Editing Basic reproduction number (section)

== Overview of <math>R_0</math> estimation methods ==

=== Compartmental models ===
{{Main|Compartmental models in epidemiology}}
[[Compartmental models in epidemiology|Compartmental models]] are a general modeling technique often applied to the [[Mathematical modelling of infectious disease|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 <math>R_0
</math>.

=== Epidemic models on networks ===
{{Main|Mathematical modelling of infectious disease}}

Epidemics can be modeled as diseases spreading over [[Complex network|networks]] of contact and disease transmission between people.<ref>{{Cite book |url=http://networksciencebook.com/ |title=Network Science by Albert-László Barabási}}</ref> 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 [[Degree distribution|average excess degree]] of the transmission network such that:

<math display="block">R_0 = \frac{\beta}{\beta+\gamma} \frac{{\langle k^2 \rangle} -{\langle k \rangle}}{{\langle k \rangle}},</math>

where 
<math> {\beta} </math>
is the per-edge transmission rate, 
<math> {\gamma} </math>
is the recovery rate, 
<math>
{\langle k \rangle}
</math> is the mean-degree (average degree) of the network and <math>
{\langle k^2 \rangle}
</math> is the second [[Moment (mathematics)|moment]] of the transmission network [[degree distribution]].

=== Heterogeneous populations ===
In populations that are not homogeneous, the definition of <math>R_0</math> 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 <math>R_0</math> must account for this difference. An appropriate definition for <math>R_0</math> in this case is "the expected number of secondary cases produced, in a completely susceptible population, produced by a typical infected individual".<ref name="Diekmann">{{cite journal | vauthors = Diekmann O, Heesterbeek JA, Metz JA | title = On the definition and the computation of the basic reproduction ratio R0 in models for infectious diseases in heterogeneous populations | journal = Journal of Mathematical Biology | volume = 28 | issue = 4 | pages = 365–82 | date = 1990 | pmid = 2117040 | doi = 10.1007/BF00178324 | hdl-access = free | s2cid = 22275430 | hdl = 1874/8051 }}</ref>

The basic reproduction number can be computed as a ratio of known rates over time: if a  contagious individual contacts <math>\beta</math> 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 <math>\dfrac{1}{\gamma}</math>, then the basic reproduction number is just <math>R_0 = \dfrac{\beta}{\gamma}</math>.  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.