Epidemiological Models with Seasonality - Mathematical Methods and Models in Biomedicine - page 387

then

2 γ

μ η

( ) < < < Λ ,

r t for T 0 t (24)

where

μ
γη −

1

1
μ

T 0 = T 0 ( η )= ln . (25)

Furthermore,
i

e ( n ω A − c η J ) i

if t

( t ) T 0 + τ )

ω (

(

≥ = T 0 + n ω + τ < Λ (26)

w ( t ) T 0 + τ )

for any positive integer n and 0 ≤ τ ≤ ω , where c is a positive constant independent

and J is the matrix 11
00

.

of η

Proof. From Eqs. ( 17d )and( 23 )wehave

d r
d t + μ

≤ γη , ≤ < Λ .

r 0 t

− γη
μ

Suppose r ( 0 ) < 1. Applying Lemma 6.1 (i) with z 0 = 1, ε = 1 ,the

( )=

inequality ( 24 ) follows with T 0 as in Eqs. ( 17a )-( 17d ). The case r 0 1 follows by

approximation.
Since s

= − −

1 i r ,

d i
d t = β

( ) + β ( ) − γ − μ + , ≤ < Λ ,

t i t w i i F T 0 t 1 2

where, by Eqs. ( 23 )-( 24 ), | F |≤ const . η ( i + w ) ≤ const . 2 η . Hence

d i
d t
d w
d t

β 1 ( i

w

t ) − γ − μ − c ηβ 2 ( t ) − c η

≥

ξ ( t ) − ξ ( t )

i
w

=( A ( t ) − c η J ) , (27)

where c is a constant independent of η ,andEq.( 26 ) follows by integration.

Lemma 6.3. If i ( t ) ≥ η for 0 ≤ t < Λ ,forsome η > 0 ,then

) ≥ ξ 1
2

1
ξ 2 ln 2

w ( t

ξ 2 η

for ≤ t < Λ . (28)

Proof. Since

d w
d t + ξ 2 w

≥ ξ 1 i ≥ ξ 1 η ,