Epidemics models in Networks

Abstract: These lectures are based on material which was presented in the 2025 Summer school at Fundação Getulio Vargas. The aim of this series is to introduce graduate students with a little background in the field of dynamical systems and network theory to epidemic models. Our goal is to give a succinct and self-contained description of the models

• The paper demonstrates that incorporating network topology through spectral analysis reveals epidemic thresholds that vary based on graph structure and heterogeneity.
• It integrates classical compartmental models with age stratification and urban mobility data to predict outbreak dynamics and assess risk allocation.
• The study highlights that degree heterogeneity and superspreading in networks lead to near-zero thresholds, challenging traditional containment strategies.

Spectral and Structural Analysis of Epidemics in Networks

Introduction

"Epidemics models in Networks" (2512.20771) is a rigorous expository monograph encompassing both classical and contemporary mathematical frameworks for modeling epidemic dynamics within structured populations. The work bridges compartmental ODE models (SIS, SIR) and advanced network-theoretic approaches, illuminating how graph topology, degree heterogeneity, mobility, and demographic stratification critically modulate epidemic thresholds, endemic equilibria, and the potential for large-scale outbreaks. This synthesis is essential for quantitative epidemiology, particularly as population structure, inter-individual contact, and urban mobility become central factors in infectious disease transmission.

Classical Compartmental Models and Epidemic Thresholds

The SIS and SIR models are introduced as canonical systems for describing infection and recovery processes within homogeneous populations. For SIS, dynamical analysis yields an explicit logistic-type equation for the infected fraction $I$:

$\dot{I} = (\beta-\gamma)I - \beta I^2$

with endemic equilibrium $I_{\text{end}} = 1-\frac{1}{r_0}$, where $r_0 = \frac{\beta}{\gamma}$ is the biological reproduction number. Notably:

• SIS Model:
  • Disease-free equilibrium is stable for $r_0 < 1$; endemic equilibrium is stable for $r_0 > 1$.
  • Seasonality and time-dependent transmission can shift stability, and explicit solution formulas are derived via Bernoulli transformation.
• SIR Model:
  • The epidemic always terminates ($\lim_{t\to\infty} I(t) = 0$), with initial growth determined by the susceptible fraction and the basic reproduction number.
  • Herd immunity threshold is characterized as $p < 1/r_0$: if the susceptible fraction falls below this value, the epidemic self-attenuates.

Model fitting procedures are detailed for historical outbreaks (e.g., Eyam plague, 1978 H1N1 in UK, and COVID-19 in Brazil), employing early epidemic data to estimate $\beta$, $\gamma$, and $r_0$. Multiple methods (linear regime approximation, invariant quantity, numerical optimization) closely approximate observed infection peaks, emphasizing the utility of these frameworks for both retrospective analysis and real-time prediction.

Network Structure and Emergence of Zero Threshold Phenomena

Transitioning from homogeneous mixing, the monograph systematically develops the integration of network topology via adjacency matrices and spectral analysis:

• Spectral Radius ($\rho(A)$): The exponential amplification of transmission routes in a graph is dictated by the spectral radius of $A$, with epidemic thresholds scaling as $r_0 < 1/\rho(A)$.
• Regular, Star, and Hub Networks:
  • For $k$-regular graphs, epidemic threshold is independent of size.
  • In star and hub-dominated graphs, $\rho \sim \sqrt{n}$, so threshold decays as $n$ grows, making global eradication infeasible in large, heterogeneous networks.

For random graphs (Erdős-Rényi, Chung-Lu models) and power-law (scale-free) networks, the spectral radius is bounded between $\tilde{w}$ (degree-variance scaled average degree) and $\sqrt{m}$ (root of maximal degree), with precise conditions derived. Crucially, in randomized power-law networks with $2 < \gamma < 2.5$, the second moment $\langle k^2 \rangle$ diverges as $n \to \infty$, so the epidemic threshold approaches zero—a stark, contradictory result to classical theory.

• Friendship Paradox: The ratio $\langle k^2 \rangle/\langle k \rangle$ (average number of friends of friends vs. average degree) quantifies the degree heterogeneity effect closely linked to transmission amplification in networks.

Age Stratification and Contact Matrices

Expanding on simple compartments, the work incorporates age-structured contact matrices and demographic data, reflecting realistic intergenerational mixing. The linearized early epidemic growth is governed by the dominant eigenvalue of a modified next-generation matrix $K = r_0 C P - I$, with $C$ as the contact matrix and $P$ as demographic weights. Country-specific contact patterns (e.g., Germany, Italy) yield distinct eigenvalue/eigenvector profiles, explaining disparities in age-incidence and fatality patterns observed during COVID-19.

• Eigenvector-Fatality Link: The initial composition of cases and deaths in each age group can be predicted by combining the transmission eigenvector with group-specific case fatality rates.

Urban Mobility and Stratified SIR Dynamics

The monograph introduces city-level stratification, integrating mobility via detailed transfer matrices derived from geolocation data. The effective next-generation matrix $K$ generalizes network transmission dynamics to spatially coupled urban populations. Analytical results for symmetric (homogeneous) and asymmetric (heterogeneous hotspot) scenarios demonstrate:

• In homogenous settings, mobility alone does not create epidemic heterogeneity; growth rates remain uniform.
• With hotspots, eigenvector analysis reveals risk allocation to high-transmission cities, with spillover into lower-risk areas via mobility.

This framework supports targeted containment strategies and resource allocation, especially in metropoles with significant incoming commuter flux.

Implications and Prospects

The interplay between spectral graph theory, compartmental epidemiology, and statistical physics of complex networks emerges as a crucial determinant of epidemic trajectories. The results establish that:

• Degree Heterogeneity: Massive superspreaders or hubs drive near-impossibility of global epidemic containment in large networks, as zero threshold phenomena dominate.
• Mobility and Stratification: Urban and demographic structure can strongly modulate both the rate and localization of outbreaks; control measures must be dynamically adapted to these features.

Future directions include the incorporation of advanced intervention modeling (vaccination, quarantine, contact tracing, ICU sharing), time-dependent control terms, delay equations, and optimization-based approaches—necessary for operational deployment in public health policy and real-time response.

Conclusion

"Epidemics models in Networks" (2512.20771) provides a mathematically rigorous synthesis connecting epidemic threshold phenomena with the spectral properties of contact networks, age structure, and urban mobility. The analysis demonstrates that heterogeneity—whether topological, demographic, or spatial—can drastically alter both the feasibility and strategy of epidemic management. The theoretical results set the foundation for informed interventions and highlight the necessity of integrating network analysis, statistical inference, and dynamic optimization in contemporary epidemic modeling.