 Solution

The physical state space is the positively invariant set $X,Y,Z\ge 0$, $X+Y+Z=N$: adding the equations gives zero, and the vector field points inward at each coordinate boundary. At an equilibrium point, either $Y=0$, giving $(X,Y,Z)=(N,0,0)$, or $\beta X=b+r$. ThereforeA positive endemic equilibrium exists exactly when $N>N_c$.

For $N<N_c$, put $d=b+r-\beta N>0$. Since $X\le N$,The last limit follows either by evaluating the exponential integral (with a $t{e}^{-bt}$ term when $d=b$), or by splitting it into an early decaying part and a late part with uniformly small $Y$. Hence $X\to N$ for every physical initial condition. Although immune numbers need not decrease monotonically initially, both infected and immune numbers tend to zero.

At the endemic equilibrium, the Jacobian matrix of the $X,Y$ subsystem isA disturbance proportional to $e^{pt}$ therefore obeysBoth coefficients are positive. Real roots have negative sum and positive product; complex conjugate roots have real part $-(b+\beta Y_s)/2<0$. Thus the endemic equilibrium is asymptotically stable by linear stability analysis.