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Use an assortatively mixed two-risk-group SIS model with equal per-person recovery rate $\gamma$. Let $x,y$ be infectious proportions in groups of sizes $pN,(1-p)N$. Let $c_H\gg c_L$ be their fixed contact rates and $\tau$ the transmission probability per contact. Suppose a small fixed fraction $ϵ<1/2$ of high-risk contacts is with the low-risk group. Contact reciprocity in a two-group epidemic requiresFor sufficiently small $p$, both groups make most contacts within their own group, and the total cross-group contacts are of order $pN$. Define $A=\tau c_H(1-ϵ)$, $B=\tau c_Hϵ$, $D=\tau c_L$ and $C(p)=Bp/(1-p)$. The forces of infection giveHere $C(p)<D$ and ${ϵ}_L<1/2$ hold in the small-$p$ regime. High-risk per-person cross contact is order one but its group size is order $p$; low-risk per-person cross contact is order $p$. These distinctions are needed to count each cross-group partnership consistently.

Linearize about the disease-free state. In infectious fractions the transmission matrix isIn infected-individual counts the next-generation matrix is insteadThe count and fraction formulations are related by a diagonal change of coordinates, so their eigenvalues coincide. The basic reproduction number is the spectral radius $R_0=\rho (M)/\gamma$, not an average of the two group reproduction numbers. The larger real eigenvalue isThis model makes a particular reciprocal mixing convention explicit; different defensible contact laws can change the first-order coefficients.