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October 2018, 15(5): 1181-1202. doi: 10.3934/mbe.2018054

Dynamics of delayed mosquitoes populations models with two different strategies of releasing sterile mosquitoes

 1 School of Mathematics and Statistics, Xinyang Normal University, Xinyang 46400, China 2 Department of Mathematical Sciences, University of Alabama in Huntsville, Huntsville, AL 35899, USA 3 Department of Mathematics, Columbus State University, Columbus, Georgia 31907, USA

* Corresponding author: limingcai@amss.ac.cn

Received  October 06, 2017 Revised  April 14, 2018 Published  May 2018

Fund Project: This research was supported partially the National Nature Science Foundation of China grant 11371305 and Nanhu Scholars Program for Young Scholars XYNU

To prevent the transmissions of mosquito-borne diseases (e.g., malaria, dengue fever), recent works have considered the problem of using the sterile insect technique to reduce or eradicate the wild mosquito population. It is important to consider how reproductive advantage of the wild mosquito population offsets the success of population replacement. In this work, we explore the interactive dynamics of the wild and sterile mosquitoes by incorporating the delay in terms of the growth stage of the wild mosquitoes. We analyze (both analytically and numerically) the role of time delay in two different ways of releasing sterile mosquitoes. Our results demonstrate that in the case of constant release rate, the delay does not affect the dynamics of the system and every solution of the system approaches to an equilibrium point; while in the case of the release rate proportional to the wild mosquito populations, the delay has a large effect on the dynamics of the system, namely, for some parameter ranges, when the delay is small, every solution of the system approaches to an equilibrium point; but as the delay increases, the solutions of the system exhibit oscillatory behavior via Hopf bifurcations. Numerical examples and bifurcation diagrams are also given to demonstrate rich dynamical features of the model in the latter release case.

Citation: 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
References:
 [1] R. Abdul-Ghani, H. F. Farag, A. F. Allam and A. A. Azazy, Measuring resistant-genotype transmission of malaria parasites: challenges and prospects, Parasitol Res., 113 (2014), 1481-1487. doi: 10.1007/s00436-014-3789-9. [2] P. L. Alonso, G. Brown, M. Arevalo-Herrera, et al, A research agenda to underpin malaria eradication, PLoS Med., 8 (2011), e1000406. doi: 10.1371/journal.pmed.1000406. [3] L. Alphey, M. Benedict, R. Bellini, G. G. Clark, D. A. Dame, M. W. Service and S. L. Dobson, Sterile-insect methods for control of mosquito-borne diseases: An analysis, Vector Borne Zoonotic Dis., 10 (2010), 295-311. doi: 10.1089/vbz.2009.0014. [4] J. Arino, L. Wang and G. S. Wolkowicz, An alternative formulation for a delayed logistic equation, J. Theor. Biol., 241 (2006), 109-119. doi: 10.1016/j.jtbi.2005.11.007. [5] M. Q. Benedict and A. S. Robinson, The first releases of transgenic mosquitoes: An argument for the sterile insect technique, Trends Parasitol, 19 (2003), 349-355. doi: 10.1016/S1471-4922(03)00144-2. [6] E. Beretta and Y. Kuang, Geometric stability switch criteria in delay differential systems with delay dependent parameters, SIAM J. Math. Anal., 33 (2002), 1144-1165. doi: 10.1137/S0036141000376086. [7] J. G. Breman, The ears of the hippopotamus: Manifestations, determinants, and estimates of the malaria burden, Am. J. Trop. Med. Hyg., 64 (2001), 1-11. doi: 10.4269/ajtmh.2001.64.1. [8] W. G. Brogdon and J. C. McAllister, Insecticide resistance and vector control, J. Agromedicine, 