Periodic Incidence in a Discrete-Time SIS Epidemic Model - Mathematical Methods and Models in Biomedicine - page 407

[50.0, 0.04]

0.04

Infectives

Transmission rate b

50

[42.0, 0.0]

Fig. 1 In Example 4.1 , the total infective population shifts from extinction to persistence as b is
varied between 42 and 50 . On the horizontal axis, 42 ≤ b ≤ 50 and on the vertical axis, 0 ≤ i ≤ 0 . 04

4.2 Geometric Growth Recruitment Function

When the recruitment function is a geometric growth function, we use proportions
to study Model ( 5 ). That is, we introduce the new variables

S ( t )

( )=

) ,

s t

N ( t

and

I ( t )

i ( t )=

) . N ( t

)) = γ

In the new variables, Model ( 5 ) with f ( N ( t N ( t ) becomes

)= γ

μ
( γ + μ ) ,

s ( t + 1

( γ + μ ) ( φ ( β t i

( t )) s ( t )+ σ i ( t ))+

(9)

)= γ i ( t + 1

( γ + μ ) ((

1 − σ ) i ( t )+( 1 − φ ( β t i ( t ))) s ( t )) .

We note that,

1
for all t in Model ( 9 ) implies all solutions live on the positive invariant line

s ( t )+ i ( t )=

{ ( s , i ) ∈ [ 0 , ∞ ) × [ 0 , ∞ ) | s + i = 1 }.

Using the substitution s = 1 − i ,the i -equation in Model ( 9 ) reduces to the “one-

dimensional” equation

γ
( γ + μ ) ((

i ( t + 1 )= 1 − σ ) i ( t )+( 1 − φ ( β t i ( t ))) ( 1 − i ( t )))

where i ( t ) ≤ 1forall t ∈{ 0 , 1 , 2 ,...} .