Biomedical Engineering Reference
In-Depth Information
(c) Beverton-Holt Recruitment:
(
1
−
γ
)
μ
kN
f
(
N
)=
)
,
γ
((
1
−
γ
)
k
+(
μ
−
1
+
γ
)
N
where the constant
1[
7
-
11
,
20
,
27
]. Under the classic Beverton-Holt
recruitment function, zero is an unstable fixed point.
N
∞
=
μ
>
0
and, as in the case of constant recruitment, the total population is asymptotically
constant.
(d) Ricker Recruitment:
k
>
0forall
N
>
)=
(
1
−
γ
)
γ
N
e
k
−
N
f
(
N
,
where
k
is a positive constant [
8
-
11
,
23
-
25
,
27
,
29
,
30
]. Under the classic Ricker
recruitment, zero is an unstable fixed point. If 0
2
1
−
γ
<
k
<
,then
N
∞
=
k
for
all
N
0 and, as in the case of constant recruitment, the total population is
asymptotically constant. However, as
k
increases past
>
2
1
−
γ
, the positive fixed
point,
k
, undergoes period-doubling bifurcations.
The total population is
uniformly persistent
if there exists a constant
η
>
0
f
t
such that lim
t
γ
(
N
)
≥
η
for every positive initial population size;
N
>
0. The total
→
∞
f
t
population is
persistent
if lim
t
0 for every positive initial population size.
Consequently, uniform persistence implies persistence of the population. We note
that, the total population is uniformly persistent whenever the recruitment function
is constant or geometric growth or Beverton-Holt or Ricker function.
γ
(
N
)
>
→
∞
3
SIS Epidemic Model with Periodic Infection
To introduce the discrete-time susceptible-infective-susceptible (
SIS
) epidemic
model, we assume that a nonfatal disease invades and subdivides the target
population into two compartments: susceptibles (noninfectives) and infectives. Prior
to the time of disease invasion, the population dynamics are assumed to be governed
by Eq. (
1
), where the recruitment function is either constant or geometric growth or
Beverton-Holt or Ricker function. Let
S
(
t
)
denote the population of susceptibles;
I
)
denotes the total population size at generation
t
,
N
∞
denotes the demographic
positive steady state or attracting population and
N
0
the initial point on a globally
attracting cycle, when it exists.
We assume that there is no immunity from the disease and there is no disease-
induced mortality. There is no vertical transmission, and both susceptible and
infected individuals reproduce into the susceptible class. Although the SIS model
has no disease-induced mortality, the population does experience death via density
dependence. Susceptible and infected individuals survive (respectively, die) with
constant probability
(
t
)
denotes the population of the infected, assumed infectious,
N
(
t
)
≡
S
(
t
)+
I
(
t
γ
(respectively,
(
1
−
γ
)
) per generation.
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