An age-dependent hybrid system for optimal contraception control of vermin

Xin Yi; Rong Liu; Xin Yi; Rong Liu

doi:10.3934/math.2025122

AIMS Mathematics

2025, Volume 10, Issue 2: 2619-2633. doi: 10.3934/math.2025122

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An age-dependent hybrid system for optimal contraception control of vermin

Xin Yi ¹,
Rong Liu ^{2
,
,}

1.
School of Mathematics and Statistics, Taiyuan Normal University, Jinzhong, 030619, China
2.
School of Applied Mathematics, Shanxi University of Finance and Economics, Taiyuan, 030006, China

Received: 25 October 2024 Revised: 21 January 2025 Accepted: 06 February 2025 Published: 13 February 2025
MSC : 35F50, 49K15, 49K20, 92D25

This paper discusses the optimal contraception control problem for vermin. The novel model consists of a first-order partial differential equation for the age-dependent density of vermin and two ordinary differential equations for the amounts of female sterilant in the environment and in an individual. We first show that the hybrid system is well-posed by applying the fixed-point theorem. Then the structure of an optimal contraception policy is established by considering the normal cone and adjoint system. Moreover, there is a unique optimal policy by employing Ekeland's variational principle and fixed-point theory. The optimal policy that we have derived offers a rational deployment strategy for the use of sterilants as a means of efficacious pest control. These criteria guarantee that during the application of sterilants, the predetermined objectives are attained while simultaneously minimizing expenditure and environmental implications. Utilizing these optimality criteria facilitates the development of streamlined and economically viable pest management protocols.
- age structure,
- hybrid system,
- contraception control,
- vermin
Citation: Xin Yi, Rong Liu. An age-dependent hybrid system for optimal contraception control of vermin[J]. AIMS Mathematics, 2025, 10(2): 2619-2633. doi: 10.3934/math.2025122

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Abstract

This paper discusses the optimal contraception control problem for vermin. The novel model consists of a first-order partial differential equation for the age-dependent density of vermin and two ordinary differential equations for the amounts of female sterilant in the environment and in an individual. We first show that the hybrid system is well-posed by applying the fixed-point theorem. Then the structure of an optimal contraception policy is established by considering the normal cone and adjoint system. Moreover, there is a unique optimal policy by employing Ekeland's variational principle and fixed-point theory. The optimal policy that we have derived offers a rational deployment strategy for the use of sterilants as a means of efficacious pest control. These criteria guarantee that during the application of sterilants, the predetermined objectives are attained while simultaneously minimizing expenditure and environmental implications. Utilizing these optimality criteria facilitates the development of streamlined and economically viable pest management protocols.

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