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Analyzing SIS Epidemics on Open Networks with Replacements


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)..." "Through simulations we showed that the original process (i.e., with arrivals and departures)..." "In this direction, graphons appear to be a promising model to generate network topologies..."

Key Insights Distilled From

by Renato Vizue... at arxiv.org 03-26-2024

https://arxiv.org/pdf/2403.16727.pdf
SIS epidemics on open networks

Deeper Inquiries

How do different rates for arrivals and departures impact the stability results derived

Different rates for arrivals and departures can significantly impact the stability results derived in epidemic models. When the rates of arrivals and departures are not similar, the system's composition changes over time, leading to variations in the number of agents present. This dynamic shift can affect the overall behavior of the epidemic process and its stability properties. Stability analyses that assume fixed compositions may no longer hold true when there is a mismatch between arrival and departure rates. In such cases, traditional spectral radius-based stability results may need to be reevaluated or modified to account for these changing dynamics.

What are the implications of neglecting deviations from expected values when estimating variances

Neglecting deviations from expected values when estimating variances can have significant implications on the accuracy of predictions and assessments in epidemic modeling. Variance measures how much individual data points differ from the mean or expected value within a dataset. By overlooking these deviations, one risks underestimating or misrepresenting the variability present in real-world scenarios. This oversight can lead to misleading conclusions about the spread and impact of diseases within populations, potentially affecting decision-making processes related to public health interventions and control strategies.

How can graphons be effectively utilized to model complex distribution-based connections in epidemic dynamics

Graphons offer a powerful framework for effectively modeling complex distribution-based connections in epidemic dynamics. Graphons provide a flexible way to represent large-scale networks with varying structures by capturing edge probabilities between nodes as continuous functions defined on [0,1]x[0,1]. By utilizing graphons, researchers can generate network topologies that remain consistent across different time points while incorporating diverse connection patterns based on underlying distributions. This approach enables more realistic simulations of disease spread over evolving networks with intricate connectivity features influenced by specific probability distributions.
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