Editing Compartmental models (epidemiology) (section)

===Transition rates===
For the full specification of the model, the arrows should be labeled with the transition rates between compartments. Between ''S'' and ''I'', the transition rate is assumed to be <math> d(S/N)/dt = -\beta SI/N^2 </math>, where <math> N </math> is the total population, <math> \beta </math> is the average number of contacts per person per time, multiplied by the probability of disease transmission in a contact between a susceptible and an infectious subject, and <math> SI/N^2 </math> is the fraction of all possible contacts that involves an infectious and susceptible individual. (This is mathematically similar to the [[law of mass action]] in chemistry in which random collisions between molecules result in a chemical reaction and the fractional rate is proportional to the concentration of the two reactants.<ref>{{cite journal |last1=Simon |first1=Cory |title=The SIR dynamic model of infectious disease transmission and its analogy with chemical kinetics |journal=PeerJ Physical Chemistry |date=2020 |volume=2 |issue=2 |pages=e14 |doi=10.7717/peerj-pchem.14 |url=https://peerj.com/articles/pchem-14/|doi-access=free }}</ref>)

Between ''I'' and ''R'', the transition rate is assumed to be proportional to the number of infectious individuals which is <math> \gamma I </math>. If an individual is infectious for an average time period <math> D </math>, then <math> \gamma = 1 / D </math>. This is also equivalent to the assumption that the length of time spent by an individual in the infectious state is a random variable with an [[exponential distribution]]. The "classical" SIR model may be modified by using more complex and realistic distributions for the I-R transition rate (e.g. the [[Erlang distribution]]).<ref name="Krylova2013">{{cite journal | vauthors = Krylova O, Earn DJ | title = Effects of the infectious period distribution on predicted transitions in childhood disease dynamics | journal = Journal of the Royal Society, Interface | volume = 10 | issue = 84 | page = 20130098 | date = July 2013 | pmid = 23676892 | pmc = 3673147 | doi = 10.1098/rsif.2013.0098 | doi-access = free }}</ref>

For the special case in which there is no removal from the infectious compartment (<math> \gamma=0 </math>), the SIR model reduces to a very simple SI model, which has a [[logistic distribution|logistic]] solution, in which every individual eventually becomes infected.