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doi: 10.3934/dcdss.2020431

Solving a class of biological HIV infection model of latently infected cells using heuristic approach

1. 

Department of Dermatology, Stomatology, Radiology and Physical Medicine, University of Murcia, Spain

2. 

Department of Mathematics and Statistics, Hazara University, Mansehra, Pakistan

3. 

Departamento de Matemática Aplicada y Estadística, Universidad Politécnica de Cartagena, Hospital de Marina, 30203-Cartagena, Región de Murcia, Spain

4. 

Department of Electrical and Computer Engineering, COMSATS University, Islamabad, Attock Campus, Attock, Pakistan

* Corresponding author: Yolanda Guerrero–Sánchez

Received  September 2019 Revised  December 2019 Published  November 2020

The intension of the recent study is to solve a class of biological nonlinear HIV infection model of latently infected CD4+T cells using feed-forward artificial neural networks, optimized with global search method, i.e. particle swarm optimization (PSO) and quick local search method, i.e. interior-point algorithms (IPA). An unsupervised error function is made based on the differential equations and initial conditions of the HIV infection model represented with latently infected CD4+T cells. For the correctness and reliability of the present scheme, comparison is made of the present results with the Adams numerical results. Moreover, statistical measures based on mean absolute deviation, Theil's inequality coefficient as well as root mean square error demonstrates the effectiveness, applicability and convergence of the designed scheme.

Citation: Yolanda Guerrero–Sánchez, Muhammad Umar, Zulqurnain Sabir, Juan L. G. Guirao, Muhammad Asif Zahoor Raja. Solving a class of biological HIV infection model of latently infected cells using heuristic approach. Discrete & Continuous Dynamical Systems - S, doi: 10.3934/dcdss.2020431
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I. Ahmad, et al., Novel applications of intelligent computing paradigms for the analysis of nonlinear reactive transport model of the fluid in soft tissues and microvessels, Neural Computing and Applications, 31 (2019), 9041-9059. doi: 10.1007/s00521-019-04203-y.  Google Scholar

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[4]

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N. Ali and G. Zaman, Asymptotic behavior of HIV-1 epidemic model with infinite distributed intracellular delays, Springer Plus, 5 (2016), 324. doi: 10.1186/s40064-016-1951-9.  Google Scholar

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N. Ali, G. Zaman and O. Algahtani, Stability analysis of HIV-1 model with multiple delays, Advances in Difference Equations, 2016 (2016), 88. doi: 10.1186/s13662-016-0808-4.  Google Scholar

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N. AliS. AhmadS. Aziz and G. Zaman, The Adomian decomposition method for solving HIV infection model of latently infected cells, Matrix Science Mathematic, 3 (2019), 5-8.  doi: 10.26480/msmk.01.2019.05.08.  Google Scholar

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[10]

D. W. Brzezinski, Review of numerical methods for NumILPT with computational accuracy assessment for fractional calculus, Applied Mathematics and Nonlinear Sciences, 3 (2018), 487-502.  doi: 10.2478/AMNS.2018.2.00038.  Google Scholar

[11]

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[12]

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[17]

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K. Hattaf and N. Yousfi, Modeling the adaptive immunity and both modes of transmission in HIV infection, Computation, 6 (2018), 37. doi: 10.3390/computation6020037.  Google Scholar

[20]

K. Hattaf, Spatiotemporal dynamics of a generalized viral infection model with distributed delays and CTL immune response, Computation, 7 (2019), 21. doi: 10.3390/computation7020021.  Google Scholar

[21]

W. HeY. Chen and Z. Yin, Adaptive neural network control of an uncertain robot with full-state constraints, IEEE transactions on cybernetics, 46 (2015), 620-629.  doi: 10.1109/TCYB.2015.2411285.  Google Scholar

[22]

A. Khare and S. Rangnekar, A review of particle swarm optimization and its applications in solar photovoltaic system, Applied Soft Computing, 13 (2013), 2997-3006.  doi: 10.1016/j.asoc.2012.11.033.  Google Scholar

[23]

D. MangoniA. TasoraA.   and R. Garziera, A primal-dual predictor-corrector interior point method for non-smooth contact dynamics, Computer Methods in Applied Mechanics and Engineering, 330 (2018), 351-367.  doi: 10.1016/j.cma.2017.10.030.  Google Scholar

[24]

A. Mehmood, et al., Integrated intelligent computing paradigm for the dynamics of micropolar fluid flow with heat transfer in a permeable walled channel, Applied Soft Computing, 79 (2019), 139-162. doi: 10.1016/j.asoc.2019.03.026.  Google Scholar

[25]

A. Mehmood, et al., Backtracking search heuristics for identification of electrical muscle stimulation models using Hammerstein structure, Applied Soft Computing, 84 (2019), 105705. doi: 10.1016/j.asoc.2019.105705.  Google Scholar

[26]

S. Momani, Z. S. Abo-Hammour and O. M. Alsmadi, Solution of inverse kinematics problem using genetic algorithms, Applied Mathematics & Information Sciences, 10 (2016), 225. doi: 10.1016/j.ins.2014.03.128.  Google Scholar

[27]

F. PelletierC. Masson and A. Tahan, Wind turbine power curve modelling using artificial neural network, Renewable Energy, 89 (2016), 207-214.  doi: 10.1016/j.renene.2015.11.065.  Google Scholar

[28]

A. S. Perelson, Modeling the interaction of the immune system with HIV, Mathematical and Statistical Approaches to AIDS Epidemiology, Springer, Berlin, Heidelberg, 1989,350–370. doi: 10.1007/978-3-642-93454-4_17.  Google Scholar

[29]

