On the Dynamics of Deterministic Epidemic Propagation over Networks (1701.03137v1)

Abstract: In this work we review a class of deterministic nonlinear models for the propagation of infectious diseases over contact networks with strongly-connected topologies. We consider network models for susceptible-infected (SI), susceptible-infected-susceptible (SIS), and susceptible-infected-recovered (SIR) settings. In each setting, we provide a comprehensive nonlinear analysis of equilibria, stability properties, convergence, monotonicity, positivity, and threshold conditions. For the network SI setting, specific contributions include establishing its equilibria, stability, and positivity properties. For the network SIS setting, we review a well-known deterministic model, provide novel results on the computation and characterization of the endemic state (when the system is above the epidemic threshold), and present alternative proofs for some of its properties. Finally, for the network SIR setting, we propose novel results for transient behavior, threshold conditions, stability properties, and asymptotic convergence. These results are analogous to those well-known for the scalar case. In addition, we provide a novel iterative algorithm to compute the asymptotic state of the network SIR system.

• The paper introduces a novel network SI model that demonstrates monotonic convergence to full contagion without threshold effects.
• It rigorously analyzes the SIS model, proving the existence of a unique endemic state above threshold and providing an iterative algorithm for its computation.
• For the SIR model, it establishes transient dynamics and stability conditions, detailing convergence to zero infection and methods for calculating asymptotic states.

Dynamics of Deterministic Epidemic Propagation on Networks

This paper presents a systematic examination of deterministic epidemic propagation models within network frameworks, specifically focusing on Susceptible-Infected (SI), Susceptible-Infected-Susceptible (SIS), and Susceptible-Infected-Recovered (SIR) models. The authors conduct a rigorous mathematical analysis of the dynamical properties, including aspects such as equilibria, stability, and convergence within strongly connected network topologies.

Network Epidemic Models

The paper adopts a mean-field approximation approach derived from original Markov-chain models, offering a deterministic portrayal for analyzing epidemic spread in networks represented as adjacency matrices. The spectral radius or the dominant eigenvalue of these matrices plays a pivotal role in determining epidemic thresholds and outcomes.

Core Contributions

1. Network SI Model:
   • Introduced a novel network SI model and analyzed its monotonic convergence toward a full contagion state, independent of initial conditions, demonstrating no threshold phenomena.
2. Network SIS Model:
   • Reviewed classical findings and presented new computations for endemic states and formulations for endemic states near epidemic thresholds.
   • Established the existence of a unique endemic state when the basic reproduction number exceeds the threshold, with an iterative algorithm provided for its computation.
3. Network SIR Model:
   • Developed new insights into transient dynamic behaviors, threshold conditions, and stability, proving convergence of infection spread to zero over time.
   • Offered a novel iterative algorithm for calculating the asymptotic state in any arbitrary initial setting.

Results and Analytical Techniques

The paper makes notable use of modern techniques from algebraic graph theory and matrix analysis to investigate propagation dynamics. The mathematical tools applied include eigenvalue analysis, Perron-Frobenius theory, and Lyapunov methods, which enable the derivation of results such as stability conditions and the behavior of equilibria in these models.

Implications and Future Directions

The results have profound implications for understanding the intricate dynamics of epidemic spread in networks, crucial for applications ranging from public health to network security. The robust numerical results and theoretical advancements presented can inform strategic intervention policies and enhance predictive modeling capabilities. Future research directions could explore extending these deterministic models to incorporate stochastic elements and investigate their utility in heterogeneous network structures or dynamically changing networks.

This work stands as a detailed contribution to the mathematical epidemiology literature, providing a foundational basis for further explorations and refinements in deterministic modeling of epidemic propagation processes on complex network structures.