jasond@mtuxo.UUCP (j.demont) (10/30/86)
I have a question relating to the mathematics of modeling the spread of a 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
leimkuhl@uiucdcsp.cs.uiuc.edu (10/31/86)
I have heard that the course of diseases follows the pattern of an S-curve. That is, after a slow initial rise, it increases exponentially until some saturation point, and then resumes a slow changing behavior. The black plague only halted when the entire susceptible population was infected. Many people developed immunity to the disease without contracting it. Of course the presence of alternative techniques of preventing the spread of AIDs (it is not really very contagious compared to some other diseases) changes the nature of the curve. For example, safe sex and clean needles can effectively prevent the spread of the disease.
avg@navajo.STANFORD.EDU (Allen Van Gelder) (11/03/86)
In article <2188@mtuxo.UUCP> jasond@mtuxo.UUCP (j.demont) writes: >... >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.
cctimar@watrose.UUCP (Cary Timar) (11/04/86)
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.