Small-core SIS endemic expansion ($s_{\mathrm{total}}=1-p(1-\gamma /A)[1+B/(\gamma -D)]+O(p^2)$)

For fixed $A>\gamma >D$, let $x_0=1-\gamma /A$. The rare high-risk group's SIS model has $x=x_0+p{B}^2\gamma /[A^2(\gamma -D)]+O(p^2)$, whereas low-risk infection is $y=pB{x}_0/(\gamma -D)+O(p^2)$. Substitution into both equilibrium equations proves these coefficients. The leading basic reproduction number is $A/\gamma$ with correction $p{B}^2/[\gamma (A-D)]$, 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.