# On the Dynamics of Deterministic Epidemic Propagation over Networks

## TL;DR

This paper reviews deterministic nonlinear models for infectious disease spread over contact networks, analyzing equilibria, stability, and thresholds for SI, SIS, and SIR models, with new results on endemic states and transient behavior.

## Contribution

It provides novel analytical results and algorithms for the stability, endemic states, and transient dynamics of network-based epidemic models.

## Key findings

- Established equilibria and stability for SI model
- Characterized endemic states and thresholds for SIS model
- Proposed an iterative algorithm for SIR asymptotic states

## 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.