An unusual stochastic order relation with some applications in sampling and epidemic theory

1993 ◽  
Vol 25 (1) ◽  
pp. 63-81 ◽  
Author(s):  
Claude Lefevre ◽  
Philippe Picard
Keyword(s):  
Stochastic Order ◽  
Infection Intensity ◽  
Order Relation ◽  
Final Size ◽  
Factorial Moments ◽  
Comparison Results ◽  
Global Cost ◽  
Frost Model ◽  
Sampling Procedures ◽  
Total Damage

One expects, intuitively, that the total damage caused by an epidemic increases, in a certain sense, with the infection intensity exerted by the infectives during their lifelength. The original object of the present work is to make precise in which probabilistic terms such a statement does indeed hold true, when the spread of the disease is described by a collective Reed–Frost model and the global cost is represented by the final size and severity. Surprisingly, this problem leads us to introduce an order relation for -valued random variables, unusual in the literature, based on the descending factorial moments. Further applications of the ordering occur when comparing certain sampling procedures through the number of un-sampled individuals. In particular, it is used to reinforce slightly comparison results obtained earlier for two such samplings.

An unusual stochastic order relation with some applications in sampling and epidemic theory

1993 ◽  
Vol 25 (01) ◽  
pp. 63-81 ◽  
Author(s):  
Claude Lefevre ◽  
Philippe Picard
Keyword(s):  
Stochastic Order ◽  
Infection Intensity ◽  
Order Relation ◽  
Final Size ◽  
Factorial Moments ◽  
Comparison Results ◽  
Global Cost ◽  
Frost Model ◽  
Sampling Procedures ◽  
Total Damage

One expects, intuitively, that the total damage caused by an epidemic increases, in a certain sense, with the infection intensity exerted by the infectives during their lifelength. The original object of the present work is to make precise in which probabilistic terms such a statement does indeed hold true, when the spread of the disease is described by a collective Reed–Frost model and the global cost is represented by the final size and severity. Surprisingly, this problem leads us to introduce an order relation for -valued random variables, unusual in the literature, based on the descending factorial moments. Further applications of the ordering occur when comparing certain sampling procedures through the number of un-sampled individuals. In particular, it is used to reinforce slightly comparison results obtained earlier for two such samplings.

scholarly journals Stochastic Epidemic Models in Structured Populations Featuring Dynamic Vaccination and Isolation

2007 ◽  
Vol 44 (03) ◽  
pp. 571-585 ◽  
Author(s):  
Frank Ball ◽  
Philip D. O'Neill ◽  
James Pike
Keyword(s):  
Exposure Period ◽  
Structured Populations ◽  
Infectious Period ◽  
Stochastic Comparison ◽  
Threshold Parameter ◽  
Final Size ◽  
Period Distribution ◽  
Comparison Results ◽  
Stochastic Epidemic ◽  
Large Populations

We consider a stochastic model for the spread of an SEIR (susceptible → exposed → infective → removed) epidemic among a population of individuals partitioned into households. The model incorporates both vaccination and isolation in response to the detection of cases. When the infectious period is exponential, we derive an explicit formula for a threshold parameter, and analytic results that enable computation of the probability of the epidemic taking off. These quantities are found to be independent of the exposure period distribution. An approximation for the expected final size of an epidemic that takes off is obtained, evaluated numerically, and found to be reasonably accurate in large populations. When the infectious period is not exponential, but has an increasing hazard rate, we obtain stochastic comparison results in the case where the exposure period is fixed. Our main result shows that as the exposure period increases, both the severity of the epidemic in a single household and the threshold parameter decrease, under certain assumptions concerning isolation. Corresponding results for infectious periods with decreasing hazard rates are also derived.

