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3- Single Replicating Species

I now describe a very simple model of a single species A propagating in some given environment. Suppose that each cell in the world is either occupied by A or empty. If A occupies a particular cell, it will try to replicate itself into a randomly-chosen neighboring cell. The replication attempt succeeds only if the neighbor is not already occupied by A. The average total rate at which an occupied cell attempts replication (summed over all neighboring cells) is the birth rate . Occasionally, at some death rate , an occupied cell will become empty. Births and deaths associated with various cells are uncorrelated; they occur asynchronously at random times generated according to an exponential distribution. This is well-known to epidemiologists as the SIS (susceptible infected susceptible) model [15]. In this case, A is some infectious agent, and the cells are hosts among which it spreads. Upon being cured, (i.e. after a death), a host immediately becomes susceptible again (i.e. there is no immunity). Of course, we need not think of A as a biological or computer virus; A could be a palm tree or a variety of other biological or digital replicators. Nevertheless, for the sake of consistency I shall refer to cells occupied with A as ``infected'' cells.

I shall now examine the behavior of this model (which I shall refer to as ``Model 1'') on several different topologies. Since my specification of the propagation rules does not refer explicitly to the topology, the topological effects can be isolated very cleanly, as will be seen.

• 3.1- Homogeneous Mixing
• 3.2- Random Graph
• 3.3- Local Topologies

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