Mathematical analysis of a vector model using larvicides and <i>Wolbachia</i>-infected male mosquitoes: an optimal and impulsive control approach

Wendpanga Birba; Boureima Sangaré; Abdoulaye Kaboré; Wendpanga Birba; Boureima Sangaré; Abdoulaye Kaboré

doi:10.3934/mbe.2026078

Mathematical Biosciences and Engineering

2026, Volume 23, Issue 7: 2132-2177. doi: 10.3934/mbe.2026078

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Research article

Mathematical analysis of a vector model using larvicides and Wolbachia-infected male mosquitoes: an optimal and impulsive control approach

1.
Laboratoire de Mathématiques Informatique et Applications (LaMIA), Université Nazi BONI, 01 BP 1091, Burkina Faso
2.
Université Lédéa Bernard Ouédraogo, 01 BP 346 Ouahigouya 01, Burkina Faso

Received: 11 April 2026 Revised: 24 May 2026 Accepted: 04 June 2026 Published: 15 July 2026

In this paper, we develop and analyze a mathematical model for mosquito population control, incorporating the release of Wolbachia-infected male mosquitoes and the application of larvicides. Two complementary approaches are considered: An optimal control model with continuous-time control functions, and an impulsive control model accounting for periodic releases of Wolbachia-infected male mosquitoes and larvicide. The main objective is to minimize the population of fertilized females, which are responsible for the transmission of vector-borne diseases, while reducing the associated biological, social, and economic costs. Using the Pontryagin maximum principle, we characterize the optimal controls and demonstrate the existence of admissible solutions. Within the framework of the impulsive model, the existence and stability conditions for periodic solutions are rigorously derived, allowing the identification of critical thresholds for the larvicide concentration and the minimum density of Wolbachia-infected mosquitoes required to achieve a significant reduction in the target population. Numerical simulations illustrate the enhanced effectiveness of the combined intervention strategy, demonstrating both faster response times and a greater impact on population suppression. These findings emphasize the necessity of a scientifically grounded, optimized, and integrated approach in the design and implementation of vector control programs.
- mathematical model,
- optimal control,
- Pontryagin maximum principle,
- impulsive control,
- Wolbachia,
- numerical analysis
Citation: Wendpanga Birba, Boureima Sangaré, Abdoulaye Kaboré. Mathematical analysis of a vector model using larvicides and Wolbachia-infected male mosquitoes: an optimal and impulsive control approach[J]. Mathematical Biosciences and Engineering, 2026, 23(7): 2132-2177. doi: 10.3934/mbe.2026078

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Abstract

In this paper, we develop and analyze a mathematical model for mosquito population control, incorporating the release of Wolbachia-infected male mosquitoes and the application of larvicides. Two complementary approaches are considered: An optimal control model with continuous-time control functions, and an impulsive control model accounting for periodic releases of Wolbachia-infected male mosquitoes and larvicide. The main objective is to minimize the population of fertilized females, which are responsible for the transmission of vector-borne diseases, while reducing the associated biological, social, and economic costs. Using the Pontryagin maximum principle, we characterize the optimal controls and demonstrate the existence of admissible solutions. Within the framework of the impulsive model, the existence and stability conditions for periodic solutions are rigorously derived, allowing the identification of critical thresholds for the larvicide concentration and the minimum density of Wolbachia-infected mosquitoes required to achieve a significant reduction in the target population. Numerical simulations illustrate the enhanced effectiveness of the combined intervention strategy, demonstrating both faster response times and a greater impact on population suppression. These findings emphasize the necessity of a scientifically grounded, optimized, and integrated approach in the design and implementation of vector control programs.

References

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