scholarly journals Dynamical system of a mosquito population with distinct birth-death rates

2021 ◽  
Vol 10 (4) ◽  
pp. 791-800
Author(s):  
Z.S. Boxonov ◽  
U.A. Rozikov
Keyword(s):  
Dynamical System ◽  
Mosquito Population ◽  
Death Rates ◽  
Birth Death

Geometric Markov chains

1995 ◽  
Vol 32 (02) ◽  
pp. 349-374
Author(s):  
William Rising
Keyword(s):  
Markov Chains ◽  
Stationary Distribution ◽  
Nearest Neighbor ◽  
First Passage Times ◽  
First Passage ◽  
Death Rates ◽  
Passage Times ◽  
Birth Death

A generalization of the familiar birth–death chain, called the geometric chain, is introduced and explored. By the introduction of two families of parameters in addition to the infinitesimal birth and death rates, the geometric chain allows transitions beyond the nearest neighbor, but is shown to retain the simple computational formulas of the birth–death chain for the stationary distribution and the expected first-passage times between states. It is also demonstrated that even when not reversible, a reversed geometric chain is again a geometric chain.

The Relaxation time for truncated birth-death processes

1987 ◽  
Vol 1 (4) ◽  
pp. 367-381 ◽  
Author(s):  
Julian Keilson ◽  
Ravi Ramaswamy
Keyword(s):  
Relaxation Time ◽  
Exit Time ◽  
Relaxation Times ◽  
Dual Process ◽  
Death Process ◽  
Exit Times ◽  
Initial State ◽  
Death Rates ◽  
Absorbing States ◽  
Birth Death

The relaxation time for an ergodic Markov process is a measure of the time until ergodicity is reached from its initial state. In this paper the relaxation time for an ergodic truncated birth-death process is studied. It is shown that the relaxation time for such a process on states {0,1, …, N} is the quasi-stationary exit time from the set {,2, …, N{0,1,…, N, N + 1} with two-sided absorption at states 0 and N + 1. The existence of such a dual process has been observed by Siegmund [15] for stochastically monotone Markov processes on the real line. Exit times for birth- death processes with two absorbing states are studied and an efficient algorithm for the numerical evaluation of mean exit times is presented. Simple analytical lower bounds for the relaxation times are obtained. These bounds are numerically accessible. Finally, the sensitivity of the relaxation time to variations in birth and death rates is studied.

scholarly journals Locally adaptive Bayesian birth-death model successfully detects slow and rapid rate shifts

2019 ◽  
Author(s):  
Andrew F. Magee ◽  
Sebastian Höhna ◽  
Tetyana I. Vasylyeva ◽  
Adam D. Leaché ◽  
Vladimir N. Minin
Keyword(s):  
Random Field ◽  
Markov Random Field ◽  
Death Process ◽  
Rate Variation ◽  
Rapid Rate ◽  
Birth Rates ◽  
Death Rates ◽  
Growth Of Groups ◽  
Markov Random ◽  
Birth Death

AbstractBirth-death processes have given biologists a model-based framework to answer questions about changes in the birth and death rates of lineages in a phylogenetic tree. Therefore birth-death models are central to macroevolutionary as well as phylodynamic analyses. Early approaches to studying temporal variation in birth and death rates using birth-death models faced difficulties due to the restrictive choices of birth and death rate curves through time. Sufficiently flexible time-varying birth-death models are still lacking. We use a piecewise-constant birth-death model, combined with both Gaussian Markov random field (GMRF) and horseshoe Markov random field (HSMRF) prior distributions, to approximate arbitrary changes in birth rate through time. We implement these models in the widely used statistical phylogenetic software platform RevBayes, allowing us to jointly estimate birth-death process parameters, phylogeny, and nuisance parameters in a Bayesian framework. We test both GMRF-based and HSMRF-based models on a variety of simulated diversification scenarios, and then apply them to both a macroevolutionary and an epidemiological dataset. We find that both models are capable of inferring variable birth rates and correctly rejecting variable models in favor of effectively constant models. In general the HSMRF-based model has higher precision than its GMRF counterpart, with little to no loss of accuracy. Applied to a macroevolutionary dataset of the Australian gecko family Pygopodidae (where birth rates are interpretable as speciation rates), the GMRF-based model detects a slow decrease whereas the HSMRF-based model detects a rapid speciation-rate decrease in the last 12 million years. Applied to an infectious disease phylodynamic dataset of sequences from HIV subtype A in Russia and Ukraine (where birth rates are interpretable as the rate of accumulation of new infections), our models detect a strongly elevated rate of infection in the 1990s.Author summaryBoth the growth of groups of species and the spread of infectious diseases through populations can be modeled as birth-death processes. Birth events correspond either to speciation or infection, and death events to extinction or becoming noninfectious. The rates of birth and death may vary over time, and by examining this variation researchers can pinpoint important events in the history of life on Earth or in the course of an outbreak. Time-calibrated phylogenies track the relationships between a set of species (or infections) and the times of all speciation (or infection) events, and can thus be used to infer birth and death rates. We develop two phylogenetic birth-death models with the goal of discerning signal of rate variation from noise due to the stochastic nature of birth-death models. Using a variety of simulated datasets, we show that one of these models can accurately infer slow and rapid rate shifts without sacrificing precision. Using real data, we demonstrate that our new methodology can be used for simultaneous inference of phylogeny and rates through time.

