We introduce a fractional order model for the human immunodeficiency virus (HIV) dynamics, where time-varying drug exposure and drug resistance are assumed. We derive conditions for the local and global asymptotic stability of the disease-free equilibrium. We find periodic stable endemic states for certain parameter values, for sinusoidal drug efficacies, and when considering a density-dependent decay rate for the T cells. Other classes of periodic drug efficacies are considered and the effect of the phases of these functions on the dynamics of the model is also studied. The order of the fractional derivative plays an important role in the severity of the epidemics.

References

References
1.
Elaiw
,
A.
, and
Xia
,
X.
,
2009
, “
HIV Dynamics: Analysis and Robust Multirate Mpc-Based Treatment Schedules
,”
J. Math. Anal. Appl.
,
359
(1), pp.
285
301
.
2.
Carvalho
,
A.
, and
Pinto
,
C.
,
2016
, “
Emergence of Drug-Resistance in Hiv Dynamics Under Distinct Haart Regimes
,”
Commun. Nonlinear Sci. Numer. Simul.
,
30
(1–3), pp.
207
226
.
3.
Carvalho
,
A.
, and
Pinto
,
C.
,
2017
, “
Within-Host and Synaptic Transmissions: Contributions to the Spread of HIV Infection
,”
Math. Methods Appl. Sci.
,
40
(4), pp.
1231
1264
.
4.
Jones
,
L.
, and
Perelson
,
A.
,
2005
, “
Opportunistic Infection as a Cause of Transient Viremia in Chronically Infected HIV Patients Under Treatment With Haart
,”
Bull. Math. Biol.
,
67
(6), pp.
1227
1252
.
5.
Li
,
M. Y.
, and
Wang
,
L.
,
2014
, “
Backward Bifurcation in a Mathematical Model for HIV Infection In Vivo With Anti-Retroviral Treatment
,”
Nonlinear Anal.: Real World Appl.
,
17
, pp.
147
160
.
6.
Wang
,
X.
,
Song
,
X.
,
Tang
,
S.
, and
Rong
,
L.
,
2016
, “
Dynamics of an HIV Model With Multiple Infection Stages and Treatment With Different Drug Classes
,”
Bull. Math. Biol.
,
78
(
2
), pp.
322
349
.
7.
Yang
,
Y.
, and
Xiao
,
Y.
,
2010
, “
Threshold Dynamics for an HIV Model in Periodic Environments
,”
J. Math. Anal. Appl.
,
361
(1), pp.
59
68
.
8.
Browne
,
C.
, and
Pilyugin
,
S.
,
2012
, “
Periodic Multidrug Therapy in a Within-Host Virus Model
,”
Bull. Math. Biol.
,
74
(
3
), pp.
562
589
.
9.
Caputo
,
M.
, and
Fabrizio
,
M.
,
2015
, “
A New Definition of Fractional Derivative Without Singular Kernel
,”
Prog. Fractional Differ. Appl.
,
1
(2), pp.
73
85
.http://icmis5.mobile.naturalspublishing.com/files/published/0gb83k287mo759.pdf
10.
Atangana
,
A.
, and
Baleanu
,
D.
,
2016
, “
New Fractional Derivatives With Nonlocal and Non-Singular Kernel: Theory and Application to Heat Transfer Model
,”
Therm. Sci.
,
18
, pp. 1–8.https://arxiv.org/ftp/arxiv/papers/1602/1602.03408.pdf
11.
Mainardi
,
F.
,
1996
, “
Fractional Relaxation-Oscillation and Fractional Diffusion-Wave Phenomena
,”
Chaos, Solutions Fractals
,
7
(9), pp.
1461
1477
.
12.
Baleanu
,
D.
, and
Muslih
,
S.
,
2005
, “
Lagrangian Formulation of Classical Fields Within Riemann-Liouville Fractional Derivatives
,”
Phys. Scr.
,
72
(
2–3
), pp.
119
121
.
13.
Baleanu
,
D.
,
Muslih
,
S.
, and
Rabei
,
E.
,
2008
, “
On Fractional Euler-Lagrange and Hamilton Equations and the Fractional Generalization of Total Time Derivative
,”
Nonlinear Dyn.
,
53
(
1–2
), pp.
67
74
.
14.
Baleanu
,
D.
,
2009
, “
About Fractional Quantization and Fractional Variational Principles
,”
Commun. Nonlinear Sci. Numer. Simulations
,
14
(
6
), pp.
2520
2523
.
15.
Nigmatullin
,
R.
, and
Baleanu
,
D.
,
2010
, “
Is It Possible to Derive Newtonian Equations of Motion With Memory?
,”
Int. J. Theor. Phys.
,
49
(4), pp.
701
708
.
16.
Pinto
,
C.
, and
Machado
,
J. T.
,
2010
, “
Complex Order Van Der Pol Oscillator
,”
Nonlinear Dyn.
,
65
(
3
), pp.
247
254
.
17.
Pinto
,
C.
, and
Machado
,
J. T.
,
2012
, “
Complex-Order Forced Van Der Pol Oscillator
,”
J. Vib. Control
,
18
(
14
), pp.
2201
2209
.
18.
Pinto
,
C.
