Math of Diseases
j.demont
contagious disease throughout a population. I recently read an article
where an "expert" said that the number of AIDS victim will increase
exponentially *indefinitely*.
It appears to me that for any communicable disease, that while the number
of victims is much, much smaller than the total population, the growth
curve of the number of victims could approximate an exponential curve.
However I speculate that the true curve approximates a hysteresis that
asymtotically approaches the total number of the population. My conjecture
is based (perhaps falsely) on the notion that as the number of people
with the disease increases, the chances of giving the disease to an
uninfected person decreases, but is partially offset by the fact that there
are more carriers avaiable to communicate it. It appears that after a while
most of the carriers will be infecting other carriers.
I would like to model the spread of the disease before it is apparent that
anyone even has it. Therefore I would like to add the constraints that it
is chronically contagious, non-fatal and in no way can be known or guarded
against. I would prefer that this be a discussion on modeling and not on
the peculiarities of the spread of AIDS.
Does anyone in net-land have experience with such modeling or have any
other thoughts on the matter?
Thanks,
Jason De Mont
AT&T
Lincroft, New Jersey
ihnp4!mtuxo!jasond
Allen Van Gelder
>...
>I would like to model the spread of the disease before it is apparent that
>anyone even has it. Therefore I would like to add the constraints that it
>is chronically contagious, non-fatal and in no way can be known or guarded
>against.
>Jason De Mont
>ihnp4!mtuxo!jasond
Think of each person in a population as a vertex in a graph, with edges
to the people he or she comes in contact with. In each time step,
there is a probability that an infected person infects a neighbor in
the graph. Assume one person is affected initially. Measure distance
from that person in terms of path length in the graph. The infection
will evidently tend to spread in a "sphere", but depending on the
assumed graph structure, the growth could be anything from linear
(the graph is a line) to quadratric (a grid) to exponential (a tree).
This model ignores the possibility that a person's set of neighbors
varies over time. It is a Markov process. Good luck on your analysis.
Cary Timar
>Think of each person in a population as a vertex in a graph, with edges
>to the people he or she comes in contact with. In each time step,
>there is a probability that an infected person infects a neighbor in
>the graph. Assume one person is affected initially. Measure distance
>from that person in terms of path length in the graph. The infection
>will evidently tend to spread in a "sphere", but depending on the
>assumed graph structure, the growth could be anything from linear
>(the graph is a line) to quadratric (a grid) to exponential (a tree).
It is best to consider the graph as a random graph. A standard
random graph would work alright for small population modelling.
Now, rather than iterating time, simply draw a second graph on the
same vertex set as the contact graph, representing communication.
Let the edges of the contagion graph occur with probability q if
they are in the contact graph, and probability 0 if not.
Supppose that the disease is only contagious on the tenth day after it
is caught. In this case, the recessional sequence from the first person
models the spread of the disease.
For larger populations, the usual random graph models do not indicate
the geographical closeness factor. That is, consider a graph S (for
state) consisting of a set of disjoint subgraphs T1, ... , Tn (for
towns) with some added vertices. Then for a vertex Joe in T1, the
probability of an edge in the contact graph connecting Joe to some other
vertex Ann is greter if Ann is also in T1.
This is very difficult to model, unless you have statistical knowledge
of how "clumped" the graph should be.