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Research Papers

Graphical Hopf Bifurcation of a Filippov HTLV-1 Model With Delay in Cytotoxic T Cells Response

[+] Author and Article Information
Elham Shamsara

Management and Social Determinants of Health
Research Center,
Mashhad University of Medical Sciences,
Mashhad 9133913716, Iran;
School of Mathematics,
Institute for Research in Fundamental
Sciences (IPM),
P.O. Box 19395-5746,
Tehran, Iran
e-mail: elham.shamsara@mail.um.ac.ir

Zahra Afsharnezhad

Department of Applied Mathematics,
Ferdowsi University of Mashhad,
Mashhad 9177948974, Iran
e-mail: afsharnezhad@math.um.ac.ir

Elham Javidmanesh

Department of Applied Mathematics,
Ferdowsi University of Mashhad,
Mashhad 9177948974, Iran
e-mail: javidmanesh@um.ac.ir

Contributed by the Dynamic Systems Division of ASME for publication in the JOURNAL OF DYNAMIC SYSTEMS, MEASUREMENT,AND CONTROL. Manuscript received March 13, 2017; final manuscript received February 22, 2018; published online April 9, 2018. Assoc. Editor: Sergey Nersesov.

J. Dyn. Sys., Meas., Control 140(9), 091007 (Apr 09, 2018) Paper No: DS-17-1150; doi: 10.1115/1.4039488 History: Received March 13, 2017; Revised February 22, 2018

In this paper, we present a discontinuous cytotoxic T cells (CTLs) response for HTLV-1. Moreover, a delay parameter for the activation of CTLs is considered. In fact, a system of differential equation with discontinuous right-hand side with delay is defined for HTLV-1. For analyzing the dynamical behavior of the system, graphical Hopf bifurcation is used. In general, Hopf bifurcation theory will help to obtain the periodic solutions of a system as parameter varies. Therefore, by applying the frequency domain approach and analyzing the associated characteristic equation, the existence of Hopf bifurcation by using delay immune response as a bifurcation parameter is determined. The stability of Hopf bifurcation periodic solutions is obtained by the Nyquist criterion and the graphical Hopf bifurcation theorem. At the end, numerical simulations demonstrated our results for the system of HTLV-1.

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Figures

Grahic Jump Location
Fig. 1

Behavior and phase portrait at τ=0.1392164589<τ0=0.3182800220

Grahic Jump Location
Fig. 2

The typical solutions are asymptotically stable when τ<τ0 with the initial value (23,0.6,1.5). The unit is cell/mm3 on the vertical axes and days for time t.

Grahic Jump Location
Fig. 3

Behavior and phase portrait at τ=0.5392164589>τ0=0.3182800220

Grahic Jump Location
Fig. 4

The periodic solutions when τ=0.5392164589>τ0=0.3182800220 with the initial value (23,0.6,1.5). The unit is cell/mm3 on the vertical axes and days for time t.

Grahic Jump Location
Fig. 5

Behavior and phase portrait at τ=0.6667582188>τ0=0.5667582188

Grahic Jump Location
Fig. 6

The typical solutions converge to a periodic solution when τ>τ0 with the initial value (12,0.6,1)

Grahic Jump Location
Fig. 7

Behavior and phase portrait at τ=0.0667582188<τ0=0.5667582188 with the initial value (12,0.6,1)

Grahic Jump Location
Fig. 8

The typical solutions are asymptotically stable when τ<τ0 with the initial value (12, 2, 1)

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