AIMS Mathematics

2022, Issue 9: 16519-16535. doi: 10.3934/math.2022904
Research article Special Issues

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

• 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.

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

Related Papers:

• 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.

 [1] J. Djordjevic, C. J. Silva, D. F. M. Torres, A stochastic SICA epidemic model for HIV transmission, Appl. Math. Lett., 84 (2018), 168–175. https://doi.org/10.1016/j.aml.2018.05.005 doi: 10.1016/j.aml.2018.05.005 [2] A. El Alami Laaroussi, M. Rachik, On the regional control of a reaction-diffusion system SIR, Bull. Math. Biol., 82 (2020), 1–25. https://doi.org/10.1007/s11538-019-00673-2 doi: 10.1007/s11538-019-00673-2 [3] J. Ewald, P. Sieber, R. Garde, S. N. Lang, S. Schuster, B. Ibrahim, Trends in mathematical modeling of host–pathogen interactions, Cell. Mol. Life Sci., 77 (2020), 467–480. https://doi.org/10.1007/s00018-019-03382-0 doi: 10.1007/s00018-019-03382-0 [4] J. Ge, K. I. Kim, Z. Lin, H. Zhu, A SIS reaction-diffusion-advection model in a low-risk and high-risk domain, J. Differential Equations, 259 (2015), 5486–5509. https://doi.org/10.1016/j.jde.2015.06.035 doi: 10.1016/j.jde.2015.06.035 [5] F. Hufsky, K. Lamkiewicz, A. Almeida, A. Aouacheria, C. Arighi, A. Bateman, et al., Computational strategies to combat COVID-19: useful tools to accelerate SARS-CoV-2 and coronavirus research, Brief. Bioinform., 22 (2021), 642–663. https://doi.org/10.1093/bib/bbaa232 doi: 10.1093/bib/bbaa232 [6] A. E. A. Laaroussi, R. Ghazzali, M. Rachik, S. Benrhila, Modeling the spatiotemporal transmission of Ebola disease and optimal control: a regional approach, Int. J. Dyn. Control, 7 (2019), 1110–1124. https://doi.org/10.1007/s40435-019-00525-w doi: 10.1007/s40435-019-00525-w [7] S. Linge, H. P. Langtangen, Programming for computations–-MATLAB/Octave, Texts in Computational Science and Engineering, 14, Springer, Cham, 2016. https://doi.org/10.1007/978-3-319-32452-4 [8] E. M. Lotfi, M. Mahrouf, M. Maziane, C. J. Silva, D. F. M. Torres, N. Yousfi, A minimal HIV-AIDS infection model with general incidence rate and application to Morocco data, Stat. Optim. Inf. Comput., 7 (2019), 588–603. https://doi.org/10.19139/soic.v7i3.834 doi: 10.19139/soic.v7i3.834 [9] A. Pazy, Semigroups of linear operators and applications to partial differential equations, Applied Mathematical Sciences, 44, Springer-Verlag, New York, 1983. https://doi.org/10.1007/978-1-4612-5561-1 [10] C. J. Silva, Stability and optimal control of a delayed HIV/AIDS-PrEP model, Discrete Contin. Dyn. Syst. Ser. S, 15 (2022), 639–654. https://doi.org/10.3934/dcdss.2021156 doi: 10.3934/dcdss.2021156 [11] C. J. Silva, D. F. M. Torres, A TB-HIV/AIDS coinfection model and optimal control treatment, Discrete Contin. Dyn. Syst., 35 (2015), 4639–4663. https://doi.org/10.3934/dcds.2015.35.4639 doi: 10.3934/dcds.2015.35.4639 [12] C. J. Silva, D. F. M. Torres, A SICA compartmental model in epidemiology with application to HIV/AIDS in Cape Verde, Ecol. Complex., 30 (2017), 70–75. https://doi.org/10.1016/j.ecocom.2016.12.001 doi: 10.1016/j.ecocom.2016.12.001 [13] C. J. Silva, D. F. M. Torres, Modeling and optimal control of HIV/AIDS prevention through PrEP, Discrete Contin. Dyn. Syst. Ser. S, 11 (2018), 119–141. https://doi.org/10.3934/dcdss.2018008 doi: 10.3934/dcdss.2018008 [14] C. J. Silva, D. F. M. Torres, Stability of a fractional HIV/AIDS model, Math. Comput. Simulat., 164 (2019), 180–190. https://doi.org/10.1016/j.matcom.2019.03.016 doi: 10.1016/j.matcom.2019.03.016 [15] C. J. Silva, D. F. M. Torres, On SICA models for HIV transmission, Mathematical modelling and analysis of infectious diseases, 302 (2020), 155–179. https://doi.org/10.1007/978-3-030-49896-2 doi: 10.1007/978-3-030-49896-2 [16] J. Smoller, Shock waves and reaction-diffusion equations, Grundlehren der mathematischen Wissenschaften, 258, Springer-Verlag, New York, 1994. [17] S. Vaz, D. F. M. Torres, A dynamically-consistent nonstandard finite difference scheme for the SICA model, Math. Biosci. Eng., 18 (2021), 4552–4571. https://doi.org/10.3934/mbe.2021231 doi: 10.3934/mbe.2021231 [18] I. I. Vrabie, $C_0$-semigroups and applications, North-Holland Mathematics Studies, 191, North-Holland Publishing Co., Amsterdam, 2003. [19] W. Wang, W. Ma, Z. Feng, Complex dynamics of a time periodic nonlocal and time-delayed model of reaction-diffusion equations for modeling $\rm CD4^+$ T cells decline, J. Comput. Appl. Math., 367 (2020), 112430. https://doi.org/10.1016/j.cam.2019.112430 doi: 10.1016/j.cam.2019.112430 [20] K. Wang, W. Wang, Propagation of HBV with spatial dependence, Math. Biosci., 210 (2007), 78–95. https://doi.org/10.1016/j.mbs.2007.05.004 doi: 10.1016/j.mbs.2007.05.004 [21] W. Wang, X. Wang, K. Guo, W. Ma, Global analysis of a diffusive viral model with cell-to-cell infection and incubation period, Math. Methods Appl. Sci., 43 (2020), 5963–5978. https://doi.org/10.1002/mma.6339 doi: 10.1002/mma.6339
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