In this paper, we propose a simplified bidimensional Wolbachia infestation model in a population of Aedes aegypti mosquitoes, preserving the main features associated with the biology of this species that can be found in higher-dimensional models. Namely, our model represents the maternal transmission of the Wolbachia symbiont, expresses the reproductive phenotype of cytoplasmic incompatibility, accounts for different fecundities and mortalities of infected and wild insects, and exhibits the bistable nature leading to the so-called principle of competitive exclusion. Using tools borrowed from monotone dynamical system theory, in the proposed model, we prove the existence of an invariant threshold manifold that allows us to provide practical recommendations for performing single and periodic releases of Wolbachia-carrying mosquitoes, seeking the eventual elimination of wild insects that are capable of transmitting infections to humans. We illustrate these findings with numerical simulations using parameter values corresponding to the wMelPop strain of Wolbachia that is considered the best virus blocker but induces fitness loss in its carriers. In these tests, we considered multiple scenarios contrasting a periodic release strategy against a strategy with a single inundative release, comparing their effectiveness. Our study is presented as an expository and mathematically accessible tool to study the use of Wolbachia-based biocontrol versus more complex models.
Citation: Diego Vicencio, Olga Vasilieva, Pedro Gajardo. Monotonicity properties arising in a simple model of Wolbachia invasion for wild mosquito populations[J]. Mathematical Biosciences and Engineering, 2023, 20(1): 1148-1175. doi: 10.3934/mbe.2023053
In this paper, we propose a simplified bidimensional
| [1] |
A. Hoffmann, B. Montgomery, J. Popovici, I. Iturbe-Ormaetxe, P. Johnson, F. Muzzi, et al., Successful establishment of Wolbachia in Aedes populations to suppress dengue transmission, Nature, 476 (2011), 454–457. https://doi.org/10.1038/nature10356 doi: 10.1038/nature10356
|
| [2] |
C. McMeniman, R. Lane, B. Cass, A. Fong, M. Sidhu, Y.-F. Wang, et al., Stable introduction of a life-shortening Wolbachia infection into the mosquito Aedes aegypti, Science, 323 (2009), 141–144. https://doi.org/10.1126/science.1165326 doi: 10.1126/science.1165326
|
| [3] |
L. Moreira, I. Iturbe-Ormaetxe, J. Jeffery, G. Lu, A. Pyke, L. Hedges, et al., A Wolbachia symbiont in Aedes aegypti limits infection with dengue, chikungunya, and plasmodium, Cell, 139 (2009), 1268–1278. https://doi.org/10.1016/j.cell.2009.11.042 doi: 10.1016/j.cell.2009.11.042
|
| [4] |
T. Ruang-Areerate, P. Kittayapong, Wolbachia transinfection in Aedes aegypti: a potential gene driver of dengue vectors, Proc. Natl. Acad. Sci. U.S.A., 103 (2006), 12534–12539. https://doi.org/10.1073/pnas.0508879103 doi: 10.1073/pnas.0508879103
|
| [5] |
T. Walker, P. Johnson, L. Moreira, I. Iturbe-Ormaetxe, F. Frentiu, C. McMeniman, et al., The wMel Wolbachia strain blocks dengue and invades caged Aedes aegypti populations, Nature, 476 (2011), 450–453. https://doi.org/10.1038/nature10355 doi: 10.1038/nature10355
|
