Assortatively mixed two-risk-group SIS model ($\dot{x}=(1-x)(Ax+By)-\gamma x,\dot{y}=(1-y)[Cx+(D-C)y]-\gamma y$)

For high-risk group fraction $p$, fixed total contact rates and reciprocal cross-group contacts, let $C=Bp/(1-p)$. The displayed SIS model describes within-group infectious fractions. In fractions the transmission matrix is $M=\left(\begin{array}{cc}A& B\\ C& D-C\end{array}\right)$; in infected-person counts it is $\mathrm{diag}(p,1-p)M\mathrm{diag}(p,1-p{)}^{-1}$. A common recovery rate gives basic reproduction number $\rho (M)/\gamma$. The mean population susceptible fraction does not generally equal its reciprocal.

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.