"Contacts between individuals serve as pathways where infections may propagate. These contact patterns can be represented by network structures. Static structures have been the common modeling paradigm but recent results suggest that temporal structures play different roles to regulate the spread of infections or infection-like dynamics. On temporal networks a vertex is active only at certain moments and inactive otherwise such that a contact is not continuously available. In several empirical networks, the time between two consecutive vertex-activation events typically follows heterogeneous activity (e.g. bursts). In this chapter, we present a simple and intuitive stochastic model of a temporal network and investigate how epidemics co-evolves with the temporal structures, focusing on the growth dynamics of the epidemics. The model assumes no underlying topological structure and is only constrained by the time between two consecutive events of vertex activation. The main observation is that the speed of the infection spread is different in case of heterogeneous and homogeneous temporal patterns but the differences depend on the stage of the epidemics. In comparison to the homogeneous scenario, the power law case results in a faster growth in the beginning but turns out to be slower after a certain time, taking several time steps to reach the whole network."

"Epidemics are often modelled using state-space models based on dynamical systems, observed through partial and noisy data. In this paper we develop stochastic extensions to the popular SEIR model with parameters evolving in time, in order to capture unknown influences of changing behaviors, public interventions, seasonal effects etc. Our models assign diffusion processes for the time-varying parameters, and our inferential procedure is based on the particle Markov Chain Monte Carlo algorithm, suitably adjusted to accommodate the features of this challenging nonlinear stochastic model. The performance of the proposed computational methods is validated on simulated data and the adopted model is applied to the 2009 A/H1N1 pandemic in England. In addition to estimating the trajectories of the effective contact rate, the methodology is applied in real time to provide evidence in related public health decisions."

"Social networks are rarely observed in full detail. In many situations properties are known for only a sample of the individuals in the network and it is desirable to induce global properties of the full social network from this "egocentric" network data. In the current paper we study a few different types of egocentric data, and show what global network properties are consistent with those egocentric data. Two global network properties are considered: the size of the largest connected component in the network (the giant), and secondly, the possible size of an epidemic outbreak taking place on the network, in which transmission occurs only between network neighbours, and with probability $p$. The main conclusion is that in most cases, egocentric data allow for a large range of possible sizes of the giant and the outbreak. However, there is an upper bound for the latter. For the case that the network is selected uniformly among networks with prescribed egocentric data (satisfying some conditions), the asymptotic size of the giant and the outbreak is characterised."

"Considerable attention has been paid, in recent years, to the use of networks in modeling complex real-world systems. Among the many dynamical processes involving networks, propagation processes -- in which final state can be obtained by studying the underlying network percolation properties -- have raised formidable interest. In this paper, we present a bond percolation model of multitype networks with an arbitrary joint degree distribution that allows heterogeneity in the edge occupation probability. As previously demonstrated, the multitype approach allows many non-trivial mixing patterns such as assortativity and clustering between nodes. We derive a number of useful statistical properties of multitype networks as well as a general phase transition criterion. We also demonstrate that a number of previous models based on probability generating functions are special cases of the proposed formalism. We further show that the multitype approach, by naturally allowing heterogeneity in the bond occupation probability, overcomes some of the correlation issues encountered by previous models. We illustrate this point in the context of contact network epidemiology."

"There is great interest in the dynamics of health behaviors in social networks and how they affect collective public health outcomes, but measuring population health behaviors over time and space requires substantial resources. Here, we use publicly available data from 101,853 users of online social media collected over a time period of almost six months to measure the spatio-temporal sentiment towards a new vaccine. We validated our approach by identifying a strong correlation between sentiments expressed online and CDC-estimated vaccination rates by region. Analysis of the network of opinionated users showed that information flows more often between users who share the same sentiments - and less often between users who do not share the same sentiments - than expected by chance alone. We also found that most communities are dominated by either positive or negative sentiments towards the novel vaccine. Simulations of infectious disease transmission show that if clusters of negative vaccine sentiments lead to clusters of unprotected individuals, the likelihood of disease outbreaks is greatly increased. Online social media provide unprecedented access to data allowing for inexpensive and efficient tools to identify target areas for intervention efforts and to evaluate their effectiveness."

"We consider measure-valued processes X_t that solve the following martingale problem: For a given initial measure X_0, and for all smooth, compactly supported test functions phi,
X_t(phi)= X_0 (phi)+ (1/2)int_0^t X_s(Delta phi)ds + thetaint_0^t X_s(phi) ds - int_0^t X_s(L_s phi) ds + M_t(phi). Here L_s(x) is the local time density process associated with X_t, and M_t(phi) is a martingale with quadratic variation [M(phi)]_t=int_0^t X_s(phi^2) ds. Such processes arise as scaling limits of SIR epidemic models. We show that there exist critical values theta_c(d) in (0,infty) for dimensions d=2,3 such that if theta> theta_c(d), then the solution survives forever with positive probability, but if theta< theta_c(d), then the solution dies out in finite time with probability 1. For d=1 we prove that the solution dies out almost surely for all values of theta. We also show that in dimensions d=2,3 the process dies out locally almost surely for any value of theta, that is, for any compact set K, the process X_t (K)=0 eventually."

