Let F be a probability distribution with support on the non-negative integers. Four methods for generating a simple undirected graph with (approximate) degree distribution F are described and compared. Two methods are based on the so called configuration model with modifications ensuring a simple graph, one method is an extension of the classical Erdős-Rényi graph where the edge probabilities are random variables, and the last method starts with a directed random graph which is then modified to a simple undirected graph. All methods are shown to give the correct distribution in the limit of large graph size, but under different assumptions on the degree distribution F and also using different order of operations.

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Consider a random graph, having a prespecified degree distribution F, but other than that being uniformly distributed, describing the social structure (friendship) in a large community. Suppose that one individual in the community is externally infected by an infectious disease and that the disease has its course by assuming that infected individuals infect their not yet infected friends independently with probability p. For this situation, we determine the values of R0, the basic reproduction number, and τ0, the asymptotic final size in the case of a major outbreak. Furthermore, we examine some different local vaccination strategies, where individuals are chosen randomly and vaccinated, or friends of the selected individuals are vaccinated, prior to the introduction of the disease. For the studied vaccination strategies, we determine Rv, the reproduction number, and τv, the asymptotic final proportion infected in the case of a major outbreak, after vaccinating a fraction v.

Consider a random graph, having a prespecified degree distribution F , but other than that being uniformly distributed, describing the social structure (friendship) in a large community. Suppose that one individual in the community is externally infected by an infectious disease and that the disease has its course by assuming that infected individuals infect their not yet infected friends independently with probability p. For this situation, we determine the values of R 0 , the basic reproduction number, and τ 0 , the asymptotic final size in the case of a major outbreak. Furthermore, we examine some different local vaccination strategies, where individuals are chosen randomly and vaccinated, or friends of the selected individuals are vaccinated, prior to the introduction of the disease. For the studied vaccination strategies, we determine R v , the reproduction number, and τ v , the asymptotic final proportion infected in the case of a major outbreak, after vaccinating a fraction v.

The distribution of the final size, K, in a general SIR epidemic model is considered in a situation when the critical parameter λ is close to 1. It is shown that with a ‘critical scaling’ λ ≈ 1 + a / n 1/3, m ≈ bn 1/3, where n is the initial number of susceptibles and m is the initial number of infected, then K / n 2/3 has a limit distribution when n → ∞. It can be described as that of T, the first passage time of a Wiener process to a parabolic barrier b + at − t 2/2. The proof is based on a diffusion approximation. Moreover, it is shown that the distribution of T can be expressed analytically in terms of Airy functions using the spectral representation connected with Airy's differential equation.

We show that for an Ising spin system of arbitrary spin with a ferromagnetic pair interaction and a "periodic" external magnetic field there is a unique equilibrium state if and only if the magnetization is continuous with respect to a uniform change in the external field. Hence, if the critical temperature T c is defined as the temperature where the spontaneous magnetization (which is a non-increasing function of the temperature) becomes positive, then the equilibrium state is unique for T> T~ and is non-unique for T< T c (when the external field is zero). This implies that the correlation functions have a cluster property for T > T~.We also show that for an anti-ferromagnet consisting of two sublattices there is a unique equilibrium state if and only if the staggered magnetization is continuous with respect to a change in the staggered field.

Iterative sampling procedures of a general type in a finite population are considered. They generalize the Reed-Frost process in that binomial sampling is replaced by an arbitrary symmetric sampling defined by a factorial series distribution. Threshold limit theorems are proved saying that the total number of sampled objects is either small with a certain limit distribution, or a finite fraction of the population with a Gaussian limit distribution as the size of the population gets large. These results extend earlier ones for the Reed-Frost process [1], and are proved in a more direct way than before.

The distribution of the final size, K, in a general SIR epidemic model is considered in a situation when the critical parameter λ is close to 1. It is shown that with a ‘critical scaling’ λ ≈ 1 + a / n1/3, m ≈ bn1/3, where n is the initial number of susceptibles and m is the initial number of infected, then K / n2/3 has a limit distribution when n → ∞. It can be described as that of T, the first passage time of a Wiener process to a parabolic barrier b + at − t2/2. The proof is based on a diffusion approximation. Moreover, it is shown that the distribution of T can be expressed analytically in terms of Airy functions using the spectral representation connected with Airy's differential equation.