| [6] |
L. Almeida, Y. Privat, M. Strugarek, N. Vauchelet, Optimal releases for population replacement strategies: application to Wolbachia, SIAM J. Math. Anal., 51 (2019), 3170–3194. https://doi.org/10.1137/18M1189841 doi: 10.1137/18M1189841
|
| [7] |
C. McMeniman, S. O'Neill, A virulent Wolbachia infection decreases the viability of the dengue vector Aedes aegypti during periods of embryonic quiescence, PLoS Negl. Trop. Dis., 4 (2010), e748. https://doi.org/10.1371/journal.pntd.0000748 doi: 10.1371/journal.pntd.0000748
|
| [8] |
S. Ritchie, M. Townsend, C. Paton, A. Callahan, A. Hoffmann, Application of wMelPop Wolbachia strain to crash local populations of Aedes aegypti, PLoS Negl. Trop. Dis., 9 (2015), e0003930. https://doi.org/10.1371/journal.pntd.0003930 doi: 10.1371/journal.pntd.0003930
|
| [9] |
L. Almeida, A. Haddon, C. Kermorvant, A. Léculier, Y. Privat, M. Strugarek, et al., Optimal release of mosquitoes to control dengue transmission, ESAIM: Proc. Surv., 67 (2020), 16–29. https://doi.org/10.1051/proc/202067002 doi: 10.1051/proc/202067002
|
| [10] |
J. Schraiber, A. Kaczmarczyk, R. Kwok, M. Park, R. Silverstein, F. Rutaganira, et al., Constraints on the use of lifespan-shortening Wolbachia to control dengue fever, J. Theor. Biol., 297 (2012), 26–32. https://doi.org/10.1016/j.jtbi.2011.12.006 doi: 10.1016/j.jtbi.2011.12.006
|
| [11] |
M. Turelli, Cytoplasmic incompatibility in populations with overlapping generations, Evolution, 64 (2010), 232–241. https://doi.org/10.1111/j.1558-5646.2009.00822.x doi: 10.1111/j.1558-5646.2009.00822.x
|
| [12] |
D. E. Campo-Duarte, D. Cardona-Salgado, O. Vasilieva, Establishing wMelPop Wolbachia infection among wild Aedes aegypti females by optimal control approach, Appl. Math. Inf. Sci., 11 (2017), 1011–1027. https://doi.org/10.18576/amis/110408 doi: 10.18576/amis/110408
|
| [13] | D. E. Campo-Duarte, O. Vasilieva, D. Cardona-Salgado, Optimal control for enhancement of Wolbachia frequency among Aedes aegypti females, Int. J. Pure Appl. Math., 112 (2017), 219–238. |
| [14] |
D. Contreras-Julio, P. Aguirre, J. Mujica, O. Vasilieva, Finding strategies to regulate propagation and containment of dengue via invariant manifold analysis, SIAM J. Appl. Dyn. Syst., 19 (2020), 1392–1437. https://doi.org/10.1137/20M131299X doi: 10.1137/20M131299X
|
| [15] |
A. Fenton, K. Johnson, J. Brownlie, G. Hurst, Solving the Wolbachia paradox: modeling the tripartite interaction between host, Wolbachia, and a natural enemy, Am. Nat., 178 (2011), 333–342. https://doi.org/10.1086/661247 doi: 10.1086/661247
|
| [16] |
D. E. Campo-Duarte, O. Vasilieva, D. Cardona-Salgado, M. Svinin, Optimal control approach for establishing wMelPop Wolbachia infection among wild Aedes aegypti populations, J. Math. Biol., 76 (2018), 1907–1950. https://doi.org/10.1007/s00285-018-1213-2 doi: 10.1007/s00285-018-1213-2
|
| [17] |
J. Farkas, S. Gourley, R. Liu, A.-A. Yakubu, Modelling Wolbachia infection in a sex-structured mosquito population carrying West Nile virus, J. Math. Biol., 75 (2017), 621–647. https://doi.org/10.1007/s00285-017-1096-7 doi: 10.1007/s00285-017-1096-7
|
| [18] |
