AUTHORS: Fahad Al Basir, Priti Kumar Roy

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ABSTRACT: In this article, a mathematical model of HIV infection is developed using fractional-order differential equation consisting uninfected CD4+T cells, infected CD4+T cells and CTL effectors (i.e. immune response cells). The fractional order model possesses non-negative solutions. The system has three equilibria: infectionfree equilibrium, infected equilibrium and CTL equilibrium. Stability conditions of the model system around the equilibria are derived. Numerically it is observed that the system is Global MittagLeffler stabile. Moreover, the necessary conditions for the optimality of the system are derived whose fractional derivative is described in the Riemann and Caputo sense. Using an objective functional, the fractional optimal control problem is solved with minimal dosage of anti-HIV drugs with an aim to minimize the infectious viral load and count of infected CD4+T cells. Efficient numerical technique is provided for solving the FOCP. Numerical simulation has been done to elucidate the analytical results.

KEYWORDS: HIV, CD4+T cell, Immune system, Fractional-Order Differential Equations (FODEs), Memory, Optimal Drug therapy, Fractional Optimal Control Problem (FOCP)

REFERENCES:

[1] B. Ramratnam, J.E. Mittler, L. Zhang, D. Boden, A. Hurley, F. Fang, C.A. Macken, A.S. Perelson, M. Markowitz, D.D. Ho, The decay of the latent reservoir of replication-competent HIV-1 is inversely correlated with the extent of residual viral replication during prolonged anti-retroviral therapy, Nature medicine, 6(1) (2000) 82-5.

[2] A.B. Gumel, C. Connell McCluskey, P. Van den Driessche, Mathematical study of a stagedprogression HIV model with imperfect vaccine, Bulletin of Mathematical Biology, 68(8)(2006) 2105-2128.

[3] P. K. Roy, A. N. Chatterjee,T-cell Proliferation in aMathematical Model of CTL Activity Through HIV-1 Infection, Lecture Notes in Engineering and Computer Science, 1 (2010) 615-620.

[4] Wang Yan, Yicang Zhou, Jianhong Wu, Jane Heffernan, Oscillatory viral dynamics in a delayed HIV pathogenesis model, Mathematical Biosciences 219(2), (2009) 104-112.

[5] S Chowdhury , PK Roy, Mathematical Modelling of Enfuvirtide and Protease Inhibitors as Combination Therapy for HIV. International Journal of Nonlinear Sciences and Numerical Simulation. 2016 Sep 29;17(6):259-75.

[6] M.A.C. Pinto, and A.R.M. Carvalho, Fractional complex-order model for HIV infection with drug resistance during therapy, Journal of Vibration and Control 22.9 (2016): 2222-2239.

[7] P.K. Roy, A.N. Chatterjee, X.Z. Li, The effect of vaccination to dendritic cell and immune cell interaction in HIV disease progression. International Journal of Biomathematics, 9(01), (2016), doi: 10.1142/S1793524516500054 0 2 4 6 8 10 0 0.2 0.4 0.6 0.8 1 Time (day) Optimal drug α=1 α=0.95 α=0.98 Figure 5: Optimal control for different value of α is plotted.

[8] R. Culshaw, S. Ruan and R. Spiteri, Optimal HIV treatment by maximising immune response, J. Math. Biol. 48 (2004), 545562.

[9] Diana Rochaa, Cristiana J. Silvaa and Delfim F. M. Torresa, Stability and Optimal Control of a Delayed HIV Model, Math Methods in the App Sci, (2016).

[10] R. Toledo-Hernandez, V. Rico-Ramirez, A. Gustavo, I. Silva, U. M.Diwekar, A fractional calculus approach to the dynamic optimization of biological reactive systems. Part I: Fractional models for biological reactions, Chemical Engineering Science, 117 , 217-228.

[11] Y. Ding, Z. Wang, H. Ye, Optimal Control of a Fractional-Order HIV-Immune System With Memory, IEEE transactions on control systems technology, 20(3) (2012) 763-769.

[12] Y. Ding and H. Ye, A fractional-order differential equation model of HIV infection of CD4+ T-cells, Mathematical and Computer Modelling, 2009, 50(3): 386 - 392.

[13] Y. Zhou, Basic theory of fractional differential equations, Singapore: World Scientific, Vol. 6, 2014.

[14] S. Abbas, M. Benchohra, G. M. N’Gurkata and B. A. Slimani, Darboux problem for fractionalorder discontinuous hyperbolic partial differential equations in Banach algebras, Complex Variables and Elliptic Equations: An International Journal, 2012, 57(2-4): 113 - 120.

