Biomedical Engineering Reference
In-Depth Information
[
,
]
On the closed interval
0
1
,let
γ
(
γ
+
μ
)
((
F
1
,
t
(
i
)=
1
−
σ
)
i
+(
1
−
φ
(
β
t
i
)) (
1
−
i
))
.
When the total population is under geometric growth, then the set of sequences
generated by
)=
F
1
,
t
(
i
(
t
+
1
i
(
t
))
(10)
is the set of density sequences generated by the “proportion” of infectives popula-
tion. We note that
1
)
.
γ
(
γ
+
μ
)
F
1
,
t
(
−
σ
−
β
t
φ
(
0
)=
0
Let
−
γβ
t
φ
(
0
)
ℜ
0
,
t
=
−
γ
)+
γσ
.
ℜ
D
(
1
Notice that when
ℜ
D
=
1, then the demographic effects disappear and
ℜ
0
reduces to
φ
(
−
γβ
)
0
t
ℜ
0
,
t
=
−
σ
)
.
1
−
γ
(
1
We note that
1
)
>
γ
(
γ
+
μ
)
F
1
,
t
(
−
σ
−
β
t
φ
(
0
)=
0
1
,
respectively,
1
)
<
γ
(
γ
+
μ
)
F
1
,
t
(
−
σ
−
β
t
φ
(
0
)=
0
1
)
,
is equivalent to
1.
The threshold parameter (basic reproduction number),
ℜ
0
,
t
>
1, respectively,
ℜ
0
,
t
<
t
=
0
F
1
,
t
(
0
)
,
T
−
1
ℜ
=
0
determines the long-term behavior of the disease in Model (
9
) . As in the case
of bounded asymptotic growth (Theorem
4.1
), we obtain that if
1, then the
proportion of infectives in the total population persists uniformly while if
ℜ
0
>
1
the proportion of infectives in the total population decreases to zero regardless of
initial population sizes. We collect these results in the following theorem.
ℜ
0
<
Theorem 4.2.
Consider Model
(
9
)
.
(a) If
.
That is, the proportion of infectives in the total population goes to zero while
the total population is increasing at a geometric rate.
ℜ
D
>
1
and
ℜ
0
<
1
,then
lim
t
→
∞
(
s
(
t
)
,
i
(
t
)) = (
1
,
0
)
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