On the asymptotic final size distribution of epidemics in heterogeneous populations

1990 ◽  
pp. 189-196 ◽  
Author(s):  
Gianpaolo Scalia-Tomba
Keyword(s):  
Size Distribution ◽  
Final Size ◽  
Heterogeneous Populations

Final size distribution for epidemics

1975 ◽  
Vol 23 (1-2) ◽  
pp. 33-46 ◽  
Author(s):  
Donald Ludwig
Keyword(s):  
Size Distribution ◽  
Final Size

A non-standard family of polynomials and the final size distribution of Reed-Frost epidemic processes

1990 ◽  
Vol 22 (1) ◽  
pp. 25-48 ◽  
Author(s):  
Claude Lefevre ◽  
Philippe Picard
Keyword(s):  
Size Distribution ◽  
Final Size ◽  
Asymptotic Results ◽  
Several Variables ◽  
Standard Family

This paper provides a global treatment of the final size distribution of Reed–Frost epidemic processes. Exact and asymptotic results are derived for both single and multipopulation situations. The key tool is a non-standard family of polynomials, introduced initially by Gontcharoff (1937) for one variable, revisited and extended here for several variables. The attractiveness of these polynomials will be enhanced in forthcoming works in the epidemic context as well as in other fields.

Colloidal Silica Particles with Bimodal Final Size Distribution: Ion-Induced Nucleation Mechanism

2007 ◽  
Vol 121-123 ◽  
pp. 105-108 ◽  
Author(s):  
K. Zhang ◽  
Z.F. Song ◽  
Y. Yan ◽  
Q.M. Chen
Keyword(s):  
Ionic Strength ◽  
Size Distribution ◽  
Colloidal Silica ◽  
Silica Particles ◽  
Hydrolysis Reaction ◽  
Hydrolysis Rate ◽  
Nucleation Process ◽  
Final Size ◽  
Bimodal Size Distribution ◽  
Experimental Conditions

Colloidal silica particles, Bimodal size distribution, Nucleation Abstract. Colloidal silica particles with bimodal size distribution have been prepared by the hydrolysis of tetraethylorthosilicate in alcoholic solutions of water and ammonia as catalyst. Experimental conditions such as concentration of NaCl, amount of water and reaction temperature were investigated to reveal the formation mechanism of the colloidal silica particles. The nucleation process of colloidal silica particles with bimodal size distribution depends on the hydrolysis rate of TEOS and the ionic strength of reaction media. The hydrolysis reaction is the rate-limiting step during the nucleation process. Nucleation involving background ions generated by TEOS hydrolysis reaction and addition of electrolyte(NaCl) is another potentially important factor for nucleation process of electrically charged clusters. A critical value of ionic strength exists in the reaction to form bimodal size distribution particles.

Asymptotic final size distribution of the multitype Reed–Frost process

1986 ◽  
Vol 23 (03) ◽  
pp. 563-584
Author(s):  
Gianpaolo Scalia-Tomba
Keyword(s):  
Size Distribution ◽  
Total Population ◽  
Large Population ◽  
Asymptotic Distributions ◽  
Reproduction Rate ◽  
Final Size ◽  
Basic Reproduction Rate ◽  
Infection Pattern ◽  
Large Numbers ◽  
A Chain

The asymptotic final size distribution of a multitype Reed–Frost process, a chain-binomial model for the spread of an infectious disease in a finite, closed multitype population, is derived, as the total population size grows large. When all subgroups are of comparable size, the infection pattern irreducible and the epidemic started by a small number of initial infectives, the classical threshold behaviour is obtained, depending on the basic reproduction rate of the disease in the population, and the asymptotic distributions for small and large outbreaks can be found. The same techniques can then be used to study other asymptotic situations, e.g. small groups in an otherwise large population, large numbers of initial infectives and reducible infection patterns.

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

1994 ◽  
Vol 26 (03) ◽  
pp. 656-670
Author(s):  
Steven M. Butler
Keyword(s):  
Size Distribution ◽  
Random Number ◽  
Asymptotic Properties ◽  
Initial Distribution ◽  
Heterogeneous Population ◽  
Initial Number ◽  
Final Size ◽  
Final State ◽  
The Mean ◽  
The Relationship

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.

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.

SIR epidemics on a Bernoulli random graph

2003 ◽  
Vol 40 (03) ◽  
pp. 779-782 ◽  
Author(s):  
Peter Neal
Keyword(s):  
Size Distribution ◽  
Random Graph ◽  
Final Size ◽  
Asymptotic Results ◽  
Stochastic Epidemic

We consider a generalized stochastic epidemic on a Bernoulli random graph. By constructing the epidemic and graph in unison, the epidemic is shown to be a randomized Reed–Frost epidemic. Hence, the exact final-size distribution and extensive asymptotic results can be derived.

SIR epidemics on a Bernoulli random graph

2003 ◽  
Vol 40 (3) ◽  
pp. 779-782 ◽  
Author(s):  
Peter Neal
Keyword(s):  
Size Distribution ◽  
Random Graph ◽  
Final Size ◽  
Asymptotic Results ◽  
Stochastic Epidemic

We consider a generalized stochastic epidemic on a Bernoulli random graph. By constructing the epidemic and graph in unison, the epidemic is shown to be a randomized Reed–Frost epidemic. Hence, the exact final-size distribution and extensive asymptotic results can be derived.

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