Dynamics and optimal control of a stochastic Zika virus model with spatial diffusion

Minna Shao; Hongyong Zhao; Minna Shao; Hongyong Zhao

doi:10.3934/mbe.2023778

Mathematical Biosciences and Engineering

2023, Volume 20, Issue 9: 17520-17553. doi: 10.3934/mbe.2023778

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Dynamics and optimal control of a stochastic Zika virus model with spatial diffusion

Minna Shao ^1,3,
Hongyong Zhao ^{1,2
,
,}

1.
College of Mathematics, Nanjing University of Aeronautics and Astronautics, Nanjing 211106, China
2.
Key Laboratory of Mathematical Modelling and High Performance Computing of Air Vehicles (NUAA), MIIT, Nanjing 211106, China
3.
College of Mathematics and Statistics, Ningxia University, Yinchuan 750021, China

Academic Editor: Yang Kuang

Received: 14 July 2023 Revised: 15 August 2023 Accepted: 29 August 2023 Published: 13 September 2023

Zika is an infectious disease with multiple transmission routes, which is related to severe congenital disabilities, especially microcephaly, and has attracted worldwide concern. This paper aims to study the dynamic behavior and optimal control of the disease. First, we establish a stochastic reaction-diffusion model (SRDM) for Zika virus, including human-mosquito transmission, human-human sexual transmission, and vertical transmission of mosquitoes, and prove the existence, uniqueness, and boundedness of the global positive solution of the model. Then, we discuss the sufficient conditions for disease extinction and the existence of a stationary distribution of positive solutions. After that, three controls, i.e. personal protection, treatment of infected persons, and insecticides for spraying mosquitoes, are incorporated into the model and an optimal control problem of Zika is formulated to minimize the number of infected people, mosquitoes, and control cost. Finally, some numerical simulations are provided to explain and supplement the theoretical results obtained.
- stochastic Zika virus model,
- extinction,
- stationary distribution,
- optimal control,
- spatial diffusion
Citation: Minna Shao, Hongyong Zhao. Dynamics and optimal control of a stochastic Zika virus model with spatial diffusion[J]. Mathematical Biosciences and Engineering, 2023, 20(9): 17520-17553. doi: 10.3934/mbe.2023778

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Abstract

Zika is an infectious disease with multiple transmission routes, which is related to severe congenital disabilities, especially microcephaly, and has attracted worldwide concern. This paper aims to study the dynamic behavior and optimal control of the disease. First, we establish a stochastic reaction-diffusion model (SRDM) for Zika virus, including human-mosquito transmission, human-human sexual transmission, and vertical transmission of mosquitoes, and prove the existence, uniqueness, and boundedness of the global positive solution of the model. Then, we discuss the sufficient conditions for disease extinction and the existence of a stationary distribution of positive solutions. After that, three controls, i.e. personal protection, treatment of infected persons, and insecticides for spraying mosquitoes, are incorporated into the model and an optimal control problem of Zika is formulated to minimize the number of infected people, mosquitoes, and control cost. Finally, some numerical simulations are provided to explain and supplement the theoretical results obtained.

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