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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.

The high-contact assumptions give $A>D$ away from a degeneracy. Since $C(p)=Bp+O(p^2)$, perturbing the larger eigenvalue in the assortatively mixed two-risk-group SIS model givesFor the biologically useful high-risk-core regime $A>\gamma >D$, put $x_0=1-\gamma /A$. Substituting $x=x_0+p{x}_1+O(p^2)$ and $y=p{y}_1+O(p^2)$ into the two equilibrium equations givesThus the group susceptible fractions and their population-weighted total areThe small-core SIS endemic expansion shows why the homogeneous SIS susceptible fraction relation fails for a mixed population: as the high-risk fraction tends to zero, $s_{\mathrm{total}}\to 1$ while $R_0\to A/\gamma >1$. A small high-risk core sustains infection even though almost everyone in the population remains susceptible. Even $s_H$ need not equal $1/R_0$ at first order.

For completeness, if $A,D<\gamma$ with a fixed gap from threshold, small $p$ gives disease-free equilibrium and total susceptibility one. If $D>\gamma$, the low-risk group already sustains infection. Put $y_0=1-\gamma /D$ and let $x_0\in (0,1)$ be the positive root of $(1-x_0)(A{x}_0+B{y}_0)=\gamma x_0$. The first-order coefficients areAgain it tends to $\gamma /D$ rather than $\gamma /A=1/R_0(0)$. These regular expansions require fixed nonzero gaps. At $A=D$ the spectral splitting is generally order $\sqrt{p}$; at $D=\gamma <A$, low-risk infection is also order $\sqrt{p}$, so a regular first-order Taylor expansion is inappropriate. An exact endemic replacement for the scalar identity is that the susceptible-weighted next-generation matrix has spectral radius one.