Poisson integral type quarantine in a stochastic SIR system.

Area of Science:

• Epidemiology
• Mathematical Biology
• Stochastic Processes

Background:

• Compartmental models are crucial for understanding infectious disease dynamics.
• Stochasticity plays a significant role in disease transmission and control.
• Quarantine is a key intervention strategy for managing epidemics.

Purpose of the Study:

• To develop and analyze a stochastic Susceptible-Infected-Recovered (SIR) model incorporating a Poisson measure for quarantine.
• To mathematically establish the conditions for stochastic stability of the disease-free equilibrium.
• To investigate the long-term behavior of the system when the basic reproduction number (R0) is greater than 1.

Main Methods:

• Formulation of a stochastic SIR model with a Poisson process for quarantine.
• Derivation of an inequality related to the transmission rate to prove stochastic stability.
• Analysis of the system's long-run behavior using stability criteria.

Main Results:

• The proposed SIR model with quarantine demonstrates stochastic stability under specific conditions.
• An inequality involving the transmission rate term is derived to ensure the stability of the disease-free equilibrium.
• When R0 > 1, the system's long-term dynamics are shown to converge to a neighborhood of the equilibrium in the deterministic counterpart.

Conclusions:

• The inclusion of a Poisson measure for quarantine effectively models disease control in stochastic epidemic systems.
• The study provides a rigorous mathematical framework for analyzing the stability of disease-free states in stochastic SIR models.
• The findings suggest that interventions like quarantine can maintain disease control, even in the presence of random fluctuations, when R0 indicates potential for widespread transmission.