Here, the quantities S, I and R denote the numbers of susceptible, infectious and recovered individuals. The total population size, N, is constant. Births and deaths are assumed to be unimportant in this form of the model: such an assumption is appropriate if the timescale on which the epidemic plays out is short compared to the demographic timescale.
In the well-mixed setting, the transmission process is described by the the mass-action term, ftcSI/N. Here, the parameter c depicts the rate at which any single individual makes contacts and the parameter ft is the probability that infection would be transmitted during any one such contact. The simplest description of recovery assumes that infectious individuals recover at a constant rate, 7. We remark that this description of recovery implies that the duration of infectiousness is exponentially distributed with average 1/7. (This distribution is somewhat unrealistic biologically.)
The corresponding network model can be formulated in an analogous way. The simplest description of infection assumes that there is a constant rate (i.e. probability per unit time) at which an infective can infect a given susceptible with whom they interact, and that this rate is identical for each edge in the network. Writing this rate as ft, and noting that the interpretation of this parameter is slightly different in the network setting, the probability of transmission along a given edge over a short period of time, dt, is equal to ft dt. Taking the recovery rate to be constant, as above, implies that an infectious individual has probability 7 dt of recovering over the time interval of length dt.
More general descriptions of infection and recovery are possible, such as allowing for a delay—known as the exposed period—between acquisition of infection and the start of infectiousness, or the inclusion of non-exponential distributions of infectiousness.
If the system is to be studied over a long time period, it may be necessary to include some description of the demographics (births and deaths) of the population. Deaths can be simulated by removing nodes from the network, births by adding nodes to the network.
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