We consider the propagation of a stochastic SIR-type epidemic in two connected populations: a relatively small local population of interest which is surrounded by a much larger external population. External infectives can temporarily enter the small population and contribute to the spread of the infection inside this population. The rules for entry of infectives into the small population as well as their length of stay are modeled by a general Markov queueing system. Our main objective is to determine the distribution of the total number of infections within both populations. To do this, the approach we propose consists of deriving a family of martingales for the joint epidemic processes and applying classical stopping time or convergence theorems. The study then focuses on several particular cases where the external infection is described by a linear branching process and the entry of external infectives obeys certain specific rules. Some of the results obtained are illustrated by numerical examples.

The paper discusses the risk of ruin in insurance coverage of an epidemic in a closed population. The model studied is an extended susceptible–infective–removed (SIR) epidemic model built by Lefèvre and Simon (Methodology Comput. Appl. Prob. 22, 2020) as a block-structured Markov process. A fluid component is then introduced to describe the premium amounts received and the care costs reimbursed by the insurance. Our interest is in the risk of collapse of the corresponding reserves of the company. The use of matrix-analytic methods allows us to determine the distribution of ruin time, the probability of ruin, and the final amount of reserves. The case where the reserves are subjected to a Brownian noise is also studied. Finally, some of the results obtained are illustrated for two particular standard SIR epidemic models.

We discuss a continuous-time Markov branching model in which each individual can trigger an alarm according to a Poisson process. The model is stopped when a given number of alarms is triggered or when there are no more individuals present. Our goal is to determine the distribution of the state of the population at this stopping time. In addition, the state distribution at any fixed time is also obtained. The model is then modified to take into account the possible influence of death cases. All distributions are derived using probability-generating functions, and the approach followed is based on the construction of families of martingales.

In this paper we consider the integral functionals of the general epidemic model up to its extinction. We develop a new approach to determine the exact Laplace transform of such integrals. In particular, we obtain the Laplace transform of the duration of the epidemic T, the final susceptible size ST, the area under the trajectory of the infectives AT, and the area under the trajectory of the susceptibles BT. The method relies on the construction of a family of martingales and allows us to solve simple recursive relations for the involved parameters. The Laplace transforms are then expanded in terms of a special class of polynomials. The analysis is generalized in part to Markovian epidemic processes with arbitrary state-dependent rates.

Soil CO2 flux measurement is a key method that can be used to monitor the hazards in an active volcanic area. In order to determine accurately the variations of the CO2 soil emission we propose an approach based on the radiocarbon (14C) deficiency recorded in the plants grown in and around the Solfatara (Naples, Italy). We twice sampled selected poaceae plants in 17 defined sites around the Solfatara volcano. 14C measurements by liquid scintillation counting (LSC) were achieved on the grass samples. The 14C deficiency determined in the sampled plants, compared to the atmosphere 14C activity, ranged from 6.6 to 51.6%. We then compared the proportion of magmatic CO2 inferred to the instantaneous measurements of CO2 fluxes from soil performed by the accumulation chamber CO2 degassing measurement at the moment of the sampling at each site. The results show a clear correlation (r=0.88) between soil CO2 fluxes and 14C activity. The determination of the plants 14C deficiency provides an estimate of the CO2 rate within a few square meters, integrating CO2 soil degassing variations and meteorological incidences over a few months. It can therefore become an efficient bio-sensor and can be used as a proxy to cartography of the soil CO2 and to determine its variations through time

The classical SIR epidemic model is generalized to incorporate a detection process of infectives in the course of time. Our purpose is to determine the exact distribution of the population state at the first detection instant and the following ones. An extension is also discussed that allows the parameters to change with the number of detected cases. The followed approach relies on simple martingale arguments and uses a special family of Abel–Gontcharoff polynomials.

In this paper we aim to apply simple actuarial methods to build an insurance plan protecting against an epidemic risk in a population. The studied model is an extended SIR epidemic in which the removal and infection rates may depend on the number of registered removals. The costs due to the epidemic are measured through the expected epidemic size and infectivity time. The premiums received during the epidemic outbreak are measured through the expected susceptibility time. Using martingale arguments, a method by recursion is developed to calculate the cost components and the corresponding premium levels in this extended epidemic model. Some numerical examples illustrate the effect of removals and the premium calculation in an insurance plan.

