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International audience; Motivated by the study of a parasite infection in a cell line, we introduce a general class of Markov processes for the modelling of population dynamics.The population process evolves as a diffusion with positive jumps whose rate is a function of the population size. It also undergoes catastrophic events which kill a fraction of the population, at a rate depending on the population state. We study the long time behaviour of this class of processes.

The continuous time Markov process considered in this paper belongs to a class of population models with linear growth and catastrophes. There, the catastrophes happen at the arrival times of a Poisson process, and at each catastrophe time, a randomly selected portion of the population is eliminated. For this population process, we derive an asymptotic upper bound for the maximum value and prove the local large deviation principle., 22 pages

The setting considered in this paper concerns a discrete-time multivariate population process under Markov modulation. Our objective is to estimate the model parameters, based on periodic observations of the network population vector. These parameters relate to the arrival, routing and departure processes, but also to the (unobservable) Markovian background process. When opting for the classical likelihood-based approach, the evaluation of the likelihood is problematic. We show however, how an accurate saddlepoint approximation can be used. Numerical experiments illustrate our method and show that even under relatively complicated conditions the parameters are estimated relatively precisely.

Population dynamic models combine density dependence and environmental effects. Ignoring sampling uncertainty might lead to biased estimation of the strength of density dependence. This is typically addressed using state‐space model approaches, which integrate sampling error and population process estimates. Such models seldom include an explicit link between the sampling procedures and the true abundance, which is common in capture–recapture settings. However, many of the models proposed to estimate abundance in the presence of capture heterogeneity lead to incomplete likelihood functions and cannot be straightforwardly included in state‐space models. We assessed the importance of estimating sampling error explicitly by taking an intermediate approach between ignoring uncertainty in abundance estimates and fully specified state‐space models for density‐dependence estimation based on autoregressive processes. First, we estimated individual capture probabilities based on a heterogeneity model for a closed population, using a conditional multinomial likelihood, followed by a Horvitz–Thompson estimate for abundance. Second, we estimated coefficients of autoregressive models for the log abundance. Inference was performed using the methodology of integrated nested Laplace approximation (INLA). We performed an extensive simulation study to compare our approach with estimates disregarding capture history information, and using R‐package VGAM, for different parameter specifications. The methods were then applied to a real data set of gray‐sided voles Myodes rufocanus from Northern Norway. We found that density‐dependence estimation was improved when explicitly modeling sampling error in scenarios with low process variances, in which differences in coverage reached up to 8% in estimating the coefficients of the autoregressive processes. In this case, the bias also increased assuming a Poisson distribution in the observational model. For high process variances, the differences between methods were small and it appeared less important to model heterogeneity., We assessed the importance of including capture history information in the estimation of density dependence from capture–recapture data. We performed an extensive simulation study, evaluating the performance of different methods in the estimation of different population processes. We found that disregarding sampling error can bias the density‐dependence estimates in low variance autoregressive population models.

We define and study an open stochastic SIR (Susceptible -- Infected -- Removed) model on a graph in order to describe the spread of an epidemic on a cattle trade network with epidemiological and demographic dynamics occurring over the same time scale. Population transition intensities are assumed to be density-dependent with a constant component, the amplitude of which determines the overall scale of the population process. Standard branching approximation results for the epidemic process are first given, along with a numerical computation method for the probability of a major epidemic outbreak. This procedure is illustrated using real data on trade-related cattle movements from a densely populated livestock farming region in western France (Finist\`ere) and epidemiological parameters corresponding to an infectious epizootic disease. Then we exhibit an exponential lower bound for the extinction time and the total size of the epidemic in the stable endemic case as a scaling parameter goes to infinity using results inspired by the Freidlin-Wentzell theory of large deviations from a dynamical system., Comment: 31 pages

