Inspired by empirical studies of networked systems such as the Internet, social networks, and biological networks, researchers have in recent years developed a variety of techniques and models to help us understand or predict the behavior of these systems. Here we review developments in this field, including such concepts as the small-world effect, degree distributions, clustering, network correlations, random graph models, models of network growth and preferential attachment, and dynamical processes taking place on networks.

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The Structure and Function of Complex Networks

The study of networks pervades all of science, from neurobiology to statistical physics. The most basic issues are structural: how does one characterize the wiring diagram of a food web or the Internet or the metabolic network of the bacterium Escherichia coli? Are there any unifying principles underlying their topology? From the perspective of nonlinear dynamics, we would also like to understand how an enormous network of interacting dynamical systems-be they neurons, power stations or lasers-will behave collectively, given their individual dynamics and coupling architecture. Researchers are only now beginning to unravel the structure and dynamics of complex networks.

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Exploring complex networks

We will review some of the major results in random graphs and some of the more challenging open problems. We will cover algorithmic and structural questions. We will touch on newer models, including those related to the WWW.

Random graphs

Convergence of Probability Measures. By P. Billingsley. Chichester, Sussex, Wiley, 1968. xii, 253 p. 9 1/4“. 117s.

Convergence of Probability Measures

The author's preface gives an outline: "This book is about weakconvergence methods in metric spaces, with applications sufficient to show their power and utility. The Introduction motivates the definitions and indicates how the theory will yield solutions to problems arising outside it. Chapter 1 sets out the basic general theorems, which are then specialized in Chapter 2 to the space C[0, l ] of continuous functions on the unit interval and in Chapter 3 to the space D [0, 1 ] of functions with discontinuities of the first kind. The results of the first three chapters are used in Chapter 4 to derive a variety of limit theorems for dependent sequences of random variables. " The book develops and expands on Donsker's 1951 and 1952 papers on the invariance principle and empirical distributions. The basic random variables remain real-valued although, of course, measures on C[0, l ] and D[0, l ] are vitally used. Within this framework, there are various possibilities for a different and apparently better treatment of the material. More of the general theory of weak convergence of probabilities on separable metric spaces would be useful. Metrizability of the convergence is not brought up until late in the Appendix. The close relation of the Prokhorov metric and a metric for convergence in probability is (hence) not mentioned (see V. Strassen, Ann. Math. Statist. 36 (1965), 423-439; the reviewer, ibid. 39 (1968), 1563-1572). This relation would illuminate and organize such results as Theorems 4.1, 4.2 and 4.4 which give isolated, ad hoc connections between weak convergence of measures and nearness in probability. In the middle of p. 16, it should be noted that C*(S) consists of signed measures which need only be finitely additive if 5 is not compact. On p. 239, where the author twice speaks of separable subsets having nonmeasurable cardinal, he means "discrete" rather than "separable." Theorem 1.4 is Ulam's theorem that a Borel probability on a complete separable metric space is tight. Theorem 1 of Appendix 3 weakens completeness to topological completeness. After mentioning that probabilities on the rationals are tight, the author says it is an

Convergence of probability measures

Despite various levels of preventive measures, in 2020 many countries have suffered severely from the coronavirus 2019 (COVID-19) pandemic caused by the severe acute respiratory syndrome coronavirus 2 (SARS-CoV-2) virus We show that population heterogeneity can significantly impact disease-induced immunity as the proportion infected in groups with the highest contact rates is greater than in groups with low contact rates We estimate that if R0 = 25 in an age-structured community with mixing rates fitted to social activity then the disease-induced herd immunity level can be around 43%, which is substantially less than the classical herd immunity level of 60% obtained through homogeneous immunization of the population Our estimates should be interpreted as an illustration of how population heterogeneity affects herd immunity, rather than an exact value or even a best estimate

https://science.sciencemag.org/content/sci/early/2020/06/22/science.abc6810.full.pdf

A mathematical model reveals the influence of population heterogeneity on herd immunity to SARS-CoV-2.

