Analyzing the control of dengue by releasing <em>Wolbachia</em>-infected male mosquitoes through a delay differential equation model

Bo Zheng; Lihong Chen; Qiwen Sun; Bo Zheng; Lihong Chen; Qiwen Sun

doi:10.3934/mbe.2019275

Mathematical Biosciences and Engineering

2019, Volume 16, Issue 5: 5531-5550. doi: 10.3934/mbe.2019275

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Analyzing the control of dengue by releasing Wolbachia-infected male mosquitoes through a delay differential equation model

1.
College of Mathematics and Information Sciences, Guangzhou University, Guangzhou, 510006, China
2.
Center for Applied Mathematics, Guangzhou University, Guangzhou, 510006, China

Received: 22 April 2019 Accepted: 12 June 2019 Published: 14 June 2019

To date, an innovative strategy to control dengue is to release Wolbachia-infected male mosquitoes into wild areas to sterilize wild female mosquito vectors by cytoplasmic incompatibility (CI). To investigate the efficacy of Wolbachia in blocking dengue virus transmission, we develop a deterministic mathematical model of human and mosquito populations in which one dengue serotype circulates. The delay differential equation model captures the respective extrinsic and intrinsic incubation periods (EIP and IIP) in the mosquito and human, as well as the maturation delay between mating and emergence of adult mosquitoes, which have received relatively little attention. We analyze the existence and stability of disease-free equilibria, and obtain a sufficient and necessary condition on the existence of the disease-endemic equilibrium. We also determine two threshold values of the release ratio $\theta$, denoted by $\theta_1^*$ and $\theta_2^*$ with $\theta_1^*>\theta_2^*$. When $\theta>\theta_1^*$, the mosquito population will be eradicated eventually. When $\theta_2^* < \theta < \theta_1^*$, a complete mosquito eradication becomes impossible, but virus eradication is ensured at the meantime. When $\theta < \theta_2^*$, the disease-endemic equilibrium emerges that allows dengue virus to circulate between humans and mosquitoes. We carry out sensitivity analysis of the threshold values in terms of the model parameters, and simulate several possible control strategies with different release ratios, which confirm the public awareness that reducing mosquito bites and killing adult mosquitoes are the most effective strategy to control the epidemic. Our model provides new insights on the effectiveness of Wolbachia in reducing dengue at a population level.
- dengue,
- Wolbachia,
- release ratio,
- sensitivity analysis,
- cytoplasmic incompatibility
Citation: Bo Zheng, Lihong Chen, Qiwen Sun. Analyzing the control of dengue by releasing Wolbachia-infected male mosquitoes through a delay differential equation model[J]. Mathematical Biosciences and Engineering, 2019, 16(5): 5531-5550. doi: 10.3934/mbe.2019275

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Abstract

To date, an innovative strategy to control dengue is to release Wolbachia-infected male mosquitoes into wild areas to sterilize wild female mosquito vectors by cytoplasmic incompatibility (CI). To investigate the efficacy of Wolbachia in blocking dengue virus transmission, we develop a deterministic mathematical model of human and mosquito populations in which one dengue serotype circulates. The delay differential equation model captures the respective extrinsic and intrinsic incubation periods (EIP and IIP) in the mosquito and human, as well as the maturation delay between mating and emergence of adult mosquitoes, which have received relatively little attention. We analyze the existence and stability of disease-free equilibria, and obtain a sufficient and necessary condition on the existence of the disease-endemic equilibrium. We also determine two threshold values of the release ratio $\theta$, denoted by $\theta_1^*$ and $\theta_2^*$ with $\theta_1^*>\theta_2^*$. When $\theta>\theta_1^*$, the mosquito population will be eradicated eventually. When $\theta_2^* < \theta < \theta_1^*$, a complete mosquito eradication becomes impossible, but virus eradication is ensured at the meantime. When $\theta < \theta_2^*$, the disease-endemic equilibrium emerges that allows dengue virus to circulate between humans and mosquitoes. We carry out sensitivity analysis of the threshold values in terms of the model parameters, and simulate several possible control strategies with different release ratios, which confirm the public awareness that reducing mosquito bites and killing adult mosquitoes are the most effective strategy to control the epidemic. Our model provides new insights on the effectiveness of Wolbachia in reducing dengue at a population level.