Representations of population models in terms of countable systems of particles are constructed, in which each particle has a `type', typically recording both spatial position and genetic type, and a level. For finite intensity models, the levels are distributed on $[0,\lambda ]$, whereas in the infinite intensity limit $\lambda\rightarrow\infty$, at each time $t$, the joint distribution of types and levels is conditionally Poisson, with mean measure $\Xi (t)\times \ell$ where $\ell$ denotes Lebesgue measure and $\Xi (t)$ is a measure-valued population process. The time-evolution of the levels captures the genealogies of the particles in the population. Key forces of ecology and genetics can be captured within this common framework. Models covered incorporate both individual and event based births and deaths, one-for-one replacement, immigration, independent `thinning' and independent or exchangeable spatial motion and mutation of individuals. Since birth and death probabilities can depend on type, they also include natural selection. The primary goal of the paper is to present particle-with-level or lookdown constructions for each of these elements of a population model. Then the elements can be combined to specify the desired model. In particular, a non-trivial extension of the spatial $\Lambda$-Fleming-Viot process is constructed.

The Moran model with recombination is considered, which describes the evolution of the genetic composition of a population under recombination and resampling. There are $n$ sites (or loci), a finite number of letters (or alleles) at every site, and we do not make any scaling assumptions. In particular, we do not assume a diffusion limit. We consider the following marginal ancestral recombination process. Let $S = \{1,...,n\}$ and $\mathcal A=\{A_1, ..., A_m\}$ be a partition of $S$. We concentrate on the joint probability of the letters at the sites in $A_1$ in individual $1$, $...$, and at the sites in $A_m$ in individual $m$, where the individuals are sampled from the current population without replacement. Following the ancestry of these sites backwards in time yields a process on the set of partitions of $S$, which, in the diffusion limit, turns into a marginalised version of the $n$-locus ancestral recombination graph. With the help of an inclusion-exclusion principle, we show that the type distribution corresponding to a given partition may be represented in a systematic way, with the help of so-called recombinators and sampling functions. The same is true of correlation functions (known as linkage disequilibria in genetics) of all orders. We prove that the partitioning process (backward in time) is dual to the Moran population process (forward in time), where the sampling function plays the role of the duality function. This sheds new light on the work of Bobrowski, Wojdyla, and Kimmel (2010). The result also leads to a closed system of ordinary differential equations for the expectations of the sampling functions, which can be translated into expected type distributions and expected linkage disequilibria., 29 pages, 6 figures

For Markov chain in stochastic process model, bilateral birth and death process is a typical Markov chain. To endow the track structure of birth and death with clear probability direct meaning and make in-depth research on the quantity, it is extremely significant to research the probability property of minimal process. This article has mainly discussed the probability property of minimal process corresponding to bilateral birth and death matrix.

The aim of this chapter is to take the limit in the renormalized version of the model of the previous chapter, i.e., let \(N \rightarrow \infty\) in the models of sections 6.5 and 6.6, respectively. In section 7.1, we shall take the limit in the model for the evolution of the population size, as a function of the two parameters x (the ancestral population size) and t (time). Since we want to stick to our rather minimal assumptions on the function f, checking tightness requires some care. In section 7.2, we will take the limit in the renormalized contour process of section 6.6 Here there are two difficulties. One is the fact that since we do not want to restrict ourself to the (sub)critical case, it is not clear whether the contour process will accumulate an arbitrary amount of local time at level 0. In order to circumvent this difficulty, we use as in chapter 5 a trick due to Delmas [16] which consists in considering the population process killed at an arbitrary time a, which amounts to reflect the contour process below a. The behavior of the contour process below a (and of its local time accumulated below level a) is described by the solution of the corresponding equation reflected below any a′ > a. The second difficulty comes from the fact that f′ is not assumed to be bounded from below. We will prove our convergence result by a combination of Theorem 7 and Girsanov’s theorem. For that sake, we shall first consider the case where | f′ | is bounded, and then the general case.

We study Markovian models for population processes in continuous time, addressing questions concerning the behaviour of ensembles of individuals (equilibrium, quasi-equilibrium and time-dependent behaviour) and, in particular, what can be deduced from models for individual behaviour.

