# American Institute of Mathematical Sciences

September  2009, 12(2): 305-321. doi: 10.3934/dcdsb.2009.12.305

## Mathematical modelling of internal HIV dynamics

 1 Department of Statistics and Modelling Science, Livingstone Tower, 26 Richmond Street, Glasgow G1 1XH, United Kingdom, United Kingdom

Received  October 2008 Revised  April 2009 Published  July 2009

We study a mathematical model for the viral dynamics of HIV in an infected individual in the presence of HAART. The paper starts with a literature review and then formulates the basic mathematical model. An expression for $R_0$, the basic reproduction number of the virus under steady state application of HAART, is derived followed by an equilibrium and stability analysis. There is always a disease-free equilibrium (DFE) which is globally asymptotically stable for $R_0 < 1$. Deterministic simulations with realistic parameter values give additional insight into the model behaviour.
Citation: Nirav Dalal, David Greenhalgh, Xuerong Mao. Mathematical modelling of internal HIV dynamics. Discrete and Continuous Dynamical Systems - B, 2009, 12 (2) : 305-321. doi: 10.3934/dcdsb.2009.12.305
 [1] Nicolas Bacaër, Xamxinur Abdurahman, Jianli Ye, Pierre Auger. On the basic reproduction number $R_0$ in sexual activity models for HIV/AIDS epidemics: Example from Yunnan, China. Mathematical Biosciences & Engineering, 2007, 4 (4) : 595-607. doi: 10.3934/mbe.2007.4.595 [2] Miaomiao Gao, Daqing Jiang, Tasawar Hayat, Ahmed Alsaedi, Bashir Ahmad. Dynamics of a stochastic HIV/AIDS model with treatment under regime switching. Discrete and Continuous Dynamical Systems - B, 2022, 27 (6) : 3177-3211. doi: 10.3934/dcdsb.2021181 [3] Cristiana J. Silva. Stability and optimal control of a delayed HIV/AIDS-PrEP model. Discrete and Continuous Dynamical Systems - S, 2022, 15 (3) : 639-654. doi: 10.3934/dcdss.2021156 [4] Benjamin H. Singer, Denise E. Kirschner. Influence of backward bifurcation on interpretation of $R_0$ in a model of epidemic tuberculosis with reinfection. Mathematical Biosciences & Engineering, 2004, 1 (1) : 81-93. doi: 10.3934/mbe.2004.1.81 [5] Toshikazu Kuniya, Mimmo Iannelli. $R_0$ and the global behavior of an age-structured SIS epidemic model with periodicity and vertical transmission. Mathematical Biosciences & Engineering, 2014, 11 (4) : 929-945. doi: 10.3934/mbe.2014.11.929 [6] Cameron J. Browne, Sergei S. Pilyugin. Minimizing $\mathcal R_0$ for in-host virus model with periodic combination antiviral therapy. Discrete and Continuous Dynamical Systems - B, 2016, 21 (10) : 3315-3330. doi: 10.3934/dcdsb.2016099 [7] Rachid Ouifki, Gareth Witten. A model of HIV-1 infection with HAART therapy and intracellular delays. Discrete and Continuous Dynamical Systems - B, 2007, 8 (1) : 229-240. doi: 10.3934/dcdsb.2007.8.229 [8] Hongyong Zhao, Peng Wu, Shigui Ruan. Dynamic analysis and optimal control of a three-age-class HIV/AIDS epidemic model in China. Discrete and Continuous Dynamical Systems - B, 2020, 25 (9) : 3491-3521. doi: 10.3934/dcdsb.2020070 [9] 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 [10] C. Burgos, J.-C. Cortés, L. Shaikhet, R.-J. Villanueva. A delayed nonlinear stochastic model for cocaine consumption: Stability analysis and simulation using real data. Discrete and Continuous Dynamical Systems - S, 2021, 14 (4) : 1233-1244. doi: 10.3934/dcdss.2020356 [11] Ténan Yeo. Stochastic and deterministic SIS patch model. Discrete and Continuous Dynamical Systems - B, 2021, 26 (12) : 6173-6184. doi: 10.3934/dcdsb.2021012 [12] Cristiana J. Silva, Delfim F. M. Torres. A TB-HIV/AIDS coinfection model and optimal control treatment. Discrete and Continuous Dynamical Systems, 2015, 35 (9) : 4639-4663. doi: 10.3934/dcds.2015.35.4639 [13] Hisashi Inaba. The Malthusian parameter and $R_0$ for heterogeneous populations in periodic environments. Mathematical Biosciences & Engineering, 2012, 9 (2) : 313-346. doi: 10.3934/mbe.2012.9.313 [14] Christine K. Yang, Fred Brauer. Calculation of $R_0$ for age-of-infection models. Mathematical Biosciences & Engineering, 2008, 5 (3) : 585-599. doi: 10.3934/mbe.2008.5.585 [15] Songbai Guo, Wanbiao Ma. Global behavior of delay differential equations model of HIV infection with apoptosis. Discrete and Continuous Dynamical Systems - B, 2016, 21 (1) : 103-119. doi: 10.3934/dcdsb.2016.21.103 [16] M. Hadjiandreou, Raul Conejeros, Vassilis S. Vassiliadis. Towards a long-term model construction for the dynamic simulation of HIV infection. Mathematical Biosciences & Engineering, 2007, 4 (3) : 489-504. doi: 10.3934/mbe.2007.4.489 [17] Gigi Thomas, Edward M. Lungu. A two-sex model for the influence of heavy alcohol consumption on the spread of HIV/AIDS. Mathematical Biosciences & Engineering, 2010, 7 (4) : 871-904. doi: 10.3934/mbe.2010.7.871 [18] Yayun Zheng, Xu Sun. Governing equations for Probability densities of stochastic differential equations with discrete time delays. Discrete and Continuous Dynamical Systems - B, 2017, 22 (9) : 3615-3628. doi: 10.3934/dcdsb.2017182 [19] Ana I. Muñoz, José Ignacio Tello. Mathematical analysis and numerical simulation of a model of morphogenesis. Mathematical Biosciences & Engineering, 2011, 8 (4) : 1035-1059. doi: 10.3934/mbe.2011.8.1035 [20] Claude-Michel Brauner, Xinyue Fan, Luca Lorenzi. Two-dimensional stability analysis in a HIV model with quadratic logistic growth term. Communications on Pure and Applied Analysis, 2013, 12 (5) : 1813-1844. doi: 10.3934/cpaa.2013.12.1813

2020 Impact Factor: 1.327