Epidemiological Models with Seasonality - Mathematical Methods and Models in Biomedicine - page 392

Proceeding similarly step-by-step, we construct a sequence { t m } such that

m ( t m ) ≥ c 2 η , 1 < t m − t m − 1 ≤ c ( η )+ c 3 ( m = 3 , 4 ,... ) .

Using once more Lemma 6.5 ,weget

e − B ( c ( η )+ c 3 ) for all t m ( t ) ≥ c 3 η > t 1 ,

and this completes the proof of Theorem 4.2 with sufficiently small δ .

7 Proof of Theorem 4.3

We shall use the following Horn's Fixed Point Theorem [ 5 ]; see also [ 4 ] where this
theorem is applied to epidemic models with seasonal contact rate.

Theorem 7.1 ([ 5 ]). Let X 0 ⊂ X 2 be nonempty convex sets in a Banach space X

such that X 0 and X 2 are compact and X 1 is open relative to X 2 . Let W be a continuous
mapping X

X 1 ⊂

X such that, for some positive integer m, W j → ( X 1 ) ⊂ X 2 for 1 ≤ j ≤

1 , and W j 1 ,whereW j m − ( X 1 ) ⊂ X 0 for m ≤ j ≤ 2 m − = W ◦ W ◦···◦ W .ThenW

j − times

has a fixed point in X 0 .
Proof. We take the Banach space X to be

4

R with points z =( s , i , w , r , ) ,and

introduce the convex sets

X 2 = Ω ( Ω as in ( 4 . 2 )) ,

i

1
2 δ ,

1
2 δ

X 1 = Ω ∩ > w > ,

X 0 = Ω ∩{ i ≥ δ , w ≥ δ },

where is any sufficiently small positive number. Clearly X 0 , X 1 , X 2 are convex

sets, X 0 and X 2 are compact, and X 1 is open relative to X 2 .Forany z 0 in X we define
z

δ

( t ) to be the solution of (4.3) with z ( 0 )= z 0 , and introduce the mapping

W : z 0 → z ( ω ) .

maps X into X and, clearly, W j

Then W ( X 1 ) ⊂ X 2 for all j = 1 , 2 , 3 ,... .By

T 0 δ , 2 where T 0 ( η , Theorem 4.2 , W j

is the function
defined in Eq. ( 38 ). Hence the conditions in Horn's theorem are satisfied with m
sufficiently large. We conclude that W has a fixed point in X 0 , which is the periodic
solution asserted in Theorem 4.3 .

( X 1 ) ⊂ X 0 if j ω > m ( 0 ))