Epidemiological Models with Seasonality - Mathematical Methods and Models in Biomedicine - page 388

ε = ξ

η

1

Lemma 6.1 (ii) with z 0 = 0, yields the asserted inequality.

2 ξ 2

Lemma 6.4. If

w ( t ) ≥ η for 0 ≤ t < Λ (29)

for some η > 0 ,then

i ( t ) ≥ c η for t 0 ≤ t < Λ , (30)

where c is a positive constant independent of η , and

ln 2

1
μ +

1
γ + μ

t 0 =

1 +

. (31)

2 α

( )

Proof. From the equation for s t ,Eq.( 17a ),

d s
d t +( μ +

) ≥ μ ,

2 α s

2

so that, by Lemma 6.1 (ii) with z 0 = 0, ε = μ / 2 ( μ + 2 α 2 ) ,

μ

s ( t ) ≥ if t 0 ≤ t < Λ , (32)

2 ( μ + 2 α 2 )

where

ln 2
μ +

2 .
Substituting Eqs. ( 28 )and( 32 ) into Eq. ( 17b ), we get

t 0 =

2 α

d i
d t +( γ + μ )

α μη 1

i ≥ α 1 ws ≥

α 2 ) .

2 ( μ + 2

Applying once more Lemma 6.1 (ii) with z 0 = 0and

α μη

1

ε =

α 2 ) , 4 ( γ + μ )( μ + 2

we obtain the inequality ( 30 )for t 0 ≤ t < Λ with t 0 as in Eq. ( 31 ).

Lemma 6.5. Set B = γ + μ + ξ 2 .Then

e − B ( t − τ ) , e − B ( t − τ )

i ( t ) ≥ i ( τ ) w ( t ) ≥ w ( τ )

for all 0 < τ < t < ∞ .

Indeed, this follows from the inequalities

d i
d t ≥− ( γ + μ )

d w
d t ≥− ξ

i , w .