Backward bifurcation of an HTLV-I model with immune response

Sumei  Li; Yicang  Zhou

doi:10.3934/dcdsb.2016.21.863

Article Contents

2016, Volume 21, Issue 3: 863-881. Doi: 10.3934/dcdsb.2016.21.863

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Backward bifurcation of an HTLV-I model with immune response

Sumei Li^1, and
Yicang Zhou^2,

1.
School of Mathematics and Statistics, Xi'an Jiaotong University, Xi'an 710049
2.
School of Mathematics and Statistics, Xi'an Jiaotong University, Xi'an, 710049

Received: April 30, 2015

Revised: August 31, 2015

Published: April 2016

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Abstract

Human T-cell Lymphotropic virus type 1(HTLV-I) causes HAM/T SP and other illnesses. HTLV-I mainly infects $CD4^+$ T cells and activates HTLV-I-specific immune response. In this paper, we formulate a mathematical model of HTLV-I to investigate the role of selective mitotic transmission, Tax expression, and CTL response in vivo. We define two parameters ($R_0$ and $R_1$) to study the model dynamics. The unique infection-free equilibrium $P_0$ is globally asymptomatic stable if $R_0<1$. There exists the chronic-infection equilibrium $P_1$ if $R_1 < 1 < R_0$. There exists a unique chronic-infection equilibrium $P_2$ if $R_1 > 1$. There is a backward bifurcation of chronic-infection equilibria with CTL response if $R_1 < 1 < R_0$. The numerical simulations shown that the existence of backward bifurcation may lead to the existence of periodic solutions.

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Mathematics Subject Classification: Primary: 34K18, 34K60; Secondary: 34K20, 92D30.

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