Monotonicity properties arising in a simple model of <i>Wolbachia</i> invasion for wild mosquito populations

Diego Vicencio; Olga Vasilieva; Pedro Gajardo; Diego Vicencio; Olga Vasilieva; Pedro Gajardo

doi:10.3934/mbe.2023053

Mathematical Biosciences and Engineering

2023, Volume 20, Issue 1: 1148-1175. doi: 10.3934/mbe.2023053

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Monotonicity properties arising in a simple model of Wolbachia invasion for wild mosquito populations

1.
Departmento de Matemática, Universidad Técnica Federico Santa María, Avenida España 1680, Valparaíso, 2390123, Chile
2.
Departmento de Matemáticas, Universidad del Valle, Cl. 13,100-00, Cali, Valle del Cauca 760032, Colombia

Received: 20 September 2022 Revised: 20 September 2022 Accepted: 11 October 2022 Published: 26 October 2022

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.
- Wolbachia symbiont,
- monotone dynamics,
- competitive model,
- minimum viable population size
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

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Abstract

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.

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