6 (1999), 41-58. [9] L. Cai, S. Ai and J. Li, Dynamics of mosquitoes populations with different strategies for releasing sterile mosquitoes, SIAM, J. Appl. Math., 74 (2014), 1786-1809. doi: 10.1137/13094102X. [10] K. Cooke, P. van den Driessche and X. Zou, Interaction of maturation delay and nonlinear birth in population and epidemic models, J. Math. Biol., 39 (1999), 332-352. doi: 10.1007/s002850050194. [11] H. Diaz, A. A. Ramirez, A. Olarte and C. Clavijo, A model for the control of malaria using genetically modified vectors, J. Theor. Biol., 276 (2011), 57-66. doi: 10.1016/j.jtbi.2011.01.053. [12] J. Dieudonné, Foundations of Modern Analysis, Academic Press, New York, 1960. [13] 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. [14] V. A. Dyck, J. Hendrichs and A. S. Robinson, Sterile insect technique -principles and practice in area-wide integrated pest management, Springer, The Netherlands, 2005. [15] C. Dye, Models for the population dynamics of the yellow fever mosquito, Aedes aegypti, J. Anim. Ecol., 53 (1984), 247-268. doi: 10.2307/4355. [16] L. Esteva and H. M. Yang, Assessing the effects of temperature and dengue virus load on dengue transmission, J. Biol. Syst., 23 (2015), 527-554. doi: 10.1142/S0218339015500278. [17] 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. [18] J. E. Gentile, S. Rund and G. R Madey, Modelling sterile insect technique to control the population of Anopheles gambiae, Malaria J., 14 (2015), 92-103. doi: 10.1186/s12936-015-0587-5. [19] J. K. Hale and S. M. Verduyn Lunel, Introduction to Functional Differential Equation, Springer, New York, 1993. doi: 10.1007/978-1-4612-4342-7. [20] J. Ito, A. Ghosh, L. A. Moreira, E. A. Wilmmer and M. Jacobs-Lorena, Transgenic anopheline mosquitoes impaired in transmission of a malria parasite, Nature, 417 (2002), 452-455. [21] M. Jankovic and S. Petrovskii, Are time delays always destabilizing? Revisiting the role of time delays and the Allee effect, Theor Ecol., 7 (2014), 335-349. doi: 10.1007/s12080-014-0222-z. [22] E. F. Knipling, Possibilities of insect control or eradication through the use of sexually sterile males, J. Econ. Entomol., 48 (1955), 459-462. [23] Y. Kuang, Delay Differential Equations with Applications in Population Dynamics, Academic Press, New York, 1993. [24] S. S. Lee, R. E. Baker, E. A. Gaffney and S. M. White, Modelling Aedes aegypti mosquito control via transgenic and sterile insect techniques: Endemics and emerging outbreaks, J. Theor. Biol., 331 (2013), 78-90. doi: 10.1016/j.jtbi.2013.04.014. [25] M. A. Lewis and P. van den Driessche, Waves of extinction from sterile insect release, Math. Biosci., 116 (1993), 221-247. doi: 10.1016/0025-5564(93)90067-K. [26] J. Li, New revised simple models for interactive wild and sterile mosquito populations and their dynamics, J. Biol. Dyna., 11 (2017), 316-333. doi: 10.1080/17513758.2016.1216613. [27] J. Li, L. Cai and Y. Li, Stage-structured wild and sterile mosquito population models and their dynamics, J. Biol.Dyna., 11 (2017), 79-101. doi: 10.1080/17513758.2016.1159740. [28] J. Lu and J. Li, Dynamics of stage-structured discrete mosquito population, J. Appl. Anal. Comput., 1 (2011), 53-67. [29] G. J. Lycett and F. C. Kafatos, Anti-malaria mosquitoes?, Nautre, 417 (2002), 387-388. [30] C. W. Morin and A. C. Comrie, Regional and seasonal response of a West Nile virus vector to climate change, PNAS, 110 (2013), 15620-15625. doi: 10.1073/pnas.1307135110. [31] W. W. Murdoch, C. J. Briggs and R. M. Nisbet, Consumer-resource dynamics, Princeton University Press, New Jersey, USA, 2003. [32] H. K. Phuc, M. H. Andreasen, et al, Late-acting dominant lethal genetic systems and mosquito control, BMC. Biol., 5 (2007), 