On the formulation of discrete-time epidemic models

Comparing sums of exchangeable Bernoulli random variables

The paper is first concerned with a comparison of the partial sums associated with two sequences of n exchangeable Bernoulli random variables. It then considers a situation where such partial sums are obtained through an iterative procedure of branching type stopped at the first-passage time in a linearly decreasing upper barrier. These comparison results are illustrated with applications to certain urn models, sampling schemes and epidemic processes. A key tool is a non-standard hierarchical class of stochastic orderings between discrete random variables valued in {0, 1,· ··, n}.

A unified analysis of the final size and severity distribution in collective Reed-Frost epidemic processes

An extended version, called collective, of the randomized Reed-Frost processes is considered where each infective during his survival time fails to transmit the infection within any given set of susceptibles with a probability depending only on the size of that set. Our purpose is to provide a unified analysis of the distribution of the final size and severity, the two main components of the cost generated by the infection process. The method developed relies on the construction of a family of martingales and the use of a family of polynomials studied recently by the authors (Lefèvre and Picard (1990)). The results generalize a number of earlier ones and are derived in a more direct and systematic way than before.

The Final State of an Epidemic in a Large Heterogeneous Population with a Large Initial Number of Infectives

We describe some asymptotic properties of a general S–I–R epidemic process in a large heterogeneous population. We assume that the infectives behave independently, that each infective has a generally distributed random number of contacts with the others in the population, and that among the initial susceptibles there is an arbitrary initial distribution of susceptibility. For the case of a large number of initial infectives, we demonstrate the asymptotic normality of the final size distribution as well as convergence of the final distribution of susceptibility as the population size approaches infinity. The relationship between the mean of the limiting final size distribution and the initial heterogeneity of susceptibility is explored, for a parametric example.

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

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.

The Final State of an Epidemic in a Large Heterogeneous Population with a Large Initial Number of Infectives

We describe some asymptotic properties of a general S–I–R epidemic process in a large heterogeneous population. We assume that the infectives behave independently, that each infective has a generally distributed random number of contacts with the others in the population, and that among the initial susceptibles there is an arbitrary initial distribution of susceptibility. For the case of a large number of initial infectives, we demonstrate the asymptotic normality of the final size distribution as well as convergence of the final distribution of susceptibility as the population size approaches infinity. The relationship between the mean of the limiting final size distribution and the initial heterogeneity of susceptibility is explored, for a parametric example.

The Early and Final States of an Epidemic in a Large Heterogeneous Population by a Small Initial Number of Infectives

This paper describes the early and final properties of a general S–I–R epidemic process in which the infectives behave independently, each infective has a random number of contacts with the others in the population, and individuals vary in their susceptibility to infection. For the case of a large initial number of susceptibles and a small (finite) initial number of infectives, we derive the threshold behavior and the limiting distribution for the final state of the epidemic. Also, we show strong convergence of the epidemic process over any finite time interval to a birth and death process, extending the results of Ball (1983). These complement some results due to Butler (1994), who considers the case of a large initial number of infectives.

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

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.

The Early and Final States of an Epidemic in a Large Heterogeneous Population by a Small Initial Number of Infectives

This paper describes the early and final properties of a general S–I–R epidemic process in which the infectives behave independently, each infective has a random number of contacts with the others in the population, and individuals vary in their susceptibility to infection. For the case of a large initial number of susceptibles and a small (finite) initial number of infectives, we derive the threshold behavior and the limiting distribution for the final state of the epidemic. Also, we show strong convergence of the epidemic process over any finite time interval to a birth and death process, extending the results of Ball (1983). These complement some results due to Butler (1994), who considers the case of a large initial number of infectives.

A unified analysis of the final size and severity distribution in collective Reed-Frost epidemic processes

An extended version, called collective, of the randomized Reed-Frost processes is considered where each infective during his survival time fails to transmit the infection within any given set of susceptibles with a probability depending only on the size of that set. Our purpose is to provide a unified analysis of the distribution of the final size and severity, the two main components of the cost generated by the infection process. The method developed relies on the construction of a family of martingales and the use of a family of polynomials studied recently by the authors (Lefèvre and Picard (1990)). The results generalize a number of earlier ones and are derived in a more direct and systematic way than before.

On the impact of epidemic severity on network immunization algorithms

There has been much recent interest in the prevention and mitigation of epidemics spreading through contact networks of host populations. Here, we investigate how the severity of epidemics, measured by its infection rate, influences the efficiency of well-known vaccination strategies. In order to assess the impact of severity, we simulate the SIR model at different infection rates on various real and model immunized networks. An extensive analysis of our simulation results reveals that immunization algorithms, which efficiently reduce the nodes’ average degree, are more effective in the mitigation of weak and slow epidemics, whereas vaccination strategies that fragment networks to small components, are more successful in suppressing severe epidemics.

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Interaction of the immunization algorithms, epidemic trials and network structure is investigated.

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Immunization algorithms are applied to various real and model networks.

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Degree-based immunization algorithms are more efficient in mitigation of weak epidemics.

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Network largest component size is a vital element in spreading of severe epidemics.

Network theory and SARS: predicting outbreak diversity

Many infectious diseases spread through populations via the networks formed by physical contacts among individuals. The patterns of these contacts tend to be highly heterogeneous. Traditional "compartmental" modeling in epidemiology, however, assumes that population groups are fully mixed, that is, every individual has an equal chance of spreading the disease to every other. Applications of compartmental models to Severe Acute Respiratory Syndrome (SARS) resulted in estimates of the fundamental quantity called the basic reproductive number R0--the number of new cases of SARS resulting from a single initial case--above one, implying that, without public health intervention, most outbreaks should spark large-scale epidemics. Here we compare these predictions to the early epidemiology of SARS. We apply the methods of contact network epidemiology to illustrate that for a single value of R0, any two outbreaks, even in the same setting, may have very different epidemiological outcomes. We offer quantitative insight into the heterogeneity of SARS outbreaks worldwide, and illustrate the utility of this approach for assessing public health strategies.

Comparing sums of exchangeable Bernoulli random variables

The paper is first concerned with a comparison of the partial sums associated with two sequences of n exchangeable Bernoulli random variables. It then considers a situation where such partial sums are obtained through an iterative procedure of branching type stopped at the first-passage time in a linearly decreasing upper barrier. These comparison results are illustrated with applications to certain urn models, sampling schemes and epidemic processes. A key tool is a non-standard hierarchical class of stochastic orderings between discrete random variables valued in {0, 1,· ··, n}.

Collective epidemic models

A number of models have been proposed to describe the spread of infectious diseases of the S-I-R type. Most of them account for variable infectivity levels, and very few incorporate variable susceptibility levels. In the present work, a new epidemic model, called a collective model, is constructed that combines both variabilities in a general way. It is then established how to determine the exact distribution of the final state and the severity of the epidemic when the infection process stops.

A mathematical model of rinderpest infection in cattle populations

A mathematical model for the epidemiology of rinderpest was developed, starting from a simplified descriptive analysis of the disease. A formula for the calculation of the probability of infection of a susceptible animal was first established. A deterministic failure threshold of the infection was then deduced. Deterministic and stochastic approaches were adopted using iterative methods on a computer. These allowed a description of the spread and the variability of an infection process in a population to be made. An illustration of the use of this model showed that, in some cases, variability effects due to stochastic factors were very important. In these particular conditions, the use of the deterministic model alone was not adequate for a good description of the infection. Consequently, improvements of the model were proposed in order to make it more realistic and to allow its use for the evaluation of the efficiency of field operations.