Threshold dynamics in an SEIRS model with latency and temporary immunity.

Area of Science:

• Epidemiology
• Mathematical Biology
• Infectious Disease Dynamics

Background:

• SEIRS models are crucial for understanding disease transmission dynamics.
• Incorporating distributed delays in latent and immune periods offers a more realistic representation of disease spread.
• Understanding the basic reproduction number (R0) and equilibrium stability is key to predicting disease persistence.

Purpose of the Study:

• To investigate a SEIRS model with distributed delays in the latent and temporary immune periods.
• To analyze the impact of probability distributions on the basic reproduction number (R0).
• To examine the stability of disease-free and endemic equilibrium points.

Main Methods:

• Development of a SEIRS model incorporating distributed delays.
• Analysis of the basic reproduction number (R0) and its dependence on probability distributions.
• Mathematical analysis of the global asymptotic stability of the disease-free equilibrium.
• Investigation of the existence and stability properties of the endemic equilibrium under various distribution functions.

Main Results:

• The basic reproduction number (R0) depends on the latent period's probability distribution but not on the temporary immunity distribution.
• The disease-free equilibrium is globally asymptotically stable when R0 < 1.
• The disease persists when R0 < 1, with the endemic equilibrium exhibiting varying stability properties based on distribution choices.
• The endemic steady state is at least locally asymptotically stable under specific conditions (decreasing exponential immunity distribution, fixed or exponentially decreasing latency).
• Oscillatory behavior in the endemic steady state is possible with constant delays in temporary immunity.

Conclusions:

• The distribution of the latent period significantly influences disease transmission thresholds (R0).
• The model predicts disease persistence and varying endemic equilibrium stability based on delay distributions.
• Numerical simulations support the theoretical findings, validating the model's predictions for disease dynamics.