= Solution
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 $\epsilon<1/2$ of high-risk contacts is with the low-risk group. <Contact reciprocity in a two-group epidemic> requires
$$
pNc_H\epsilon=(1-p)Nc_L\epsilon_L,
\qquad \epsilon_L=\frac{pc_H\epsilon}{(1-p)c_L}.
$$
For sufficiently small $p$, both groups make most contacts within their own group, and the total cross-group contacts are of order $pN$. Define $A=\tau c_H(1-\epsilon)$, $B=\tau c_H\epsilon$, $D=\tau c_L$ and $C(p)=Bp/(1-p)$. The <forces of infection> give
$$
\boxed{\begin{aligned}
\dot x&=(1-x)(Ax+By)-\gamma x,\\
\dot y&=(1-y)[C(p)x+(D-C(p))y]-\gamma y.
\end{aligned}}
$$
Here $C(p)<D$ and $\epsilon_L<1/2$ 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> is
$$
M(p)=\begin{pmatrix}A&B\\C(p)&D-C(p)\end{pmatrix}.
$$
In infected-individual counts the <next-generation matrix> is instead
$$
K=\frac1\gamma\begin{pmatrix}A&C(p)\\B&D-C(p)\end{pmatrix}.
$$
The 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> is
$$
\lambda_+(p)=\frac{A+D-C(p)+\sqrt{[A-D+C(p)]^2+4BC(p)}}2.
$$
This model makes a particular reciprocal mixing convention explicit; different defensible contact laws can change the first-order coefficients.
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