Epidemic model with time delays and fertility/mortality rates

Anastasia Mozokhina; Ivan Popravka; Masoud Saade; Vitaly Volpert; Anastasia Mozokhina; Ivan Popravka; Masoud Saade; Vitaly Volpert

doi:10.3934/math.20251143

AIMS Mathematics

2025, Volume 10, Issue 11: 25849-25878. doi: 10.3934/math.20251143

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Epidemic model with time delays and fertility/mortality rates

1.
RUDN University, 6 Miklukho-Maklaya St, 117198 Moscow, Russia
2.
Institut Camille Jordan, UMR 5208 CNRS, University Lyon 1, 69622 Villeurbanne, France

Received: 19 August 2025 Revised: 23 October 2025 Accepted: 28 October 2025 Published: 10 November 2025
MSC : 34K20, 92D30

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.
- epidemic-model,
- time delay,
- fertility,
- mortality
Citation: Anastasia Mozokhina, Ivan Popravka, Masoud Saade, Vitaly Volpert. Epidemic model with time delays and fertility/mortality rates[J]. AIMS Mathematics, 2025, 10(11): 25849-25878. doi: 10.3934/math.20251143

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Abstract

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.

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