Mathematical analysis, forecasting and optimal control of HIV/AIDS spatiotemporal transmission with a reaction diffusion SICA model

Houssine Zine; Abderrahim El Adraoui; Delfim F. M. Torres; Houssine Zine; Abderrahim El Adraoui; Delfim F. M. Torres

doi:10.3934/math.2022904

AIMS Mathematics

2022, Volume 7, Issue 9: 16519-16535. doi: 10.3934/math.2022904

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Mathematical analysis, forecasting and optimal control of HIV/AIDS spatiotemporal transmission with a reaction diffusion SICA model

1.
Center for Research and Development in Mathematics and Applications (CIDMA), Department of Mathematics, University of Aveiro, 3810-193 Aveiro, Portugal
2.
Laboratory of Analysis Modeling and Simulation (LAMS), Department of Mathematics and Computer Science, Faculty of Sciences Ben M'Sik, Hassan Ⅱ University of Casablanca, Morocco

Received: 04 May 2022 Revised: 18 June 2022 Accepted: 04 July 2022 Published: 08 July 2022
MSC : 49J15, 49K15, 76R50, 92D30

We propose a mathematical spatiotemporal epidemic SICA model with a control strategy. The spatial behavior is modeled by adding a diffusion term with the Laplace operator, which is justified and interpreted both mathematically and physically. By applying semigroup theory on the ordinary differential equations, we prove existence and uniqueness of the global positive spatiotemporal solution for our proposed system and some of its important characteristics. Some illustrative numerical simulations are carried out that motivate us to consider optimal control theory. A suitable optimal control problem is then posed and investigated. Using an effective method based on some properties within the weak topology, we prove existence of an optimal control and develop an appropriate set of necessary optimality conditions to find the optimal control pair that minimizes the density of infected individuals and the cost of the treatment program.
- HIV/AIDS epidemiology,
- reaction-diffusion,
- spatiotemporal SICA model,
- optimal control strategies,
- necessary optimality conditions
Citation: Houssine Zine, Abderrahim El Adraoui, Delfim F. M. Torres. Mathematical analysis, forecasting and optimal control of HIV/AIDS spatiotemporal transmission with a reaction diffusion SICA model[J]. AIMS Mathematics, 2022, 7(9): 16519-16535. doi: 10.3934/math.2022904

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Abstract

We propose a mathematical spatiotemporal epidemic SICA model with a control strategy. The spatial behavior is modeled by adding a diffusion term with the Laplace operator, which is justified and interpreted both mathematically and physically. By applying semigroup theory on the ordinary differential equations, we prove existence and uniqueness of the global positive spatiotemporal solution for our proposed system and some of its important characteristics. Some illustrative numerical simulations are carried out that motivate us to consider optimal control theory. A suitable optimal control problem is then posed and investigated. Using an effective method based on some properties within the weak topology, we prove existence of an optimal control and develop an appropriate set of necessary optimality conditions to find the optimal control pair that minimizes the density of infected individuals and the cost of the treatment program.

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