LINEAR BIRTH/IMMIGRATION-DEATH PROCESS WITH BINOMIAL CATASTROPHES

2015 ◽  
Vol 30 (1) ◽  
pp. 79-111 ◽  
Author(s):  
Stella Kapodistria ◽  
Tuan Phung-Duc ◽  
Jacques Resing
Keyword(s):  
Relaxation Time ◽  
Random Environment ◽  
Equilibrium Distribution ◽  
Stochastic Order ◽  
Multidimensional Systems ◽  
Time Dependent ◽  
Death Process ◽  
Factorial Moments ◽  
System Parameters ◽  
Number Of Individuals

In this paper, we study birth/immigration-death processes under mild (binomial) catastrophes. We obtain explicit expressions for both the time-dependent (transient) and the limiting (equilibrium) factorial moments, which are then used to construct the transient and equilibrium distribution of the population size. We demonstrate that our approach is also applicable to multidimensional systems such as stochastic processes operating under a random environment and other variations of the model at hand. We also obtain various stochastic order results for the number of individuals with respect to the system parameters, as well as the relaxation time.

Stochastic comparisons of interfailure times under a relevation replacement policy

2017 ◽  
Vol 54 (1) ◽  
pp. 134-145 ◽  
Author(s):  
Miguel A. Sordo ◽  
Georgios Psarrakos
Keyword(s):  
Stochastic Order ◽  
Replacement Policy ◽  
Convex Order ◽  
Stochastic Comparisons ◽  
Increasing Convex Order ◽  
Failure Times ◽  
Usual Stochastic Order ◽  
Comparison Results ◽  
Excess Wealth Order ◽  
Cumulative Residual

AbstractWe provide some results for the comparison of the failure times and interfailure times of two systems based on a replacement policy proposed by Kapodistria and Psarrakos (2012). In particular, we show that when the first failure times are ordered in terms of the dispersive order (or, the excess wealth order), then the successive interfailure times are ordered in terms of the usual stochastic order (respectively, the increasing convex order). As a consequence, we provide comparison results for the cumulative residual entropies of the systems and their dynamic versions.

scholarly journals Stochastic Epidemic Models in Structured Populations Featuring Dynamic Vaccination and Isolation

2007 ◽  
Vol 44 (3) ◽  
pp. 571-585 ◽  
Author(s):  
Frank Ball ◽  
Philip D. O'Neill ◽  
James Pike
Keyword(s):  
Exposure Period ◽  
Structured Populations ◽  
Infectious Period ◽  
Stochastic Comparison ◽  
Threshold Parameter ◽  
Final Size ◽  
Period Distribution ◽  
Comparison Results ◽  
Stochastic Epidemic ◽  
Large Populations

We consider a stochastic model for the spread of an SEIR (susceptible → exposed → infective → removed) epidemic among a population of individuals partitioned into households. The model incorporates both vaccination and isolation in response to the detection of cases. When the infectious period is exponential, we derive an explicit formula for a threshold parameter, and analytic results that enable computation of the probability of the epidemic taking off. These quantities are found to be independent of the exposure period distribution. An approximation for the expected final size of an epidemic that takes off is obtained, evaluated numerically, and found to be reasonably accurate in large populations. When the infectious period is not exponential, but has an increasing hazard rate, we obtain stochastic comparison results in the case where the exposure period is fixed. Our main result shows that as the exposure period increases, both the severity of the epidemic in a single household and the threshold parameter decrease, under certain assumptions concerning isolation. Corresponding results for infectious periods with decreasing hazard rates are also derived.

scholarly journals Further Results of the TTT Transform Ordering of Order n

Symmetry ◽  
2021 ◽  
Vol 13 (10) ◽  
pp. 1960
Author(s):  
Lei Yan ◽  
Diantong Kang ◽  
Haiyan Wang
Keyword(s):  
Reliability Analysis ◽  
Partial Order ◽  
Stochastic Models ◽  
Stochastic Order ◽  
Order Relation ◽  
Partial Order Relation ◽  
Coherent Systems ◽  
Closure Properties ◽  
Distributed Components ◽  
Distribution Class

To compare the variability of two random variables, we can use a partial order relation defined on a distribution class, which contains the anti-symmetry. Recently, Nair et al. studied the properties of total time on test (TTT) transforms of order n and examined their applications in reliability analysis. Based on the TTT transform functions of order n, they proposed a new stochastic order, the TTT transform ordering of order n (TTT-n), and discussed the implications of order TTT-n. The aim of the present study is to consider the closure and reversed closure of the TTT-n ordering. We examine some characterizations of the TTT-n ordering, and obtain the closure and reversed closure properties of this new stochastic order under several reliability operations. Preservation results of this order in several stochastic models are investigated. The closure and reversed closure properties of the TTT-n ordering for coherent systems with dependent and identically distributed components are also obtained.

