This model was for the first time proposed by William Ogilvy Kermack and Anderson Gray McKendrick as a special case of what we now call Kermack–McKendrick theory, and followed work McKendrick had done with Ronald Ross. This system is non-linear, however it is possible to derive its analytic solution in implicit form. Other numerical tools include Monte Carlo methods, such as the Gillespie algorithm. Firstly note that from:  it follows that:  expressing in mathematical terms the constancy of population formula_8. Note that the above relationship implies that one need only study the equation for two of the three variables. Secondly, we note that the dynamics of the infectious class depends on the following ratio:  the so-called basic reproduction number (also called basic reproduction ratio). This ratio is derived as the expected number of new infections (these new infections are sometimes called secondary infections) from a single infection in a population where all subjects are susceptible. This idea can probably be more readily seen if we say that the typical time between contacts is formula_10, and the typical time until removal is formula_11. From here it follows that, on average, the number of contacts by an infectious individual with others "before" the infectious has been removed is: formula_12  By dividing the first differential equation by the third, separating the variables and integrating we get  where formula_14 and formula_15 are the initial numbers of, respectively, susceptible and removed subjects. Writing formula_16 for the initial proportion of susceptible individuals, and formula_17 and  formula_18 for the proportion of susceptible and removed individuals respectively in the limit formula_19 one has   (note that the infectious compartment empties in this limit). This transcendental equation has a solution in terms of the Lambert function, namely   This shows that at the end of an epidemic, unless formula_22, not all individuals of the population have been removed, so some must remain susceptible. This means that the end of an epidemic is caused by the decline in the number of infectious individuals rather than an absolute lack of susceptible subjects. The role of the basic reproduction number is extremely important. In fact, upon rewriting the equation for infectious individuals as follows:  it yields that if:  then:  i.e., there will be a proper epidemic outbreak with an increase of the number of the infectious (which can reach a considerable fraction of the population). On the contrary, if  then  i.e., independently from the initial size of the susceptible population the disease can never cause a proper epidemic outbreak.