Page 10 - IJOCTA-15-1

P. Kumar / IJOCTA, Vol.15, No.1, pp.1-13 (2025)

α 2 βΛ and
β 1 + I − − (α + ν + δ)
δ δ Q 0 glΛ
s − s 0 + g
+

α λΛ dI g 0 g 0 δ
−λB 1 + I − B = 0, (10) = > 0
δ δ dB (B 1 ,0) s 1 − glα
g 0 d B 1
s 2 Q 0 glΛ
K B − s − s 0 + g + B s 1 g 0 δ
g 0 g 0 δ if B 1 < . (14)
I = . (11) glα
s 1 − glα B
g 0 δ
dI s 1 g 0 δ
(iv) → ∞ when B = .
From (10), we get the following results: dB glα
From the above given conditions, a positive in-
¯ ¯
tersecting point (B, I) will be received under the
(i) For the case B = 0, I = 0 or I =
βΛ − (α + ν + δ) inequality (14), i.e. when
δ = I 1 (say), we notice that
α
β 1 + ) K glΛ s 1 g 0 δ
δ Q 0
βΛ s − s 0 + g + < . (15)
I 1 < 0, when < (ν + α + δ) and I 1 > 0 s g 0 δ glα
δ g 0
otherwise.
¯
¯
Then, E and P can be estimated from (9) as
Λ
¯
Λ I < . □
(ii) B → ∞ when I → . α
α + δ ∗
Theorem 2. The equilibrium point E is unsta-
1
∗
ble and E is stable provided α 3 (α 2 α 1 − α 0 α 3 ) −
(iii) At (0, 0), the slope: 2 2
α > 0, where α 0 , α 1 , α 2 and α 3 are defined in
1
the given proof.

dI
= Proof. Firstly, we define the variational matrix
dB
M associated to the model (7), given by

−λΛ
δ
 βΛ
 < 0, if > (ν + α + δ),  
βΛ − (ν + α + δ) δ A βI + λB λ(P − I) 0
δ
 βΛ  −α −δ 0 0 
> 0, if < (ν + α + δ). M =  

δ  s 1 0 s − s 0 + gE − 2sB gB 
(12) K
0 l 0 −g 0
(16)
Also, at (0, I 1 ), the slope is
where A = βP − 2βI − λB − (α + ν + δ).
Similarly, the corresponding variational ma-

dI
= trix M 1 to the model (7) at equilibria
dB Q 0 + lΛ
Λ
∗
E 0, , 0, δ is defined by
λ(ν + α + δ) βΛ g 0
 1 δ
< 0, if > (ν + α + δ),

βΛ δ
β( − (ν + α + δ))  βΛ 
δ − (ν + α + δ) 0 λΛ 0
 βΛ δ δ
> 0, if < (ν + α + δ).
δ  −α −d 0 0 
 
(13) M 1 =  Q 0 + lΛ  .
 s 1 0 s − s 0 + g δ 0 
g 0
0 l 0 −g 0
From (11), we have the following results: (17)
Then, the two eigen values are −δ and −g 0 and
the other values can be calculated from the fol-
(i) I = 0, in the case B = 0 or
lowing quadratic equation
K Q 0 glΛ
B = s − s 0 + g + = B 1 > 0, which
s g 0 g 0 δ lΛ
is positive. ψ − βΛ − (ν + α + δ) + s − s 0 + g Q 0 + δ ψ
2
s 1 g 0 δ δ g 0
(ii) I → ∞ when B = .
glα βΛ Q 0 + lΛ
(iii) + − (ν + α + δ) s − s 0 + g δ
δ g 0
Q 0 glΛ
s − s 0 + g + Λ
dI g 0 g 0 δ − λs 1 = 0.

= − < 0 δ
dB s 1 (18)
(0,0)
4