Biomedical Engineering Reference
In-Depth Information
[
,
∞
)
→
[
,
]
Let
φ
:
0
0
1
be a
monotone concave probability function with
,
φ
(
φ
(
φ
(
)=
)
<
)
≥
∈
[
,
∞
)
.
At generation
t
,we
assume the susceptible individuals become infected with nonlinear probability
(
0
1
N
0and
N
0forall
N
0
−
φ
(
β
(
)
/
(
)))
1
t
I
t
N
t
per generation and infected individuals recover with constant
σ
∈
(
,
)
−
probability
0
1
, where the transmission constant
β
t
is
T
periodic
.That
=
β
.
∈{
,
,...,
−
}
is,
β
For each
t
0
1
T
1
,
β
t
is positive and models the impact
t
+
T
t
of prevalence on
φ
. When infections are modeled as Poisson processes, then the
e
−
(
β
t
I
(
t
))
/
N
(
t
)
[
8
-
11
,
17
,
18
,
29
,
30
].
The periodically forced frequency-dependent discrete-time SIS model implicitly
assumes the ordering of events. At the end of each generation, susceptibles
become infected while infected recover; both susceptibles and infected reproduce
into the susceptible class; a fraction of each class is removed. This important
assumptions distinguish our discrete-time model from a similar continuous-time
differential equation model. Typically, continuous-time differential equation models
with similar well-defined distinct temporal phases are non-autonomous. Taking into
account the temporal ordering of events, we derive our model in the following
three steps.
φ
(
β
(
)
/
(
)) =
“escape” function,
t
I
t
N
t
1. Disease transmission and recovery
⎫
⎬
β
t
I
(
t
)
N
S
S
1
(
t
)=
φ
(
t
)+
σ
I
(
t
)
,
(
t
)
1
β
t
I
(
t
)
N
S
(2)
⎭
I
1
(
t
)=(
1
−
σ
)
I
(
t
)+
−
φ
(
t
)
.
(
t
)
That is, after disease transmission and recovery,
S
1
(
t
)
denotes the susceptible
denotes the infected.
2. Reproduction (both
S
and
I
reproduce into
S
)
individuals and
I
1
(
t
)
S
2
(
t
)=
S
1
(
t
)+
f
(
S
1
(
t
)+
I
1
(
t
))
,
I
2
(
t
)=
I
1
(
t
)
.
That is,
β
t
I
(
t
)
N
(
t
)
S
⎫
⎬
(
)=
φ
(
)+
σ
(
)+
(
(
))
,
S
2
t
t
I
t
f
N
t
1
β
t
I
(
t
)
N
(
t
)
S
(3)
⎭
I
2
(
t
)=(
1
−
σ
)
I
(
t
)+
−
φ
(
t
)
.
That is, after disease transmission, recovery from the disease and reproduction,
S
2
(
t
)
denotes the susceptible individuals and
I
2
(
t
)
denotes the infected.
3. Death/survival
β
t
I
(
t
)
N
S
⎫
⎬
S
3
(
t
)=
γ
S
2
(
t
)=
γ
φ
(
t
)+
σ
I
(
t
)+
f
(
N
(
t
))
,
(
t
)
1
β
t
I
(
t
)
N
S
(4)
⎭
I
3
(
t
)=
γ
I
2
(
t
)=
γ
(
1
−
σ
)
I
(
t
)+
−
φ
(
t
)
.
(
t
)
After disease transmission, recovery, reproduction and survival (death),
S
3
(
t
)
denotes the susceptible individuals and
I
3
(
t
)
denotes the infected.
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