11–16. doi: 10.1186/1741-7007-5-11. [33] E. P. Pliego, J. Vel$\acute{a}$zquez-Castro and A. F. Collar, Seasonality on the life cycle of Aedes aegypti mosquito and its statistical relation with dengue outbreaks, Appl. Math. Model., 50 (2017), 484-496. doi: 10.1016/j.apm.2017.06.003. [34] M. Rafikov, L. Bevilacqua and A. P. Wyse, Optimal control strategy of malaria vector using genetically modified mosquitoes, J. Theor. Biol., 258 (2009), 418-425. doi: 10.1016/j.jtbi.2008.08.006. [35] S. J. Schreiber, Allee effect, extinctions, and chaotic transients in simple population models, Theor. Popul. Biol., 64 (2003), 201-209. doi: 10.1016/S0040-5809(03)00072-8. [36] J. Smith, M. Amador and R. Barrera, Seasonal and habitat effects on dengue and West Nile Virus Vectors in San Juan, Puerto Rico, J. Am. Mosq. Control. Assoc., 25 (2009), 38-46. doi: 10.2987/08-5782.1. [37] H. Townson, SIT for African malaria vectors: Epilogue, Malar. J., 8 (2009), S10. doi: 10.1186/1475-2875-8-S2-S10. [38] WHO, 10 facts on malaria, http://www.who.int/features/factfiles/malaria/en/. [39] J. Wu, H. R. Thieme, Y. Lou and G. Fan, Stability and persistence in ODE models for populations with many stages, Math. Biosc. Eng., 12 (2015), 661-686. doi: 10.3934/mbe.2015.12.661. [40] B. Zheng, M. Tang and J. Yu, Modeling Wolbachia spread in mosquitoes through delay differential equations, SIAM J. Appl. Math., 74 (2014), 743-770. doi: 10.1137/13093354X. [41] B. Zheng, M. Tang, J. Yu and J. Qiu, Wolbachia spreading dynamics in mosquitoes with imperfect maternal transmission, J. Math. Biol., 76 (2018), 235-263. doi: 10.1007/s00285-017-1142-5.

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References:
 [1] R. Abdul-Ghani, H. F. Farag, A. F. Allam and A. A. Azazy, Measuring resistant-genotype transmission of malaria parasites: challenges and prospects, Parasitol Res., 113 (2014), 1481-1487. doi: 10.1007/s00436-014-3789-9. [2] P. L. Alonso, G. Brown, M. Arevalo-Herrera, et al, A research agenda to underpin malaria eradication, PLoS Med., 8 (2011), e1000406. doi: 10.1371/journal.pmed.1000406. [3] L. Alphey, M. Benedict, R. Bellini, G. G. Clark, D. A. Dame, M. W. Service and S. L. Dobson, Sterile-insect methods for control of mosquito-borne diseases: An analysis, Vector Borne Zoonotic Dis., 10 (2010), 295-311. doi: 10.1089/vbz.2009.0014. [4] J. Arino, L. Wang and G. S. Wolkowicz, An alternative formulation for a delayed logistic equation, J. Theor. Biol., 241 (2006), 109-119. doi: 10.1016/j.jtbi.2005.11.007. [5] M. Q. Benedict and A. S. Robinson, The first releases of transgenic mosquitoes: An argument for the sterile insect technique, Trends Parasitol, 19 (2003), 349-355. doi: 10.1016/S1471-4922(03)00144-2. [6] E. Beretta and Y. Kuang, Geometric stability switch criteria in delay differential systems with delay dependent parameters, SIAM J. Math. Anal., 33 (2002), 1144-1165. doi: 10.1137/S0036141000376086. [7] J. G. Breman, The ears of the hippopotamus: Manifestations, determinants, and estimates of the malaria burden, Am. J. Trop. Med. Hyg., 64 (2001), 1-11. doi: 10.4269/ajtmh.2001.64.1. [8] W. G. Brogdon and J. C. McAllister, Insecticide resistance and vector control, J. Agromedicine, 6 (1999), 41-58. [9] L. Cai, S. Ai and J. Li, Dynamics of mosquitoes populations with different strategies for releasing sterile mosquitoes, SIAM, J. Appl. Math., 74 (2014), 1786-1809. doi: 10.1137/13094102X. [10] K. Cooke, P. van den Driessche and X. Zou, Interaction of maturation delay and nonlinear birth in population and epidemic models, J. Math. Biol., 39 (1999), 332-352. doi: 10.1007/s002850050194. [11] H. Diaz, A. A. Ramirez, A. Olarte and C. Clavijo, A model for the control of malaria using genetically modified