# American Institute of Mathematical Sciences

December  2019, 9(4): 393-399. doi: 10.3934/naco.2019038

## Numerical solution with analysis of HIV/AIDS dynamics model with effect of fusion and cure rate

 Department of Mathematics, National Institute of Technology Silchar, Cachar, Assam-788010, INDIA

* Corresponding author: pkguptaitbhu@gmail.com

The reviewing process of the paper is handled by Gafurjan Ibragimov, Siti Hasana Sapar and Siti Nur Iqmal Ibrahim

Received  December 2017 Revised  July 2018 Published  August 2019

Fund Project: The first author is supported by Science, Technology & Innovation Scheme and CPDA grant.

The main objective of this manuscript is to study the dynamical behaviour and numerical solution of a HIV/AIDS dynamics model with fusion effect and cure rate. Local and global asymptotic stability of the model is established by Routh-Hurwitz criterion and Lyapunov functional method for infection-free equilibrium point. The numerical solutions of the model has also examined for support of analysis, through Mathematica software.

Citation: Praveen Kumar Gupta, Ajoy Dutta. Numerical solution with analysis of HIV/AIDS dynamics model with effect of fusion and cure rate. Numerical Algebra, Control and Optimization, 2019, 9 (4) : 393-399. doi: 10.3934/naco.2019038
##### References:
 [1] D. Burg, L. Rong, A. U. Neumann and H. Dahari, Mathematical modeling of viral kinetics under immune control during primary HIV-1 infection,, Journal of Theoretical Biology, 259 (2009), 751-759.  doi: 10.1016/j.jtbi.2009.04.010. [2] P. V. Driessche and J. Watmough, Reproduction numbers and sub-threshold endemic equilibria for compartmental models of disease transmission,, Mathematical Biosciences, 180 (2002), 29-48.  doi: 10.1016/S0025-5564(02)00108-6. [3] A. Dutta and P. K. Gupta, A mathematical model for transmission dynamics of HIV/AIDS with effect of weak CD4+ T cells,, Chinese Journal of Physics, 56 (2018), 1045-1056. [4] P. Essunger and A. S. Perelson, Modeling HIV infection of CD4+ T-cell subpopulations,, Journal of Theoretical Biology, 170 (1994), 367-391. [5] G. Haas, A. Hosmalin, F. Hadida, J. Duntze, P. Debré and B. Autran, Dynamics of HIV variants and specific cytotoxic T-cell recognition in non-progressors and progressors,, Immunology Letter, 57 (1997), 63-68. [6] H. F. Huo, R. Chen and X. Y. Wang, Modelling and stability of HIV/AIDS epidemic model with treatment,, Applied Mathematical Modelling, 40 (2016), 6550-6559.  doi: 10.1016/j.apm.2016.01.054. [7] Y. Liu and C. Chen, Role of nanotechnology in HIV/AIDS vaccine development,, Advanced Drug Delivery Reviews, 103 (2016), 76-89. [8] J. Luo, W. Wang, H. Chen and R. Fu, Bifurcations of a mathematical model for HIV dynamics,, Journal of Mathematical Analysis and Applications, 434 (2016), 837-857.  doi: 10.1016/j.jmaa.2015.09.048. [9] H. J. Marquez, Nonlinear Control Systems Analysis and Design, Wiley, 2003. [10] A. Mojaver and H. Kheiri, Mathematical analysis of a class of HIV infection models of CD4+ T-cells with combined antiretroviral therapy,, Applied Mathematics and Computation, 259 (2015), 258-270.  doi: 10.1016/j.amc.2015.02.064. [11] L. Rong, M. A. Gilchrist, Z. Feng and A. S. Perelson, Modeling within host HIV-1 dynamics and the evolution of drug resistance: Tradeoffs between viral enzyme function and drug susceptibility,, Journal of Theoretical Biology, 247 (2007), 804-818.  doi: 10.1016/j.jtbi.2007.04.014. [12] P. K. Srivastava and P. Chandra, Modeling the dynamic of HIV and CD4+ T cells during primary infection,, Nonlinear Analysis: Real World Applications, 11 (2010), 612-618.  doi: 10.1016/j.nonrwa.2008.10.037. [13] S. Wiggins, Introduction to Applied Nonlinear Dynamical Systems and Chaos, New York City: Springer Verlag, 2003.

