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Journal of Applied Nonlinear Dynamics 12(1) (2023) 147--169 | DOI:10.5890/JAND.2023.03.011

Bhagya Jyoti Nath$^1$, Hemanta Kumar Sarmah$^2$

$^1$ Department of Mathematics, Barnagar College, Sorbhog, 781317, Barpeta, Assam, India

$^{2}$ Department of Mathematics, Gauhati University, Guwahati, 781014, Assam, India

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Abstract

In this paper, a delayed differential equation model for HIV infection of CD4$^+$ T cells with full logistic proliferation of healthy CD4$^+$ T cells, cure rate of HIV infected CD4$^+$ T cells, and fusion effect is investigated. At first, we have proved the basic properties of the model like non-negativity and uniform boundedness of solutions. It is found that our model exhibits two equilibria: HIV infection-free equilibrium and HIV-infected equilibrium point, later is obtained whenever basic reproduction number is greater than one. The stability criteria for both equilibria are investigated and basic reproduction number is found to be a threshold parameter. Our study indicates that delay can destabilize HIV infected equilibrium and lead to occurrence of Hopf bifurcation. Also, we have calculated the length of delay to preserve stability by using Nyquist criterion. Moreover, explicit formulae are derived using the normal form theory and center manifold argument in order to determine the direction, stability, and period of periodic bifurcating solutions. Finally, numerical simulations are done to illustrate the analytical results.

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