= Solution
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> gives
$$
\boxed{R_0=\frac A\gamma+
\frac{B^2}{\gamma(A-D)}p+O(p^2).}
$$
For the biologically useful high-risk-core regime $A>\gamma>D$, put $x_0=1-\gamma/A$. Substituting $x=x_0+px_1+O(p^2)$ and $y=py_1+O(p^2)$ into the two <equilibrium> equations gives
$$
0=Bx_0+(D-\gamma)y_1,
\qquad
0=-(A-\gamma)x_1+\frac{\gamma B}{A}y_1.
$$
Thus the group susceptible fractions and their population-weighted total are
$$
\boxed{\begin{aligned}
s_H&=\frac\gamma A-\frac{B^2\gamma}{A^2(\gamma-D)}p+O(p^2),\\
s_L&=1-\frac{Bx_0}{\gamma-D}p+O(p^2),\\
s_{\rm total}&=1-px_0\left(1+\frac B{\gamma-D}\right)+O(p^2).
\end{aligned}}
$$
The <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, $s_{\rm total}\to1$ 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 $(1-x_0)(Ax_0+By_0)=\gamma x_0$. The first-order coefficients are
$$
y_1=\frac{B\gamma(x_0-y_0)}{D(D-\gamma)},\quad
x_1=\frac{B(1-x_0)y_1}{\gamma-A(1-2x_0)+By_0},\quad
s_{\rm total}=1-y_0+p(y_0-x_0-y_1)+O(p^2).
$$
Again 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.
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