Analysis and optimal control of an HIV model with logistic growth and infected cells in eclipse phase

Abstract

A mathematical model of the human immunodeficiency virus infection with logistic growth, infected cells in eclipse phase and therapy is investigated. The model includes four nonlinear differential equations describing the evolution of uninfected CD4⁺ T cells, infected CD4⁺ T cells in latent stage, productively infected CD4⁺ T cells and free virus. Two types of treatments are incorporated into the model; the purpose of the first one consists to block the viral proliferation while the role of the second is to prevent new infections. The positivity and boundedness of solutions are established. The local stability of the disease free steady state and the infection steady states are studied. An optimal control problem is proposed and investigated. Numerical simulations are performed, confirming stability of the free and endemic equilibria and illustrating the effectiveness of the two incorporated treatments via an efficient optimal control.

Keywords

• HIV infection,logistic growth,treatment,stability,optimal control

References

1. Chesney, M.A., et al, Adherence: a necessity for successful HIV combination therapy, 13, (1999), S271-8.
2. Pankavich, S., The Effects of Latent Infection on the Dynamics of HIV, Differential Equations and Dynamical Systems 24(3), (2016), 281-303.
3. Allali, K., Harroudi, S. and Torres, D. F. M. , Analysis and optimal control of an intracellular delayed HIV model with CTL immune response, Math. Comput. Sci 12(2), (2018) , 111-127.
4. Rocha, D., Silva, C. J. and Torres, D. F. M., Stability and Optimal Control of a Delayed HIV Model, Math. Methods Appl. Sci 41(6), (2018), 2251-2260.
5. Allali, K., Tabit, Y. and Harroudi, S., On HIV Model with Adaptive Immune Response, Two Saturated Rates and Therapy, textitMathematical Modelling of Natural Phenomena 12(5), (2017), 1-14.
6. Nowak, M.A., Bonhoeffer, S., Shaw, G.M., May, R.M., Anti-viral drug treatment: Dynamics of resistance in free virus and infected cell populations, Journal of theoretical biology 184(2), (1997), 203-217.
7. Allali, K., Danane, J. and Kuang, Y., Global Analysis for an HIV Infection Model with CTL Immune Response and Infected Cells in Eclipse Phase, Applied Sciences 7(8), (2017), 861.
8. Rong, L., Gilchrist, M. A., Feng, Z. and Perelson, A. S., Modeling within-host HIV-1 dynamics and the evolution of drug resistance: trade-offs between viral enzyme function and drug susceptibility, textitJournal of Theoretical biology 247(4), (2007), 804-818.

9. Rong, L. and Perelson, A. S., Asymmetric division of activated latently infected cells may explain the decay kinetics of the HIV-1 latent reservoir and intermittent viral blips, Mathematical biosciences 217(1), (2009), 77-87.
10. Buonomo, B. and Vargas-De-León, C., Global stability for an HIV-1 infection model including an eclipse stage of infected cells, textitJournal of Mathematical Analysis and Applications 385(2), (2012), 709-720.
11. Sun, Q., Min, L. and Kuang, Y., Global stability of infection-free state and endemic infection state of a modified human immunodeficiency virus infection model, IET systems biology 9(3), (2015), 95-103.
12. Selinger, C. and Katze, M. G., Mathematical models of viral latency, Current opinion in virology 3(4), (2013), 402-407. Orellana : Orellana, J.M., Optimal drug scheduling for HIV therapy efficiency improvement, Biomedical Signal Processing and Control 6(4), ( 2011), 379-386.
13. Wang, Y., Zhou, Y., Wu, J. and Heffernan, J., Oscillatory viral dynamics in a delayed HIV pathogenesis model, Mathematical Biosciences 219(2), (2009), 104-112.
14. Wang, L., Li and M. Y., Mathematical analysis of the global dynamics of a model for HIV infection of CD4+ T cells, textitMathematical Biosciences 200(1), (2006), 44-57.
15. Culshaw, R. V., and Ruan, S., A delay-differential equation model of HIV infection of CD4+ T-cells, Mathematical biosciences 165(1), (2000), 27-39.
16. Smith, H. L. and De Leenheer, P., Virus dynamics: A global analysis, SIAM Journal on Applied Mathematics 63(4), (2003), 1313-1327.
17. Van den Driessche, P., and Watmough, J., Reproduction numbers and sub-threshold endemic equilibria for compartmental models of disease transmission, Mathematical biosciences 180(1-2), (2002), 29-48.
18. Fleming, W. H. and Rishel, R. W., Deterministic and stochastic optimal control (Vol. 1), Springer Science and Business Media, (2012).
19. Lukes, D. L., Differential equations, ser, Mathematics in Science and Engineering. London: Academic Press Inc.[Harcourt Brace Jovanovich Publishers] 162, (1982).
20. Pontryagin, L. S., Boltyanskii, V. G., Gamkrelidze, R. V. and Mishchenko, E. F., The mathematical theory of optimal processes, (1962).
21. Lai, X. and Zou, X., Modeling cell-to-cell spread of HIV-1 with logistic target cell growth, Journal of Mathematical Analysis and Applications 426(1), (2015), 563-584.
22. Jones, L. E. and Perelson, A. S., Transient viremia, plasma viral load, and reservoir replenishment in HIV-infected patients on antiretroviral therapy, Journal of acquired immune deficiency syndromes (1999) 45(5), (2007), 483.
23. Meskaf, A., Allali, K. and Tabit, Y., Optimal control of a delayed hepatitis B viral infection model with cytotoxic T-lymphocyte and antibody responses, International Journal of Dynamics and Control 5(3), (2017), 893-902.