Modeling the dynamics of Wolbachia-infected and uninfected Aedes aegypti populations by delay differential equations

A.S. Benedito¹ ; C.P. Ferreira¹ ; M. Adimy²

¹ Inria, Université de Lyon, Université Lyon 1, 69200 Villeurbanne, France
² São Paulo State University (UNESP), Institute of Biosciences, 18618-689 Botucatu, SP, Brazil

Mathematical modelling of natural phenomena, Tome 15 (2020), article no. 76.

Résumé

Starting from an age structured partial differential model, constructed taking into account the mosquito life cycle and the main features of the Wolbachia-infection, we derived a delay differential model using the method of characteristics, to study the colonization and persistence of the Wolbachia-transinfected Aedes aegypti mosquito in an environment where the uninfected wild mosquito population is already established. Under some conditions, the model can be reduced to a Nicholson-type delay differential system; here, the delay represents the duration of mosquito immature phase that comprises egg, larva and pupa. In addition to mortality and oviposition rates characteristic of the life cycle of the mosquito, other biological features such as cytoplasmic incompatibility, bacterial inheritance, and deviation on sex ratio are considered in the model. The model presents three equilibriums: the extinction of both populations, the extinction of Wolbachia-infected population and persistence of uninfected one, and the coexistence. The conditions of existence for each equilibrium are obtained analytically and have been interpreted biologically. It is shown that the increase of the delay can promote, through Hopf bifurcation, stability switch towards instability for the nonzero equilibriums. Overall, when the delay increases and crosses predetermined thresholds, the populations go to extinction.

Détail

DOI : 10.1051/mmnp/2020041

Bibliographie

[1] T.H. Ant, C.S. Herd, V. Geoghegan, A.A. Hoffmann, S.P. Sinkins The wolbachia strain wau provides highly efficient virus transmission blocking in aedes aegypti PLoS Pathogens 2018 e1006815

[2] Z.A. Awrahman, F. Champion De Crespigny, N. Wedell The impact of wolbachia, male age and mating history on cytoplasmic incompatibility and sperm transfer in drosophila simulans J. Evolut. Biol 2014 1 10

[3] J.K. Axford, P.A. Ross, H.L. Yeap, A.G. Callahan, A.A. Hoffmann Fitness of walbb wolbachia infection in aedes aegypti: parameter estimates in an outcrossed background and potential for population invasion Am. J. Trop. Med. Hygiene 2016 507 516

[4] E. Beretta, Y. Kuang Geometric stability switch criteria in delay differential systems with delay dependent parameters SIAM J. Math. Anal 2002 1144 1165

[5] L. Berezansky, E. Braverman, L. Idels Nicholson’s blowflies differential equations revisited: Main results and open problems Appl. Math. Model 2010 1405 1417

[6] L. Berezansky, L. Idels, L. Troib Global dynamics of nicholson-type delay systems with applications Nonlinear Anal.: Real World Appl 2011 436 445

[7] P.-A. Bliman, M.S. Aronna, F.C. Coelho, M.A. H.B. Da Silva Ensuring successful introduction of wolbachia in natural populations of aedes aegypti by means of feedback control J. Math. Biol 2018 1269 1300

[8] P.-A. Bliman, D. Cardona-Salgado, Y. Dumont, O. Vasilieva Implementation of control strategies for sterile insect techniques Math. Biosci 2019 43 60

[9] S.R. Bordenstein, S.R. Bordenstein Temperature affects the tripartite interactions between bacteriophage wo, wolbachia, and cytoplasmic incompatibility PLoS ONE 2011 e29106

[10] E. Braverman, D. Kinzebulatov Nicholson’s blowflies equation with a distributed delay Can. Appl. Math. Quart 2006 107 128

[11] R.A. Costello, Effects of environmental and physiological factors on the acoustic behavior of Aedes aegypti (L.) (Diptera: Culicidae). PhD thesis, University of Manitoba, Canada (1974).

[12] C. Dye Models for the population dynamics of the yellow fever mosquito, Aedes aegypti J. Animal Ecol. 1984 247 268

[13] J.Z. Farkas, P. Hinow Structured and unstructured continuous models for wolbachia infections Bull. Math. Biol 2010 2067 2088

[14] J.Z. Farkas, S.A. Gourley, R. Liu, A.A. Yakubu Modelling wolbachia infection in a sex-structured mosquito population carrying west nile virus J. Math. Biol 2017

[15] C.P. Ferreira Aedes aegypti and wolbachia interaction: population persistence in a changing environment Theor. Ecol. 2019

[16] C.P. Ferreira, H.M. Yang, L. Esteva Assessing the suitability of sterile insect technique applied to Aedes aegypti J. Biol. Syst. 2008 565 577

[17] D.J. Gubler The global emergence/resurgence of arboviral diseases as public health problems Arch. Med. Res 2002 330 342

[18] N.D. Hayes Roots of the transcendental equation associated with a certain difference-differential equation J. London Math. Soc 1950 226 232

[19] S.P. Hernandez, A.M. Loaiza, C.A.A. Minoli A reaction-diffusion model for controlling the Aedes aegypti with wolbachia Int. J. Contemp. Math. Sci. 2016 385 394