The deltaic plain of the Petite Camargue which constitutes the western part of the Rhone Delta, began its main progradation around 2000 yr ago. Several delta lobes follow each other and have participated in the deltaic evolution. The deltaic lobes have distinct morphologies which reflect the dynamic fluvial and marine processes under the influence of climatic and human controls. Two delta lobe systems were built by the Daladel and Peccaïs channels, after which a deflected wave-influenced delta lobe was formed by the La Ville and Saint-Roman channels. The latest channel, the Rhone Vif channel, is skewed because this channel was completely canalized and engineered up to its mouth in the beginning of the 16th century. Since the avulsion of this channel about 1550 A.D., the coastline of the Petite Camargue has been especially affected by the influence of waves and currents. The spits replaced the beach ridges which juxtaposed themselves and have migrated westward since the 16th century. The formation of the western part of the delta in the last 2000 yr is affected by not only the fluvial sedimentary fluxes and the coastal dynamics to the mouth but also climatic change and human influence.

In this paper we are concerned with a stochastic model for the spread of an epidemic in a closed homogeneously mixing population when an infective can go through several stages of infection before being removed. The transitions between stages are governed by either a Markov process or a semi-Markov process. An infective of any stage makes contacts amongst the population at the points of a Poisson process. Our main purpose is to derive the distribution of the final epidemic size and severity, as well as an approximation by branching, using simple matrix analytic methods. Some illustrations are given, including a model with treatment discussed by Gani (2006).

The purpose of this work is to construct a bridge between two classical topics in applied probability: the finite-time ruin probability in insurance and the final outcome distribution in epidemics. The two risk problems are reformulated in terms of the joint right-tail and left-tail distributions of order statistics for a sample of uniforms. This allows us to show that the hidden algebraic structures are of polynomial type, namely Appell in insurance and Abel–Gontcharoff in epidemics. These polynomials are defined with random parameters, which makes their mathematical study interesting in itself.

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 tindependent 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.

Our first theorem states that the convolution of two symmetric densities which are k-monotone on (0,∞) is again (symmetric) k-monotone provided 0 < k ≤ 1. We then apply this result, together with an extremality approach, to derive sharp moment and exponential bounds for distributions having such shape constrained densities.

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.

The atmospheric response to the 11-year solar cycle is studied using the fully interactive 3-D coupled chemistry-general circulation model LMDz-REPROBUS with a complete seasonal cycle. We will show results concerning a comparison between two series of 20-year runs, one in maximum of activity and the other in minimum. The stratosphere-troposphere system shows partly significant response to a solar cycle enhancement of UV radiation. We show how the changes in stratospheric ozone, temperature and zonal wind are connected.

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.

In this paper we consider the problem of first-crossing from above for a partial sums process m+S t , t ≥ 1, with the diagonal line when the random variables X t , t ≥ 1, are independent but satisfying nonstationary laws. Specifically, the distributions of all the X t s belong to a common parametric family of arithmetic distributions, and this family of laws is assumed to be stable by convolution. The key result is that the first-crossing time distribution and the associated ballot-type formula rely on an underlying polynomial structure, called the generalized Abel-Gontcharoff structure. In practice, this property advantageously provides simple and efficient recursions for the numerical evaluation of the probabilities of interest. Several applications are then presented, for constant and variable parameters.

This article is concerned with a loading-dependent model of cascading failure proposed recently by Dobson, Carreras, and Newman [6]. The central problem is to determine the distribution of the total number of initial components that will have finally failed. A new approach based on a closed connection with epidemic modeling is developed. This allows us to consider a more general failure model in which the additional loads caused by successive failures are arbitrarily fixed (instead of being constant as in [6]). The key mathematical tool is provided by the partial joint distributions of order statistics for a sample of independent uniform (0,1) random variables.

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.