Biomedical Engineering Reference
In-Depth Information
[
]
can be used to estimate the basic
reproduction number
R
0
of Eq. (
10
)forthesameDFE
x
0
.
Examples of epidemiological models for periodically occurring diseases were
given in [
1
,
2
,
4
,
8
-
10
,
12
], and the references therein. In [
1
,
12
] there are examples
where
R
0
Then the question arises whether
R
0
>
[
]
<
[
]
R
0
,andin[
8
] there is an example where
R
0
R
0
. For a malaria
model [
3
]itwasshownthat
[
]
<
<
,
if
R
0
1then
R
0
1
(4a)
if
[
R
0
]
>
1then
R
0
>
1
.
(4b)
This means that
can be used to determine the stability or instability of the DFE
for Eq. (
3
). Such a result is very useful since it is much more difficult to compute
R
0
than to compute
[
R
0
]
.
In this chapter we prove the statements in Eqs. (
4a
)and(
4b
) in the context of a
waterborne disease model. Some of the arguments overlap with those in [
3
]. In the
concluding section we discuss the general features of our proofs and illustrate how
the same methods can be applied to other disease models.
[
R
0
]
2
The Basic Reproduction Number for Autonomous Systems
A fundamental concept in epidemiological studies of infectious diseases is the
concept of the basic reproduction number,
R
0
. It is defined as the expected number
of secondary infections produced by an infective individual in a completely healthy
but susceptible population. If
R
0
<
1,
then the disease will not die out and will persist. For autonomous epidemic models,
R
0
<
1 then the disease will die out, whereas if
R
0
>
1 implies that all the eigenvalues of the Jacobian matrix about the DFE have
negative real parts; if
R
0
>
1, then at least one eigenvalue has a positive real part.
A general autonomous compartmental model was developed in [
11
]. The model
includes
m
disease compartments with population densities
x
1
,...,
x
m
and
n
−
m
non-disease compartments with population densities
x
m
+
1
,...,
x
n
. The dynamical
system has the form
d
x
i
d
t
=
F
i
(
x
)
−V
i
(
x
)
(
1
≤
i
≤
m
)
,
(5a)
d
x
j
d
t
=
(
)
(
+
≤
≤
)
,
g
j
x
m
1
j
n
(5b)
where
x
varies in the space
n
+
=
{
R
x
=(
x
1
,...,
x
n
)
,
x
i
≥
0forall
i
}.
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