[20] A.A. Hoffmann, B.L. Montgomery, J. Popovici, I. Iturbe-Ormaetxe, P.H. Johnson, F. Muzzi, M. Greenfield, M. Durkan, Y.S. Leonga, Y. Dong, H. Cook, J. Axford, A.G. Callahan, N. Kenny, C. Omodei, E.A. Mcgraw, P.A. Ryan, S.A. Ritchie, M. Turelli, S.L. O’Neill Successful establishment of wolbachia in aedes populations to suppress dengue transmission Nature 2011 454 457

[21] M. Huang, M.X. Tang, J.S. Yu Wolbachia infection dynamics by reaction-diffusion equations Sci. China Math 2015 77 96

[22] M. Huang, J. Luo, L. Hu, B. Zheng, J. Yu Assessing the efficiency of wolbachia driven aedes mosquito suppression by delay differential equations J. Theor. Biol 2018

[23] M.G. Huang, M.X. Tang, J.S. Yu, B. Zheng The impact of mating competitiveness and incomplete cytoplasmic incompatibility on wolbachia-driven mosquito population suppression Math. Biosci. Eng 2019 4741 4757

[24] H. Hughes, N. Britton Modelling the use of wolbachia to control dengue fever transmission Bull. Math. Biol 2013

[25] L. Idels, M Kipnis Stability criteria for a nonlinear nonautonomous system with delays Appl. Math. Model 2009 2293 2297

[26] S. Lunel and J. Hale, Introduction to functional differential equations. In Vol. 99 of Applied Mathematical Sciences. Springer-Verlag (1993).

[27] M. Keeling, F.M. Jiggins, J.M. Read The invasion and coexistence of competing wolbachia strains Heredity 2003 382 388

[28] J.G. King, C. Souto-Maior, L.M. Sartori, R.M. De Freitas, M. Gomes Variation in wolbachia effects on aedes mosquitoes as a determinant of invasiveness and vectorial capacity Nat. Commun 2018

[29] X. Ling, A.M. Carrie, T. Panpim, M.H. James Two-sex mosquito model for the persistence of wolbachia J. Biol. Dyn 2017 216 237

[30] C.J. Mcmeniman, R.V. Lane, B.N. Cass, A.W.C. Fong, M. Sidhu, Y.-F. Wang, S.L. O’Neill Stable introduction of a life-shortening wolbachia infection into the mosquito Aedes aegypti Science 2009 141 144

[31] M. Ndii, R. Hickson, G. Mercer Modelling the introduction of wolbachia into Aedes aegypti mosquitoes to reduce dengue transmission ANZIAM J. 2012 213 227

[32] Z. Qu, L. Xue, J. Hyman Modeling the transmission of wolbachia in mosquitoes for controlling mosquito-borne diseases SIAM J. Appl. Math 2018 826 852

[33] M. Rafikov, E. Rafikova, H.M. Yang Optimization of the Aedes aegypti control strategies for integrated vector management J. Appl. Math. 2015 918194

[34] J.M. Reinhold, C.R. Lazzari, C. Lahondère Effects of the environmental temperature on Aedes aegypti and Aedes albopictus mosquitoes: a review Insects 2018 158

[35] P.A. Ross, I. Wiwatanaratanabutr, J.K. Axford, V.L. White, N.M. Endersby-Harshman and A.A. Hoffmann, Wolbachia infections in aedes aegypti differ markedly in their response to cyclical heatstress (2017).

[36] I.E. Leonard T. Hillen and H. Van Roessel Partial Differential Equations: Theory and Completely Solved Problems. Wiley (2012).

[37] Z. Veneti, M.E. Clark, T.L. Karr, C. Savakis, K. Bourtzis Heads or tails: Host-parasite interactions in the drosophila-wolbachia system Appl. Environ. Microbiol 2004 5366 5372

[38] P.F. Viana-Medeiros, D.F. Bellinato, A.J. Martins, D. Valle Insecticide resistance, associated mechanisms and fitness aspects in two Brazilian Stegomyia aegypti ( Med. Veterin. Entomol 2017 340 350

[39] T. Walker, P.H. Johnson, L.A. Moreira, I. Iturbe-Ormaetxe, F.D. Frentiu, C.J. Mcmeniman, Y.S. Leong, Y. Dong, J. Axford, P. Kriesner, A.L. Lloyd, S.A. Ritchie, S.L. O’Neill, A.A. Hoffmann The WMEL wolbachia strain blocks dengue and invades caged Aedes aegypti populations Nature 2011 450 453

[40] Z. Xi, C.C. Khoo, S.L. Dobson Wolbachia establishment and invasion in an Aedes aegypti laboratory population Science 2005 326 328

[41] H.M. Yang, C.P. Ferreira Assessing the effects of vector control on dengue transmission Appl. Math. Comput 2008 401 413

[42] H.M. Yang, M.L. Macoris, K.C. Galvani, M.T. Andrighetti, D.M. Wanderley Assessing the effects of temperature on the population of Aedes aegypti, the vector of dengue Epidemiol. Infection 2009 1188 1202

[43] H.L. Yeap, P. Mee, T. Walker, A.R. Weeks, S.L. O’Neill, P. Johnson, S.A. Ritchie, K.M. Richardson, C. Noteg, N.M. Endersby, A.A. Hoffmann Dynamics of the ‘popcorn’ wolbachia infection in outbred Aedes aegypti informs prospects for mosquito vector control Genetics 2011 583 595