We study a discrete-time SEIR epidemic model with demographic turnover, latency, temporary immunity, and saturating exponential incidence. The total population satisfies an exact Beverton–Holt recursion; consequently the epidemic dynamics reduce, after the demographic transient, to an invariant constant-population simplex of size N∗. For the reduced map we compute the discrete next-generation operator and obtain an explicit basic reproduction number R0. We prove a sharp threshold alternative: if R0≤1, then the disease-free equilibrium is globally asymptotically stable; if R0>1, then the disease-free equilibrium is repelling on the infection-present state space, the infected classes are uniformly persistent, there is a unique endemic equilibrium, and this endemic equilibrium is globally asymptotically stable in the interior of the simplex. The Lyapunov function used for the endemic equilibrium is the classical Volterra form. The main technical point is not the formal choice of this function, but an order-free concavity estimate for the saturating incidence term. This estimate yields the endemic marginal inequality R0∗(X∗)I∗. Numerical phase portraits, threshold sensitivity calculations, and two data-oriented illustrations are included to show how the threshold quantities enter biologically and computationally.
Saber Elaydi (Sat,) studied this question.