Nonlinear Dynamics and Chaos in a Fractional-Order HIV Model

Haiping Ye,Yongsheng Ding

doi:10.1155/2009/378614

Abstract

We introduce fractional order into an HIV model. We consider the effect of viral diversity on the human immune system with frequency dependent rate of proliferation of cytotoxic T-lymphocytes (CTLs) and rate of elimination of infected cells by CTLs, based on a fractional-order differential equation model. For the one-virus model, our analysis shows that the interior equilibrium which is unstable in the classical integer-order model can become asymptotically stable in our fractional-order model and numerical simulations confirm this. We also present simulation results of the chaotic behaviors produced from the fractional-order HIV model with viral diversity by using an Adams-type predictor-corrector method.

Highlights

An important part of the human immune response against viral infections is cytotoxic T lymphocytes CTLs 1
Our analysis shows that the interior equilibrium which is unstable in the classical integer order model can become asymptotically stable in our fractional order model
We have proposed a fractional order HIV model, as a generalization of an integer order model, developed by Iwami et al 4

Summary

Introduction

An important part of the human immune response against viral infections is cytotoxic T lymphocytes CTLs 1. They recognize and kill cells which are infected by virus. This model has been important in the field of mathematical modelling of HIV infection In their model, there is no interaction among different types of CTL Zj. Iwami et al 4 assumed that the correlation is incorporated as a function of the frequency that the specific CTLs Zj encounter in the specific infected cells Ij. In a similar manner, they considered the rate of elimination of specific infected cells Ij by the specific CTLs Zj to be proportional to this frequency. We find that chaos does exist in the fractional order HIV model with viral diversity

Model Derivation

One-Virus Model

Two-Virus Model

20 CTL 2 a

Conclusions