C. Ferreira, Aedes aegypti and Wolbachia interaction: population persistence in an environment changing, Theor. Ecol., 13 (2020), 137–148. https://doi.org/10.1007/s12080-019-00435-9 doi: 10.1007/s12080-019-00435-9
|
| [19] |
B. Zheng, M. Tang, J. Yu, Modeling Wolbachia spread in mosquitoes through delay differential equations, SIAM J. Appl. Math., 74 (2014), 743–770. https://doi.org/10.1137/13093354X doi: 10.1137/13093354X
|
| [20] |
A. Adekunle, M. Meehan, E. McBryde, Mathematical analysis of a Wolbachia invasive model with imperfect maternal transmission and loss of Wolbachia infection, Infect. Dis. Model., 4 (2019), 265–285. https://doi.org/10.1016/j.idm.2019.10.001 doi: 10.1016/j.idm.2019.10.001
|
| [21] |
L. Almeida, M. Duprez, Y. Privat, N. Vauchelet, Mosquito population control strategies for fighting against arboviruses, Math. Biosci. Eng., 16 (2019), 6274–6297. https://doi.org/10.3934/mbe.2019313 doi: 10.3934/mbe.2019313
|
| [22] |
P.-A. Bliman, M. S. Aronna, F. Coelho, M. da Silva, Ensuring successful introduction of Wolbachia in natural populations of Aedes aegypti by means of feedback control, J. Math. Biol., 76 (2018), 1269–1300. https://doi.org/10.1007/s00285-017-1174-x doi: 10.1007/s00285-017-1174-x
|
| [23] |
L. Xue, C. Manore, P. Thongsripong, J. Hyman, Two-sex mosquito model for the persistence of Wolbachia, J. Biol. Dyn., 11 (2017), 216–237. https://doi.org/10.1080/17513758.2016.1229051 doi: 10.1080/17513758.2016.1229051
|
| [24] |
J. Farkas, P. Hinow, Structured and unstructured continuous models for Wolbachia infections, Bull. Math. Biol., 72 (2010), 2067–2088. https://doi.org/10.1007/s11538-010-9528-1 doi: 10.1007/s11538-010-9528-1
|
| [25] |
I. Dorigatti, C. McCormack, G. Nedjati-Gilani, N. Ferguson, Using Wolbachia for dengue control: insights from modelling, Trends Parasitol., 34 (2018), 102–113. https://doi.org/10.1016/j.pt.2017.11.002 doi: 10.1016/j.pt.2017.11.002
|
| [26] |
H. Dutra, M. Rocha, F. Dias, S. Mansur, E. Caragata, L. Moreira, Wolbachia blocks currently circulating Zika virus isolates in Brazilian Aedes aegypti mosquitoes, Cell Host Microbe, 19 (2016), 771–774. https://doi.org/10.1016/j.chom.2016.04.021 doi: 10.1016/j.chom.2016.04.021
|
| [27] | N. Ferguson, D. Kien, H. Clapham, R. Aguas, V. Trung, T. Chau, et al., Modeling the impact on virus transmission of Wolbachia-mediated blocking of dengue virus infection of Aedes aegypti, Sci. Transl. Med., 7 (2015), 279ra37–279ra37. |
| [28] |
M. Woolfit, I. Iturbe-Ormaetxe, J. Brownlie, T. Walker, M. Riegler, A. Seleznev, et al., Genomic evolution of the pathogenic Wolbachia strain, wMelPop, Genome Biol. Evol., 5 (2013), 2189–2204. https://doi.org/10.1093/gbe/evt169 doi: 10.1093/gbe/evt169
|
| [29] |
H. Yeap, P. Mee, T. Walker, A. Weeks, S. O'Neill, P. Johnson, et al., Dynamics of the "popcorn" Wolbachia infection in outbred Aedes aegypti informs prospects for mosquito vector control, Genetics, 187 (2011), 583–595. https://doi.org/10.1534/genetics.110.122390 doi: 10.1534/genetics.110.122390
|
| [30] |
S.-B. Hsu, H. Smith, P. Waltman, Competitive exclusion and coexistence for competitive systems on ordered Banach spaces, Trans. Am. Math. Soc., 348 (1996), 4083–4094. https://doi.org/10.1090/S0002-9947-96-01724-2 doi: 10.1090/S0002-9947-96-01724-2
