2- SIS Model on a Random GraphDue to its structural simplicity and the relative ease with which it can be analyzed, the first type of graph on which we shall study virus propagation is a random graph, illustrated in Fig. 1. We construct a random graph of N nodes by making random, independent decisions about whether to include each of the N(N-1) possible directional edges which can connect two nodes. If p is the probability for a particular edge to be included in the graph, the expected total number of edges is pN(N-1).
In our version of the SIS model, each edge has
associated with it an infection rate
Our goal is to study the behavior of the SIS model on a typical member of the class of random graphs with N nodes and edge probability p. In the remainder of this section we explore several different techniques for doing so: the deterministic approximation, approximate probabilistic analysis, and simulation.
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, and the probability per unit time that node 2 will be
cured is
. If node 4 becomes infected by either node 2 or node 6, its probability per
unit time of being cured will be
.
(also referred to as the birth rate
of the virus) at which an infected node j can infect an uninfected neighbor k. Similarly, each
node j has a cure rate
(or death rate of the virus) at which it will become
cured if it is infected. The probabilities per unit time of infection along any edge and of cure
of any node are independent. Once an individual is cured, it is immediately capable of being
reinfected. The standard homogeneous interaction version of the SIS model is easily recovered from
this more general model by connecting all possible pairs of nodes and making all infection and cure
rates identical.