On the Dynamics of Deterministic Epidemic Propagation over Networks

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.

💡 Research Summary

The paper provides a comprehensive review and original contributions on deterministic nonlinear epidemic models defined on strongly‑connected weighted digraphs. It focuses on three classic compartmental structures—SI, SIS, and SIR—extending the well‑known scalar (well‑mixed) formulations to a network setting where each node represents either a single individual or a homogeneous sub‑population.

For the SI model, the authors write the dynamics as (\dot x_i = \beta (1-x_i)\sum_j a_{ij}x_j) and prove that, on any strongly‑connected graph, every trajectory starting from a non‑zero infection vector remains in the invariant set (