Compartmental models Compartmental models are a general modeling technique often applied to the
mathematical modeling of infectious diseases. In these models, population members are assigned to 'compartments' with labels – for example, S, I, or R, (Susceptible, Infectious, or Recovered). These models can be used to estimate R_0 .
Epidemic models on networks Epidemics can be modeled as diseases spreading over
networks of contact and disease transmission between people. Nodes in these networks represent individuals and links (edges) between nodes represent the contact or disease transmission between them. If such a network is a locally tree-like network, then the basic reproduction can be written in terms of the
average excess degree of the transmission network such that: R_0 = \frac{\beta}{\beta+\gamma} \frac{{\langle k^2 \rangle} -{\langle k \rangle}}, where {\beta} is the per-edge transmission rate, {\gamma} is the recovery rate, {\langle k \rangle} is the mean-degree (average degree) of the network and {\langle k^2 \rangle} is the second
moment of the transmission network
degree distribution.
Heterogeneous populations In populations that are not homogeneous, the definition of R_0 is more subtle. The definition must account for the fact that a typical infected individual may not be an average individual. As an extreme example, consider a population in which a small portion of the individuals mix fully with one another while the remaining individuals are all isolated. A disease may be able to spread in the fully mixed portion even though a randomly selected individual would lead to fewer than one secondary case. This is because the typical infected individual is in the fully mixed portion and thus is able to successfully cause infections. In general, if the individuals infected early in an epidemic are on average either more likely or less likely to transmit the infection than individuals infected late in the epidemic, then the computation of R_0 must account for this difference. An appropriate definition for R_0 in this case is "the expected number of secondary cases produced, in a completely susceptible population, produced by a typical infected individual". The basic reproduction number can be computed as a ratio of known rates over time: if a contagious individual contacts \beta other people per unit time, if all of those people are assumed to contract the disease, and if the disease has a mean infectious period of \dfrac{1}{\gamma}, then the basic reproduction number is just R_0 = \dfrac{\beta}{\gamma}. Some diseases have multiple possible latency periods, in which case the reproduction number for the disease overall is the sum of the reproduction number for each transition time into the disease. ==Effective reproduction number==