A stochastic model for two interacting populations

1970 ◽  
Vol 7 (3) ◽  
pp. 544-564 ◽  
Author(s):  
Niels G. Becker
Keyword(s):  
Stochastic Model ◽  
Linear Models ◽  
Death Process ◽  
Linear Component ◽  
Birth And Death Process ◽  
Linear Interaction ◽  
Interacting Populations ◽  
Non Linear ◽  
The Subject ◽  
Interaction Component

To explain the growth of interacting populations, non-linear models need to be proposed and it is this non-linearity which proves to be most awkward in attempts at solving the resulting differential equations. A model with a particular non-linear component, initially proposed by Weiss (1965) for the spread of a carrier-borne epidemic, was solved completely by different methods by Dietz (1966) and Downton (1967). Immigration parameters were added to the model of Weiss and the resulting model was made the subject of a paper by Dietz and Downton (1968). It is the aim here to further generalize the model by introducing birth and death parameters so that the result is a linear birth and death process with immigration for each population plus the non-linear interaction component.

A stochastic model for two interacting populations

1970 ◽  
Vol 7 (03) ◽  
pp. 544-564 ◽  
Author(s):  
Niels G. Becker
Keyword(s):  
Stochastic Model ◽  
Linear Models ◽  
Death Process ◽  
Linear Component ◽  
Birth And Death Process ◽  
Linear Interaction ◽  
Interacting Populations ◽  
Non Linear ◽  
The Subject ◽  
Interaction Component

To explain the growth of interacting populations, non-linear models need to be proposed and it is this non-linearity which proves to be most awkward in attempts at solving the resulting differential equations. A model with a particular non-linear component, initially proposed by Weiss (1965) for the spread of a carrier-borne epidemic, was solved completely by different methods by Dietz (1966) and Downton (1967). Immigration parameters were added to the model of Weiss and the resulting model was made the subject of a paper by Dietz and Downton (1968). It is the aim here to further generalize the model by introducing birth and death parameters so that the result is a linear birth and death process with immigration for each population plus the non-linear interaction component.

scholarly journals An approximate stochastic model for phage reproduction in a bacterium

1962 ◽  
Vol 2 (4) ◽  
pp. 478-483 ◽  
Author(s):  
J. Gani
Keyword(s):  
Stochastic Model ◽  
Recent Article ◽  
Deterministic Model ◽  
Approximate Model ◽  
Death Process ◽  
Reproductive Process ◽  
Previous Note ◽  
The Subject ◽  
Birth Death

The stochastic birth-death process considered in this paper provides an approximate model for phage reproduction in a bacterium. In a recent paper, Hershey [1] has discussed reproduction and recombination in phage crosses, and a deterministic model for the reproductive process has been the subject of a previous note by the author [2]. A very readable account of the process is given by Sanders [3] in his recent article, “The life of viruses”.

On the asymptotic distribution of the maximum number of infectives in epidemic models by immigration

1994 ◽  
Vol 31 (3) ◽  
pp. 606-613 ◽  
Author(s):  
V. M. Abramov
Keyword(s):  
Asymptotic Distribution ◽  
Epidemic Model ◽  
Epidemic Models ◽  
Death Process ◽  
Birth And Death Process ◽  
Initial Number ◽  
Martingale Approach ◽  
Distribution Of The Maximum ◽  
Non Linear

This paper considers the asymptotic distribution of the maximum number of infectives in an epidemic model by showing that, as the initial number of susceptibles converges to infinity, the process of infectives converges almost surely to a birth and death process. The model studied here is more general than usual (see e.g. Bailey (1975), Bharucha-Reid (1960), Keilson (1979)) in that it incorporates immigration and the limiting birth and death process is non-linear. The main novelty of the present paper is the martingale approach used to prove the above-mentioned convergence.

On the asymptotic distribution of the maximum number of infectives in epidemic models by immigration

1994 ◽  
Vol 31 (03) ◽  
pp. 606-613
Author(s):  
V. M. Abramov
Keyword(s):  
Asymptotic Distribution ◽  
Epidemic Model ◽  
Epidemic Models ◽  
Death Process ◽  
Birth And Death Process ◽  
Initial Number ◽  
Martingale Approach ◽  
Distribution Of The Maximum ◽  
Non Linear

This paper considers the asymptotic distribution of the maximum number of infectives in an epidemic model by showing that, as the initial number of susceptibles converges to infinity, the process of infectives converges almost surely to a birth and death process. The model studied here is more general than usual (see e.g. Bailey (1975), Bharucha-Reid (1960), Keilson (1979)) in that it incorporates immigration and the limiting birth and death process is non-linear. The main novelty of the present paper is the martingale approach used to prove the above-mentioned convergence.

