Threshold dynamics of a time-delayed SEIRS model with pulse vaccination

Area of Science:

• Epidemiology
• Mathematical Biology
• Infectious Disease Dynamics

Background:

• The spread of infectious diseases is often modeled using compartmental models, but incorporating factors like time delays and vaccination strategies is complex.
• Previous models, such as Gao et al. (2007), identified the basic reproduction number (R0) but left open the precise threshold for disease eradication versus uniform persistence in delayed, impulsive models.
• Understanding these dynamics is crucial for effective public health interventions.

Purpose of the Study:

• To develop and analyze a delayed SEIRS (Susceptible-Exposed-Infectious-Recovered-Susceptible) epidemic model with pulse vaccination and a varying population.
• To derive the basic reproduction number (R0) and establish a sharp threshold that distinguishes between disease eradication and uniform persistence.
• To investigate the impact of pulse vaccination and time delays on disease transmission dynamics.

Main Methods:

• Formulation of a delayed SEIRS mathematical model incorporating periodic pulse vaccination and a time-varying total population.
• Derivation of the basic reproduction number (R0) using established epidemiological modeling techniques.
• Mathematical analysis to determine the global stability of the disease-free periodic solution and conditions for uniform persistence.
• Numerical simulations to visualize analytical findings and explore parameter sensitivities.

Main Results:

• The basic reproduction number (R0) was successfully derived for the proposed model.
• A sharp threshold was identified: if R0 < 1, the disease-free periodic solution is globally attractive (eradication); if R0 > 1, the disease is uniformly persistent.
• These results provide a definitive criterion for disease control, improving upon prior research that lacked a sharp threshold.
• Numerical simulations confirmed the analytical results and highlighted the significant influence of pulse vaccination timing and the duration of the delay on disease spread.

Conclusions:

• The study successfully established sharp threshold dynamics for a delayed, impulsive SEIRS model with pulse vaccination.
• The findings provide crucial insights into disease control strategies, emphasizing the importance of vaccination timing and managing time delays in transmission.
• This work represents a significant advancement in the mathematical modeling of infectious diseases, particularly for impulsive vaccination strategies.