We consider epidemics with removal (SIR epidemics) in populations that mix at two levels: global and local. We develop a general modelling framework for such processes, which allows us to analyze the conditions under which a large outbreak is possible, the size of such outbreaks when they can occur and the implications for vaccination strategies, in each case comparing our results with the simpler homogeneous mixing case. More precisely, we consider models in which each infectious individual i has a global probability $p_G$ for infecting each other individual in the population and a local probability $p_L$, typically much larger, of infecting each other individual among a set of neighbors $\mathscr{N}(i)$. Our main concern is the case where the population is partitioned into local groups or households, but our approach also applies to cases where neighborhoods do not form a partition, for instance, to spatial models with a mixture of local (e.g., nearest-neighbor) and global contacts. We use a variety of theoretical approaches: a random graph framework for the initial exposition of the simple case where an individual's contacts are independent; branching process approximations for the general threshold result; and an embedding representation for rigorous results on the final size of outbreaks. From the applied viewpoint the key result is that, compared with the homogeneous mixing model in which individuals make contacts simply with probability $p_G$, the local infectious contacts have an "amplification" effect. The basic reproductive ratio of the epidemic is increased from its individual-to-individual value $R_G$ in the absence of local infections to a group-to-group value $R_* = \mu R_G$, where $\mu$ is the mean size of an outbreak, started by a randomly chosen individual, in which only local infections count. Where the groups are large and the within-group epidemics are above threshold, this amplification can permit an outbreak in the whole population at very low levels of $p_G$, for instance, for $p_G = O(1/Nn)$ in a population of N divided into groups of size n. The implication of these results for control strategies is that vaccination should be directed preferentially toward reducing $\mu$; we discuss the conditions under which the equalizing strategy, aimed at leaving unvaccinated sets of neighbors of equal sizes, is optimal. We also discuss the estimation of our threshold parameter $R_*$ from data on epidemics among households.

/pdf/epidemics-with-two-levels-of-mixing-3sn0112buz.pdf

Epidemics with two levels of mixing

Strong approximations for epidemic models

This paper is concerned with a general stochastic model for susceptible→infective→removed epidemics, among a closed finite population, in which during its infectious period a typical infective makes both local and global contacts. Each local contact of a given infective is with an individual chosen independently according to a contact distribution ‘centred’ on that infective, and each global contact is with an individual chosen independently and uniformly from the whole population. The asymptotic situation in which the local contact distribution remains fixed as the population becomes large is considered. The concepts of local infectious clump and local susceptibility set are used to develop a unified approach to the threshold behaviour of this class of epidemic models. In particular, a threshold parameter R* governing whether or not global epidemics can occur, the probability that a global epidemic occurs and the mean proportion of initial susceptibles ultimately infected by a global epidemic are all determined. The theory is specialised to (i) the households model, in which the population is partitioned into households and local contacts are chosen uniformly within an infective’s household; (ii) the overlapping groups model, in which the population is partitioned in several ways, with local uniform mixing within the elements of the partitions; and (iii) the great circle model, in which individuals are equally spaced on a circle and local contacts are nearest-neighbour.

A general model for stochastic SIR epidemics with two levels of mixing.

We provide a unified probabilistic approach to the distribution of total size and total area under the trajectory of infectives for a general stochastic epidemic with any specified distribution of the infectious period. The key tool is a Wald&s identity for the epidemic process. The generalisation of our results to epidemics spreading amongst a heterogeneous population is straightforward.

https://www.jstor.org/stable/pdfplus/1427301.pdf

Frank Ball

Papers

A mathematical model reveals the influence of population heterogeneity on herd immunity to SARS-CoV-2.

Epidemics with two levels of mixing

Strong approximations for epidemic models

A general model for stochastic SIR epidemics with two levels of mixing.

A Unified Approach to the Distribution of Total Size and Total Area under the Trajectory of Infectives in Epidemic Models