Abstract

The aim of this paper is to develop a fractional order mathematical model for describing the spread of hepatitis B virus (HBV). We also provide a rigorous mathematical analysis of the stability of the disease-free equilibrium (DFE) and the endemic equilibrium of the system based on the basic reproduction number. Here, the infectious disease HBV model is described mathematically in a nonlinear system of differential equations in a caputo sense, and hence, Jacobi collocation method is used to reduce into a system of nonlinear equations. Finally, Newton Raphson method is used for the systems of nonlinear equations to arrive at an approximate solution and matlab 2018 has helped us to simulate the nature of each compartment and effects of the possible control strategies (i.e., vaccination and isolation).

References

References
1.
Podlubny
,
I.
,
1999
,
Fractional Differential Equations: An Introduction to Fractional Derivatives, Fractional Differential Equations, Some Methods of Their Solution and Some of Their Applications
,
Academic Press
,
San Diego, CA
.
2.
Ross
,
B.
,
1977
, “
Fractional Calculus
,”
Math. Mag.
,
50
(
3
), pp.
115
122
.10.1080/0025570X.1977.11976630
3.
Alaria
,
A.
,
Khan
,
A.
,
Suthar
,
D.
, and
Kumar
,
D.
,
2019
, “
Application of Fractional Operators in Modelling for Charge Carrier Transport in Amorphous Semiconductor With Multiple Trapping
,”
Int. J. Appl. Comput. Math.
,
5
(
6
), pp.
1
10
.10.1007/s40819-019-0750-8
4.
Hilfer
,
R.
,
2000
,
Applications of Fractional Calculus in Physics
,
Word Scientific Publishing Company
,
Singapore
.
5.
Mainardi
,
F.
,
1997
,
Fractional Calculus: Some Basic Problems in Continuum and Statistical Mechanics
,
Springer
,
Vienna, Austria
.
6.
Mistry
,
L.
,
Khan
,
A.
,
Suthar
,
D.
, and
Kumar
,
D.
,
2019
, “
A New Numerical Method to Solve Non-Linear Fractional Differential Equations
,”
Int. J. Innovative Technol. Exploring Eng.
,
8
(
12
), pp.
1
6
.10.35940/ijitee.L2741.1081219
7.
Naik
,
P.
,
Zu
,
J.
, and
Owolabi
,
K.
,
2020
, “
Modeling the Mechanics of Viral Kinetics Under Immune Control During Primary Infection of HIV-1 With Treatment in Fractional Order
,”
Phys. A: Stat. Mech. Its Appl.
,
545
, p.
123816
.10.1016/j.physa.2019.123816
8.
Naik
,
P.
,
Zu
,
J.
, and
Owolabi
,
K.
,
2020
, “
Global Dynamics of a Fractional Order Model for the Transmission of HIV Epidemic With Optimal Control
,”
Chaos, Solitons Fractals
,
138
, p.
109826
.10.1016/j.chaos.2020.109826
9.
Owolabi
,
K.
,
2016
, “
Numerical Solution of Diffusive HBV Model in a Fractional Medium
,”
Springer Plus
, 5, p. 1643.10.1186/s40064-016-3295-x
10.
Owolabi
,
K.
, and
Atangana
,
A.
,
2019
, “
Mathematical Analysis and Computational Experiments for an Epidemic System With Nonlocal and Nonsingular Derivative
,”
Chaos, Solitons Fractals
,
126
, pp.
41
49
.10.1016/j.chaos.2019.06.001
11.
Owolabi
,
K.
, and
Shikongo
,
A.
,
2020
, “
Fractional Operator Method on a Multi-Mutation and Intrinsic Resistance Model
,”
Alexandria Eng. J.
, 59(4), pp.
1999
2013
.10.1016/j.aej.2019.12.033
12.
Ramani
,
P.
,
Khan
,
A.
, and
Suthar
,
D.
,
2019
, “
Revisiting Analytical-Approximate Solution of Time Fractional Rosenau-Hyman Equation Via Fractional Reduced Differential Transform Method
,”
Int. J. Emerg. Technol.
,
10
(
2
), pp.
403
409
.https://www.researchtrend.net/ijet/pdf/Revisiting%20Analytical-Approximate%20Solution%20of%20Time%20Fractional%20Rosenau-Hyman%20Equation%20via%20Fractional%20Reduced%20Differential%20Transform%20Method%20ARIF%20M.pdf
13.
De Espndola
,
J.
,
Bavastri
,
C.
, and
De Oliveira Lopes
,
E.
,
2008
, “
Design of Optimum Systems of Viscoelastic Vibration Absorbers for a Given Material Based on the Fractional Calculus Model
,”
J. Vib. Control
,
14
(
9
), pp.
1607
1630
.10.1177/1077546308087400
14.
Duarte
,
F.
, and
Machado
,
J.
,
2006
, “
Fractional Dynamics in the Describing Function Analysis of Nonlinear Friction
,”
Proceedings of the Second IFAC Workshop on Fractional Differentiation and Its Applications
, Vol.
2
, July.
15.
Torvik
,
P.
, and
Bagley
,
R.
,
1984
, “
