Biomedical Engineering Reference
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a constraint on the applicability of the model. Stochastic transmission (including a
Poisson process) in a small population (close to extinction), for example, would not
be described by the SIS model.
From
N
, we obtain that the total host population
dynamics in the presence of the disease are described by Model (
1
). In the absence of
the disease, the susceptible host population dynamics are described by the following
single species model:
(
t
+
1
)=
S
(
t
+
1
)+
I
(
t
+
1
)
S
(
t
+
1
)=
f
γ
(
S
(
t
)) =
γ
(
S
(
t
)+
f
(
S
(
t
)))
.
When the recruitment function is either constant or geometric growth or Beverton-
Holt or Ricker function, then
f
(
S
)
>
0 whenever
S
>
0. Hence, in Model (
5
),
S
(
0
)
>
0and
I
(
0
)=
0 implies that
S
(
t
)
>
0and
I
(
t
)=
0for
t
=
1
,
2
,....
In the present chapter, we use Model (
5
) to study the relationship between the
demographic and disease dynamics in a periodically forced frequency-dependent
discrete-time SIS model. When the transmission rate is constant, then Model (
5
)
reduces to that of Castillo-Chavez and Yakubu [
8
-
11
]. Others have studied discrete-
time SIS models with constant transmission and periodic survival rates [
2
,
3
,
17
,
18
,
29
,
30
].
4
Disease Extinction Versus Disease Persistence
Classical theory of disease epidemics usually involves computation of an epidemic
threshold parameter, the basic reproductive number
ℜ
0
[
2
-
4
]. In this section, we
introduce
ℜ
0
for Model (
5
), the SIS model with periodic disease transmission. In
this section, we study Model (
5
), where the recruitment function is either constant
or described by the Beverton-Holt model.
4.1
Asymptotically Constant Recruitment Functions
k
(
1
−
γ
)
γ
When the recruitment function is either
f
(
N
)=
(constant recruitment) or
(Beverton-Holt model), then
the total population is asymptotically constant. That is, long-term demographic
effects disappear in Model (
1
) and lim
t
f
(
N
)=((
1
−
γ
)
μ
kN
)
/
(
γ
((
1
−
γ
)
k
+(
μ
−
1
+
γ
)
N
))
N
(
t
)=
N
∞
>
0. In this case, letting
S
(
t
)=
→
∞
N
∞
−
(
)
≥
−
equation
of Model (
5
) gives rise to the following one-
dimensional equivalent non-autonomous “limiting system” [
6
,
18
,
32
].
I
t
0inthe
I
1
β
t
I
(
t
)
I
(
t
+
1
)=
γ
(
1
−
σ
)
I
(
t
)+
−
φ
(
N
∞
−
I
(
t
))
,
(7)
N
∞
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