Epidemiological Models with Seasonality - Mathematical Methods and Models in Biomedicine - page 386

[ ] <

where n is any positive integer. Since R 0 1, both eigenvalues of A have negative

< − λ

real parts, say ,sothat

const. e − λ t

| ( ) |≤ .

z t

From Eq. ( 21 ) we then conclude that ( i ( t ) , w ( t )) → 0as t → ∞ ;thenalso r ( t ) → 0

as t → ∞ , and the assertion of Theorem 4.1 follows.

6 Proof of Theorem 4.2

We shall need the following lemma.

Lemma 6.1. Let a, b be any positive numbers.

(i) If

d z
d t +

az ≤ b for 0 < t < Λ

b
a ,thenz

b

and 0 ≤ z ( 0 ) ≤ z 0 , 0 < ε < z 0 − ( t ) ≤

a + ε

for all T 0 < t < Λ where

− /

1
a

z 0 b a

T 0 = ln .

ε

(ii) If

d z
d t +

az ≥ b for 0 < t < Λ

b
a −

b
a − ε

and z ( 0 ) ≥ z 0 ≥ 00 < ε < z 0 ,thenz ( t ) ≥ for all T 1 < t < Λ where

b

a −

z 0

1
a

T 1 = ln .

ε

The proof follows immediately by integration.
In the sequel we shall use the bounds

0 < α 1 ≤ β i ( t ) ≤ α 2 ( i = 1 , 2 ) , 0 < ξ 1 ≤ ξ ( t ) ≤ ξ 2

Lemma 6.2. Let η be any small positive number such that η < μ / ( 2 γ ) .If

i ( t ) < η for 0 < t < Λ , (23)