DYNAMICAL BEHAVIOR IN A FRACTIONAL ORDER EPIDEMIC MODEL A. George Maria Selvam Sacred heart College, Tirupattur – 635 601. S. India. D. Abraham Vianny Knowledge Institute of Technology, Kakapalayam – 637 504. S. India. S. Britto Jacob Sacred heart College, Tirupattur – 635 601. S. India. Original Research Paper Mathematics Fractional order SIR epidemic model is considered for dynamical analysis. The basic reproductive number is established and an analysis is carried out to study the stability of the equilibrium points. The time plots and phase portraits are provided for different sets of parameter values. Numerical simulations are presented to illustrate the stability analysis using Generalized Euler method. ABSTRACT KEYWORDS : Fractional order, SIR model, Differential equations, Stability, Generalized Euler Method. Volume - 7 | Issue - 7 | July - 2017 | 4.894 ISSN - 2249-555X | IF : | IC Value : 79.96 464 INDIAN JOURNAL OF APPLIED RESEARCH
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DYNAMICAL BEHAVIOR IN A FRACTIONAL ORDER ......[6] Z.M. Odibat and Shaher Moamni, An algorithm for the Numerical solution of differential equations of fractional order, J. Appl. Math.
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DYNAMICAL BEHAVIOR IN A FRACTIONAL ORDER EPIDEMIC MODEL
A. George Maria Selvam
Sacred heart College, Tirupattur – 635 601. S. India.
D. Abraham Vianny
Knowledge Institute of Technology, Kakapalayam – 637 504. S. India.
S. Britto Jacob Sacred heart College, Tirupattur – 635 601. S. India.
Original Research Paper
Mathematics
Fractional order SIR epidemic model is considered for dynamical analysis. The basic reproductive number is established and an analysis is carried out to study the stability of the equilibrium points. The time plots and phase portraits are provided
for different sets of parameter values. Numerical simulations are presented to illustrate the stability analysis using Generalized Euler method.
the fractional derivative order 0.9a = . For these
parameters the corresponding eigen values are
1 0.03l = - , and 2,3 0.0208 0.0614il = - ± for 1E ..
Also 1,2,3arg( ) 3.1416 1.41372
pl a= > = and
0 1.3889 1R = > . Then the endemic equilibrium is
locally asymptotically stable. See Figure 3.
Figure 3. Time series and Phase diagram of endemic
equilibrium 1E
and Different Fractional Derivatives
( ' )sa with 0 1R > .
468 INDIAN JOURNAL OF APPLIED RESEARCH
Volume - 7 | Issue - 7 | July - 2017 | 4.894ISSN - 2249-555X | IF : | IC Value : 79.96
BIFURCATION
Bifurcation diagrams provide information about
abrupt changes in the qualitative behavior in the
dynamics of the system. The parameter values at
which these changes occur are called bifurcation
points. If the qualitative change occurs in a
neighborhood of an equilibrium point or periodic
solution, it is local bifurcation. In this section, we
give the bifurcation diagrams of the systems (3). See
Figure 4-7.
Figure 4. The bifurcation of Susceptible population,
Infected population with initial values
0 0( , ) (0.95,0.05), 0.8, 0.2, 0.1, 1.0, [0.0,1.0]S I b d hd b= = = = = Î
and 0.5a = .
Figure 5. The bifurcation of Susceptible population,
Infected population with initial values
0 0( , ) (0.95,0.05), 4.0, 0.2, 0.12, 0.11, [2.0,3.5]S I b d hb d= = = = = Î
and 0.5a = .
Figure 6. The bifurcation of Susceptible population,
Infected population with initial values
0 0( , ) (0.95,0.05), 1.2, 0.4, 0.45, 0.24, 7.0S I b d hb d= = = = = =
and [0.1,0.6]aÎ .
INDIAN JOURNAL OF APPLIED RESEARCH 469
Figure 7. The bifurcation of Susceptible population,
Infected population with initial values
0 0( , ) (0.95,0.05), 1.2, 0.4, 0.45, 0.24, [6.0,9.0]S I b d hb d= = = = = Î
and 0.5a = .
References:[1] Fred Brauer, Mathematical Epidemiology, Springer, 2008.[2] Matt J. Keeling and Pejman Rohani, Modeling Infectious Diseases, Princeton
University Press, 2008. [3] K. S. Miller and B. Ross, An introduction to the Fractional Calculus and Fractional
Differential Equations, John Wiley & Sons, INC 1993.[4] Moustafa El-Shahed and Ahmed Alsaedi, The Fractional SIRC Model and Influenza A,
Mathematical Problems in Engineering, Volume 2011, Article ID 480378, 9 pages, doi:10.1155/2011/480378.
[5] Ivo Petras, Fractional Order Nonlinear Systems- Modelling, Analysis and Simulation, Higher Education Press, Springer International Edition, April 2010.
[6] Z.M. Odibat and Shaher Moamni, An algorithm for the Numerical solution of differential equations of fractional order, J. Appl. Math. and Informatics, 26(2008), 15-27.
[7] Z. Odibat and N. Shawagfeh, Generalized Taylor’s formula, Appl. Math. Comput. 186 (2007), 286-293.
[8] James Holland Jones, Notes on R0, Department of Anthropological Science, Standford University, May 1, 2007.
[9] E. Ahmed, A.M.A. El-Sayed, H.A.A. El-Saka, Equilibrium points, Stability and Numerical solutions of Fractional order Predator-Prey and rabies models, Journal of Mathematical Analysis and Applications, 325 (2007), 542-553.
Volume - 7 | Issue - 7 | July - 2017 | 4.894ISSN - 2249-555X | IF : | IC Value : 79.96