Approximating Quasi-Stationary Behaviour in Network-Based SIS Dynamics.

Area of Science:

• Epidemiology
• Mathematical Biology
• Computational Science

Background:

• Deterministic Susceptible-Infectious-Susceptible (SIS) models typically predict a stable endemic steady-state.
• Stochastic SIS models lack a true endemic steady-state but can display approximately stable behavior.
• Relating deterministic approximations to stochastic dynamics is challenging.

Purpose of the Study:

• To bridge the gap between deterministic approximations and stochastic dynamics in SIS models.
• To introduce a novel approximation method based on the quasi-stationary distribution (QSD).
• To provide a more robust link between approximate and stochastic epidemic models.

Main Methods:

• Developed a system of ordinary differential equations (ODEs) to approximate infected individuals in the QSD.
• Applied the QSD framework to arbitrary contact networks and parameter values.
• Compared the proposed QSD approximations with existing deterministic approximation methods.

Main Results:

• The proposed ODE system accurately approximates the number of infected individuals in the QSD.
• At high epidemic levels, QSD approximations align with existing methods.
• Near the epidemic threshold, QSD approximations diverge from existing methods, which incorrectly predict an all-susceptible state.

Conclusions:

• The QSD provides a more accurate representation of approximate stable behavior in stochastic SIS models.
• The developed ODE system offers a robust link to stochastic dynamics, particularly near the epidemic threshold.
• This approach enhances the reliability of deterministic approximations for understanding epidemic spread.