In 1927 Kermack and McKendrick formulated a model to describe an epidemic outbreak in a demographically closed population. A misconception maintains that their paper is only about the SIR model. In fact it allows for a general expected contribution $A(\tau)$ to the prevailing force of infection by individuals that were infected tau units of time ago. The mathematical formulation takes the form of a nonlinear renewal equation with kernel A. This equation easily provides an expression for the basic reproduction number $R_0$ and equations for the Malthusian parameter $r$ and the final epidemic size.

Erik Volz and Joel Miller extended the SIR model to the configuration network in the infinite population limit. The first aim of the talk is to similarly extend the general K-McK model. This leads once again to a nonlinear renewal equation and next to useful characterizations of $R_0$, $r$ and the final size.

The second aim is to introduce some dynamic variants of the configuration network. Concerning epidemic spread, we now limit ourselves to the SIR model. We find that the situation is much easier when only edges come and go than when there is turnover of vertices as well. The intuitive explanation involves the 'mean field at distance one' of the title.

The third aim is to admit a serious weakness: the considered networks lack (short) cycles! This confession mainly serves to raise a question: is it possible to constructively define an infinite population network that admits both cycles and a statistical description of its spatio-temporal properties that sustains an analytical treatment of epidemic spread across the network?

The lecture is based on joint work with KaYin Leung (now at Stockholm University) and Mirjam Kretzschmar (Julius Center, Utrecht, and RIVM, Bilthoven).

Most epidemic models assume that the spreading process takes place on a single level (be it a single population, a meta-population system or a network of contacts). The latter results from our current limited knowledge about the interplay among the various scales involved in the transmission of infectious diseases at the global scale. Therefore, pressing problems rooted at the interdependency of multi-scales call for the development of a whole new set of theoretical and simulation approaches. In this talk, we show that the recently developed framework of multilayer networks allows to tackle many of the existing challenges in the study of multi-scale diseases, ranging from interacting diseases to new phenomena like disease localization. Specifically, we (1) characterize analytically the epidemic thresholds of two interacting diseases for different scenarios and numerically compute the temporal evolution characterizing the unfolding dynamics; and (2) we present a continuous formulation of epidemic spreading on multilayer networks using a tensorial representation, showing the existence of disease localization and the emergence of two or more transitions, which are characterized analytically and numerically through the inverse participation ratio. Our findings show the importance of considering the multilayer nature of many real systems, as this interdependency usually gives raise to new phenomenology.

We present a general metastability result for the contact process on a sequence of growing finite graphs. We also give a uniform lower bound on the extinction time, of exponential order, when the infection parameter is large enough. This extends previous results of Mountford-Mourrat-Valesin-Yao which proved the same results with an additional bounded degree assumption. Joint work with Daniel Valesin.

The talk is joint work with Glauco Valle of UFRJ. We take as our point of depart a paper of Blondel, Cancrini, Martinelli, Toberto and Toninelli giving some rates of convergence to equilibrium for the model where parameter $q$ exceeds $\frac{1}{2}$ for classes of "polynomial growing" graphs. We show exponential convergence for a wider class of graphs but assuming that parameter $q$ is close to $1$.

The number of reported early syphilis cases in San Francisco has increased steadily since 2005. It is not yet clear what factors are responsible for such an increase. A recent analysis of the sexual contact network of men who have sex with men with syphilis in San Francisco has discovered a large connected component, members of which have a significantly higher chance of syphilis and HIV compared to non-member individuals. This presentation shows how to exploit the existence of the largest connected component to design new notification strategies that can potentially contribute to reduce the number of cases. We develop a model capable of incorporating multiple types of notification strategies and compare the corresponding incidence of syphilis. Through extensive simulations, we show that notifying the community of the infection state of few central nodes appears to be the most effective approach, balancing the cost of notification and the reduction of syphilis incidence. Additionally, among the different measures of centrality, the eigenvector centrality reveals to be the best to reduce the incidence in the long term.

Financial contagion is an epidemic-like phenomenon whereby financial distress can propagate across a network of banks connected by credit relationships, possibly leading to the collapse of the entire system. In order to estimate the systemic risk of financial contagion, the knowledge of the entire interbank network is required. However, due to confidentiality issues, banks only disclose their total exposure towards the aggregate of all other banks, and not their individual exposures towards each bank. A similar problem is encountered in epidemiology. Is it possible to statistically reconstruct the hidden structure of a network in such a way that privacy is protected, but at the same time higher-order properties are correctly predicted? In this talk, I will present a general maximum-entropy approach to the problem of network reconstruction and systemic risk estimation. I will illustrate the power of the method when applied to various economic, social, and biological systems. Then, as a counter-example, I will show how the Dutch interbank network started to depart from its reconstructed counterpart in the three years preceding the 2008 crisis. Over this period, many topological properties of the network showed a gradual transition to the crisis, suggesting their usefulness as early-warning signals. By definition, these signals are undetectable if the network is reconstructed from partial bank-specific information.

