Page 11 - IJOCTA-15-1

A nonlinear mathematical model to describe the transmission dynamics of the citrus canker epidemic

From the above-given expression, the constant The constraints first and second are explicit, and
quantity and the coefficient of ψ are not positive, if the third exists, then the fourth also exists as
∗
respectively. So by the Routh-Hurwitz criteria, α 0 > 0. Thus, the equilibria E is locally asymp-
2
this equilibrium point is unstable. totically stable under the given constraints. □
Now the variational matrix M 2 related to equilib-
∗ ¯ ¯
¯
¯
ria E (I, P, B, E) is written by
2
3. Fractional-order model
 ¯ ¯ ¯ ¯ 
B βI + λB λ(P − I) 0
3.1. Preliminaries
 −α −d 0 0 
 
M 2 =  s ¯ s 1 I ¯  Here we remind the given definitions from the the-
 0 − B + gB
s 1 K ¯
¯ 
 B  ory of fractional calculus:
0 l 0 −g 0
(19) Definition 1. 16 The Riemann-Liouville (RL)
¯ ¯
¯
where B = −(βI + λPB ). fractional integral of a function K (t) ∈
I ¯ C η (η ≥ −1) is defined by
The respective characteristic polynomial is ex-
Z t
1
pressed by J K(t) = (t − s) γ−1 K(s)ds,
γ
Γ(γ) 0
0
2
3
4
ψ + α 3 ψ + α 2 ψ + α 1 ψ + α 0 = 0, (20) J K (t) = K(t). (22)
where
Definition 2. 16 The definition of Caputo deriva-
¯ ¯
λPB s s 1 I ¯ tive of order γ > 0 for a function K(t) : (0, ∞) →
¯
¯
α 3 = βI + + δ + B + + g 0 ,
¯
I ¯ K B R is given as follows:
¯ ¯
λPB s s 1 I ¯
¯
¯
α 2 = βI + δ + B + + g 0 Z t
¯
I ¯ K B γ D K (t) = 1 (t − s) m−γ−1 K (s)ds,
m
C
¯ ¯ t Γ(m − γ)
s s 1 I s s 1 I 0
¯
¯
+ d B + + g 0 + B + g 0 (23)
¯
¯
K B K B
where m = 1 + ⌊γ⌋ and ⌊γ⌋ is the integer-part of
¯
¯
¯
¯
+ α(βI + λB) − λ(P − I)s 1 ,
γ.
¯ ¯
λPB s s 1 I ¯
¯
¯
α 1 = βI + δ B + + g 0
¯
I ¯ K B Now we redefine the given integer-order model
¯ ¯
λPB s s 1 I ¯ into Caputo-type fractional-order form to include
¯
¯
+ βI + + δ B + g 0 memory in the system. We know that the mem-
¯
I ¯ K B
ory effects cannot be well captured in the integer-
s s 1 I ¯
¯
¯
¯
+ α(βI + λB) B + + g 0 order cases but can be easily observed by us-
¯
K B ing fractional-order operators. When we gen-
¯
¯
− λ(P − I)s 1 (δ + g 0 ), eralise any model from integer-order sense to
¯ ¯ ¯ fractional-order, the time dimension of the pa-
λPB s s 1 I
¯
¯
α 0 = βI + δ B + g 0 rameters changes from time −1 to time −γ (γ is the
¯
I ¯ K B
order of the fractional operator). Therefore, the
¯
s s 1 I
¯ ¯ ¯ generalised form of the proposed classical model
+ α(βI + λB)g 0 B + ¯
K B (7) is derived as follows:
¯
¯
¯
− λ(P − I)(δs 1 g 0 − αlgB) > 0.
 γ C γ γ γ γ γ
 D I(t) = (β I + λ B)(P − I) − (ν + α + δ )I,
t
By Murata, 35 the conditions for local stability of  C γ γ γ

γ
 D P(t) = Λ − δ P − α I,

the model are t B
γ
γ
γ
γ
γ
C
 D B(t) = s B 1 − + s I − s B + g BE,
t
K
 1 0



C
0  γ D E(t) = Q 0 + lP − g 0 E,
α 3 α 1 t
α 3 α 1
α 3 > 0, > 0, 1 α 2 α 0 > 0, (24)

1
α 2
0
α 3 α 1 γ C
where D is the Caputo-type derivative operator
t
α 3 α 1
0 0 of fractional order γ. To check its existence and

1 α 2 α 0 0 uniqueness and to perform other numerical anal-
> 0. (21)
0 0
α 3 α 1 yses on the newly proposed model 24, we rewrite

0 1
α 2 α 0 it in the following short form:
5