A birth, death and migration process

1969 ◽  
Vol 6 (03) ◽  
pp. 687-691 ◽  
Author(s):  
S. R. Adke
Keyword(s):  
Time Dependent ◽  
Death Process ◽  
Migration Process ◽  
Death Rates ◽  
And Migration ◽  
Birth Death

A model proposed by Bailey (1968) for migratory individuals which reproduce according to a simple birth-death process is generalized to include time dependent birth and death rates.

scholarly journals A Multitype Birth–Death Model for Bayesian Inference of Lineage-Specific Birth and Death Rates

2020 ◽  
Vol 69 (5) ◽  
pp. 973-986 ◽  
Author(s):  
Joëlle Barido-Sottani ◽  
Timothy G Vaughan ◽  
Tanja Stadler
Keyword(s):  
Bayesian Inference ◽  
Simulated Data ◽  
Pathogen Transmission ◽  
Type Model ◽  
Model Parameters ◽  
Data Sets ◽  
Death Rates ◽  
Joint Inference ◽  
Extinction Rates ◽  
Birth Death

Abstract Heterogeneous populations can lead to important differences in birth and death rates across a phylogeny. Taking this heterogeneity into account is necessary to obtain accurate estimates of the underlying population dynamics. We present a new multitype birth–death model (MTBD) that can estimate lineage-specific birth and death rates. This corresponds to estimating lineage-dependent speciation and extinction rates for species phylogenies, and lineage-dependent transmission and recovery rates for pathogen transmission trees. In contrast with previous models, we do not presume to know the trait driving the rate differences, nor do we prohibit the same rates from appearing in different parts of the phylogeny. Using simulated data sets, we show that the MTBD model can reliably infer the presence of multiple evolutionary regimes, their positions in the tree, and the birth and death rates associated with each. We also present a reanalysis of two empirical data sets and compare the results obtained by MTBD and by the existing software BAMM. We compare two implementations of the model, one exact and one approximate (assuming that no rate changes occur in the extinct parts of the tree), and show that the approximation only slightly affects results. The MTBD model is implemented as a package in the Bayesian inference software BEAST 2 and allows joint inference of the phylogeny and the model parameters.[Birth–death; lineage specific rates, multi-type model.]

scholarly journals Families of birth-death processes with similar time-dependent behaviour

2000 ◽  
Vol 37 (03) ◽  
pp. 835-849 ◽  
Author(s):  
R. B. Lenin ◽  
P. R. Parthasarathy ◽  
W. R. W. Scheinhardt ◽  
E. A. van Doorn
Keyword(s):  
Death Rate ◽  
General Setting ◽  
Time Dependent ◽  
Death Process ◽  
Similar Time ◽  
Death Rates ◽  
Transition Functions ◽  
Time Dependent Behaviour ◽  
Dependent Behaviour ◽  
Birth Death

We consider birth-death processes taking values in but allow the death rate in state 0 to be positive, so that escape from is possible. Two such processes with transition functions are said to be similar if, for all there are constants c ij such that for all t ≥ 0. We determine conditions on the birth and death rates of a birth-death process for the process to be a member of a family of similar processes, and we identify the members of such a family. These issues are also resolved in the more general setting in which the two processes are called similar if there are constants c ij and ν such that for all t ≥ 0.

Conditions for exponential ergodicity and bounds for the decay parameter of a birth-death process

1985 ◽  
Vol 17 (3) ◽  
pp. 514-530 ◽  
Author(s):  
Erik A. Van Doorn
Keyword(s):  
Complete Solution ◽  
Sufficient Conditions ◽  
Rational Functions ◽  
Difficult Problem ◽  
Upper And Lower Bounds ◽  
Death Process ◽  
Decay Parameter ◽  
Death Rates ◽  
Exponential Ergodicity ◽  
Birth Death

This paper is concerned with two problems in connection with exponential ergodicity for birth-death processes on a semi-infinite lattice of integers. The first is to determine from the birth and death rates whether exponential ergodicity prevails. We give some necessary and some sufficient conditions which suffice to settle the question for most processes encountered in practice. In particular, a complete solution is obtained for processes where, from some finite state n onwards, the birth and death rates are rational functions of n. The second, more difficult, problem is to evaluate the decay parameter of an exponentially ergodic birth-death process. Our contribution to the solution of this problem consists of a number of upper and lower bounds.

Conditions for exponential ergodicity and bounds for the decay parameter of a birth-death process

1985 ◽  
Vol 17 (03) ◽  
pp. 514-530 ◽  
Author(s):  
Erik A. Van Doorn
Keyword(s):  
Complete Solution ◽  
Sufficient Conditions ◽  
Rational Functions ◽  
Difficult Problem ◽  
Upper And Lower Bounds ◽  
Death Process ◽  
Decay Parameter ◽  
Death Rates ◽  
Exponential Ergodicity ◽  
Birth Death

This paper is concerned with two problems in connection with exponential ergodicity for birth-death processes on a semi-infinite lattice of integers. The first is to determine from the birth and death rates whether exponential ergodicity prevails. We give some necessary and some sufficient conditions which suffice to settle the question for most processes encountered in practice. In particular, a complete solution is obtained for processes where, from some finite state n onwards, the birth and death rates are rational functions of n. The second, more difficult, problem is to evaluate the decay parameter of an exponentially ergodic birth-death process. Our contribution to the solution of this problem consists of a number of upper and lower bounds.

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