In this paper we present various approaches for the transient analysis of a Markovian population process with total catastrophes. We discuss the pros and the cons of these methodologies and point out how they lead to different tractable extensions. As an illustrating example, we consider the nonhomogeneous Poisson process with total catastrophes. The extension of the probabilistic methodologies for analyzing models with binomial catastrophes is also discussed.

Markovian population models are used in conservation biology to find an accurate estimate of a population's extinction probability. Such models require handling of large transition matrices and calculations are thus extremely time-consuming when large populations have to be studied. To accomplish these problems, some authors have suggested to group together several states/sizes of the population. Unfortunately, this so-called binning frequently results in errors in estimates obtained. The main problem with binning is that it assumes that grouped states behave nearly identical with respect to the underlying stochastic population process and that so far binning methods implicitly violate this assumption. In this paper, we present a new binning method based on self-organizing Kohonen neural networks for time-homogeneous Markovian metapopulation models. The neural networks are used to analyse one-step transitions of the Markov chain in order to group only nearly identical states. We show that the new method is more qualified for the use in conservation than deterministic methods that we had discussed in a previous paper (first order Fibonacci binning, pairs binning). It reveals more accurate and more reliable estimates than these methods. Errors in estimated extinction probabilities that were introduced by binning did not exceed one order of magnitude and errors in the global population size did not exceed 30%. These errors in estimates correspond to a low inaccuracy in model parameter values of population growth and migration ranging from ±1 to ±10%. The reduction in the state space of the studied metapopulations ranged from 21 to 33% per subpopulation. The resulting decrease in computing time caused by our binning method is substantial particularly with regard to simulations tasks such as comparing the extinction risk of several populations or performing a detailed sensitivity analysis for model parameters assumed. The successful estimation of the extinction risk of eight natural butterfly populations demonstrates the applicability of our new binning method in conservation biology. A comparison of extinction probabilities and mean population sizes estimated by Monte Carlo simulations [Griebeler, E.M., Seitz, A., 2002. An individual based model for the conservation of the endangered Large Blue butterfly, Maculinea arion (Lepidoptera: Lycaenidae). Ecol. Model. 134, 343–356] with those obtained from the respective Markov chains for these butterfly populations revealed similar results on the accuracy of estimates and the reduction in transition matrices that were predicted by the comparative error analysis for binning methods.

In this paper, we investigate the statistical properties of a network size in vehicular ad-hoc networks. The vehicles arrive or depart at the highway through one of the traffic entry points according to a Markov population process. Using the general immigration-death process — coined as general arrival rate and general departure rate, we derive the probability distribution of the network size, i.e., the number of vehicles in a segment.

We consider a migration process whose singleton process is a time-dependent Markov replacement process. For the singleton process, which may be treated as either open or closed, we study the limiting distribution, the distribution of the time to replacement and related quantities. For a replacement process in equilibrium we obtain a version of Little's law and we provide conditions for reversibility. For the resulting linear population process we characterize exponential ergodicity for two types of environmental behaviour, i.e. either convergent or cyclic, and finally for large population sizes a diffusion approximation analysis is provided.

Complex networks, comprised of individual elements that interact with each other through reaction channels, are ubiquitous across many scientific and engineering disciplines. Examples include biochemical, pharmacokinetic, epidemiological, ecological, social, neural, and multi-agent networks. A common approach to modeling such networks is by a master equation that governs the dynamic evolution of the joint probability mass function of the underling population process and naturally leads to Markovian dynamics for such process. Due however to the nonlinear nature of most reactions, the computation and analysis of the resulting stochastic population dynamics is a difficult task. This review article provides a coherent and comprehensive coverage of recently developed approaches and methods to tackle this problem. After reviewing a general framework for modeling Markovian reaction networks and giving specific examples, the authors present numerical and computational techniques capable of evaluating or approximating the solution of the master equation, discuss a recently developed approach for studying the stationary behavior of Markovian reaction networks using a potential energy landscape perspective, and provide an introduction to the emerging theory of thermodynamic analysis of such networks. Three representative problems of opinion formation, transcription regulation, and neural network dynamics are used as illustrative examples., Comment: 52 pages, 11 figures, for freely available MATLAB software, see http://www.cis.jhu.edu/~goutsias/CSS%20lab/software.html

