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).

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

1993 ◽  
Vol 25 (04) ◽  
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).

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.

The final outcome of an epidemic model with several different types of infective in a large population

1995 ◽  
Vol 32 (3) ◽  
pp. 579-590 ◽  
Author(s):  
Frank Ball ◽  
Damian Clancy
Keyword(s):  
Stochastic Model ◽  
Size Distribution ◽  
Epidemic Model ◽  
Infected Individual ◽  
Large Population ◽  
Asymptotic Distributions ◽  
Initial Population ◽  
Group Model ◽  
Final Size ◽  
Different Types

We consider a stochastic model for the spread of an epidemic amongst a closed homogeneously mixing population, in which there are several different types of infective, each newly infected individual choosing its type at random from those available. The model is based on the carrier-borne model of Downton (1968), as extended by Picard and Lefèvre (1990). The asymptotic distributions of final size and area under the trajectory of infectives are derived as the initial population becomes large, using arguments based on those of Scalia-Tomba (1985), (1990). We then use our limiting results to compare the asymptotic final size distribution of our model with that of a related multi-group model, in which the type of each infective is assigned deterministically.

The final outcome of an epidemic model with several different types of infective in a large population

1995 ◽  
Vol 32 (03) ◽  
pp. 579-590 ◽  
Author(s):  
Frank Ball ◽  
Damian Clancy
Keyword(s):  
Stochastic Model ◽  
Size Distribution ◽  
Epidemic Model ◽  
Infected Individual ◽  
Large Population ◽  
Asymptotic Distributions ◽  
Initial Population ◽  
Group Model ◽  
Final Size ◽  
Different Types

We consider a stochastic model for the spread of an epidemic amongst a closed homogeneously mixing population, in which there are several different types of infective, each newly infected individual choosing its type at random from those available. The model is based on the carrier-borne model of Downton (1968), as extended by Picard and Lefèvre (1990). The asymptotic distributions of final size and area under the trajectory of infectives are derived as the initial population becomes large, using arguments based on those of Scalia-Tomba (1985), (1990). We then use our limiting results to compare the asymptotic final size distribution of our model with that of a related multi-group model, in which the type of each infective is assigned deterministically.

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

1995 ◽  
Vol 32 (2) ◽  
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 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.

Functionals of random mappings: exact and asymptotic results

1991 ◽  
Vol 23 (3) ◽  
pp. 437-455 ◽  
Author(s):  
P. J. Donnelly ◽  
W. J. Ewens ◽  
S. Padmadisastra
Keyword(s):  
Frequency Spectrum ◽  
Exact Results ◽  
Asymptotic Results ◽  
Number Of Components ◽  
Random Mapping ◽  
Random Mappings ◽  
Simulation Results ◽  
Central Tool

A random mapping partitions the set {1, 2, ···, m} into components, where i and j are in the same component if some functional iterate of i equals some functional iterate of j. We consider various functionals of these partitions and of samples from it, including the number of components of ‘small' size and of size O(m) as m → ∞the size of the largest component, the number of components, and various symmetric functionals of the normalized component sizes. In many cases exact results, while available, are uniformative, and we consider various approximations. Numerical and simulation results are also presented. A central tool for many calculations is the ‘frequency spectrum', both exact and asymptotic.

The transition probabilities of the general stochastic epidemic model

1975 ◽  
Vol 12 (3) ◽  
pp. 415-424 ◽  
Author(s):  
Richard J. Kryscio
Keyword(s):  
Stochastic Model ◽  
Convenient Method ◽  
Epidemic Model ◽  
Transition Probabilities ◽  
Stochastic Epidemic ◽  
General Stochastic Model ◽  
Stochastic Epidemic Model ◽  
Forward Equations ◽  
General Stochastic

Recently, Billard (1973) derived a solution to the forward equations of the general stochastic model. This solution contains some recursively defined constants. In this paper we solve these forward equations along each of the paths the process can follow to absorption. A convenient method of combining the solutions for the different paths results in a simplified non-recursive expression for the transition probabilities of the process.

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.

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