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.

References

References
1.
Welsh
,
J. S.
,
2011
, “
Contagious Cancer
,”
Oncologist
,
16
(
1
), pp.
1
4
.
2.
Koyanagi
,
Y.
,
Itoyama
,
Y.
,
Nakamura
,
N.
,
Takamatsu
,
K.
,
Kira
,
J. I.
,
Iwamasa
,
T.
,
Goto
,
I.
, and
Yamamoto
,
N.
,
1993
, “
In Vivo Infection of Human T-Cell Leukemia Virus Type I in Non-T Cells
,”
Virology
,
196
(
1
), pp.
25
33
.
3.
Bai
,
Z.
, and
Zhou
,
Y.
,
2012
, “
Dynamics of a Viral Infection Model With Delayed CTL Response and Immune Circadian Rhythm
,”
Chaos, Solitons Fractals
,
45
(
9–10
), pp.
1133
1139
.
4.
Asquith
,
B.
, and
Bangham
,
C. R.
,
2007
, “
Quantifying HTLV-I Dynamics
,”
Immunol. Cell Biol.
,
85
(
4
), pp.
280
286
.
5.
Bangham
,
C. R.
,
2000
, “
The Immune Response to HTLV-I
,”
Curr. Opinion Immunol.
,
12
(
4
), pp.
397
402
.
6.
Lang
,
J.
, and
Li
,
M. Y.
,
2012
, “
Stable and Transient Periodic Oscillations in a Mathematical Model for CTL Response to HTLV-I Infection
,”
J. Math. Biol.
,
65
(
1
), pp.
181
199
.
7.
Bangham
,
C. R.
,
2003
, “
Human T-Lymphotropic Virus Type 1 (HTLV-1): Persistence and Immune Control
,”
Int. J. Hematol.
,
78
(
4
), pp.
297
303
.
8.
Bangham
,
C. R.
,
2003
, “
The Immune Control and Cell-to-Cell Spread of Human T-Lymphotropic Virus Type 1
,”
J. Gen. Virol.
,
84
(
Pt. 12
), pp.
3177
3189
.
9.
Fan
,
R.
,
Dong
,
Y.
,
Huang
,
G.
, and
Takeuchi
,
Y.
,
2014
, “
Apoptosis in Virus Infection Dynamics Models
,”
J. Biol. Dyn.
,
8
(
1
), pp.
20
41
.
10.
Shamsara
,
E.
,
Afsharnejad
,
Z.
, and
Mostolizadeh
,
R.
,
2017
, “
Hopf Bifurcation for a Discontinuous Htlv-1 Model
,”
FILOMAT
, 31(20), pp. 6247–6267.
11.
Nowak
,
M.
, and
May
,
R. M.
,
2000
,
Virus Dynamics: Mathematical Principles of Immunology and Virology
,
Oxford University Press
, Oxford, UK.
12.
Gómez-Acevedo
,
H.
,
Li
,
M. Y.
, and
Jacobson
,
S.
,
2010
, “
Multistability in a Model for CTL Response to HTLV-I Infection and Its Implications to HAM/TSP Development and Prevention
,”
Bull. Math. Biol.
,
72
(
3
), pp.
681
696
.
13.
Li
,
M. Y.
, and
Shu
,
H.
,
2012
, “
Global Dynamics of a Mathematical Model for HTLV-I Infection of CD4+ T Cells With Delayed CTL Response
,”
Nonlinear Anal.: Real World Appl.
,
13
(
3
), pp.
1080
1092
.
14.
Shamsara
,
E.
,
Mostolizadeh
,
R.
, and
Afsharnezhad
,
Z.
,
2016
, “
Transcritical Bifurcation of an Immunosuppressive Infection Model
,”
Iranian J. Numer. Anal. Optim.
,
6
(
2
), pp.
1
16
.
15.
Wodarz
,
D.
,
2007
,
Killer Cell Dynamics
, Vol.
32
,
Springer
, Cham, Switzerland.
16.
Balasubramaniam
,
P.
,
Tamilalagan
,
P.
, and
Prakash
,
M.
,
2015
, “
Bifurcation Analysis of HIV Infection Model With Antibody and Cytotoxic T-Lymphocyte Immune Responses and Beddington–DeAngelis Functional Response
,”
Math. Methods Appl. Sci.
,
38
(
7
), pp.
1330
1341
.
17.
Egami
,
C.
,
2009
, “
Bifurcation Analysis of the Nowak–Bangham Model in CTL Dynamics
,”
Math. Biosci.
,
221
(
1
), pp.
33
42
.
18.
Hethcote
,
H. W.
,
2000
, “
The Mathematics of Infectious Diseases
,”
SIAM Rev.
,
42
(
4
), pp.
599
653
.
19.
Z.
Hu
,
X. Z.
,
Liu
,
H. X.
,
Wang
, and
Ma
,
W.
,
2010
, “
Analysis of the Dynamics of a Delayed HIV Pathogenesis Model
,”
J. Comput. Appl. Math.
,
234
(
2
), pp.
461
476
.
20.
Li
,
M. Y.
, and
Shu
,
H.
,
2011
, “
Multiple Stable Periodic Oscillations in a Mathematical Model of CTL Response to HTLV-I Infection
,”
Bull. Math. Biol.
,
73
(
8
), pp.
1774
1793
.
21.
Muroya
,
Y.
,
Enatsu
,
Y.
, and
Li
,
H.
,
2013
, “
Global Stability of a Delayed HTLV-I Infection Model With a Class of Nonlinear Incidence Rates and CTLs Immune Response
,”
Appl. Math. Comput.