We present a tool called SHAVE that approximates the transient distribution of a continuous-time Markov population process by combining moment-based and state-based representations of probability distributions. As an intermediate step, SHAVE constructs a stochastic hybrid model from the original process which is then solved numerically.

A birth-and-death (BD) process is a continuous-time Markov chain on the nonnegative integers whose transitions are restricted to its nearest neighbors. These processes have enjoyed wide popularity in a diverse set of fields. This article formally defines the BD process, reviews basic results for its transient and asymptotic analysis, discusses some special cases, and describes two applications in queueing theory. Keywords: birth-and-death process; Markov process; population models; queueing; level dependent

We consider a bivariate Markov counting process with transition probabilities having a particular structure, which includes a number of useful population processes. Using a suitable random time-scale transformation, we derive some probability statements about the process and some asymptotic results. These asymptotic results are also derived using martingale methods. Further, it is shown that these methods and results can be used for inference on the rate parameters for the process. The general epidemic model and the square law conflict model are used as illustrative examples.

We discuss various statistical distributions of earthquake numbers. Previously we derived several discrete distributions to describe earthquake numbers for the branching model of earthquake occurrence: these distributions are the Poisson, geometric, logarithmic, and the negative binomial (NBD). The theoretical model is the `birth and immigration' population process. The first three distributions above can be considered special cases of the NBD. In particular, a point branching process along the magnitude (or log seismic moment) axis with independent events (immigrants) explains the magnitude/moment-frequency relation and the NBD of earthquake counts in large time/space windows, as well as the dependence of the NBD parameters on the magnitude threshold (magnitude of an earthquake catalog completeness). We discuss applying these distributions, especially the NBD, to approximate event numbers in earthquake catalogs. There are many different representations of the NBD. Most can be traced either to the Pascal distribution or to the mixture of the Poisson distribution with the gamma law. We discuss advantages and drawbacks of both representations for statistical analysis of earthquake catalogs. We also consider applying the NBD to earthquake forecasts and describe the limits of the application for the given equations. In contrast to the one-parameter Poisson distribution so widely used to describe earthquake occurrence, the NBD has two parameters. The second parameter can be used to characterize clustering or over-dispersion of a process. We determine the parameter values and their uncertainties for several local and global catalogs, and their subdivisions in various time intervals, magnitude thresholds, spatial windows, and tectonic categories., Comment: 50 pages,15 Figs, 3 Tables

This paper provides a countable representation for a class of infinite-dimensional diffusions which extends the infinitely-many-neutral-alleles model and is related to the two-parameter Poisson-Dirichlet process. By means of Gibbs sampling procedures, we define a reversible Moran-type population process. The associated process of ranked relative frequencies of types is shown to converge in distribution to the two-parameter family of diffusions, which is stationary and ergodic with respect to the two-parameter Poisson-Dirichlet distribution. The construction provides interpretation for the limiting process in terms of individual dynamics.

The use of Markovian models is an established way for deriving the complete distribution of the size of a population and the probability of extinction. However, computationally impractical transition matrices frequently result if this mathematical approach is applied to natural populations. Binning, or aggregating population sizes, has been used to permit a reduction in the dimensionality of matrices. Here, we present three deterministic binning methods and study the errors due to binning for a metapopulation model. Our results indicate that estimation errors of the investigated methods are not consistent and one cannot make generalizations about the quality of a method. For some compared output variables of populations studied, binning methods that caused a strong reduction in dimensionality of matrices resulted in better estimations than methods that produced a weaker reduction. The main problem with deterministic binning methods is that they do not properly take into account the stochastic population process itself. Straightforward usage of binning methods may lead to substantial errors in population-dynamical predictions.

