Modeling Infectious Diseases in Humans and Animals
Matt J. Keeling & Pejman Rohani

SIS model on a network (page 280)

Networks provide a unified way to think about the interaction between individuals or populations, and are especially useful when each individual is in direct contact with only a small proportion of the population. The primary advantage of network models is their ability to capture complex individual-level structure in a simple framework. To specify all the connections within a network, we can form a matrix from all the interaction strengths which we expect to be sparse with the majority of values being zero. Usually, for simplicity, two individuals (or populations) are either assumed to be connected with a fixed interaction strength or unconnected (and therefore have an interaction strength of zero). In such cases, the network of contacts is specified by a graph matrix G, where Gij is 1 if individuals i and j are connected, or 0 otherwise. We generally assume that the matrix is symmetric such that a connection allows infection to pass in both directions.
Networks have many similarities with individual-based spatial models (program 7.5 & 7.6), in that spatial interactions can be defined in terms of a kernel. However, in networks, contacts tend to be of equal strength and limited in number. This can be used to considerable advantage in simulations:

where Ij is one if and only if individual j is infectious.

is the population size, randomly distributed in 2-D for visualisation purposes
is the average number of connections per individual in the population
is the transmission rate across a contact
is the recovery rate for infectious individuals
is the type of network to be created , either 'Random', 'Lattice', 'SmallWorld' or 'Spatial'

All parameters must be positive.


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