Biology Reference
In-Depth Information
700
An interesting feature of this model is that for certain values of the
parameters (and for certain initial conditions) convergence to a steady
periodic cycle is possible. This is demonstrated in Figure 2-14
by presenting the trajectories of I(t) and S(t) and a phase diagram of
I versus S. This behavior of the solution may be interpreted in the
following way. For many infectious diseases, periods of acute epidemic
outbreaks may be separated by relatively quiet periods, such as in the
measles data depicted in Figure 2-2. A mathematical model capable of
describing such events should have oscillating trajectories. Note that in
simulations involving delay, initial conditions for some of the
functions must be provided for an entire interval of length D
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B. Epidemic Models with an Intermediate State
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For the SIR and the SIS models, we assumed that an infected person
becomes instantaneously infectious. For many infectious diseases (such
as measles and AIDS), a latent or incubation period exists, during which
the person is infected but is not yet infectious. One way to modify the
SIR model to incorporate this scenario is to introduce a new group E for
individuals who are infected but not yet infectious. When an individual
is infected, he or she is moved to group E and remains there until he or
she becomes infectious.
FIGURE 2-14.
Numerical solution of the SIR model with delay
described by Eq. (2-10) that exhibits a robust
periodicity for large values of t for a ¼ 0.002,
b ¼ 0.55, D ¼ 10, and l(t) ¼ l(0) for t in the
interval [D, 0]. Panel A shows the time
trajectories for S and I for large t while panel B
depicts the (S, I)-phase trajectory.
If the incubation period does not vary significantly, we may assume that
all individuals from group E move to the group of infectious I after a
time period D, where D represents the approximate value of the
incubation or latent period. This will result in a model with delay. For
some diseases, this assumption will be close to the truth. For example,
the incubation period for measles varies from 8 to 13 days, with the bulk
of cases being close to 9 or 10 days. On the other hand, this assumption
will clearly be unjustified if we consider a disease where the length of
delay varies widely. For example, in AIDS the incubation periods range
from months to decades.
Where the duration of the incubation period varies significantly, we may
assume that a certain proportion of the exposed group becomes
infectious over a unit time interval. This is similar to the assumption in
the SIR model regarding the flow of infectives into group R. We leave
the detailed mathematical description of these models as an exercise.
E
XERCISE
2-9
Modify the SIR model by adding a new group E that consists of infected
individuals who are not yet infectious. Give the block diagram and the
mathematical equations. Consider the following two cases separately:
(a) All infectives spend a fixed time D
>
0 in group E and are moved
to the group of infectious I after that.
