Epidemiological Models with Seasonality - Mathematical Methods and Models in Biomedicine - page 391

Then, by Lemma 6.5 ,

e − Bt i ( t + t 1 ) > η for all t > 0 ,

and by Lemma 6.3 we deduce that

t 1 ) > ξ

1
ξ 2 ln 2 ξ 2 e − Bt 0

1

t 0 +

where t 0 = w ( 1 + .

2

Setting t 1 =

t 0 +

t 1 , it follows that

t 1 ) ≥ ξ

1

t 1 ) ≥ η e − Bt 0 e − Bt 0 i ( , w (

ξ 2 η

. (40)

2

We next consider the case

w ( t 1 ) > η .

By Lemma 6.5 ,

e − Bt w ( t + t 1 ) > η for all t > 0 ,

and, by Lemma 6.4 ,

e − Bt 0
where t 0 is defined by Eq. ( 31 ). Hence, setting t 1 =

i ( t 0 + t 1 ) > c η

t 0 + t 1 we get,

t 1 ) > η e − Bt 0 t 1 ) ≥ e − Bt 0 w ( , i ( c η . (41)

We have thus proved that Eq. ( 39 ) implies that either Eq. ( 40 )orEq.( 41 )must
hold, and, by Lemma 6.5 , we conclude that

t 1 , t 1 }, m ( t 1 ) ≥ c 2 η , t 1 = max { (42)

and, since we can take t 2 > 1 + t 1 ,

1 < t 1 − t 1 < c 3 , (43)

where the constants c 2 , c 3 are independent of .

We now repeat the previous argument starting at t

η

= t 1 with m ( 0 ) replaced by

m ( t 1 ) . We then need to replace T 1 ( η , m ( 0 )) by T 1 ( η , m ( t 1 )) . Note that whereas

T 1 ( η , m ( 0 )) → ∞ if m ( 0 ) → 0, T 1 ( η , m ( t 1 )) is bounded from above by a constant

c ( η ) which depends only on η . We conclude that there exists a point t 2 such that

m ( t 2 ) > c 2 η , 1 < t 2 − t 1 < c ( η )+ c 3 ,

where the constants c 2 , c 3 are the same constants as in Eqs. ( 42 )and( 43 ).