Epidemiological Models with Seasonality - Mathematical Methods and Models in Biomedicine - page 390

Hence, from Eq. ( 36 ),

e n ωλ 1 0
0

Qz

Qz ( T 0 + n ω + τ )= ( T 0 + τ ) . (37)

nw λ 2

(

We next observe that q 11 , q 12 ) is an eigenvector of E corresponding to λ

1 .One

> λ >

can verify directly that, since 0, q 11 and q 12 have the same sign; this

also follows from the general theory mentioned in the paragraph following Theorem
2.1. Since the i -th component of the right-hand side of Eq. ( 37 ) is equal to

λ

2 ,

λ

1 1

e nw λ 1

[ q i 1 i ( T 0 + τ )+ q i 2 w ( T 0 + τ )] ,

the assertion ( 34 ) then follows from Eqs. ( 35 )and( 37 ).

Lemma 6.7. For any sufficiently small η (say 0 < η < η ) and arbitrarily small

0

m ( 0 )( m ( 0 ) > 0 ) ,set

BT 0 ( η )+

1

+

1
λ 1 ω

η
c 1 m

T 1 = T 1 ( η , m ( 0 )) = T 0 ( η )+ ln + , (38)

( 0 )

where T 0 ( η ) is defined by Eq. ( 25 ). Then there exists a first time t 1 , 0 < t 1 ≤ T 1

such that

M ( t 1 ) > η . (39)

Proof. We proceed by contradiction, assuming that Eq. ( 39 ) is not true. Then M ( t ) <

η T 1 . Hence we can apply

Lemma 6.6 to deduce that Eq. ( 34 ) holds, and, by Lemma 6.6 ,wethenhave

for all 0 ≤ t ≤ T 1 and, in particular, i ( t ) < η for 0 ≤ t ≤

c 1 e λ 1 n ω m c 1 e λ 1 n ω e − BT 0 m M ( T 0 + n ω ) ≥ ( T 0 ) ≥ ( 0 ) .

But the right-hand side is larger than η if

BT 0 + +

η
c 1 m

λ 1 n ω > ln ,

( 0 )

i.e., M ( t ) > η for t = T 1 , which is a contradiction.

Proof of Theorem 4.2 . By L emm a 6.7 there exists a t 1 , t 1 ≤ T 1 ( η , m ( 0 )) such that

Eq. ( 39 ) holds, i.e., max ( i ( t 1 ) , w ( t 1 )) > η . Consider first the case

i ( t 1 ) > η .