References

[1]	S. Bhatt, P. Gething, O. Brady, et al., The global distribution and burden of dengue, Nature, 496 (2013), 504–507.
[2]	Y. Wang, X. Liu, C. Li, et al., A survey of insecticide resistance in Aedes albopictus (Diptera: Culicidae) during a 2014 dengue fever outbreak in Guangzhou, China, J. Econ. Entomol., 110 (2017), 239–244.
[3]	F. Baldacchino, B. Caputo, F. Chandre, et al., Control methods against invasive Aedes mosquitoes in Europe: a review, Pest. Manag. Sci., 71 (2015), 1471–1485.
[4]	J. Werren, Biology of Wolbachia, Ann. Rev. Entomol., 42 (1997), 587–609.
[5]	G. Bian, D. Joshi, Y. Dong, et al., Wolbachia invades Anopheles stephensi populations and induces refractoriness to plasmodium infection, Science, 340 (2013), 748–751.
[6]	Z. Xi, C. Khoo and S. Dobson, Wolbachia establishment and invasion in an Aedes aegypti labora-tory population, Science, 310 (2005), 326–328.
[7]	E. Waltz, US reviews plan to infect mosquitoes with bacteria to stop disease, Nature, 89 (2016), 450–451.
[8]	M. Keeling, F. Jiggins and J. Read, The invasion and coexistence of competing Wolbachia strains, Heredity, 91 (2003), 382–388.
[9]	B. Zheng, M. Tang and J. Yu, Modeling Wolbachia spread in mosquitoes through delay differential equations, SIAM J. Appl. Math., 74 (2014), 743–770.
[10]	B. Zheng, M. Tang, J. Yu, et al., Wolbachia spreading dynamics in mosquitoes with imperfect maternal transmission, J. Math. Biol., 76 (2018), 235–263.
[11]	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.
[12]	L. Cai, S. Ai and G. Fan, Dynamics of delayed mosquitoes populations models with two different strategies of releasing sterile mosquitoes, Math. Biosci. Eng., 15 (2018), 1181–1202.
[13]	M. Huang, J. Luo, L. Hu, et al., Assessing the efficiency of Wolbachia driven Aedes mosquito suppression by delay differential equations, J. Theor. Biol., 440 (2018), 1–11.
[14]	Y. Li and X. Liu, An impulsive model for Wolbachia infection control of mosquito-borne diseases with general birth and death rate functions, Nonlinear Anal: RWA, 37 (2017), 412–432.
[15]	L. Hu, M. Huang, M. Tang, et al., Wolbachia spread dynamics in stochastic environments, Theor. Popul. Biol., 106 (2015), 32–44.
[16]	M. Turelli and N. Barton, Deploying dengue-suppressing Wolbachia: Robust models predict slow but effective spatial spread in Aedes aegypti, Theor. Popul. Biol., 115 (2017), 45–60.
[17]	E. Newton and P. Reiter, A model of the transmission of dengue fever with an evaluation of the impact of ultra-low volume (ULV) insecticide applications on dengue epidemics, Am. J. Trop. Med. Hyg., 47 (1992), 709–720.
[18]	M. Atkinson, Z. Su, N. Alphey, et al., Analyzing the control of mosquito-borne diseases by a dominant lethal genetic system, Proc. Natl. Acad. Sci. USA, 104 (2007), 9540–9545.
[19]	S. Pinho, C. Ferreira, L. Esteva, et al., Modelling the dynamics of dengue real epidemics, Phil. Trans. R. Soc. A, 368 (2010), 5679–5693.
[20]	L. Xue, X. Fang and J. Hyman, Comparing the effectiveness of different strains of Wolbachia for controlling chikungunya, dengue fever, and zika, PLoS Neglect. Trop. D, 12 (2018), e0006666.
[21]	J. Zhang, C. Yang, Z. Jin, et al., Dynamics analysis of SIR epidemic model with correlation coef-ficients and clustering coefficient in networks, J. Theor. Biol., 449 (2018), 1–13.
[22]	L. Zou, J. Chen, X. Feng, et al., Analysis of a dengue model with vertical transmission and ap-plication to the 2014 dengue outbreak in Guangdong Province, China, B. Math. Biol., 80 (2018), 2633–2651.
[23]	C. Craig, The Etiology of Dengue Fever, J. Am. Med. Assoc., 75 (1920), 1171–1176.
[24]	C. Dye, Models for the population dynamics of the yellow fever mosquito, Aedes aegypti, J. Anim. Ecol., 53 (1984), 247–268.
[25]	F. Coutinhoa, M. Burattinia, L. Lopeza, et al., Threshold conditions for a non-autonomous epi-demic system describing the population dynamics of Dengue, Bull. Math. Biol., 68 (2006), 2263–2282.
[26]	M. Guzman and E. Harris, Dengue, Lancet, 385 (2015), 453–465.
[27]	M. Chan and M. Johansson, The incubation periods of dengue viruses, PLoS One, 7 (2012), e50972.
[28]	H. Smith, An Introduction to Delay Differential Equations with Applications to the Life Sciences, Volume 57, Springer, New York, 2011.
[29]	National Bureau of Statistics of China, National Data, Accessed 20 Jan 2019. Available from: http://data.stats.gov.cn/.
[30]	T. Scott, A. Morrison, L. Lorenz, et al., Longitudinal studies of Aedes aegypti (Diptera: Culicidae) in Thailand and Puerto Rico: population dynamics, J. Med. Entomol., 37 (2000), 77–88.
[31]	C. Paupy, B. Ollomo, B. Kamgang, et al., Comparative role of Aedes albopictus and Aedes ae-gypti in the emergence of Dengue and Chikungunya in central Africa, Vector-Borne and Zoonotic Diseases, 10 (2010), 259–266.
[32]	Y. Ye, S. Chenoweth, A. Carrasco, et al., Evolutionary potential of the extrinsic incubation period of dengue virus in Aedes aegypti, Evolution, 70 (2016), 2459–2469.
[33]	J. Yu, Modeling mosquitoes population suppression based on delay differential equations, SIAM J. Appl. Math., 78 (2018), 3168–3187.

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