn. Syst., Ser. B, 4 (2004), 1173-1202.  doi: 10.3934/dcdsb.2004.4.1173. [17] G. A. Ngwa, On the population dynamics of the malaria vector, Bull. Math. Biol., 68 (2006), 2161-2189.  doi: 10.1007/s11538-006-9104-x. [18] G. A. Ngwa and W. S. Shu, A mathematical model for endemic malaria with variable human and mosquito populations, Math. Comp. Modelling, 32 (2000), 747-763.  doi: 10.1016/S0895-7177(00)00169-2. [19] R. C. A. Thome, H. M. Yang and L. Esteva, Optimal control of Aedes aegypti mosquitoes by the sterile insect technique and insecticide, Math. Biosci., 223 (2010), 12-23.  doi: 10.1016/j.mbs.2009.08.009. [20] WHO, Malaria, Fact Sheets, 2017, http://www.who.int/mediacentre/factsheets/fs094/en. [21] Wikipedia, Sterile Insect Technique, 2017, http://en.wikipedia.org/wiki/Sterile_insect_technique.

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##### References:
 [1] L. Alphey, M. Benedict, R. Bellini, G. G. Clark, D. A. Dame, M. W. Service and S. L. Dobson, Steril-insect methods of mosquito-borne diseases: An analysis, Vector-Borne Zoonotic Dis., 10 (2010), 295-311. [2] R. M. Anderson and R. M. May, Infectious Diseases of Humans, Dynamics and Control, Oxford Univ. Press, Oxford, 1991. [3] H. J. Barclay, Mathematical models for the use of sterile insects, in Sterile Insect Technique. Principles and Practice in Area-Wide Integrated Pest Management, (V. A. Dyck, J. Hendrichs, and A. S. Robinson, Eds.), Springer, Heidelberg, (2005), 147-174. doi: 10.1007/1-4020-4051-2_6. [4] A. C. Bartlett and R. T. Staten, The steril insect release method and other genetic control strategies, in Radcliffe's IPM world Textbook, 1996, Available from: https://ipmworld.umn.edu/bartlett. [5] A. Berman and R. J. Plemmons, Nonnegative Matrices in the Mathematical Sciences, Academic Press, New York, 1979. [6] L. Cai, S. Ai and J. Li, Dynamics of mosquitoes populations with different strategies for models for releasing sterile mosquitoes, SIAM J. Appl. Math., 74 (2014), 1786-1809.  doi: 10.1137/13094102X. [7] CDC, Malaria Facts, 2017, Available from: http://www.cdc.gov/malaria/about/facts.html. [8] Y. Dumont and J. M. Tchuenche, Mathematical studies on the sterile insect technique for the Chikungunya disease and Aedes albopictus, J. Math. Biol., 65 (2012), 809-854.  doi: 10.1007/s00285-011-0477-6. [9] L. Esteva and H. M. Yang, Mathematical model to assess the control of Aedes aegypti mosquitoes by the sterile insect technique, Math. Biosci., 198 (2005), 132-147.  doi: 10.1016/j.mbs.2005.06.004. [10] J. M. Hyman and J. Li, The Reproductive aumber for an HIV model with differential infectivity and staged progression, Linear Algebra Appl., 398 (2005), 101-116.  doi: 10.1016/j.laa.2004.07.017. [11] J. Li, Malaria models with partial immunity in humans, Math. Biol. Eng., 5 (2008), 789-801.  doi: 10.3934/mbe.2008.5.789. [12] J. Li, Malaria model with stage-structured mosquitoes, Math. Biol. Eng., 8 (2011), 753-768.  doi: 10.3934/mbe.2011.8.753. [13] J. Li, Modeling of transgenic mosquitoes and impact on malaria transmission, J. Biol. Dynam., 5 (2011), 474-494.  doi: 10.1080/17513758.2010.523122. [14] J. Li and Z. Yuan, Modeling releases of sterile mosquitoes with different strategies, J. Biol. Dynam., 9 (2015), 1-14.  doi: 10.1080/17513758.2014.977971. [15] J. Li, New revised simple models for interactive wild and sterile mosquito populations and their dynamics, J. Biol. Dynam., 11 (2017), 316-333.  doi: 10.1080/17513758.2016.1216613. [16] G. A. Ngwa, Modelling the dynamics of endemic malaria in growing populations, Discrete Contin. Dyn. Syst., Ser. B, 4 (2004), 1173-1202.  doi: 10.3934/dcdsb.2004.4.1173. [17] G. A. Ngwa, On the population dynamics of the malaria vector, Bull. Math. Biol., 68 (2006), 2161-2189.  doi: 10.1007/s11538-006-9104-x. [18] G. A. Ngwa and W. S. Shu, A mathematical model for endemic malaria with variable human and mosquito populations, Math. Comp. Modelling, 32 (2000), 747-763.  doi: 10.1016/S0895-7177(00)00169-2. [19] R. C. A. Thome, H. M. Yang and L. Esteva, Optimal control of Aedes aegypti mosquitoes by the sterile insect technique and insecticide, Math. Biosci., 223 (2010), 12-23.  doi: 10.1016/j.mbs.2009.08.009. [20] WHO, Malaria, Fact Sheets, 2017, http://www.who.int/mediacentre/factsheets/fs094/en. [21] Wikipedia, Sterile Insect Technique, 2017, http://en.wikipedia.org/wiki/Sterile_insect_technique.
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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