On a stochastic model of an epidemic

The epidemic model considered here, first given by Bartlett (see for example [2]), provides for the immigration of new susceptibles and infectives, as well as describing the spread of infection to susceptibles already present and the removal of infectives. The epidemic curve, relating the numbers of susceptibles and infectives, has been studied for certain cases by Bartlett [1], Kendall [6] and others, and provides a motivation for the results given here. With the aid of criteria given by Reuter [8], [9], the main question considered is the asymptotic behaviour of the mean duration of the epidemic. The behaviour of the limits of the transition probabilities pij(t) as t → ∞ is also investigated.

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The transition probabilities of the general stochastic epidemic model

Journal of Applied Probability ◽

10.2307/3212856 ◽

1975 ◽

Vol 12 (3) ◽

pp. 415-424 ◽

Cited By ~ 9

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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On population-size-dependent branching processes

Advances in Applied Probability ◽

10.2307/1427223 ◽

1984 ◽

Vol 16 (1) ◽

pp. 30-55 ◽

Cited By ~ 53

Author(s):

F. C. Klebaner

Keyword(s):

Stochastic Model ◽

Asymptotic Behaviour ◽

Rate Of Convergence ◽

Population Size ◽

Gamma Distribution ◽

Branching Processes ◽

Independent Random Variables ◽

Size Dependent ◽

Offspring Distribution ◽

The Mean

We consider a stochastic model for the development in time of a population {Zn} where the law of offspring distribution depends on the population size. We are mainly concerned with the case when the mean mk and the variance of offspring distribution stabilize as the population size k grows to ∞, The process exhibits different asymptotic behaviour according to m < l, m = 1, m> l; moreover, the rate of convergence of mk to m plays an important role. It is shown that if m < 1 or m = 1 and mn approaches 1 not slower than n–2 then the process dies out with probability 1. If mn approaches 1 from above and the rate of convergence is n–1, then Zn/n converges in distribution to a gamma distribution, moreover a.s. both on a set of non-extinction and there are no constants an, such that Zn/an converges in probability to a non-degenerate limit. If mn approaches m > 1 not slower than n–α, α > 0, and do not grow to ∞ faster than nß, β <1 then Zn/mn converges almost surely and in L2 to a non-degenerate limit. A number of general results concerning the behaviour of sums of independent random variables are also given.

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On population-size-dependent branching processes

Advances in Applied Probability ◽

10.1017/s0001867800022333 ◽

1984 ◽

Vol 16 (01) ◽

pp. 30-55 ◽

Cited By ~ 21

Author(s):

F. C. Klebaner

Keyword(s):

Stochastic Model ◽

Asymptotic Behaviour ◽

Rate Of Convergence ◽

Population Size ◽

Gamma Distribution ◽

Branching Processes ◽

Independent Random Variables ◽

Size Dependent ◽

Offspring Distribution ◽

The Mean

We consider a stochastic model for the development in time of a population {Z n } where the law of offspring distribution depends on the population size. We are mainly concerned with the case when the mean mk and the variance of offspring distribution stabilize as the population size k grows to ∞, The process exhibits different asymptotic behaviour according to m < l, m = 1, m> l; moreover, the rate of convergence of mk to m plays an important role. It is shown that if m < 1 or m = 1 and mn approaches 1 not slower than n –2 then the process dies out with probability 1. If mn approaches 1 from above and the rate of convergence is n –1, then Zn /n converges in distribution to a gamma distribution, moreover a.s. both on a set of non-extinction and there are no constants an , such that Zn /an converges in probability to a non-degenerate limit. If mn approaches m > 1 not slower than n– α, α > 0, and do not grow to ∞ faster than nß , β <1 then Zn /mn converges almost surely and in L 2 to a non-degenerate limit. A number of general results concerning the behaviour of sums of independent random variables are also given.

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The transition probabilities of the general stochastic epidemic model

Journal of Applied Probability ◽

10.1017/s0021900200048221 ◽

1975 ◽

Vol 12 (03) ◽

pp. 415-424 ◽

Cited By ~ 5

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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The global dynamics of stochastic Holling type II predator-prey models with non constant mortality rate

Filomat ◽

10.2298/fil1718811z ◽

2017 ◽

Vol 31 (18) ◽

pp. 5811-5825

Author(s):

Xinhong Zhang

Keyword(s):

Mortality Rate ◽

Stochastic Model ◽

Sufficient Conditions ◽

Positive Periodic Solution ◽

Global Dynamics ◽

Type Ii ◽

Predator Prey ◽

Persistence In The Mean ◽

The Mean ◽

Predator Prey Models

In this paper we study the global dynamics of stochastic predator-prey models with non constant mortality rate and Holling type II response. Concretely, we establish sufficient conditions for the extinction and persistence in the mean of autonomous stochastic model and obtain a critical value between them. Then by constructing appropriate Lyapunov functions, we prove that there is a nontrivial positive periodic solution to the non-autonomous stochastic model. Finally, numerical examples are introduced to illustrate the results developed.

