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

Pairwise SIS approximation model (page 285)

The simplest form of a pair-based approximation model is used to capture disease spread through a network of contacts (see Section 7.6 of this chapter). In its most basic form, this pair-wise model assumes an equal number of contacts per individual and no clustering; this approximation therefore corresponds most closely to the random network, although adaptations that capture clustering or heterogeneities are possible.
The method operates by modelling the number of susceptible-infected pairs in the network as well as simply the number of individuals. In principle to calculate the behaviour of pairs we need to know about triples:

where the arrows indicate the direction of transmission. However, by making a suitable approximation,

we can formulate a set of equations that capture the spatial correlations due to network structure and yet only contain one extra state variable:

is the number of connections per individual in the population
is the transmission rate across a contact
is the recovery rate for infectious individuals
is the  initial number of infected individuals in the population. Obviously [X](0)=N-[Y](0)
is the initial number of X-Y pairs; we set [XY](0)=n[X][Y]/N
is the number of individuals in the population.

All parameters must be positive.

Python ProgramMATLAB Code.

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