"Disease awareness in infection dynamics can be modeled with adaptive contact networks whose rewiring rules reflect the attempt by susceptibles to avoid infectious contacts. Simulations of this type of models show an active phase with constant infected node density in which the interplay of disease dynamics and link rewiring prompts the convergence towards a well defined degree distribution, irrespective of the initial network topology. We develop a method to study this dynamic equilibrium and give an analytic description of the structure of the characteristic degree distributions and other network measures. The method applies to a broad class of systems and can be used to determine the steady-state topology of many other adaptive networks."

17 pp. review paper. "Infectious disease remains, despite centuries of work to control and mitigate its effects, a major problem facing humanity. This paper reviews the mathematical modelling of infectious disease epidemics on networks, starting from the simplest Erdos-Renyi random graphs, and building up structure in the form of correlations, heterogeneity and preference, paying particular attention to the links between random graph theory, percolation and dynamical systems representing transmission. Finally, the problems posed by networks with a large number of short closed looks are discussed."

"Despite its importance for rumors or innovations propagation, peer-to-peer collaboration, social networking, or marketing, the dynamics of information spreading is not well understood. Since the diffusion depends on the heterogeneous patterns of human behavior and is driven by the participants’ decisions, its propagation dynamics shows surprising properties not explained by traditional epidemic or contagion models. Here we present a detailed analysis of our study of real viral marketing campaigns where tracking the propagation of a controlled message allowed us to analyze the structure and dynamics of a diffusion graph involving over 31 000 individuals. We found that information spreading displays a non-Markovian branching dynamics that can be modeled by a two-step Bellman-Harris branching process that generalizes the static models known in the literature and incorporates the high variability of human behavior. It explains accurately all the features of information propagation under the “tipping point” and can be used for prediction and management of viral information spreading processes."

Even the abstract was tl;dr.

"In this paper we present a new ODE based framework for modelling disease transmission on dynamic contact networks. We adapt and extend the effective degree model for a static network to account for the random creation and deletion of links between individuals. The resulting set of ODEs is solved numerically and results are compared to those obtained using individual-based stochastic network simulations. We show that the ODEs display excellent agreement for the evolution of both the disease and the network, and is able to accurately capture the epidemic threshold for a wide range of parameters. Using the proposed model we show that mild epidemics can be controlled while keeping the contact network well connected, and this is in contrast with severe epidemics, where successful control via link removal leads to a disconnected network."

"Studies of disease spreading on complex networks have provided a deep insight into the conditions of onset, dynamics and prevention of epidemics in human populations and malicious software propagation in computer networks. Identifying nodes which, when initially infected, infect the largest part of the network and ranking them according to their epidemic impact is a priority for public health policies. In simulations of the disease spreading in SIR model on studied empirical complex networks, it is shown that the ranking depends on the dynamical regime of the disease spreading. A possible mechanism leading to this dynamical dependence is illustrated in an analytically tractable example. A measure called epidemic centrality, averaging the epidemic impact over all possible disease spreading regimes, is introduced as a basis of epidemic ranking. Contrary to standard notion, the epidemic centrality of nodes with high degree, k-cores value or betweenness, which are structurally central, is comparable to epidemic centrality of structurally peripheral nodes. These findings indicate that the impact of an epidemic starting at structurally peripheral nodes may be considerably underestimated. Network periphery should gain a more prominent role in the allocation of resources in future epidemic preparedness plans."

There's an arxiv version as well...

"We propose a class of random scale-free spatial networks with nested community structures and analyze Reed-Frost epidemics with community related independent transmissions. We show that the epidemic threshold may be trivial or not depending on the relation among community sizes, distribution of the number of communities and transmission rates"

"For a broad range single-seed contagion processes acting on generalized random networks, we derive a unifying analytic expression for the possibility of global spreading events in a straightforward, physically intuitive fashion. Our reasoning lays bare a direct mechanical understanding of an archetypal spreading phenomena that is not evident in circuitous extant mathematical approaches."  (Are they really that circuitous?)

I'll have to revise this (in light of arxiv:1004.4704, no less!), but it was a very fun lecture to write and give, and covers the essential points. (& of course blew most of the kids minds.)

Re-purposing standard epidemic models (SIR, etc.) as descriptions of zombie outbreaks. (Conclusion: we're DOOMED!)

A very nice attempt by Durrett to bring some proper mathematical hygiene to studying networks more interesting than the Erdos-Renyi (-Rappoport-Solomonoff) model. Particularly nice on using branching processes to prove results about network growth processes (as might be expected given Durrett's earlier work on particle systems).

"We study the spreading of an infection within an SIS epidemiological model on a network. Susceptible agents are given the opportunity of breaking their links with infected agents. Broken links are either permanently removed or reconnected with the rest of the population. Thus, the network coevolves with the population as the infection progresses. We show that a moderate reconnection frequency is enough to completely suppress the infection. A partial, rather weak isolation of infected agents suffices to eliminate the endemic state."