|
| [31] | O. E. Escobar-Lasso, O. Vasilieva, A simplified monotone model of Wolbachia invasion encompassing Aedes aegypti mosquitoes, Stud. Appl. Math., 146 (2021), 565–585. |
| [32] | H. Smith, Monotone Dynamical Systems: An Introduction to the Theory of Competitive and Cooperative Systems, vol. 41 of Mathematical Surveys and Monographs, American Mathematical Society, Providence RI, USA, 1995. |
| [33] |
J. Jiang, X. Liang, X.-Q. Zhao, Saddle-point behavior for monotone semiflows and reaction–diffusion models, J. Differ. Equ., 203 (2004), 313–330. https://doi.org/10.1016/j.jde.2004.05.002 doi: 10.1016/j.jde.2004.05.002
|
| [34] | U. Boscain, B. Piccoli, Optimal syntheses for control systems on 2-D manifolds, vol. 43 of Mathématiques & Applications, Springer-Verlag, Berlin, Germany, 2004. |
| [35] |
H. Sussmann, Regular synthesis for time-optimal control of single-input real analytic systems in the plane, SIAM J. Control Optim., 25 (1987), 1145–1162. https://doi.org/10.1137/0325062 doi: 10.1137/0325062
|
| [36] |
H. Sussmann, The structure of time-optimal trajectories for single-input systems in the plane: the $c^{\infty}$ nonsingular case, SIAM J. Control Optim., 25 (1987), 433–465. https://doi.org/10.1137/0325025 doi: 10.1137/0325025
|
| [37] |
H. Sussmann, The structure of time-optimal trajectories for single-input systems in the plane: the general real analytic case, SIAM J. Control Optim., 25 (1987), 868–904. https://doi.org/10.1137/0325048 doi: 10.1137/0325048
|
| [38] |
P.-A. Bliman, A feedback control perspective on biological control of dengue vectors by Wolbachia infection, Eur. J. Contr., 59 (2021), 188–206. https://doi.org/10.1016/j.ejcon.2020.09.006 doi: 10.1016/j.ejcon.2020.09.006
|
| [39] |
H. Aida, H. Dieng, T. Satho, A. Nurita, M. Salmah, F. Miake, et al., The biology and demographic parameters of Aedes albopictus in northern peninsular Malaysia, Asian Pac. J. Trop. Biomed., 1 (2011), 472–477. https://doi.org/10.1016/S2221-1691(11)60103-2 doi: 10.1016/S2221-1691(11)60103-2
|
| [40] |
D. Angeli, E. Sontag, Monotone control systems, IEEE Trans. Automat. Contr., 48 (2003), 1684–1698. https://doi.org/10.1109/TAC.2003.817920 doi: 10.1109/TAC.2003.817920
|
| [41] |
J. H. Arias-Castro, H. J. Martinez-Romero, O. Vasilieva, Biological and chemical control of mosquito population by optimal control approach, Games, 11 (2020), 62. https://doi.org/10.3390/g11040062 doi: 10.3390/g11040062
|
| [42] |
P.-A. Bliman, D. Cardona-Salgado, Y. Dumont, O. Vasilieva, Implementation of control strategies for sterile insect techniques, Math. Biosci., 314 (2019), 43–60. https://doi.org/10.1016/j.mbs.2019.06.002 doi: 10.1016/j.mbs.2019.06.002
|
| [43] |
M. H. Chan, P. S. Kim, Modelling a Wolbachia invasion using a slow–fast dispersal reaction–diffusion approach, Bull. Math. Biol., 75 (2013), 1501–1523. https://doi.org/10.1007/s11538-013-9857-y doi: 10.1007/s11538-013-9857-y
|
| [44] |
D. Cianci, J. Van Den Broek, B. Caputo, F. Marini, D. Torre, H. Heesterbeek, et al., Estimating mosquito population size from mark–release–recapture data, J. Med. Entomol., 50 (2013), 533–542. https://doi.org/10.1603/ME12126 doi: 10.1603/ME12126