vectors, J. Theor. Biol., 276 (2011), 57-66. doi: 10.1016/j.jtbi.2011.01.053. [12] J. Dieudonné, Foundations of Modern Analysis, Academic Press, New York, 1960. [13] 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. [14] V. A. Dyck, J. Hendrichs and A. S. Robinson, Sterile insect technique -principles and practice in area-wide integrated pest management, Springer, The Netherlands, 2005. [15] C. Dye, Models for the population dynamics of the yellow fever mosquito, Aedes aegypti, J. Anim. Ecol., 53 (1984), 247-268. doi: 10.2307/4355. [16] L. Esteva and H. M. Yang, Assessing the effects of temperature and dengue virus load on dengue transmission, J. Biol. Syst., 23 (2015), 527-554. doi: 10.1142/S0218339015500278. [17] 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. [18] J. E. Gentile, S. Rund and G. R Madey, Modelling sterile insect technique to control the population of Anopheles gambiae, Malaria J., 14 (2015), 92-103. doi: 10.1186/s12936-015-0587-5. [19] J. K. Hale and S. M. Verduyn Lunel, Introduction to Functional Differential Equation, Springer, New York, 1993. doi: 10.1007/978-1-4612-4342-7. [20] J. Ito, A. Ghosh, L. A. Moreira, E. A. Wilmmer and M. Jacobs-Lorena, Transgenic anopheline mosquitoes impaired in transmission of a malria parasite, Nature, 417 (2002), 452-455. [21] M. Jankovic and S. Petrovskii, Are time delays always destabilizing? Revisiting the role of time delays and the Allee effect, Theor Ecol., 7 (2014), 335-349. doi: 10.1007/s12080-014-0222-z. [22] E. F. Knipling, Possibilities of insect control or eradication through the use of sexually sterile males, J. Econ. Entomol., 48 (1955), 459-462. [23] Y. Kuang, Delay Differential Equations with Applications in Population Dynamics, Academic Press, New York, 1993. [24] S. S. Lee, R. E. Baker, E. A. Gaffney and S. M. White, Modelling Aedes aegypti mosquito control via transgenic and sterile insect techniques: Endemics and emerging outbreaks, J. Theor. Biol., 331 (2013), 78-90. doi: 10.1016/j.jtbi.2013.04.014. [25] M. A. Lewis and P. van den Driessche, Waves of extinction from sterile insect release, Math. Biosci., 116 (1993), 221-247. doi: 10.1016/0025-5564(93)90067-K. [26] J. Li, New revised simple models for interactive wild and sterile mosquito populations and their dynamics, J. Biol. Dyna., 11 (2017), 316-333. doi: 10.1080/17513758.2016.1216613. [27] J. Li, L. Cai and Y. Li, Stage-structured wild and sterile mosquito population models and their dynamics, J. Biol.Dyna., 11 (2017), 79-101. doi: 10.1080/17513758.2016.1159740. [28] J. Lu and J. Li, Dynamics of stage-structured discrete mosquito population, J. Appl. Anal. Comput., 1 (2011), 53-67. [29] G. J. Lycett and F. C. Kafatos, Anti-malaria mosquitoes?, Nautre, 417 (2002), 387-388. [30] C. W. Morin and A. C. Comrie, Regional and seasonal response of a West Nile virus vector to climate change, PNAS, 110 (2013), 15620-15625. doi: 10.1073/pnas.1307135110. [31] W. W. Murdoch, C. J. Briggs and R. M. Nisbet, Consumer-resource dynamics, Princeton University Press, New Jersey, USA, 2003. [32] H. K. Phuc, M. H. Andreasen, et al, Late-acting dominant lethal genetic systems and mosquito control, BMC. Biol., 5 (2007), 11–16. doi: 10.1186/1741-7007-5-11. [33] E. P. Pliego, J. Vel$\acute{a}$zquez-Castro and A. F. Collar, Seasonality on the life cycle of Aedes aegypti mosquito and its statistical relation with dengue outbreaks, Appl. Math. Model., 50 (2017), 484-496. doi: 10.1016/j.apm.2017.06.003. [34] M. Rafikov, L. Bevilacqua and A. P. Wyse, Optimal control strategy of malaria vector using genetically modified mosquitoes, J. Theor. Biol., 258 (2009), 418-425. doi: 10.1016/j.jtbi.2008.08.006. [35] S. J. Schreiber, Allee effect, extinctions, and