[15] A. G¨okdoan, A. Yildirim, M. Merdan, Solving a fractional order model of HIV infection of CD4+ T cells. Mathematical and Computer Modelling, 54(9)(2011) 2132-8.

[16] Yan, Y. and Kou, C., Stability analysis for a fractional differential model of HIV infection of CD4+ T-cells with time delay. Mathematics and Computers in Simulation, 82(9) (2012) 1572- 1585.

[17] OZalp, Nuri and Demirci, Elif A fractional order ¨ SEIR model with vertical transmission. Mathematical and Computer Modelling. 2011 Jul 31;54(1):1-6.

[18] F. Haq, K. Shah, M. Shahzad, Numerical analysis of fractional order model of HIV-1 infection of CD4+ T-cells. Computational Methods for Differential Equations, 5(1) (2017) 1-11.

[19] Z. Liu, P.Lu, Stability analysis for HIV infection of CD4+ T-cells by a fractional differential time-delay model with cure rate. Advances in Difference Equations, 2014(1),(2014), doi:10.1186/1687-1847-2014-298

[20] F. Al Basir, A. M. Elaiw, Dipak Kesh, Priti Kumar Roy, Optimal Control of a Fractional-order Enzyme Kinetic Model, Control and Cybernetics, vol. 44 (4), (2015).

[21] P. K. Roy, S. Saha, F. Al Basir, Effect of awareness programs in controlling the disease HIV/AIDS: an optimal control theoretic approach, Advances in Difference Equations, (2015): 217, doi: 10.1186/s13662-015-0549-9

[22] P.K. Roy, Mathematical models for therapeutic approaches to control HIV disease transmission, Springer, (2015).

[23] A.A.M. Arafa, S.Z. Rida, M. Khalil, A fractional-order model of HIV infection with drug therapy effect. Journal of the Egyptian Mathematical Society, 22(3) (2014) 538-543.

[24] E. Ahmed and A. S. Elgazzar, On fractional order differential equations model for nonlocal epidemics, Phys. A, vol. 379, pp. 607614,2007.

[25] Sourav Rana, Sabyasachi Bhattacharya, Joydeep Pal, Gaston M. NGurkata, Joydev Chattopadhyay, Paradox of enrichment: A fractional differential approach with memory, Physica A 392 (2013) 3610-3621.

[26] Ghosh, M.K., Pal, J. and Roy, P.K., How Memory Regulates Drug Resistant Pathogenic Bacteria? A Mathematical Study, International Journal of Applied and Computational Mathematics, (2017) 1-27.

[27] F. A. Rihan, D. Baleanu, S. Lakshmanan and R. Rakkiyappan, On Fractional SIRC Model with Salmonella Bacterial Infection, Abstract and Applied Analysis, Article ID 136263, (2014).

[28] V.D. Djordjevi, J. Jari, B. Fabry, J.J. Fredberg, D. Stamenovi, Fractional derivatives embody essential features of cell rheological behavior, Annals of Biomedical Engineering, 31 (2003) 692- 699.

[29] C. Li, F. Zeng, Numerical Methods for Fractional Calculus, CRC Press, Taylor and Francis Group, 2015, 24.

[30] Lin, W., Global existence theory and chaos control of fractional differential equations, JMAA, 332(2007) 709-726.

[31] Y. Li, Y. Chen, I. Podlubny, Stability of fractional-order nonlinear dynamic systems: Lyapunov direct method and generalized MittagLeffler stability, Comput Math Appl, 59 (2010) 1810-1821.

[32] W. Deng, C. Li, J. L, Stability analysis of linear fractional differential system with multiple timedelays, Nonlinear Dyn, 48 (2007) 409416.

[33] B. Chen, J. Chen, Razumikhin-type stability theorems for functional fractional-order differential equations and applications, Appl. Math. Comput. 254 (2015) 6369.

[34] H. L. Li, Y. L. Jiang, Z. Wang, L. Zhang, Z. Teng, Global MittagLeffler stability of coupled system of fractional-order differential equations on network, Applied Mathematics and Computation, 270, (2015) 269-277.

[35] W. H. Fleming and R. W. Rishel, Deterministic and Stochastic Optimal Control, Springer Verlag (1975).

[36] Xianbing, CAO and Abhirup, DATTA and Fahad, AL BASIR and Kumar, ROY Priti, Fractional-order model of the disease Psoriasis: A control based mathematical approach, Journal of Systems Science and Complexity, 29(6),(2016) 1565-1584.