We will discuss a new method to predict the covariance structure of the metastable distribution of an SIS epidemic on an arbitrary graph with general infection and healing rates. This method is motivated by a clustering method of the infection matrix using non-negative matrix factorisation, and then by approximating the clustered network by a continuous vector valued SDE. It turns out that the resulting covariance structure works remarkably well, even when we don’t actually use the factorisation or the clustering. Furthermore, once we have an idea of the covariance structure, we can use this to improve the mean field approximation of the infection probability of each node.

An avalanche or cascade occurs when one event causes one or more subsequent events, which in turn may cause further events in a chain reaction. Avalanching dynamics are studied in many disciplines, with a recent focus on average avalanche shapes, i.e., the temporal profiles of avalanches of fixed duration. At the critical point of the dynamics the rescaled average avalanche shapes for different durations collapse onto a single universal curve. We apply Markov branching process theory to derive an equation governing the average avalanche shape for cascade dynamics on networks. Analysis of the equation at criticality demonstrates that nonsymmetric average avalanche shapes (as observed in some experiments) occur for certain combinations of dynamics and network topology. We give examples using numerical simulations of models for information spreading, neural dynamics, and behaviour adoption and we propose simple experimental tests to quantify whether cascading systems are in the critical state. This is joint work with Rick Durrett.

The epidemic spreading has been widely studied in homogeneous cases, where each node may get infected by an infected neighbor with the same rate. However, the infection rate between a pair of nodes, which may depend on e.g. their interaction frequency, is usually heterogeneous and even correlated with their nodal degrees in the contact network. In this talk, we will discuss the influence of heterogeneous infection rates on the Susceptible-Infected-Susceptible (SIS) epidemic spreading. Two types of heterogeneous infection rates are considered: the infection rate of each node pair is independently identically distributed or depends on their nodal degrees. Employing the classic homogeneous SIS model as the benchmark, we explore the influence of heterogeneous infections rates on the average fraction of infected nodes in the metastable state. Our findings as well as models of the infection rates are validated in real-world networks.

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The availability of novel digital data streams that can be used as proxy for monitoring infectious disease incidence is ushering in a new era for real-time forecast approaches to disease spreading. We propose a general framework based on stochastic, spatially structured mechanistic models (individual level microsimulation) initialized with geo-localized data streams. We show the results achieved for seasonal flu in the US, Italy and Spain in the seasons 2014/15 and 2015/16. In all countries studied, the proposed framework provides reliable results with leads of up to 6 weeks that became more stable and accurate with progression of the season. The results for the United States have been generated in real-time in the context of the Centers for Disease Control and Prevention "Forecasting the Influenza Season Challenge". A characteristic feature of the mechanistic modeling approach is in the explicit estimate of key epidemiological parameters relevant for public health decision-making that cannot be achieved with statistical models that do not consider the disease dynamic.

The stochastic SIS logistic epidemic is a well-known birth-and-death chain, used to model the spread of an epidemic within a population of a given size $N$. We study the behaviour of the process as the population size $N$ tends to infinity. Our main results concern the "barely subcritical" regime, where the recovery rate exceeds the infection rate by an amount that tends to $0$ as the population size tends to infinity but not too fast. We derive precise asymptotics for the distribution of the extinction time and the total number of cases throughout the subcritical regime.

We further consider a simple stochastic model for the spread of a disease caused by two virus strains in a closed homogeneously mixing population of size $N$. In our model, the spread of each strain is described by the stochastic logistic SIS epidemic process in the absence of the other strain, and we assume that there is perfect cross-immunity between the two virus strains, that is, individuals infected by one strain are temporarily immune to re-infections and infections by the other strain. For the case where one strain has a strictly larger basic reproductive ratio than the other, and the stronger strain on its own is supercritical (that is, its basic reproductive ratio is larger than $1$), we derive precise asymptotic results for the distribution of the time when the weaker strain disappears from the population, that is, its extinction time. We further consider what happens when the difference between the two reproductive ratios may tend to $0$.

(Joint works with various co-authors)

It has been widely witnessed the interdependence of interactions among human populations and epidemic dynamics in the literature of complex network science, where generally such temporal impacts on both humans and epidemic evolution have been simplified and neglected. In this talk, we will report our continuous efforts to uncover the hidden temporal interactions of social popoulations and epidemic dynamics, and discuss some open questions left for the coming years.

We propose a new approximation framework that unifies and generalizes a number of existing mean-field approximation methods for the SIS epidemic model on complex networks. We derive the framework, which we call the Universal Mean-Field Framework (UMFF), as a set of approximations of the exact Markovian SIS equations. Our main novelty is that we describe the mean-field approximations from the perspective of the isoperimetric problem, which results in bounds on the UMFF approximation error. These new bounds provide insight in the accuracy of existing mean-field methods, such as the N-Intertwined Mean-Field Approximation (NIMFA) and Heterogeneous Mean-Field method (HMF) which are contained by UMFF. Additionally, the isoperimetric inequality relates the UMFF approximation accuracy to the regularity notions of Szemerédi's regularity lemma, which yields a prediction about the behavior of the SIS process on large graphs.