Epidemiological Models with Seasonality - Mathematical Methods and Models in Biomedicine - page 389

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Until now we have not used the assumption that R 0 1. We shall now use this

= −

assumption. Setting E A c η J , c as in Eq. ( 27 ), the eigenvalues of the constant

= λ ( η )

matrix E , λ , satisfy i i

< λ , λ >

λ 0 2 1 1

if η is sufficiently small, say η < η 0 .

We shall use the notation

M ( t )= max { i ( t ) , w ( t ) }, m ( t )= min { i ( t ) , w ( t ) }. (33)

Lemma 6.6. Suppose Eq. ( 23 ) holds with 0 < η < min { η 0 , μ / 2 γ } for arbitrarily

small η 0 > 0 .Then

c 1 e λ 1 n ω m

M ( T 0 + n ω + τ ) ≥ ( T 0 + τ ) if T 0 + n ω + τ < Λ , (34)

where c 1 is a positive constant independent of η and n is any positive integer.

Proof. Denote by z ( t ) the solution of

d z
d t =

E ( t ) z , z ( T 0 + τ )=( i ( T 0 + τ ) , w ( T 0 + τ )) ,

( )= ( ) −

where E J . By the proof of Lemma 6.2 the inequality ( 27 ) holds, so

that, by a comparison argument as in Sect. 5 ,

t A t c η

( ) ≤ ( ) , ( ) ≤ ( ) + τ < < Λ .

z 1 t i t z 2 t w t if T 0 t (35)

Next,

⎛ ⎞

T 0 + n ω + τ

⎝ ⎠ z

e n ω E z z ( T 0 + nw + τ )= exp E ( s ) d s ( T 0 + τ )= ( T 0 + τ ) . (36)

T 0 + τ

Let ( Q =( q ij ) be a matrix such that

λ 1 0
0

Q − 1 Q E = D , D = .

λ 2

Then

e n ωλ 1 0
0

e n ω Q E Q − 1

Q e n ω E Q − 1 e n ω D = = = .

n ωλ 2