, and
Machado
,
J. T.
,
2013
, “
Fractional Model for Malaria Transmission Under Control Strategies
,”
Comput. Math. Appl.
,
66
(
5
), pp.
908
916
.
19.
Pinto
,
C.
, and
Carvalho
,
A.
,
2017
, “
The Role of Synaptic Transmission in a HIV Model With Memory
,”
Appl. Math. Comput.
,
292
, pp.
76
95
.
20.
Pinto
,
C.
, and
Carvalho
,
A.
,
2014
, “
New Findings on the Dynamics of HIV and TB Coinfection Models
,”
Appl. Math. Comput.
,
242
, pp.
36
46
.
21.
Doha
,
E.
,
Bhrawy
,
A.
,
Baleanu
,
D.
, and
Hafez
,
R.
,
2014
, “
A New Jacobi Rational-Gauss Collocation Method for Numerical Solution of Generalized Pantograph Equations
,”
Appl. Numer. Math.
,
77
, pp.
43
54
.
22.
Baleanu
,
D.
,
Magin
,
R.
,
Bhalekar
,
S.
, and
Daftardar-Gejji
,
V.
,
2015
, “
Chaos in the Fractional Order Nonlinear Bloch Equation With Delay
,”
Commun. Nonlinear Sci. Numer. Simul.
,
25
(
1–3
), pp.
41
49
.
23.
Atangana
,
A.
, and
Koca
,
I.
,
2016
, “
Chaos in a Simple Nonlinear System With Atangana-Baleanu Derivatives With Fractional Order
,”
Chaos, Solutions Fractals
,
89
, pp.
447
454
.
24.
Kumar
,
D.
,
Singh
,
J.
, and
Baleanu
,
D.
,
2016
, “
Numerical Computation of a Fractional Model of Differential-Difference Equation
,”
ASME J. Comput. Nonlinear Dyn.
,
11
(6), p.
061004
.
25.
Herzallah
,
M.
, and
Baleanu
,
D.
,
2010
, “
On Abstract Fractional Order Telegraph Equation
,”
ASME J. Comput. Nonlinear Dyn.
,
5
(2), p.
021008
.
26.
Razminia
,
A.
, and
Baleanu
,
D.
,
2013
, “
Fractional Hyperchaotic Telecommunication Systems: A New Paradigm
,”
ASME J. Comput. Nonlinear Dyn.
,
8
(
3
), p.
031012
.
27.
Diethelm
,
K.
,
2013
, “
A Fractional Calculus Based Model for the Simulation of an Outbreak of Dengue Fever
,”
Nonlinear Dyn.
,
71
(
4
), pp.
613
619
.
28.
Pinto
,
C.
, and
Carvalho
,
A.
,
2015
, “
Fractional Modeling of Typical Stages in HIV Epidemics With Drug-Resistance
,”
Prog. Fractional Differentiation Appl.
,
1
(
2
), pp.
111
122
.http://www.naturalspublishing.com/files/published/55816j9k59zxso.pdf
29.
Pinto
,
C.
, and
Carvalho
,
A.
,
2015
, “
Fractional Complex-Order Model for HIV Infection With Drug Resistance During Therapy
,”
J. Vib. Control
,
22
(
9
), pp.
2222
2239
.
30.
Pinto
,
C.
, and
Carvalho
,
A.
,
2017
, “
A Latency Fractional Order Model for HIV Dynamics
,”
J. Comput. Appl. Math.
,
312
(1), pp.
240
256
.
31.
Sardar
,
T.
,
Rana
,
S.
,
Bhattacharya
,
S.
,
Al-Khaled
,
K.
, and
Chattopadhyay
,
J.
,
2015
, “
A Generic Model for a Single Strain Mosquito-Transmitted Disease Memory on the Host and the Vector
,”
Math. Biosci.
,
263
, pp.
18
36
.
32.
Tavazoei
,
M.
, and
Haeri
,
M.
,
2008
, “
Chaotic Attractors in Incommensurate Fractional Order Systems
,”
Phys. D
,
237
(20), pp.
2628
2637
.
33.
Driessche
,
P.
, and
Watmough
,
P.
,
2002
, “
Reproduction Numbers and Sub-Threshold Endemic Equilibria for Compartmental Models of Disease Transmission
,”
Math. Biosci.
,
180
(1–2), pp.
29
48
.
34.
LaSalle
,
J.
,
1976
,
The Stability of Dynamical Systems
,
SIAM
,
Philadelphia, PA
.
35.
Chitnis
,
N.
,
Hyman
,
J.
, and
Cushing
,
J.
,
2008
, “
Determining Important Parameters in the Spread of Malaria Through the Sensitivity Analysis of a Mathematical Model
,”
Bull. Math. Biol.
,
70
, pp.
1272
1296
.
36.
Diethelm
,
K.
, and
Freed
,
A.
,
1998
,
The Frac PECE Subroutine for the Numerical Solution of Differential Equations of Fractional Order
,
S.
Heinzel
and
T.
Plesser
, eds.,
Forschung und Wissenschaftliches Rechnen
, Gesellschaft für wissenschaftliche Datenverarbeitung mbH Göttingen, GÖttingen, Germany.
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