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.

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.

scholarly journals The mathematics of multiple lockdowns

2021 ◽  
Vol 11 (1) ◽  
Author(s):  
Antonio Scala
Keyword(s):  
Beneficial Effect ◽  
Epidemic Model ◽  
Final Size ◽  
Optimal Response ◽  
Support Strategy ◽  
The Impact ◽  
New Strategies

AbstractWhile vaccination is the optimal response to an epidemic, recent events have obliged us to explore new strategies for containing worldwide epidemics, like lockdown strategies, where the contacts among the population are strongly reduced in order to slow down the propagation of the infection. By analyzing a classical epidemic model, we explore the impact of lockdown strategies on the evolution of an epidemic. We show that repeated lockdowns have a beneficial effect, reducing the final size of the infection, and that they represent a possible support strategy to vaccination policies.

VARIABILITY FOR CARRIER-BORNE EPIDEMICS AND REED–FROST MODELS INCORPORATING UNCERTAINTIES AND DEPENDENCIES FROM SUSCEPTIBLES AND INFECTIVES

2010 ◽  
Vol 24 (2) ◽  
pp. 303-328 ◽  
Author(s):  
Eva María Ortega ◽  
Laureano F. Escudero
Keyword(s):  
Risk Assessment ◽  
Epidemic Model ◽  
Epidemic Models ◽  
Random Environments ◽  
Random Parameters ◽  
Infection Rates ◽  
Disease Propagation ◽  
Increasing Concave Order ◽  
Special Cases ◽  
Frost Model

This article provides analytical results on which are the implications of the statistical dependencies among certain random parameters on the variability of the number of susceptibles of the carrier-borne epidemic model with heterogeneous populations and of the number of infectives under the Reed–Frost model with random infection rates. We consider dependencies among the random infection rates, among the random infectious times, and among random initial susceptibles of several carrier-borne epidemic models. We obtain conditions for the variability ordering between the number of susceptibles for carrier-borne epidemics under two different random environments, at any time-scale value. These results are extended to multivariate comparisons of the random vectors of populations in the strata. We also obtain conditions for the increasing concave order between the number of infectives in the Reed–Frost model under two different random environments, for any generation. Variability bounds are obtained for different epidemic models from modeling dependencies for a range of special cases that are useful for risk assessment of disease propagation.

scholarly journals From damage models to SIR epidemics and cascading failures

2009 ◽  
Vol 41 (1) ◽  
pp. 247-269 ◽  
Author(s):  
Maude Gathy ◽  
Claude Lefèvre
Keyword(s):  
Branching Process ◽  
Damage Model ◽  
Exact Distribution ◽  
Partial Sums ◽  
Mathematical Tool ◽  
Close Approximation ◽  
Final Size ◽  
Damage Models ◽  
Starting Point ◽  
Made In

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.

A note on the moments of the final size of the general epidemic model

1980 ◽  
Vol 17 (2) ◽  
pp. 532-538 ◽  
Author(s):  
Ross Dunstan
Keyword(s):  
Population Size ◽  
Epidemic Model ◽  
Asymptotic Series ◽  
Algebraic Methods ◽  
Series Expansions ◽  
Final Size

In the general epidemic model we study the first two moments of the final size. Beginning with the backwards equation, algebraic methods are used to find their asymptotic series expansions as the population size increases.

First crossing of basic counting processes with lower non-linear boundaries: A unified approach through pseudopolynomials (I)

1996 ◽  
Vol 28 (03) ◽  
pp. 853-876 ◽  
Author(s):  
Philippe Picard ◽  
Claude Lefèvre
Keyword(s):  
General Framework ◽  
Counting Process ◽  
Lower Boundary ◽  
Exact Distribution ◽  
Applied Probability ◽  
Renewal Processes ◽  
Counting Processes ◽  
Unified Approach ◽  
Non Linear ◽  
Birth Processes

The paper is concerned with the distribution of the levelNof the first crossing of a counting process trajectory with a lower boundary. Compound and simple Poisson or binomial processes, gamma renewal processes, and finally birth processes are considered. In the simple Poisson case, expressing the exact distribution ofNrequires the use of a classical family of Abel–Gontcharoff polynomials. For other cases convenient extensions of these polynomials into pseudopolynomials with a similar structure are necessary. Such extensions being applicable to other fields of applied probability, the central part of the present paper has been devoted to the building of these pseudopolynomials in a rather general framework.

The final outcome and temporal solution of a carrier-borne epidemic model

1995 ◽  
Vol 32 (02) ◽  
pp. 304-315 ◽  
Author(s):  
Frank Ball ◽  
Damian Clancy
Keyword(s):  
Stochastic Model ◽  
Explicit Expression ◽  
Exponential Distribution ◽  
Epidemic Model ◽  
Joint Distribution ◽  
Time Dependent ◽  
Infectious Period ◽  
Final Size ◽  
Probabilistic Structure ◽  
Dependent State

We consider a stochastic model for the spread of a carrier-borne epidemic amongst a closed homogeneously mixing population, in which a proportion 1 − π of infected susceptibles are directly removed and play no part in spreading the infection. The remaining proportion π become carriers, with an infectious period that follows an arbitrary but specified distribution. We give a construction of the epidemic process which directly exploits its probabilistic structure and use it to derive the exact joint distribution of the final size and severity of the carrier-borne epidemic, distinguishing between removed carriers and directly removed individuals. We express these results in terms of Gontcharoff polynomials. When the infectious period follows an exponential distribution, our model reduces to that of Downton (1968), for which we use our construction to derive an explicit expression for the time-dependent state probabilities.

scholarly journals Coupling of Two SIR Epidemic Models with Variable Susceptibilities and Infectivities

2007 ◽  
Vol 44 (01) ◽  
pp. 41-57 ◽  
Author(s):  
Peter Neal
Keyword(s):  
Size Distribution ◽  
Random Graph ◽  
Epidemic Model ◽  
Epidemic Models ◽  
Final Size ◽  
Sir Epidemic ◽  
Novel Approach ◽  
Stochastic Epidemic ◽  
Sir Epidemic Models ◽  
Variable Susceptibility

The variable generalised stochastic epidemic model, which allows for variability in both the susceptibilities and infectivities of individuals, is analysed. A very different epidemic model which exhibits variable susceptibility and infectivity is the random-graph epidemic model. A suitable coupling of the two epidemic models is derived which enables us to show that, whilst the epidemics are very different in appearance, they have the same asymptotic final size distribution. The coupling provides a novel approach to studying random-graph epidemic models.

The final size and severity of a generalised stochastic multitype epidemic model

1993 ◽  
Vol 25 (4) ◽  
pp. 721-736 ◽  
Author(s):  
Frank Ball ◽  
Damian Clancy
Keyword(s):  
Stochastic Model ◽  
Epidemic Model ◽  
Exact Results ◽  
Final Size ◽  
Asymptotic Results ◽  
Current Group ◽  
Total Size ◽  
Original Group

We consider a stochastic model for the spread of an epidemic amongst a population split into m groups, in which infectives move among the groups and contact susceptibles at a rate which depends upon the infective's original group, its current group, and the group of the susceptible. The distributions of total size and total area under the trajectory of infectives for such epidemics are analysed. We derive exact results in terms of multivariate Gontcharoff polynomials by treating our model as a multitype collective Reed–Frost process and slightly adapting the results of Picard and Lefèvre (1990). We also derive asymptotic results, as each of the group sizes becomes large, by generalising the method of Scalia-Tomba (1985), (1990).

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