A. S. PerelsonD. E. Kirschner and R. De Boer, Dynamics of HIV infection of CD4+ T cells. Mathematical biosciences, Mathematical Biosciences, 114 (1993), 81-125.   Google Scholar

[30]

M. Prague, Use of dynamical models for treatment optimization in HIV infected patients: A sequential Bayesian analysis approach, Journal de la Societe Francaise de Statistique, 157 (2016), 20.  Google Scholar

[31]

M. A. Z. RajaF. H. ShahM. Tariq and I. Ahmad, Design of artificial neural network models optimized with sequential quadratic programming to study the dynamics of nonlinear Troesch's problem arising in plasma physics, Neural Computing and Applications, 29 (2018), 83-109.  doi: 10.1007/s00521-016-2530-2.  Google Scholar

[32]

M. A. Z. RajaJ. A. Khan and T. Haroon, Stochastic numerical treatment for thin film flow of third grade fluid using unsupervised neural networks, Journal of the Taiwan Institute of Chemical Engineers, 48 (2015), 26-39.  doi: 10.1016/j.jtice.2014.10.018.  Google Scholar

[33]

M. A. Z. RajaU. FarooqN. I. ChaudharyA. M. Wazwaz and M. A.  , Stochastic numerical solver for nanofluidic problems containing multi-walled carbon nanotubes, Applied Soft Computing, 38 (2016), 561-586.  doi: 10.1016/j.asoc.2015.10.015.  Google Scholar

[34]

M. A. Z. RajaJ. MehmoodZ. SabirA. K. Nasab and M. A. Manzar, Numerical solution of doubly singular nonlinear systems using neural networks-based integrated intelligent computing, Neural Computing and Applications, 31 (2019), 793-812.  doi: 10.1007/s00521-017-3110-9.  Google Scholar

[35]

M. A. Z. Raja, M. Umar, Z. Sabir, J. A. Khan and D. Baleanu, A new stochastic computing paradigm for the dynamics of nonlinear singular heat conduction model of the human head, The European Physical Journal Plus, 133 (2018), 364. doi: 10.1140/epjp/i2018-12153-4.  Google Scholar

[36]

M. A. Z. Raja, Solution of the one-dimensional Bratu equation arising in the fuel ignition model using ANN optimised with PSO and SQP, Connection Science, 26 (2014), 195-214.  doi: 10.1080/09540091.2014.907555.  Google Scholar

[37]

M. A. Z. RajaM. S. AslamN. I. ChaudharyM. Nawaz and S. M. Shah, Design of hybrid nature-inspired heuristics with application to active noise control systems, Neural Computing and Applications, 31 (2019), 2563-2591.  doi: 10.1007/s00521-017-3214-2.  Google Scholar

[38]

M. A. Z. RajaU. AhmedA. ZameerA. K. Kiani and N. I. Chaudhary, Bio-inspired heuristics hybrid with sequential quadratic programming and interior-point methods for reliable treatment of economic load dispatch problem, Neural Computing and Applications, 31 (2019), 447-475.  doi: 10.1007/s00521-017-3019-3.  Google Scholar

[39]

M. A. Z. RajaM. S. AslamN. I. Chaudhary and W. U. Khan, Bio-inspired heuristics hybrid with interior-point method for active noise control systems without identification of secondary path, Frontiers of Information Technology & Electronic Engineering, 19 (2018), 246-259.  doi: 10.1631/FITEE.1601028.  Google Scholar

[40]

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Figure 1.  Graphical illustration of presented scheme for HIV infection model of latently infected cells
Figure 2.  Pseudo code using PSO-IPA
Figure 3.  Trained weights or decision variables of ANN on the basis of best fitness achieved
Figure 4.  Results for HIV infection spread model
Figure 5.  Comparative study on AE of the presented solutions using 5 neurons with the Adams results
Figure 6.  Analysis on MAD for convergence along with the histograms for 5 neurons
Figure 7.  Analysis on RMSE for convergence along with the histograms for 5 neurons
Figure 8.  Analysis on TIC for convergence along with the histograms for 5 neurons
Figure 9.  Statistics based results of Problem 1 for $ x(t) $ and $ w(t) $
Figure 10.  Statistics based results of Problem 1 for $ y(t) $ and $ \nu(t) $
Table 1.  List of parameter and setting used for reported study of HIV infection model
Index Description Settings [8]
$ S_{1} $ Initial value of uninfected CD4+T cells 7
$ S_{2} $ Initial value of infected CD4+T cells 2
$ S_{3} $ Initial value of Virus free cells 1
$ S_{4} $ Initial value of latently infected cells 4
$ \mu $ Rate of uninfected CD4+T cells 0.4
$ \lambda $ Recovery Rate of infected cells 0.3
$ d $ Death rate of uninfected CD4+T cells 0.01
$ \alpha $ Rate of infection spread 0.04
$ q $ Rate of removal of recombinants 0.1
$ a $ Death rate of virus free cells 0.2
$ u $ Death rate of latently infected cells 0.03
Index Description Settings [8]
$ S_{1} $ Initial value of uninfected CD4+T cells 7
$ S_{2} $ Initial value of infected CD4+T cells 2
$ S_{3} $ Initial value of Virus free cells 1
$ S_{4} $ Initial value of latently infected cells 4
$ \mu $ Rate of uninfected CD4+T cells 0.4
$ \lambda $ Recovery Rate of infected cells 0.3
$ d $ Death rate of uninfected CD4+T cells 0.01
$ \alpha $ Rate of infection spread 0.04
$ q $ Rate of removal of recombinants 0.1
$ a $ Death rate of virus free cells 0.2
$ u $ Death rate of latently infected cells 0.03
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