Periodic Incidence in a Discrete-Time SIS Epidemic Model - Mathematical Methods and Models in Biomedicine - page 409

(b) If ℜ D > 1 and ℜ 0 > 1 , then the proportion of infectives persists uniformly.

.
That is, the proportion of infectives in the total population goes to zero while
the total population is decreasing at a geometric rate.
(d) If

(c) If ℜ D < 1 and ℜ 0 < 1 ,then lim

t → ∞ (

s ( t ) , i ( t )) = ( 1 , 0 )

1 , then the proportion of infectives persists uniformly.
That is, the proportion of infectives in the total population persists uniformly
while the total population is decreasing at a geometric rate.

ℜ D < 1 and ℜ 0 >

)= γ Proof. Since f ( N N , N increases geometrically when ℜ D > 1 while it

decreases geometrically when ℜ D < 1. To establish the result, we prove that if in

addition ℜ 0 < 1, then all solutions ( s ( t ) , i ( t )) of Model ( 9 ) converge to the disease-

free equilibrium point 1 we proceed exactly as in

the proof of Theorem 4.1 and use Theorem 4.6 in [ 18 ] to prove that as t

( 1 , 0 ) as t → ∞ .However,if ℜ 0 >

→ ∞ the

proportion of infective population persists uniformly.

5 Ricker Recruitment Function

If new recruits are governed by the Ricker model,

)= (

1 − γ )

γ

N e k − N

f ( N ,

2

and 0 < k < ,then N ∞ = k for all N > 0 and the total population is asymptotically

− γ

1

constant. If, in addition ℜ 0 > 1, then by Theorem 4.1 , the infective population

persists uniformly. However, if ℜ 0 < 1 , then the infective population goes extinct

(Theorem 4.1 ).
As k increases past

2

, the demographic dynamics becomes periodic and
Theorem 1 no longer applies. In [ 15 ], Elaydi and Yakubu proved that non-trivial
periodic orbits are not globally attracting in autonomous models such as Model
( 1 ). Next, we use the following example, Example 5.1 , to illustrate that when the
infective population is nonzero and the demographic N -dynamics undergoes period-
doubling bifurcations, then the N -dynamics can “drive” both the S -dynamics and
I -dynamics in Model ( 5 ) (see Figs. 2 - 4 ).

1 − γ

Example 5.1. In Model ( 5 ), let

β t I
N

)= (

1 − γ )

γ

e − β t N N e k − N

f ( N and φ = ,

where

t

k ∈ [ 1 , 6 ] , β t = a + b × ( 1 +( − 1 ) ) , a = 20 , b = 10 , γ = 0 . 1and σ = 0 . 01.