Stochastic Comparison of Some Wear Processes

1993 ◽  
Vol 7 (3) ◽  
pp. 421-435 ◽  
Author(s):  
Franco Pellerey ◽  
Moshe Shaked
Keyword(s):  
Stochastic Order ◽  
Random Variables ◽  
Order Relation ◽  
Stochastic Comparison ◽  
Counting Processes

Let [NA(t), t≥0] and [NB(t), t≥0] be two counting processes with independent interarrivals Ai and Bi, respectively, i ∈ + ≡ [1,2,… ], and let [Xi, i ∈ +] and [Yi, i ∈ +] be two sequences of independent non-negative random variables. Consider the two compound processes [U(t), t ≥ 0] and [V(t), t ≥ 0] defined as U(t) = Xi, t ≥ 0, and V(t) = Yi, t ≥ 0. Then it is well known (and not hard to verify) that if the Ai's are independent of the Xi's, and if the Bi's are independent of the yi's, and if Ai ≥st Bi and Xi ≤st Yi, for all i ∈ +, then [U(t), t ≥ 0] ≤st [V(t), t ≥ 0]. In this paper we provide conditions under which this stochastic order relation holds even if Ai ≤st Bi and/or Xi ≥st Yi for all i ∈ +. In fact, we derive the stochastic order relationships among processes that are much more general than the cumulative ones described above. For example, it is not necessarily assumed that the Ai's are independent of the Xi's or that the Bi's are independent of the Yi's. Examples that illustrate the applications of the theory are included.

scholarly journals Nonstationarity and randomization in the Reed-Frost epidemic model

2005 ◽  
Vol 42 (4) ◽  
pp. 950-963 ◽  
Author(s):  
Claude Lefèvre ◽  
Philippe Picard
Keyword(s):  
Wide Class ◽  
Epidemic Model ◽  
Exact Distribution ◽  
Unified Approach ◽  
Final Size ◽  
Final Number ◽  
Nonstationary Model ◽  
Frost Model ◽  
Resistance To Infection

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.

Stochastic properties of a cumulative damage threshold crossing model

1999 ◽  
Vol 36 (3) ◽  
pp. 720-732 ◽  
Author(s):  
Nader Ebrahimi
Keyword(s):  
Poisson Process ◽  
Failure Rate ◽  
Stochastic Order ◽  
Sufficient Conditions ◽  
Damage Threshold ◽  
Survival Functions ◽  
Proposed Model ◽  
Threshold Crossing ◽  
Stochastic Properties ◽  
Total Damage

In this paper we describe a model for survival functions. Under this model a system is subject to shocks governed by a Poisson process. Each shock to the system causes a random damage that grows in time. Damages accumulate additively and the system fails if the total damage exceeds a certain capacity or threshold. Various properties of this model are obtained. Sufficient conditions are derived for the failure rate (FR) order and the stochastic order to hold between the random lifetimes of two systems whose failures can be described by our proposed model.

scholarly journals Nonstationarity and randomization in the Reed-Frost epidemic model

2005 ◽  
Vol 42 (04) ◽  
pp. 950-963 ◽  
Author(s):  
Claude Lefèvre ◽  
Philippe Picard
Keyword(s):  
Wide Class ◽  
Epidemic Model ◽  
Exact Distribution ◽  
Unified Approach ◽  
Final Size ◽  
Final Number ◽  
Nonstationary Model ◽  
Frost Model ◽  
Resistance To Infection

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.

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