chaotic transients in simple population models, Theor. Popul. Biol., 64 (2003), 201-209. doi: 10.1016/S0040-5809(03)00072-8. [36] J. Smith, M. Amador and R. Barrera, Seasonal and habitat effects on dengue and West Nile Virus Vectors in San Juan, Puerto Rico, J. Am. Mosq. Control. Assoc., 25 (2009), 38-46. doi: 10.2987/08-5782.1. [37] H. Townson, SIT for African malaria vectors: Epilogue, Malar. J., 8 (2009), S10. doi: 10.1186/1475-2875-8-S2-S10. [38] WHO, 10 facts on malaria, http://www.who.int/features/factfiles/malaria/en/. [39] J. Wu, H. R. Thieme, Y. Lou and G. Fan, Stability and persistence in ODE models for populations with many stages, Math. Biosc. Eng., 12 (2015), 661-686. doi: 10.3934/mbe.2015.12.661. [40] B. Zheng, M. Tang and J. Yu, Modeling Wolbachia spread in mosquitoes through delay differential equations, SIAM J. Appl. Math., 74 (2014), 743-770. doi: 10.1137/13093354X. [41] B. Zheng, M. Tang, J. Yu and J. Qiu, Wolbachia spreading dynamics in mosquitoes with imperfect maternal transmission, J. Math. Biol., 76 (2018), 235-263. doi: 10.1007/s00285-017-1142-5.
Bistable phenomena still occur in (6). Here, parameter values $a_0 = 10, \mu_0 = 0.1, \mu_1 = 0.5, \mu_2 = 0.4, $$b = 21, \xi_1 = 0.5, \xi_2 = 0.4, \tau = 0.2. For b = 21 < b_0 = 21.93, there exists three nontrivial equilibria. Boundary equilibrium E_0(0, 6.76) is a locally asymptotically stable node. Positive equilibrium E_1^* (7.84,4.07) is a saddle point, and positive equilibrium E_2^* (8.70,3.87) is a locally asymptotically stable. The effect of time delay \tau in (6) on the level of the positive equilibria shown in the above figure. All other parameters are the same as in Figure 1 except \tau being varied. The effect of time delay \tau on stability of the positive equilibrium E_2^* in system (15). A phase portrait indicates that there is a stable periodic solution for \tau = 4.9. Parameter values are chosen to be a_0 = 30, \mu_0 = 0.1,\ \mu_1 = 0.5,\ \mu_2 = 1.5,\$$b = 2,\ \xi_1 = 4, \xi_2 = 0.51$. Initial conditions is $(w,g) = (10,5)$ for delay $\tau = 1.5$ and $\tau = 4.9$. For delay $\tau = 0$, we have to change initial conditions to $(1,1)$ to obtain a solution converging to the interior equilibrium (while a solution starting at (10, 5) will converges to the trivial equilibrium (0, 0) instead).
Two dimensional bifurcation diagram in parameter space $(\tau, b)$. On the graph, the torus bifurcation curve is very close to Fold-Hopf bifurcation curve. To have a better view, we include a zoomed figure.
A bifurcation diagram of genetically-modified mosquito population $g(t)$ using delay $\tau$ as a bifurcation parameter in model (15).
The existence of bi-stability in the form of two stable periodic solutions for $\tau = 5.18$. The solid line corresponds the periodic solution with initial values $(w,g) = (0.5,1.5)$ and the dotted line corresponds to the periodic solution with initial value $(w,g) = (10,5)$. Here, parameter values $a = 30, \mu_0 = 0.1,\ \mu_1 = 0.5,\$$\mu_2 = 1.5,\ b = 2,\ \xi_1 = 4, \xi_2 = 0.51$.
The existence of bi-stability in the form of two stable periodic solutions for $\tau = 6.55$. The solid line corresponds to the periodic solution with initial values $(w,g) = (0.5,1.5)$ and the dotted line corresponds to the periodic solution with initial value $(w,g) = (0.71,0.11)$.
One dimensional bifurcation diagram of periodic solutions in delay $\tau$. Vertical axis is the amplitude of periodic solutions or equilibria.
Stability change of periodic solutions as delay $\tau$ varies. Vertical axis is the amplitude of periodic solutions.
The plotting of the profile of periodic solutions along torus bifurcation points.
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