We present a central limit theorem for the population process of a simple genetic algorithm. This theory approximates the discrete population process with finite population size by a continuous Gaussian — process. As a special application we consider a search space with only two elements. For this case we present a complete solution of the differential equation arising in the calculation of the Gaussian — process. This analysis gives first insight into the theory of the general case. Some applications of the theory and extensions to the general case are mentioned, too.

The processes by which disease spreads in a population of individuals are inherently stochastic. The master equation has proven to be a useful tool for modeling such processes. Unfortunately, solving the master equation analytically is possible only in limited cases (e.g., when the model is linear), and thus numerical procedures or approximation methods must be employed. Available approximation methods, such as the system size expansion method of van Kampen, may fail to provide reliable solutions, whereas current numerical approaches can induce appreciable computational cost. In this paper, we propose a new numerical technique for solving the master equation. Our method is based on a more informative stochastic process than the population process commonly used in the literature. By exploiting the structure of the master equation governing this process, we develop a novel technique for calculating the exact solution of the master equation – up to a desired precision – in certain models of stochastic epidemiology. We demonstrate the potential of our method by solving the master equation associated with the stochastic SIR epidemic model. MATLAB software that implements the methods discussed in this paper is freely available as Supporting Information S1.

SUMMARY It is known that in the limit of large population sizes certain classes of Markov population processes can be approximated by a deterministic process plus a noise term. In the usual approximation procedures the noise term is proportional to a Wiener process. This allows us to derive an estimation procedure for Markov population processes that are observed periodically at times A, 2A,.... 1. INTRODIUCTION In this paper we are concerned with the general problem of estimating parameters in a time homogeneous Markov population process. If the trajectory of the process can be observed in continuous time, then parameter estimation is straightforward in principle, for it is possible to write down the likelihood explicitly, and even if explicit maximum likelihood estimators for the required parameters are not obtainable they can be obtained numerically. We refer to Billingsley (1975) for a general approach to the problem, and to Moran (1951, 1953), Bartlett, Brennan & Pollock (1971), Bailey & Thomas (1971) and Keiding (1974) for detailed analyses in particular cases. In many situations, it is not possible to observe the process continuously but rather, observations are sampled at regular or irregular epochs. In this case the likelihood involves a product of factors, where a typical factor is the conditional distribution of the process at time t + A, say, given its value at time t. In most applications these distributions are intractable, and some approximation is required. Fortunately, there is a straightforward Gaussian approximation which is valid whenever the number of jumps between observation epochs is large. There have been a number of recent investigations into the asymptotic behaviour of Markov population processes under these conditions (Barbour, 1972, 1974; Kurtz, 1971; McNeil, 1972; McNeil & Schach, 1973; van Kampen, 1961, 1973). In ?? 2 and 3, below, we specify the class of processes to which our results will be usefully applied, and outline the general estimation procedure. In the remaining sections the procedure is illustrated using the example of the linear growth birth and death process, and an example is given in which a problem of identifiability of parameters arises because of the Gaussian approximation used.

