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

Summary Background Since December, 2019, Wuhan, China, has experienced an outbreak of coronavirus disease 2019 (COVID-19), caused by the severe acute respiratory syndrome coronavirus 2 (SARS-CoV-2). Epidemiological and clinical characteristics of patients with COVID-19 have been reported but risk factors for mortality and a detailed clinical course of illness, including viral shedding, have not been well described. Methods In this retrospective, multicentre cohort study, we included all adult inpatients (≥18 years old) with laboratory-confirmed COVID-19 from Jinyintan Hospital and Wuhan Pulmonary Hospital (Wuhan, China) who had been discharged or had died by Jan 31, 2020. Demographic, clinical, treatment, and laboratory data, including serial samples for viral RNA detection, were extracted from electronic medical records and compared between survivors and non-survivors. We used univariable and multivariable logistic regression methods to explore the risk factors associated with in-hospital death. Findings 191 patients (135 from Jinyintan Hospital and 56 from Wuhan Pulmonary Hospital) were included in this study, of whom 137 were discharged and 54 died in hospital. 91 (48%) patients had a comorbidity, with hypertension being the most common (58 [30%] patients), followed by diabetes (36 [19%] patients) and coronary heart disease (15 [8%] patients). Multivariable regression showed increasing odds of in-hospital death associated with older age (odds ratio 1·10, 95% CI 1·03–1·17, per year increase; p=0·0043), higher Sequential Organ Failure Assessment (SOFA) score (5·65, 2·61–12·23; p Interpretation The potential risk factors of older age, high SOFA score, and d-dimer greater than 1 μg/mL could help clinicians to identify patients with poor prognosis at an early stage. Prolonged viral shedding provides the rationale for a strategy of isolation of infected patients and optimal antiviral interventions in the future. Funding Chinese Academy of Medical Sciences Innovation Fund for Medical Sciences; National Science Grant for Distinguished Young Scholars; National Key Research and Development Program of China; The Beijing Science and Technology Project; and Major Projects of National Science and Technology on New Drug Creation and Development.

Clinical course and risk factors for mortality of adult inpatients with COVID-19 in Wuhan, China: a retrospective cohort study

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

Complex networks arise in a wide range of biological and sociotechnical systems. Epidemic spreading is central to our understanding of dynamical processes in complex networks, and is of interest to physicists, mathematicians, epidemiologists, and computer and social scientists. This review presents the main results and paradigmatic models in infectious disease modeling and generalized social contagion processes.

Epidemic processes in complex networks

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.

Percolation theory is a part of statistical physics that deals with the slow flow of liquid through porous media, and is more generally extended to consider the formation of long-range connectivity in a random system. It has been suggested that this theory might apply to the spread of infectious diseases in certain conditions, but no natural examples have been reported until now. A disease that does behave in this way is plague (Yersinia pestis infection) among great gerbils in Central Asia. The flea dispersal movements carrying plague from one family group of great gerbils to another are small compared to the vast areas of the desert habitat. This equates to a system in which plague percolates through an area only if the landscape is sufficiently filled with family groups of hosts. This paper presents an example of a percolation threshold for infectious disease, namely the plague among great gerbils in Central Asia. The threshold arises from the spatial constraints on flea dispersal movements that carry plague from one family group of great gerbils to another. These movements are small compared to the vast contiguous areas of desert that have been colonized by the great gerbil. The result is a percolation system wherein plague is only able to percolate through the landscape if it is sufficiently 'filled' with family groups of hosts. Percolation theory is most commonly associated with the slow flow of liquid through a porous medium, with applications to the physical sciences1. Epidemiological applications have been anticipated for disease systems where the host is a plant or volume of soil2,3, and hence is fixed in space. However, no natural examples have been reported. The central question of interest in percolation theory4, the possibility of an infinite connected cluster, corresponds in infectious disease to a positive probability of an epidemic. Archived records of plague (infection with Yersinia pestis) in populations of great gerbils (Rhombomys opimus) in Kazakhstan have been used to show that epizootics only occur when more than about 0.33 of the burrow systems built by the host are occupied by family groups5. The underlying mechanism for this abundance threshold is unknown. Here we present evidence that it is a percolation threshold, which arises from the difference in scale between the movements that transport infectious fleas between family groups and the vast size of contiguous landscapes colonized by gerbils. Conventional theory predicts that abundance thresholds for the spread of infectious disease arise when transmission between hosts is density dependent such that the basic reproduction number (R0) increases with abundance, attaining 1 at the threshold. Percolation thresholds, however, are separate, spatially explicit thresholds that indicate long-range connectivity in a system and do not coincide with R0 = 1. Abundance thresholds are the theoretical basis for attempts to manage infectious disease by reducing the abundance of susceptibles, including vaccination and the culling of wildlife6,7,8. This first natural example of a percolation threshold in a disease system invites a re-appraisal of other invasion thresholds, such as those for epidemic viral infections in African lions (Panthera leo), and of other disease systems such as bovine tuberculosis (caused by Mycobacterium bovis) in badgers (Meles meles).

/pdf/the-abundance-threshold-for-plague-as-a-critical-percolation-3p5a41rrbb.pdf

The abundance threshold for plague as a critical percolation phenomenon

Five challenges for spatial epidemic models.

http://wrap.warwick.ac.uk/65797/1/WRAP_1-s2.0-S1755436514000334-main.pdf

Eight challenges for network epidemic models.

This paper is concerned with a stochastic SIR (susceptible → infective → removed) model for the spread of an epidemic amongst a population of individuals, with a random network of social contacts, that is also partitioned into households. The behaviour of the model as the population size tends to infinity in an appropriate fashion is investigated. A threshold parameter which determines whether or not an epidemic with few initial infectives can become established and lead to a major outbreak is obtained, as are the probability that a major outbreak occurs and the expected proportion of the population that are ultimately infected by such an outbreak, together with methods for calculating these quantities. Monte Carlo simulations demonstrate that these asymptotic quantities accurately reflect the behaviour of finite populations, even for only moderately sized finite populations. The model is compared and contrasted with related models previously studied in the literature. The effects of the amount of clustering present in the overall population structure and the infectious period distribution on the outcomes of the model are also explored.

/pdf/analysis-of-a-stochastic-sir-epidemic-on-a-random-network-2jabzlb1s7.pdf

Pieter Trapman

Papers

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

The abundance threshold for plague as a critical percolation phenomenon

Five challenges for spatial epidemic models.

Eight challenges for network epidemic models.

Analysis of a stochastic SIR epidemic on a random network incorporating household structure.