SIS model ($\dot{I}=\beta I(1-I)-\gamma I$)

An SIS model returns recovered hosts directly to susceptibility, so immunity does not accumulate. For homogeneous frequency-dependent transmission, a positive endemic state exists at $I_{\ast }=1-1/{\mathcal{R}}_0$ when ${\mathcal{R}}_0=\beta /\gamma >1$. This epidemiological model is distinct from the galaxy-dynamics abbreviation SIS for a singular isothermal sphere.

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.

For $\dot{i}=\beta i(1-i)-\gamma i$, a positive endemic equilibrium requires $R_0=\beta /\gamma >1$ and gives susceptible fraction $s_{\ast }=\gamma /\beta$. Below threshold the disease-free susceptible fraction is one. The reciprocal relation is a property of this homogeneous scalar SIS model, not a universal identity for heterogeneous populations.

For two otherwise symmetric SIS models with recovery rate $\gamma$, primary transmission $\beta$ and secondary susceptibility multiplier $q$, split hosts into uninfected, singly infected and doubly infected classes. The displayed conditions give stable disease-free and joint endemic states, with an intermediate saddle: a strong Allee effect. The endemic susceptible fraction solves $(q-1)S^2-qS+{\mathcal{R}}_0^{-2}=0$. The smaller root is the stable high-prevalence state. Coinfected hosts recover each infection independently, so recovery from coinfection leads to a single-infection class rather than directly to complete susceptibility.