Biology Reference
In-Depth Information
model is supposed to predict. Here we will be interested only in the question of how
the proportion of infected individuals in the population changes over time.
SIRmodels come in a variety of flavors; in particular, there are a lot of details to con-
sider that differ from disease to disease. In actual modeling, these details are inferred
from the available data and the model is constructed by deriving suitable assumptions
from the data. Here we do not have the space to consider actual data for a disease;
instead let us consider some modeling assumptions that one might have deduced:
1.
An infected individual will be infectious for a time period
T
0
with mean value
E
and very small standard deviation.
2.
The time lag between infection and becoming infectious is very small relative
to
E
(
T
0
)
so that it may be negligible.
3.
After ceasing to be infectious, an individual will remain immune to the disease
for a time period
T
1
with
E
(
T
0
)
(
T
1
)
≈
pE
(
T
0
)
where
p
is a positive integer given
of
T
1
is small.
4.
Contact between an infected and a susceptible individual during the time inter-
val of infectiousness will result in a new infection with probability
q
.
by the data and the standard deviation
σ
These are all the assumptions we will be using here. But of course, these assump-
tions already gloss over a lot of details that might (or might not) significantly influence
the disease dynamics. Before reading about howwe build a model from these assump-
tions, you may want to take a few minutes to do the following:
Exercise 6.16.
List some details that are being ignored by the assumptions above
but may significantly influence how the disease will spread in the population.
Since the length of time an individual remains infectious has “very small standard
deviation” and the time lag between the moment of transmission and becoming infec-
tions may be negligible, we might try to work with a discrete-time model where the
unit of time is chosen as
T
0
. Now consider a network
N
,
1
=
,
with
V
D
=[
]
D
p
n
and constant
p
=
(
p
,...,
p
)
. Let us interpret each
i
∈[
n
]
as an individual of a fixed
population, interpret state
s
i
(
t
)
=
0 as individual
i
being infectious at time
t
, state
s
i
(
t
)
=
p
as individual
i
being susceptible to infection and state 0
<
s
i
(
t
)<
p
as
individual
i
being immune to infection. An arc
A
D
indicates that individual
i
interacts with individual
j
. It is natural to assume here that
A
D
is symmetric, that
is, either both arcs
i
,
j
∈
are in
A
D
or neither of them is, but as we will see in
Project 8 [
1
] this assumption is not actually needed.
Will the dynamics of
N
correspond to the dynamics of the disease? Possibly. Notice
that
s
i
(
i
,
j
,
j
,
i
, which nicely corresponds to being
immune from the disease for a time period of
pT
0
.
However, in the above interpretation of
N
an individual
j
that is susceptible at
time
t will
become infectious at time
t
t
)
=
0 implies
s
i
(
t
+
τ)
=
0for
τ
∈[
p
]
A
D
with individual
i
being infectious at time
t
. This would imply that the transmission
probability
q
is 1 and either
i
+
1 whenever there is an arc
i
,
j
∈
,
j
always interact during a time interval of length
T
0
(when
A
D
). These
assumptions are too extreme to be realistic. In practice, there will always be an element
i
,
j
,
j
,
i
∈
A
D
), or
i
,
j
never interact (when
i
,
j
,
j
,
i
∈
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