Epidemic model with time delays and fertility/mortality rates

We developed a delayed SIR (Susceptible-Infected-Recovered) model incorporating infectious/immune periods and demographics (fertility and mortality rates), proving the existence, nonnegativity, and uniqueness of solutions for the system under demographic equilibrium. Analysis confirmed a threshold at ${\mathfrak{R}}_0=1$, with an endemic equilibrium emerging when ${\mathfrak{R}}_0>1$. Crucially, the stability of this endemic state was governed by a critical mortality rate ( ${\mu }_c$). High-mortality populations ( $\mu >{\mu }_c$) exhibited a stable endemic state, whereas low-mortality populations ( $\mu <{\mu }_c$) experienced instability and sustained oscillations. For these low-mortality populations, critical thresholds for the transmission rate ( ${\beta }_c$) and disease duration ( ${\tau }_{1c}$) were identified, beyond which destabilization occurred. This demonstrated a fundamental dual dependence of long-term disease dynamics on both demographic (e.g., life expectancy) and epidemiological (e.g., transmission rate, disease duration) parameters. Consequently, public health strategies (like vaccination targets) may need adjustment based on a population's demographic structure, not just its immediate epidemiological characteristics.