On the Appearance of the Fractional Derivative in the Behaviour of Real Materials
,”
ASME J. Appl. Mech.
,
51
(
2
), pp.
294
298
.10.1115/1.3167615
16.
Gutierrez
,
R.
,
Rosario
,
J.
, and
Machado
,
J.
,
2010
, “
Fractional Order Calculus: Basic Concepts and Engineering Applications
,”
Math. Probl. Eng.
,
2010
, pp.
1
19
.10.1155/2010/375858
17.
Djidjou Demasse
,
R.
,
Tewa
,
J.
,
Bowong
,
S.
, and
Emvudu
,
Y.
,
2016
, “
Optimal Control for an Age-Structured Model for the Transmission of Hepatitis B
,”
J. Math. Biol.
,
73
(
2
), pp.
305
333
.10.1007/s00285-015-0952-6
18.
Agarwal
,
R.
,
Purohit
,
S.
, and
Kritika
,
2019
, “
A Mathematical Fractional Model With Nonsingular Kernel for Thrombin Receptor Activation in Calcium Signalling
,”
Math. Methods Appl. Sci.
,
42
(
18
), pp.
7160
7171
.10.1002/mma.5822
19.
Baleanu
,
D.
,
Jajarmi
,
A.
,
Mohammadi
,
H.
, and
Rezapour
,
S.
,
2020
, “
A New Study on the Mathematical Modelling of Human Liver With CaputoFabrizio Fractional Derivative
,”
Chaos, Solitons Fractals
,
134
(
18
), p.
109705
.10.1016/j.chaos.2020.109705
20.
Baleanu
,
D.
,
Mohammadi
,
H.
, and
Rezapour
,
S.
,
2020
, “
A Mathematical Theoretical Study of a Particular System of CaputoFabrizio Fractional Differential Equations for the Rubella Disease Model
,”
Adv. Differ. Eq.
,
2020
, p.
184
.10.1186/s13662-020-02614-z
21.
Khan
,
M.
,
Hammouch
,
Z.
, and
Baleanu
,
D.
,
2019
, “
Modelling the Dynamics of Hepatitis E Via the Caputo Fabrizio Derivative
,”
Math. Model. Nat. Phenom.
,
14
(
3
), p.
311
.10.1051/mmnp/2018074
22.
Kumar
,
D.
,
Singh
,
J.
,
Qurashi
,
M.
, and
Baleanu
,
D.
,
2019
, “
A New Fractional SIRS-SI Malaria Disease Model With Application of Vaccines, Antimalarial Drugs, and Spraying
,”
Adv. Differ. Eq.
,
2019
, p. 278.10.1186/s13662-019-2199-9
23.
Rihan
,
F.
,
Baleanu
,
D.
,
Lakshmanan
,
S.
, and
Rakkiyappan
,
R.
,
2014
, “
On Fractional SIRC Model With Salmonella Bacterial Infection
,”
Abstract Appl. Anal.
,
2014
, p.
136263
.10.1155/2014/136263
24.
Ahmed
,
E.
,
El-Sayed
,
A.
, and
El-Saka
,
H.
,
2007
, “
Equilibrium Points, Stability and Numerical Solutions of Fractional-Order Predator-Prey and Rabies Models
,”
J. Math. Anal. Appl.
,
325
(
1
), pp.
542
553
.10.1016/j.jmaa.2006.01.087
25.
Cardoso
,
L.
,
Dos Santos
,
F.
, and
Camargo
,
R.
,
2018
, “
Analysis of Fractional-Order Models for Hepatitis B
,”
Comput. Appl. Math.
,
37
(
4
), pp.
4570
4586
.10.1007/s40314-018-0588-4
26.
Ciupe
,
S.
,
Ribeiro
,
R.
,
Nelson
,
P.
,
Dusheik
,
G.
, and
Perelson
,
A.
,
2007
, “
The Role of Cells Refractory to Productive Infection in Acute Hepatitis B Viral Dynamics
,”
Proc. Natl. Acad. Sci.
,
104
(
12
), pp.
5050
5055
.10.1073/pnas.0603626104
27.
Diethelm
,
K.
,
Ford
,
N.
,
Freed
,
A.
, and
Luchko
,
Y.
,
2005
, “
Algorithms for the Fractional Calculus Selection of Numerical Methods
,”
Comput. Methods Appl. Mech. Eng.
,
194
(
6–8
), pp.
743
773
.10.1016/j.cma.2004.06.006
28.
Forde
,
J.
,
Ciupe
,
S.
,
Arias
,
A.
, and
Lenhart
,
S.
,
2016
, “
Optimal Control of Drug Therapy in a Hepatitis B Model
,”
Appl. Sci.
,
6
(
8
), pp.
1
18
.10.3390/app6080219
29.
Zhou
,
X.
, and
Sun
,
Q.
,
2014
, “
Stability Analysis of a Fractional-Order HBV Infection Model
,”
Int. J. Adv. App. Math. Mech.
,
2
(
2
), pp.
1
6
.http://www.ijaamm.com/uploads/2/1/4/8/21481830/v2n2p1.pdf
30.
Bhrawy
,
A.
,
2016
, “
A Jacobi Spectral Collocation Method for Solving Multi-Dimensional Nonlinear Fractional Sub-Diffusion Equations
,”
Numer Algor.
,
73
(
1
), pp.
91
113
.10.1007/s11075-015-0087-2
31.
Doha
,
E.
,
Bhrawy
,
A.
, and
Ezz-Eldien
,
S.
,
2012
, “
A New Jacobi Operational Matrix: An Application for Solving Fractional Differential Equations
,”
Appl. Math. Model.
,
36
(
10
), pp.
4931
4943
.10.1016/j.apm.2011.12.031
32.
Matignon
,
D.
,
1996
, “
Stability Results for Fractional Differential Equations With Applications to Control Processing
,”
Comput. Eng. Syst. Appl.
,
2
, pp.
963
968
.https://www.researchgate.net/publication/2581881_Stability_Results_For_Fractional_Differential_Equations_With_Applications_To_Control_Processing
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