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Optimal stationary strategies in leavable Markov decision processes

Journal of Applied Probability ◽

10.1017/s0021900200038481 ◽

1990 ◽

Vol 27 (01) ◽

pp. 134-145

Author(s):

Matthias Fassbender

Keyword(s):

Transition Probabilities ◽

Practical Applications ◽

Stationary Strategies ◽

Countable State ◽

Markov Decision ◽

Bias Optimality ◽

The Mean ◽

Optimal Stationary Strategies ◽

The Cost ◽

Reward Criterion

This paper establishes the existence of an optimal stationary strategy in a leavable Markov decision process with countable state space and undiscounted total reward criterion. Besides assumptions of boundedness and continuity, an assumption is imposed on the model which demands the continuity of the mean recurrence times on a subset of the stationary strategies, the so-called ‘good strategies'. For practical applications it is important that this assumption is implied by an assumption about the cost structure and the transition probabilities. In the last part we point out that our results in general cannot be deduced from related works on bias-optimality by Dekker and Hordijk, Wijngaard or Mann.

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Distribution of the number of alleles in multigene families

Journal of Applied Probability ◽

10.1017/s0021900200043655 ◽

1992 ◽

Vol 29 (04) ◽

pp. 759-769

Author(s):

R. C. Griffiths

Keyword(s):

Stochastic Model ◽

Gene Conversion ◽

Lower Bounds ◽

Multigene Families ◽

Expected Number ◽

Allele Type ◽

The Mean

The distribution of the number of alleles in samples from r chromosomes is studied. The stochastic model used includes gene conversion within chromosomes and mutation at loci on the chromosomes. A method is described for simulating the distribution of alleles and an algorithm given for computing lower bounds for the mean number of alleles. A formula is derived for the expected number of samples from r chromosomes which contain the allele type of a locus chosen at random.

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A stochastic model for polymer degradation

Journal of Applied Probability ◽

10.2307/3212635 ◽

1972 ◽

Vol 9 (1) ◽

pp. 43-53 ◽

Cited By ~ 1

Author(s):

S. K. Srinivasan ◽

K. M. Mehata

Keyword(s):

Stochastic Model ◽

General Model ◽

Polymer Degradation ◽

Mean Square ◽

Product Density ◽

The Mean ◽

Number Of Segments

The stochastic model for breaking of molecular segments proposed by Bithell is analysed and some results relating to the distribution of the number of fragments are obtained by using a slightly more general model which allows multiple ruptures. The product density technique is employed to derive the mean and mean square number of segments at any time t and the number of segments with length greater than y at time of production.

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Statistics of nascent and mature RNA fluctuations in a stochastic model of transcriptional initiation, elongation, pausing, and termination

10.1101/2020.05.13.092650 ◽

2020 ◽

Author(s):

Tatiana Filatova ◽

Nikola Popovic ◽

Ramon Grima

Keyword(s):

Experimental Data ◽

Fluorescence Microscopy ◽

Stochastic Model ◽

Explicit Dependence ◽

Transcriptional Initiation ◽

Estimate Parameter ◽

Mean And Variance ◽

The Mean ◽

Nascent Rna ◽

Parameter Values

AbstractRecent advances in fluorescence microscopy have made it possible to measure the fluctuations of nascent (actively transcribed) RNA. These closely reflect transcription kinetics, as opposed to conventional measurements of mature (cellular) RNA, whose kinetics is affected by additional processes downstream of transcription. Here, we formulate a stochastic model which describes promoter switching, initiation, elongation, premature detachment, pausing, and termination while being analytically tractable. By computational binning of the gene into smaller segments, we derive exact closed-form expressions for the mean and variance of nascent RNA fluctuations in each of these segments, as well as for the total nascent RNA on a gene. We also derive exact expressions for the first two moments of mature RNA fluctuations, and approximate distributions for total numbers of nascent and mature RNA. Our results, which are verified by stochastic simulation, uncover the explicit dependence of the statistics of both types of RNA on transcriptional parameters and potentially provide a means to estimate parameter values from experimental data.

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