Core Concepts
Analyzing SIS epidemics on open networks using replacements to approximate the process and derive upper bounds for key metrics.
Abstract
The content analyzes continuous-time SIS epidemics on open networks, focusing on replacements as an approximation method. The study considers stochastic settings with Poisson processes for arrivals and departures, deriving upper bounds for expectation and variance of aggregate infection levels. The analysis extends to stability results, time-varying networks, and the impact of mobility on disease spread. The paper explores open multi-agent systems, considering replacements in large populations with similar arrival and departure rates. It delves into the dynamics of SIS models on closed networks, stability conditions, Lyapunov functions, and spectral properties. The work also discusses the behavior of aggregate functions during replacements and provides insights into the second moment analysis. Numerical simulations illustrate the results obtained from the theoretical analysis.
I. Introduction
- Analyzing continuous-time SIS epidemics on open networks.
- Focus on approximating processes using replacements.
- Deriving upper bounds for key metrics in epidemic dynamics.
II. SIS Model on Closed Networks
- Definition and features of classical SIS model.
- Stability results based on adjacency matrix spectral radius.
- Analysis of disease-free equilibrium conditions.
III. Open SIS Epidemics
- Formulation of SIS epidemic with arrivals and departures.
- Consideration of replacement events in fixed network approximations.
IV. Replacements as an Approximation
- Analysis of aggregate function variation during replacement events.
- Upper bounds derived for expectation and variance under replacement process assumptions.
V. First Moment
- Behavior analysis of aggregate function under pure replacement process.
- Propositions regarding expected values during replacement events.
VI. Second Moment
- Examination of second moment behavior under replacement processes.
- Propositions regarding second moment expectations under specific assumptions.
VII. Numerical and Simulation Results
- Illustrative computations showing evolution of moments in a sample realization.
VIII. Conclusion
- Summary of findings related to analyzing SIS epidemics with replacements in open multi-agent systems.
Stats
Arrivals take place according to Poisson processes with rate µa = µd = µ = 7.
Recovery rate δ is constant at 1.5p ¯β while infection rate βn = ¯β/n = 0.1/n.
New agents have infection probability xj determined by random variable Θ with mean m = 1/2 and variance σ2 = 1/12.
Quotes
"The evolution of V in this open setting is given by a stochastic differential equation (SDE)..."
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