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From: cctimar@watrose.UUCP (Cary Timar)
Newsgroups: net.puzzle,sci.math,sci.med
Subject: Re: Math of Diseases
Message-ID: <8246@watrose.UUCP>
Date: Mon, 3-Nov-86 21:03:37 EST
Article-I.D.: watrose.8246
Posted: Mon Nov  3 21:03:37 1986
Date-Received: Tue, 4-Nov-86 08:14:02 EST
References: <2188@mtuxo.UUCP> <1057@navajo.STANFORD.EDU>
Reply-To: cctimar@watrose.UUCP (Cary Timar)
Organization: U of Waterloo, Ontario
Lines: 31
Xref: mnetor net.puzzle:1564 sci.math:118 sci.med:174

In article <1057@navajo.STANFORD.EDU> avg@navajo.UUCP (Allen Van Gelder) writes:
>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.