= Small-core SIS endemic expansion
{title2=$s_{\rm total}=1-p(1-\gamma/A)[1+B/(\gamma-D)]+O(p^2)$}
For fixed $A>\gamma>D$, let $x_0=1-\gamma/A$. The rare high-risk group's <SIS model> has $x=x_0+pB^2\gamma/[A^2(\gamma-D)]+O(p^2)$, whereas low-risk infection is $y=pBx_0/(\gamma-D)+O(p^2)$. Substitution into both <equilibrium> equations proves these coefficients. The leading <basic reproduction number> is $A/\gamma$ with correction $pB^2/[\gamma(A-D)]$, while global susceptibility tends to one. These expressions explain why an endemic core can coexist with a large susceptible majority; they are nonuniform when a group is near its own threshold.
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