Page 446 - Mathematical Techniques of Fractional Order Systems

Applications of Continuous-time Fractional Order Chapter | 14 431

whenever needed. It was noticed that the phase trajectories of the system
in Eq. (14.60a c) became larger as the stenosis became more deteriorated.

14.6.2 Generalized Chaotic Susceptible Infected
Recovered Epidemic Model
Mathematical modeling of epidemiological spread of diseases help predict
and control the outbreak of an epidemic. Epidemiological models are in fact
population models where the rate of growth of each group of population
(susceptible or infected) is given by a differential equation that describes the
interaction between the groups and the external factors if they exist. Ansari
et al. (2015) proposed a modified SIR model:

S βSI
α
D S 5 rS 1 2 2 ; ð14:62aÞ
k 1 1 aS
α
D I 5 βIZ 2 μI 2 γI; ð14:62bÞ
1 1 aZ
α
D Z 5 1 ð S 2 ZÞ; ð14:62cÞ
T
where S, R, and Z are the densities of the susceptible and infected within the
population and the information factor, respectively. r is the intrinsic growth
rate of susceptible, k is the carrying capacity of susceptible, a is the satura-
tion factor that measures the inhibitory effect, β is the transmission or con-
tact rate, γ is the rate of recovery from infection, and μ is the death rates.
Lastly, T is the average delay of the collected information on the disease.
The system has three equilibrium points: the trivial equilibrium point
E 1 5 ð0; 0; 0Þ, the disease free point E 2 5 ðk; 0; kÞ, and the endemic point at
^ ^ ^
E 3 5 S; I; Z where:
μ 1 γ r
^
^
^
^
^ ^
S 5 ; I 5 ð1 1 aSÞðk 2 SÞ; Z 5 S: ð14:63Þ
β 2 aðμ 1 γÞ βk
The authors proposed a synchronization scheme based on an active con-
trol method. The master system is defined as (Ansari et al., 2015):

α
D S m 5 rS m 1 2 S m 2 βS m I m ; ð14:64aÞ
k 1 1 aS m
α
D I m 5 βI m Z m 2 μI m 2 γI m ; ð14:64bÞ
1 1 aZ m
α
D Z m 5 1 ð S m 2 Z m Þ; ð14:64cÞ
T