A stage-structured SEIR model with time-dependent delays in an almost periodic environment

Area of Science:

• Mathematical epidemiology
• Dynamical systems theory
• Population dynamics

Background:

• Mathematical models are crucial for understanding infectious disease transmission.
• Incorporating time-dependent parameters like maturation and incubation periods enhances model realism.
• Stage structure and latency are key features in many epidemiological models.

Purpose of the Study:

• To propose and investigate an almost periodic SEIR model with stage structure and latency.
• To introduce and analyze threshold parameters for population and disease persistence/extinction.
• To examine the impact of time-dependent maturation and incubation periods on disease dynamics.

Main Methods:

• Development of an almost periodic SEIR model with stage structure and latency.
• Introduction of two basic reproduction ratios: $\hat{R}_{0}$ for population and $R_{0}$ for disease.
• Analytical investigation of global attractivity for population extinction and disease persistence/extinction conditions.
• Numerical simulations to verify analytical findings and explore parameter sensitivity.

Main Results:

• If $\hat{R}_{0}<1$, the population extinction state is globally attractive.
• If $\hat{R}_{0}>1$, disease dynamics depend on $R_{0}$: extinction if $R_{0}<1$, persistence if $R_{0}>1$.
• Numerical simulations confirm analytical results and highlight the influence of maturation and incubation period variability.

Conclusions:

• The proposed SEIR model provides insights into disease dynamics under time-varying conditions.
• Threshold parameters $\hat{R}_{0}$ and $R_{0}$ are critical determinants of population and disease persistence.
• Variability in maturation and incubation periods plays a significant role in disease transmission patterns.