,
219
(
21
), pp.
10559
10573
.
22.
Shamsara
,
E.
,
Afsharnezhad
,
Z.
, and
Efatti
,
S.
,
2018
, “Optimal Drug Control of a Four Dimensional HIV Infection Model,” Acta Biotheoritica, in press.
23.
Shu
,
H.
,
Wang
,
L.
, and
Watmough
,
J.
,
2013
, “
Global Stability of a Nonlinear Viral Infection Model With Infinitely Distributed Intracellular Delays and CTL Immune Responses
,”
SIAM J. Appl. Math.
,
73
(
3
), pp.
1280
1302
.
24.
Tian
,
X.
, and
Xu
,
R.
,
2014
, “
Global Stability and Hopf Bifurcation of an HIV-1 Infection Model With Saturation Incidence and Delayed CTL Immune Response
,”
Appl. Math. Comput.
,
237
, pp.
146
154
.
25.
Xie
,
Q.
,
Huang
,
D.
,
Zhang
,
S.
, and
Cao
,
J.
,
2010
, “
Analysis of a Viral Infection Model With Delayed Immune Response
,”
Appl. Math. Modell.
,
34
(
9
), pp.
2388
2395
.
26.
Afsharnezhad
,
Z.
, and
Amaleh
,
M. K.
,
2011
, “
Continuation of the Periodic Orbits for the Differential Equation With Discontinuous Right Hand Side
,”
J. Dyn. Differ. Equations
,
23
(
1
), pp.
71
92
.
27.
Akhmet
,
M.
,
2010
,
Principles of Discontinuous Dynamical Systems
,
Springer Science & Business Media
, New York.
28.
Gouzé
,
J. L.
, and
Sari
,
T.
,
2002
, “
A Class of Piecewise Linear Differential Equations Arising in Biological Models
,”
Dyn. Syst.
,
17
(
4
), pp.
299
316
.
29.
Elaiw
,
A. M.
,
2010
, “
Global Properties of a Class of HIV Models
,”
Nonlinear Anal.: Real World Appl.
,
11
(
4
), pp.
2253
2263
.
30.
Shamsara
,
E.
,
Shamsara
,
J.
, and
Afsharnezhad
,
Z.
,
2016
, “
Optimal Control Therapy and Vaccination for Special HIV-1 Model With Delay
,”
Theory Biosci.
,
135
(
4
), pp.
217
230
.
31.
Rihan
,
F. A.
, and
Rihan
,
B. F.
,
2015
, “
Numerical Modelling of Biological Systems With Memory Using Delay Differential Equations
,”
Appl. Math
.,
9
(
3
), pp.
1645
1658
.
32.
Di Bernardo
,
M.
,
Budd
,
C. J.
,
Champneys
,
A. R.
,
Kowalczyk
,
P.
,
Nordmark
,
A. B.
,
Tost
,
G. O.
, and
Piiroinen
,
P. T.
,
2008
, “
Bifurcations in Nonsmooth Dynamical Systems
,”
SIAM Rev.
,
50
(
4
), pp.
629
701
.
33.
Filippov
,
A. F.
, 1988,
Differential Equations With Discontinuous Right Hand Sides: Control Systems
, Springer, Dordrecht, The Netherlands.
34.
Kunze
,
M.
, and
Küpper
,
T.
,
2001
,
Non-Smooth Dynamical Systems: An Overview
,
Springer
, Berlin.
35.
Leine
,
R. I.
, and
Nijmeijer
,
H.
,
2013
,
Dynamics and Bifurcations of Non-Smooth Mechanical Systems
, Vol.
18
,
Springer Science & Business Media
, Berlin.
36.
Kuznetsov
,
Y. A.
,
2013
,
Elements of Applied Bifurcation Theory
, Vol.
112
,
Springer Science & Business Media
, New York.
37.
Moiola
,
J. L.
, and
Chen
,
G.
,
1996
,
Hopf Bifurcation Analysis
,
World Scientific
, Singapore.
38.
Allwright
,
D. J.
,
1977
, “
Harmonic Balance and the Hopf Bifurcation
,” Math. Proc. Cambridge Philos. Soc.,
82
(
3
), pp.
453
467
.
39.
Mees
,
A.
, and
Chua
,
L.
,
1979
, “
The Hopf Bifurcation Theorem and Its Applications to Nonlinear Oscillations in Circuits and Systems
,”
IEEE Trans. Circuits Syst.
,
26
(
4
), pp.
235
254
.
40.
Mees
,
A. I.
,
1981
,
Dynamics of Feedback Systems
,
Wiley
, New York.
41.
Xu
,
C.
,
Tang
,
X.
, and
Liao
,
M.
,
2010
, “
Frequency Domain Analysis for Bifurcation in a Simplified Tri-Neuron BAM Network Model With Two Delays
,”
Neural Networks
,
23
(
7
), pp.
872
880
.
42.
Moiola
,
J. L.
, and
Chen
,
G.
,
1993
, “
Frequency Domain Approach to Computation and Analysis of Bifurcations and Limit Cycles: A Tutorial
,”
Int. J. Bifurcation Chaos
,
3
(
4
), pp.
843
867
.
43.
Yu
,
W.
, and
Cao
,
J.
,
2007
, “
Stability and Hopf Bifurcation on a Two-Neuron System With Time Delay in the Frequency Domain
,”
Int. J. Bifurcation Chaos
,
17
(
4
), pp.
1355
1366
.
44.
Irving
,
R. S.
,
2003
,
Integers, Polynomials, and Rings: A Course in Algebra
,
Springer Science & Business Media
, New York.
You do not currently have access to this content.