show all references

##### References:
 [1] D. Burg, L. Rong, A. U. Neumann and H. Dahari, Mathematical modeling of viral kinetics under immune control during primary HIV-1 infection,, Journal of Theoretical Biology, 259 (2009), 751-759.  doi: 10.1016/j.jtbi.2009.04.010. [2] P. V. Driessche and J. Watmough, Reproduction numbers and sub-threshold endemic equilibria for compartmental models of disease transmission,, Mathematical Biosciences, 180 (2002), 29-48.  doi: 10.1016/S0025-5564(02)00108-6. [3] A. Dutta and P. K. Gupta, A mathematical model for transmission dynamics of HIV/AIDS with effect of weak CD4+ T cells,, Chinese Journal of Physics, 56 (2018), 1045-1056. [4] P. Essunger and A. S. Perelson, Modeling HIV infection of CD4+ T-cell subpopulations,, Journal of Theoretical Biology, 170 (1994), 367-391. [5] G. Haas, A. Hosmalin, F. Hadida, J. Duntze, P. Debré and B. Autran, Dynamics of HIV variants and specific cytotoxic T-cell recognition in non-progressors and progressors,, Immunology Letter, 57 (1997), 63-68. [6] H. F. Huo, R. Chen and X. Y. Wang, Modelling and stability of HIV/AIDS epidemic model with treatment,, Applied Mathematical Modelling, 40 (2016), 6550-6559.  doi: 10.1016/j.apm.2016.01.054. [7] Y. Liu and C. Chen, Role of nanotechnology in HIV/AIDS vaccine development,, Advanced Drug Delivery Reviews, 103 (2016), 76-89. [8] J. Luo, W. Wang, H. Chen and R. Fu, Bifurcations of a mathematical model for HIV dynamics,, Journal of Mathematical Analysis and Applications, 434 (2016), 837-857.  doi: 10.1016/j.jmaa.2015.09.048. [9] H. J. Marquez, Nonlinear Control Systems Analysis and Design, Wiley, 2003. [10] A. Mojaver and H. Kheiri, Mathematical analysis of a class of HIV infection models of CD4+ T-cells with combined antiretroviral therapy,, Applied Mathematics and Computation, 259 (2015), 258-270.  doi: 10.1016/j.amc.2015.02.064. [11] L. Rong, M. A. Gilchrist, Z. Feng and A. S. Perelson, Modeling within host HIV-1 dynamics and the evolution of drug resistance: Tradeoffs between viral enzyme function and drug susceptibility,, Journal of Theoretical Biology, 247 (2007), 804-818.  doi: 10.1016/j.jtbi.2007.04.014. [12] P. K. Srivastava and P. Chandra, Modeling the dynamic of HIV and CD4+ T cells during primary infection,, Nonlinear Analysis: Real World Applications, 11 (2010), 612-618.  doi: 10.1016/j.nonrwa.2008.10.037. [13] S. Wiggins, Introduction to Applied Nonlinear Dynamical Systems and Chaos, New York City: Springer Verlag, 2003.
Dynamical behaviour of the model (1) for $R_0 = 0.9406 < 1$
Dynamical behaviour of the model (1) for $R_0 = 5.6140 > 1$
List of parameters
 Parameters Explanations r Natural production rate of uninfected CD4+ T cells $\rho_1$ Fusion rate of CD4+ T-cells and virus $\rho_2$ Rate of new infection into the infective compartment $\rho_3$ Recovery rate of infected cells $\sigma_1$ Normal death rate of uninfected CD4+ T cells $\sigma_2$ Lytic death rate of infected cells $\sigma_3$ Loss rate of virus $A$ Average number of viral particles produced by an infected CD4+ T-cell
 Parameters Explanations r Natural production rate of uninfected CD4+ T cells $\rho_1$ Fusion rate of CD4+ T-cells and virus $\rho_2$ Rate of new infection into the infective compartment $\rho_3$ Recovery rate of infected cells $\sigma_1$ Normal death rate of uninfected CD4+ T cells $\sigma_2$ Lytic death rate of infected cells $\sigma_3$ Loss rate of virus $A$ Average number of viral particles produced by an infected CD4+ T-cell
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