Markovian contact processes (mcp's) include some of the commoner models for the spatial spread of population processes, such as the ‘simple’ and ‘general’ epidemics, percolation processes, and birth, death and migration processes. They are here set in a framework which relates them to each other, and to a particular basic model, the contact birth process (cbp); indeed they are defined as modifications of the cbp. The immediate advantage of this is that bounds for the velocity of the cbp, which can be obtained from the linear equation describing its expected numbers (provided the contact distribution has exponentially bounded tail) apply also to general mcp's. Existence of moments (and for some processes their time-derivatives) of St (θ), the distance to the furthest individual in an arbitrary direction θ, can also be deduced whenever the corresponding moment of the contact distribution exists. An important subclass consists of population-monotone mcp's, for which the distributions of St can be shown to be subconvolutive, so that the work of Hammersley (1974) can be applied to obtain convergence theorems for St /t; and these can be extended to convergence of the convex hull of the set of inhabited points. These results are particularly valuable because they apply to some non-linear processes, e.g. the simple epidemic. Some special results on the cbp (Section 6) emphasize the differences between linear and non-linear spatial stochastic processes. Although the paper is written throughout in terms of continuous-time processes, the results on subconvolutive distributions are actually more easily applied in the discrete-time case. (Conversely, the approach used in the accompanying paper by Biggins (pp. 62–84), which is written in terms of discrete-time processes, can be extended to deal also with continuous-time processes.)

This paper is concerned with a class of population growth processes in discrete time; the simple epidemic process is considered as a specific example. A Markov chain model is constructed and standard Markov methods are used to study the main biological concepts. A simple and explicit formula is obtained for the transient distribution of the population size. Then, the cost of the process is defined and the joint probability generating function of its components is derived. Finally, the results are extended to the case where the inter-transition periods are bounded i.i.d. random variables.

The linear birth and death process with catastrophes is formulated as a right continuous random walk on the non-negative integers which evolves in continuous time with an instantaneous jump rate proportional to the current value of the process. It is shown that distributions of the population size can be represented in terms of those of a certain Markov branching process. The ergodic theory of Markov branching process transition probabilities is then used to develop a fairly complete understanding of the behaviour of the population size of the birth-death-catastrophe process.

In contrast to the classical theory of partial sums of independent and identically distributed random variables, the maximum value taken by a component of a Markov population process x N is typically largely determined by the variation in its mean, rather than by stochastic fluctuation. A closer approximation to its distribution is found by considering the supremum of V ( t ) − N 1 2 c ( t ) for a suitable centred Gaussian process V , where c incorporates the effect of the variation in the mean of x N . Under appropriate conditions, it is shown that this has a distribution which is normally distributed, to within an error of order N − 1 2 log N , and expressions for the mean and variance of the approximating distribution are derived.

The distribution of the maximum and the extinction probability for a Markovian population is derived. Asymptotic growth is described, using the sequence of sojourn times. A regularity criterion for the processes under consideration exists under certain assumptions. For a class of processes with specific population-dependent transition rates the asymptotic behaviour is given explicitly.

This paper proposes a model or population characterized by a birth process generating individuals and a core process describing the dynamics of each individual in the population. When the core process is a finite-state continuous-time semi-Markov process, one obtains a complete description of the census process, which gives at any time the number of individuals in each state. The problem of optimally controlling the population process through its birth process is then formulated as a problem of optimal control of a jump process in the continuous-time setting. Two examples show how to use the dynamic programming equations obtained in this paper.

We study the asymptotics of a class of age structured population models when the initial population is larger and larger and the reproduction regime is properly scaled. Results of Kurtz and Wang are generalised, mainly with extensions of ideas originating from Kurtz. Newborn individuals have life-spans which are i.i.d. and independent of everything else. The initial populations are supposed to have age structures as well as residual life-span distributions that stabilise asymptotically. The infinitesimal probabilities of reproduction of a brood of size k at time t are supposed to depend smoothly on the age structure among living individuals at all timepoints preceding t . The ‘age structure’ is then thought of as an object which is scaled by a parameter of the order of the size of the initial population. By a multidimensional random clock argument, it is possible to relate such a population to a countable set of time homogeneous Poisson processes, one for each brood of size k , with points that carry k independent life-span marks for the siblings corresponding to it. The equations expressing this relation are quite complicated. First, in theorems 1 and 1′, law-of-large-numbers arguments applied to the mentioned Poisson processes yield deterministic equations for an age distribution evolution, which approximates the actual random one asymptotically. This is done in great generality, and restrictions assumed are natural. Under the assumption that no brood sizes are larger than some k 0 and some further regularity, then white noise type approximations of the marked Poisson processes are demonstrated to be relevant for the actual fluctuations of the age-structure around its limiting stable evolution.

