A stochastic SIR epidemic model with Lévy jump and media coverage

Area of Science:

• Epidemiology
• Mathematical Biology
• Stochastic Processes

Background:

• Understanding infectious disease dynamics is crucial for public health interventions.
• Stochastic models offer a more realistic representation of disease transmission compared to deterministic models.
• Factors like temporary immunity and media coverage significantly influence epidemic trajectories.

Purpose of the Study:

• To propose and analyze a stochastic susceptible-infectious-recovered (SIR) epidemic model.
• To investigate the impact of temporary immunity, media coverage, and Lévy jumps on disease dynamics.
• To establish conditions for epidemic extinction and persistence.

Main Methods:

• Development of a stochastic SIR model with temporary immunity and media coverage.
• Inclusion of Lévy jumps to account for sudden environmental changes or disease introductions.
• Mathematical analysis to prove the existence of a unique global positive solution.
• Derivation of sufficient conditions for disease extinction and persistence in the mean.
• Numerical simulations to validate theoretical findings.

Main Results:

• A unique global positive solution for the proposed epidemic model was established.
• Sufficient conditions were derived to determine whether the epidemic disease becomes extinct or persists.
• The threshold behavior governing disease persistence was analyzed.
• Numerical simulations confirmed the theoretical predictions regarding disease dynamics.

Conclusions:

• The stochastic SIR model with temporary immunity and media coverage provides valuable insights into epidemic dynamics.
• Lévy jumps play a significant role in the stochastic behavior of infectious diseases.
• The derived conditions and threshold analysis can inform public health strategies for disease control.