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.

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 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 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.

scholarly journals A general stochastic model for bivariate episodes driven by a gamma sequence

2021 ◽  
Vol 8 (1) ◽  
Author(s):  
Charles K. Amponsah ◽  
Tomasz J. Kozubowski ◽  
Anna K. Panorska
Keyword(s):  
Stochastic Model ◽  
Joint Distribution ◽  
Pareto Distribution ◽  
Integral Transforms ◽  
Bivariate Distribution ◽  
Conditional Distributions ◽  
General Stochastic Model ◽  
Heavy Tailed ◽  
Discrete Pareto Distribution ◽  
Special Case

AbstractWe propose a new stochastic model describing the joint distribution of (X,N), where N is a counting variable while X is the sum of N independent gamma random variables. We present the main properties of this general model, which include marginal and conditional distributions, integral transforms, moments and parameter estimation. We also discuss in more detail a special case where N has a heavy tailed discrete Pareto distribution. An example from finance illustrates the modeling potential of this new mixed bivariate distribution.

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.

scholarly journals Endemic Behaviour of SIS Epidemics with General Infectious Period Distributions

2014 ◽  
Vol 46 (01) ◽  
pp. 241-255 ◽  
Author(s):  
Peter Neal
Keyword(s):  
Epidemic Model ◽  
Branching Process ◽  
Endemic Equilibrium ◽  
Infectious Period ◽  
Sis Epidemic Model ◽  
Branching Process With Immigration ◽  
Sis Epidemic ◽  
Sis Epidemics ◽  
Surprising Observation

We study the endemic behaviour of a homogeneously mixing SIS epidemic in a population of size N with a general infectious period, Q, by introducing a novel subcritical branching process with immigration approximation. This provides a simple but useful approximation of the quasistationary distribution of the SIS epidemic for finite N and the asymptotic Gaussian limit for the endemic equilibrium as N → ∞. A surprising observation is that the quasistationary distribution of the SIS epidemic model depends on Q only through

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.

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