Interparticle dependence in a linear death process subjected to a random environment

1990 ◽  
Vol 27 (3) ◽  
pp. 491-498 ◽  
Author(s):  
Claude Lefèvre ◽  
György Michaletzky
Keyword(s):  
Random Environment ◽  
Death Rate ◽  
Monotone Function ◽  
Death Process ◽  
Birth And Death Process ◽  
Survival Times ◽  
Current State ◽  
Non Linear

Recently, Ball and Donnelly (1987) investigated the nature of the interparticle dependence in a death process with non-linear rates. In this note, after some remarks on their result, a similar problem is examined for a linear death process where the death rate per particle is a monotone function of the current state of a random environment. It is proved that if the exterior process involved is a homogeneous birth-and-death process valued in ℕ, then the survival times of any subset of particles are positively upper orthant dependent. A simple example shows that this property is not valid for general exterior processes.

Interparticle dependence in a linear death process subjected to a random environment

1990 ◽  
Vol 27 (03) ◽  
pp. 491-498 ◽  
Author(s):  
Claude Lefèvre ◽  
György Michaletzky
Keyword(s):  
Random Environment ◽  
Death Rate ◽  
Monotone Function ◽  
Death Process ◽  
Birth And Death Process ◽  
Survival Times ◽  
Current State ◽  
Non Linear

Recently, Ball and Donnelly (1987) investigated the nature of the interparticle dependence in a death process with non-linear rates. In this note, after some remarks on their result, a similar problem is examined for a linear death process where the death rate per particle is a monotone function of the current state of a random environment. It is proved that if the exterior process involved is a homogeneous birth-and-death process valued in ℕ, then the survival times of any subset of particles are positively upper orthant dependent. A simple example shows that this property is not valid for general exterior processes.

Using Hybrid ARIMAX-ANN Model for Simulating Rainfall - Runoff - Sediment Process Case Study

2013 ◽  
Vol 4 (2) ◽  
pp. 44-60 ◽  
Author(s):  
Vahid Nourani ◽  
Samira Roumianfar ◽  
Elnaz Sharghi
Keyword(s):  
Time Series ◽  
Linear Models ◽  
Sediment Load ◽  
Moving Average ◽  
Linear Component ◽  
Rainfall Runoff ◽  
Ann Model ◽  
Non Linear ◽  
Runoff Sediment ◽  
Artificial Neural Network Ann

The need for accurate modeling of rainfall-runoff-sediment processes has grown rapidly in the past decades. This study investigates the efficiency of black-box models including Artificial Neural Network (ANN) and Autoregressive Integrated Moving Average with eXogenous input (ARIMAX) models for forecasting the rainfall-runoff-sediment process. According to the complex behavior of the rainfall-runoff-sediment time series, they include both linear and nonlinear components; therefore, employing a hybrid model which combines the advantages of both linear and non-linear models improves the accuracy of prediction. In this paper, a hybrid of ARIMAX-ANN model is applied to rainfall-runoff-sediment modeling of a watershed. At the first step of the hybrid modeling, the ARIMAX method is applied to forecast the linear component of the rainfall-runoff process and then in the second step, an ANN model is used to find the non-linear relationship among the residuals of the fitted linear ARIMAX model. Finally, total effective time series of runoff, obtained by the hybrid ARIMAX-ANN model are imposed as input to the proposed ANN model for prediction daily suspended sediment load of the watershed. The proposed model is more appropriate, as it uses the semi-linear relation for prediction of sediment load.

The extinction time of a general birth and death process with catastrophes

1986 ◽  
Vol 23 (04) ◽  
pp. 851-858 ◽  
Author(s):  
P. J. Brockwell
Keyword(s):  
Laplace Transform ◽  
Size Distribution ◽  
Sufficient Conditions ◽  
Necessary And Sufficient Conditions ◽  
Death Process ◽  
Extinction Time ◽  
Birth And Death Process ◽  
Different Types ◽  
The Laplace Transform ◽  
Necessary And Sufficient

The Laplace transform of the extinction time is determined for a general birth and death process with arbitrary catastrophe rate and catastrophe size distribution. It is assumed only that the birth rates satisfyλ0= 0,λj> 0 for eachj> 0, and. Necessary and sufficient conditions for certain extinction of the population are derived. The results are applied to the linear birth and death process (λj=jλ, µj=jμ) with catastrophes of several different types.

G Optimal Design in Non linear Models to Increase Silicon Oxide Purity Levels and Electrical Conductivity

2018 ◽  
pp. 150-155
Author(s):  
Muklas Rivai
Keyword(s):  
Electrical Conductivity ◽  
Optimal Design ◽  
Linear Models ◽  
Silicon Oxide ◽  
Information Matrix ◽  
Relevant Information ◽  
Non Linear

Optimal design is a design which required in determining the points of variable factors that would be attempted to optimize the relevant information so that fulfilled the desired criteria. The optimal fulfillment criteria based on the information matrix of the selected model.

Non-Linear Interaction of MHD Waves with Small-Scale Ionospheric Irregularities

2004 ◽  
Vol 61 (7-12) ◽  
pp. 1055-1071
Author(s):  
N. N. Gerasimova ◽  
V. G. Sinitsin ◽  
Yu. M. Yampolski
Keyword(s):  
Ionospheric Irregularities ◽  
Mhd Waves ◽  
Small Scale ◽  
Linear Interaction ◽  
Non Linear
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