An environmental process was characterized by a stationary first order autoregressive process with Gaussian noise. This process was then linked to survivorship and reproductive success by logistic tranformations to construct a stochastic population process, related to generalized age-dependent branching processes, for projecting age-structured populations in a random environment. Unexpectedly large fluctuations of conditional mean total population size, given different realizations of the environmental process. were observed, particularly when the environmental process had a strictly positive autocorrelation function. These large fluctuations led to summarizing the results of 100 replications of projecting conditional mean total population size, given the environmental process, for 100 epochs in terms of extreme value statistics rather than the more conventional means and variances. Qualitative comparisons with fluctuations in abundance of some species of birds suggested that some choices of parameter values for the environmental process led to plausible models for the evaluation of selected bird populations

A population is considered in which the number of individuals X sub t, t = 0, 1, 2, ..., added to the population in the time interval (t, t+1) is a Markov chain with the non-negative integers as state-space. At the end of each interval one individual is removed from the population, the process coming to a stop when the population size is zero. A method is developed for finding the probability distribution of the time to extinction of such a process for given initial conditions. The model was suggested by a problem in storage theory.

Consider a population consisting of one type of individual living in a fixed region with area A. In [8], we constructed a stochastic population model in which the death rate is affected by the age of the individual and the birth rate is affected by the population density PA(t), i.e., the population size divided by the area A of the given region. In [8], we proposed a continuous deterministic model which in general is a nonlinear Volterra type integral equation and proved that under appropriate conditions the sequence PA(t) would converge to the solution P(t) of our integral equation in the sense that lim→∞Psup0⩽s⩽t|PA(s) − P(s)|>e=0 for every e > 0 . In this paper, we obtain a “central limit theorem” for the random element √A(PA(t)−P(t)). We prove that under appropriate conditions √A(PA(t)−P(t)) will converge to a Gaussian process. (See Theorem 3.4 for the explicit formula of this Gaussian process.)

Many population processes in demography, epidemiology and other fields can be represented by a time-continuous Markov chain model with a finite state space. If we have complete information on the life history of a cohort, the intensities of the Markov model may be estimated by the occurrence/ exposure rates or by nonparametric techniques. In many situations, however, we have only incomplete information. In this paper we consider the special, but important, case where the occurrences and the total exposure are known, but not the distribution of the latter over the various separate statuses. Methods for handling such data, so-called demographic incidence rates, and methods for estimating the intensities from this kind of data, are known in the literature. However, their statistical properties are only vaguely known. The present paper gives a thorough presentation of the theory of these methods, and provide rigorous proofs of their statistical properties using stochastic process theory. 1. Introduction. In demography and fields with a similar methodology, a population process on the individual level can frequently be represented by a time-continuous Markov chain with a finite state space. The states of the Markov chain represent the demographic statuses and the jumps between the states correspond to the demographic events. The time parameter of the Markov process may be the age of an individual, the time since a specific event, or some such quantity. We will call it seniority. Given this general framework, the population phenomena may be described by the transition intensities of the model. Consequently, it is of great interest to estimate these functions. In principle, this is an easy task if the individuals studied are watched continuously over some period of time. One may then use the classical methods based on occurrence/exposure rates (for a review, see Hoem, 1976), or for small populations or samples, nonparametric techniques developed by Aalen (1978) and others. Quite often, however, one does not obtain the sufficiently detailed data required to use any of these methods. This fact has inhibited the use of Markov chain models in demography and related fields. The development and study of statistical methods for situations with incomplete data is therefore of considerable interest. There seems to be no general solution to the problem of estimating the intensities of partially observed Markov chains. In the present paper we consider a special, but important, type of incomplete observation. Namely, that transitions within a subset of states K are observed in detail, while counts of transitions out