# E the pair plays this role.B.two. Simplifications of fundamental pairwise

So we expect some alter of variables to show that it can be equivalent to the compact pairwise model.NIH-PA Author o-Veratraldehyde; 5,6-DimethoxybenzaldehydeBiological Activity manuscript NIH-PA Author Manuscript NIH-PA Author ManuscriptIt was previously noted [12] that if we make the generic assumption that where [AkB] represents the number of partnerships between men and women of status A obtaining k partners and men and women of status B, then a pairwise strategy can be applied to derive an early version on the EBCM model [47]. Author manuscript; offered in PMC 2015 January 08.Miller and KissPageOur derivation on the EBCM model was based around the pairs closure. So we anticipate some change of variables to show that it is equivalent to the compact pairwise model.NIH-PA Author Manuscript NIH-PA Author Manuscript NIH-PA Author ManuscriptIt was previously noted [12] that if we make the generic assumption that where [AkB] represents the number of partnerships between men and women of status A obtaining k partners and people of status B, then a pairwise method is often employed to derive an early version with the EBCM model [47]. Generally, we count on this assumption to fail if A is either I or R. However, within the particular case where status A is susceptible, the assumption is constant: No matter the degree of an individual, it has no effect on the status of its neighbors so lengthy since it remains susceptible. We don't have to have the general form on the closure for our derivation, just the distinct case using a = S. In the derivation of your compact pairwise model, we claimed that I, the probability a companion of a susceptible person u is infected, is independent of k. This follows from the pairs closure, but we didn't prove that if we get started with the standard pairwise model and assume this probability is independent of k, then it remains independent of k at all later occasions. To address this, we turn to I k = [SkI]/k[Sk] and Sk = [SkS]/k[Sk]. We'll take the derivative of Ik, and show that if these are is initially k-independent, then its derivative is kindependent. We haveSo we see that if Ik and Sk are independent of k at a offered time, then the derivative of Ik is also independent of k. A similar calculation shows that the derivative of S k is independent of k. Thus we conclude that if at any time all Ik and all Sk are independent of k remain so for title= title= journal.pone.0159633 abstract' target='resource_window'>jivr.v8i2.812 future time.Math Model Nat Phenom. Author manuscript; readily available in PMC 2015 January 08.Miller and KissPageThis combined using the work in the principal text shows that if Ik and Sk are initially kindependent (equivalently, the pairs closure holds), then the basic pairwise model reduces to the compact pairwise model. We're now ready to derive the EBCM equations in the compact pairwise model. B.two.two. Deriving EBCM model from compact pairwise model We title= cas.12979 now derive the EBCM model from the compact pairwise model. We later derive the compact pairwise model in the EBCM model. We start our derivation using the observation that (for all k) [k] = ?k Sk [Sk]. So . We define .