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:
[1]

L. AlpheyM. BenedictR. BelliniG. G. ClarkD. A. DameM. W. Service and S. L. Dobson, Steril-insect methods of mosquito-borne diseases: An analysis, Vector-Borne Zoonotic Dis., 10 (2010), 295-311. Google Scholar

[2]

R. M. Anderson and R. M. May, Infectious Diseases of Humans, Dynamics and Control, Oxford Univ. Press, Oxford, 1991.Google Scholar

[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. Google Scholar

[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.Google Scholar

[5]

A. Berman and R. J. Plemmons, Nonnegative Matrices in the Mathematical Sciences, Academic Press, New York, 1979. Google Scholar

[6]

L. CaiS. 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. Google Scholar

[7]

CDC, Malaria Facts, 2017, Available from: http://www.cdc.gov/malaria/about/facts.html.Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[11]

J. Li, Malaria models with partial immunity in humans, Math. Biol. Eng., 5 (2008), 789-801. doi: 10.3934/mbe.2008.5.789. Google Scholar

[12]

J. Li, Malaria model with stage-structured mosquitoes, Math. Biol. Eng., 8 (2011), 753-768. doi: 10.3934/mbe.2011.8.753. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[19]

R. C. A. ThomeH. 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. Google Scholar

[20]

WHO, Malaria, Fact Sheets, 2017, http://www.who.int/mediacentre/factsheets/fs094/en.Google Scholar

[21]

Wikipedia, Sterile Insect Technique, 2017, http://en.wikipedia.org/wiki/Sterile_insect_technique.Google Scholar

show all references

References:
[1]

L. AlpheyM. BenedictR. BelliniG. G. ClarkD. A. DameM. W. Service and S. L. Dobson, Steril-insect methods of mosquito-borne diseases: An analysis, Vector-Borne Zoonotic Dis., 10 (2010), 295-311. Google Scholar

[2]

R. M. Anderson and R. M. May, Infectious Diseases of Humans, Dynamics and Control, Oxford Univ. Press, Oxford, 1991.Google Scholar

[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. Google Scholar

[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.Google Scholar

[5]

A. Berman and R. J. Plemmons, Nonnegative Matrices in the Mathematical Sciences, Academic Press, New York, 1979. Google Scholar

[6]

L. CaiS. 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. Google Scholar

[7]

CDC, Malaria Facts, 2017, Available from: http://www.cdc.gov/malaria/about/facts.html.Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[11]

J. Li, Malaria models with partial immunity in humans, Math. Biol. Eng., 5 (2008), 789-801. doi: 10.3934/mbe.2008.5.789. Google Scholar

[12]

J. Li, Malaria model with stage-structured mosquitoes, Math. Biol. Eng., 8 (2011), 753-768. doi: 10.3934/mbe.2011.8.753. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[19]

R. C. A. ThomeH. 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. Google Scholar

[20]

WHO, Malaria, Fact Sheets, 2017, http://www.who.int/mediacentre/factsheets/fs094/en.Google Scholar

[21]

Wikipedia, Sterile Insect Technique, 2017, http://en.wikipedia.org/wiki/Sterile_insect_technique.Google Scholar

Figure 1.  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 $\bar b < b \le b_c$
Figure 2.  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 $\bar b < b < b_c$
Figure 3.  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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