|
| [45] |
P. Crain, J. Mains, E. Suh, Y. Huang, P. Crowley, S. Dobson, Wolbachia infections that reduce immature insect survival: Predicted impacts on population replacement, BMC Evol. Biol., 11 (2011), 290. https://doi.org/10.1186/1471-2148-11-290 doi: 10.1186/1471-2148-11-290
|
| [46] | S. De Oliveira, D. Villela, F. Dias, L. Moreira, R. de Freitas, How does competition among wild type mosquitoes influence the performance of Aedes aegypti and dissemination of Wolbachia pipientis?, PLoS Negl. Trop. Dis., 11 (2017), e0005947. |
| [47] |
H. Delatte, G. Gimonneau, A. Triboire, D. Fontenille, Influence of temperature on immature development, survival, longevity, fecundity, and gonotrophic cycles of Aedes albopictus, vector of chikungunya and dengue in the Indian Ocean, J. Med. Entomol., 46 (2009), 33–41. https://doi.org/10.1603/033.046.0105 doi: 10.1603/033.046.0105
|
| [48] |
J.-T. Gong, Y. Li, T.-P. Li, Y. Liang, L. Hu, D. Zhang, et al., Stable introduction of plant-virus-inhibiting Wolbachia into plant hoppers for rice protection, Curr. Biol., 30 (2020), 4837–4845. https://doi.org/10.1016/j.cub.2020.09.033 doi: 10.1016/j.cub.2020.09.033
|
| [49] |
L. Gouagna, J.-S. Dehecq, D. Fontenille, Y. Dumont, S. Boyer, Seasonal variation in size estimates of Aedes albopictus population based on standard mark-release-recapture experiments in an urban area on Reunion Island, Acta Trop., 143 (2015), 89–96. https://doi.org/10.1016/j.actatropica.2014.12.011 doi: 10.1016/j.actatropica.2014.12.011
|
| [50] | L. Perko, Differential Equations and Dynamical Systems, Texts in Applied Mathematics, Springer, New York, USA, 2013. |
| [51] |
A. C. Pimentel, C. S. Cesar, M. Martins, R. Cogni, The antiviral effects of the symbiont bacteria Wolbachia in insects, Front. Immunol., 11 (2021), 3690. https://doi.org/10.3389/fimmu.2020.626329 doi: 10.3389/fimmu.2020.626329
|
| [52] | E. Pliego-Pliego, O. Vasilieva, J. Velázquez-Castro, A. Fraguela-Collar, Control strategies for a population dynamics model of Aedes aegypti with seasonal variability and their effects on dengue incidence, Appl. Math. Model., 81 (2020), 296–319. |
| [53] | H. Smith, Monotone dynamical systems: Reflections on new advances & applications, Discrete Contin. Dyn. Syst. Ser. A, 37 (2017), 485–504. |
| [54] |
L. Styer, S. Minnick, A. Sun, T. Scott, Mortality and reproductive dynamics of Aedes aegypti (Diptera: Culicidae) fed human blood, Vector Borne Zoonotic Dis., 7 (2007), 86–98. https://doi.org/10.1089/vbz.2007.0216 doi: 10.1089/vbz.2007.0216
|
| [55] | E. Suh, S. Dobson, Reduced competitiveness of Wolbachia infected Aedes aegypti larvae in intra-and inter-specific immature interactions, J. Invertebr. Pathol., 114 (2013), 173–177. |
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Diego Vicencio, Olga Vasilieva, Pedro Gajardo. Monotonicity properties arising in a simple model of Wolbachia invasion for wild mosquito populations[J]. Mathematical Biosciences and Engineering, 2023, 20(1): 1148-1175. doi: 10.3934/mbe.2023053
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