Dynamical analysis and optimal control of an multi-age-structured vector-borne disease model with multiple transmission pathways

Huihui Liu; Yaping Wang; Linfei Nie; Huihui Liu; Yaping Wang; Linfei Nie

doi:10.3934/math.20241727

AIMS Mathematics

2024, Volume 9, Issue 12: 36405-36443. doi: 10.3934/math.20241727

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Research article

Dynamical analysis and optimal control of an multi-age-structured vector-borne disease model with multiple transmission pathways

College of Mathematics and System Sciences, Xinjiang University, Urumqi 830017, China

Received: 04 November 2024 Revised: 13 December 2024 Accepted: 20 December 2024 Published: 31 December 2024
MSC : 34E10, 65C30, 92B05

Based on the diversity of transmission routes and host heterogeneity of some infectious diseases, a dynamical model with multi-age-structured, asymptomatic infections, as well as horizontal and vectorial transmission, is proposed. First, the existence and uniqueness of the global positive solution of this model is discussed and the exact expression of the basic reproduction number $ \mathcal{R}_0 $ is obtained using the linear approximation method. Further, we deduce that the disease-free steady state $ \mathcal{E}^0 $ is globally asymptotically stable for $ \mathcal{R}_0 < 1 $, the endemic steady state $ \mathcal{E}^* $ exists and the disease is persistent for $ \mathcal{R}_0 > 1 $. In addition, the locally asymptotically stability of $ \mathcal{E}^* $ is also obtained under some certain conditions. Next, our model is extended to a control problem and the existence and uniqueness of the optimal control by using the Gateaux derivative. Finally, numerical simulations are used to explain the main theoretical results and discuss the impact of age-structured parameters and control strategies on the prevention and control of vector-borne infectious diseases.
- vector-borne diseases,
- multiple routes of transmission,
- age structure,
- asymptomatic infection,
- optimal control
Citation: Huihui Liu, Yaping Wang, Linfei Nie. Dynamical analysis and optimal control of an multi-age-structured vector-borne disease model with multiple transmission pathways[J]. AIMS Mathematics, 2024, 9(12): 36405-36443. doi: 10.3934/math.20241727

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Abstract

Based on the diversity of transmission routes and host heterogeneity of some infectious diseases, a dynamical model with multi-age-structured, asymptomatic infections, as well as horizontal and vectorial transmission, is proposed. First, the existence and uniqueness of the global positive solution of this model is discussed and the exact expression of the basic reproduction number $ \mathcal{R}_0 $ is obtained using the linear approximation method. Further, we deduce that the disease-free steady state $ \mathcal{E}^0 $ is globally asymptotically stable for $ \mathcal{R}_0 < 1 $, the endemic steady state $ \mathcal{E}^* $ exists and the disease is persistent for $ \mathcal{R}_0 > 1 $. In addition, the locally asymptotically stability of $ \mathcal{E}^* $ is also obtained under some certain conditions. Next, our model is extended to a control problem and the existence and uniqueness of the optimal control by using the Gateaux derivative. Finally, numerical simulations are used to explain the main theoretical results and discuss the impact of age-structured parameters and control strategies on the prevention and control of vector-borne infectious diseases.

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