Nonstationarity and randomization in the Reed-Frost epidemic model

The purpose of this paper is to determine the exact distribution of the final size of an epidemic for a wide class of models of susceptible–infective–removed type. First, a nonstationary version of the classical Reed–Frost model is constructed that allows us to incorporate, in particular, random levels of resistance to infection in the susceptibles. Then, a randomized version of this nonstationary model is considered in order to take into account random levels of infectiousness in the infectives. It is shown that, in both cases, the distribution of the final number of infected individuals can be obtained in terms of Abel–Gontcharoff polynomials. The new methodology followed also provides a unified approach to a number of recent works in the literature.

From damage models to SIR epidemics and cascading failures

This paper is concerned with a nonstationary Markovian chain of cascading damage that constitutes an iterated version of a classical damage model. The main problem under study is to determine the exact distribution of the total outcome of this process when the cascade of damages finally stops. Two different applications are discussed, namely the final size for a wide class of SIR (susceptible → infective → removed) epidemic models and the total number of failures for a system of components in reliability. The starting point of our analysis is the recent work of Lefèvre (2007) on a first-crossing problem for the cumulated partial sums of independent parametric distributions, possibly nonstationary but stable by convolution. A key mathematical tool is provided by a nonstandard family of remarkable polynomials, called the generalised Abel–Gontcharoff polynomials. Somewhat surprisingly, the approach followed will allow us to relax some model assumptions usually made in epidemic theory and reliability. To close, approximation by a branching process is also investigated to a certain extent.

On Multiply Monotone Distributions, Continuous or Discrete, with Applications

This paper is concerned with the class of distributions, continuous or discrete, whose shape is monotone of finite integer ordert. A characterization is presented as a mixture of a minimum oftindependent uniform distributions. Then, a comparison oft-monotone distributions is made using thes-convex stochastic orders. A link is also pointed out with an alternative approach to monotonicity based on a stationary-excess operator. Finally, the monotonicity property is exploited to reinforce the classical Markov and Lyapunov inequalities. The results are illustrated by several applications to insurance.

Stein's Method and Stochastic Orderings

A stochastic ordering approach is applied with Stein's method for approximation by the equilibrium distribution of a birth-death process. The usual stochastic order and the more general s-convex orders are discussed. Attention is focused on Poisson and translated Poisson approximations of a sum of dependent Bernoulli random variables, for example, k-runs in independent and identically distributed Bernoulli trials. Other applications include approximation by polynomial birth-death distributions.

On Multiply Monotone Distributions, Continuous or Discrete, with Applications

This paper is concerned with the class of distributions, continuous or discrete, whose shape is monotone of finite integer order t. A characterization is presented as a mixture of a minimum of t independent uniform distributions. Then, a comparison of t-monotone distributions is made using the s-convex stochastic orders. A link is also pointed out with an alternative approach to monotonicity based on a stationary-excess operator. Finally, the monotonicity property is exploited to reinforce the classical Markov and Lyapunov inequalities. The results are illustrated by several applications to insurance.

Exact Lower Bounds on the Exponential Moments of Truncated Random Variables

Exact lower bounds on the exponential moments of min(y,X) andX1{X<y} are provided given the first two moments of a random variableX. These bounds are useful in work on large deviation probabilities and nonuniform Berry-Esseen bounds, when the Cramér tilt transform may be employed. Asymptotic properties of these lower bounds are presented. Comparative advantages of the so-called Winsorization min(y,X) over the truncationX1{X<y} are demonstrated. An application to option pricing is given.

Variability order of the latent and the infectious periods in a deterministic SEIR epidemic model and evaluation of control effectiveness

We use distribution theory and ordering of non-negative random variables to study the Susceptible-Exposed-Infectious-Removed (SEIR) model with two control measures, quarantine and isolation, to reduce the spread of an infectious disease. We identify that the probability distributions of the latent period and the infectious period are primary features of the SEIR model to formulate the epidemic threshold and to evaluate the effectiveness of the intervention measures. If the primary features are changed, the conclusions will be altered in an importantly different way. For the latent and infectious periods with known mean values, it is the dilation, a generalization of variance, of their distributions that ranks the effectiveness of these control measures. We further propose ways to set quarantine and isolation targets to reduce the controlled reproduction number below the threshold using observed initial growth rate from outbreak data. If both quarantine and isolation are 100% effective, one can directly use the observed growth rate for setting control targets. If they are not 100% effective, some further knowledge of the distributions is required.