# Analysis of the dynamics of a mathematical model for HIV infection

Volume 23, Issue 3, pp 181--195
Publication Date: November 01, 2020 Submission Date: July 18, 2020 Revision Date: September 15, 2020 Accteptance Date: September 17, 2020
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### Authors

Bhagya Jyoti Nath - Department of Mathematics, Barnagar College, Sorbhog, 781317, Barpeta, Assam, India. Kaushik Dehingia - Department of Mathematics, Gauhati University, Guwahati, 781014, Assam, India. Hemanta Kumar Sarmah - Department of Mathematics, Gauhati University, Guwahati, 781014, Assam, India.

### Abstract

Mathematical models are essential tools in the study of different infectious diseases. Researchers have developed other in-host models to investigate HIV dynamics in the human body. In this paper, a mathematical model for the HIV infection of $CD4^+$ T cells is analyzed. We consider the proliferation of T cells in this study. It is found that there exist two equilibrium states for this model: Infection-free equilibrium state and infected equilibrium state. Local stability is discussed for both infection-free and infected equilibrium states using Routh--Hurwitz criteria. Also, we calculate the basic reproduction number $(R_0)$ for the model with the help of next generation matrix method. The global stability of the infection-free equilibrium point is discussed using Lyapunov's second method. From the stability analysis, it is found that if basic reproduction number $R_0 \leq 1$, infection of HIV is cleared out, and if $R_0 >1$, infection of HIV persists. The conditions for global stability of the infected equilibrium point are derived using a geometric approach. We find a parameter region where the infected equilibrium point is globally stable. We carry out numerical simulations to verify the results. Also, the effects of the proliferation rate of uninfected $CD4^+$ T cells and recovery rate of infected $CD4^+$ T cells in dynamics of the T cells and free virus are studied using numerical simulations. It is found that small variations of these parameters can change the model's whole dynamics, and infection can be controlled by controlling the proliferation rate and improving the recovery rate.

### Share and Cite

##### ISRP Style

Bhagya Jyoti Nath, Kaushik Dehingia, Hemanta Kumar Sarmah, Analysis of the dynamics of a mathematical model for HIV infection, Journal of Mathematics and Computer Science, 23 (2021), no. 3, 181--195

##### AMA Style

Nath Bhagya Jyoti, Dehingia Kaushik, Sarmah Hemanta Kumar, Analysis of the dynamics of a mathematical model for HIV infection. J Math Comput SCI-JM. (2021); 23(3):181--195

##### Chicago/Turabian Style

Nath, Bhagya Jyoti, Dehingia, Kaushik, Sarmah, Hemanta Kumar. "Analysis of the dynamics of a mathematical model for HIV infection." Journal of Mathematics and Computer Science, 23, no. 3 (2021): 181--195

### Keywords

• HIV infection
• global stability
• $CD4^+$ T cells
• basic reproduction number

•  34A34
•  37N25

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