On parameter estimation in population models II: Multi-dimensional processes and transient dynamics

Ross, Joshua V.; Pagendam, Dan; Pollett, P. K.

doi:10.1016/j.tpb.2008.12.002

Cited by 39 publications

(45 citation statements)

References 19 publications

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“…This approach has become predominant in the applied metapopulation literature, because it provides a vehicle for parameter estimation [46] and permits control mechanisms to be investigated using simple optimisation tools such as dynamic programming [60,62]. Indeed, discrete-time Markov chain models predominate in the ecology literature (even in cases where they are not faithful to population dynamics), perhaps due in part to a widespread misconception that a discrete time model is needed if populations are observed (and controlled) at discrete time points [59,58]. Numerical methods and simulation are generally used to analyse discrete time metapopulation models, typically the EC case only [29,34,68], and until recently there have been few analytical studies [18,45].…”

Section: Metapopulationsmentioning

confidence: 99%

Limit theorems for discrete-time metapopulation models

Buckley¹,

Pollett²

2010

Probab. Surveys

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Abstract:We describe a class of one-dimensional chain binomial models of use in studying metapopulations (population networks). Limit theorems are established for time-inhomogeneous Markov chains that share the salient features of these models. We prove a law of large numbers, which can be used to identify an approximating deterministic trajectory, and a central limit theorem, which establishes that the scaled fluctuations about this trajectory have an approximating autoregressive structure.AMS 2000 subject classifications: Primary 60J10, 92B05; secondary 60J80.Received January 2010. MetapopulationsA metapopulation is a population confined to a network of geographically separated habitat patches that may suffer extinction locally and be recolonized through dispersal of individuals from other patches. The term was coined by Levins [41] Levins [40] was the first to provide a succinct mathematical description of a metapopulation, proposing that the number n t of occupied patches at time t in a group of N patches should follow the law of motionwith c being the colonization rate and e being the local extinction rate. This is Verhulst's model [63] for population growth and Levins used Pearl's rationale [51,52,53] to derive it. Furthermore, Levins was able to divine an explicit solution to (1) in the case where both c and e are time dependent, and he derived a diffusion approximation for n t (surprisingly, the time-inhomogeneous * This is an original survey paper. 53

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Section: Metapopulationsmentioning

confidence: 99%

Limit theorems for discrete-time metapopulation models

Buckley¹,

Pollett²

2010

Probab. Surveys

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“…The model is also often used to study the initial stages of emerging infectious disease growth in a single community (see for example [44]). We can assess the degree of uncertainty not accounted for when deterministic SI model is used in place of the stochastic model, and when there is uncertainty in the initial state of the disease process or demographic variability in the population at risk.…”

Section: Investigationmentioning

confidence: 99%

Modelling population processes with random initial conditions

Pollett

Dooley

Ross

2010

Mathematical Biosciences

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Population dynamics are almost inevitably associated with two predominant sources of variation: the first, demographic variability, a consequence of chance in progenitive and deleterious events; the second, initial state uncertainty, a consequence of partial observability and reporting delays and errors. Here we outline a general method for incorporating random initial conditions in population models where a deterministic model is sufficient to describe the dynamics of the population. Additionally, we show that for a large class of stochastic models the overall variation is the sum of variation due to random initial conditions and variation due to random dynamics, and thus we are able to quantify the variation not accounted for when random dynamics are ignored.Our results are illustrated with reference to both simulated and real data.

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“…Theorem 3.9 implies that the distribution of the population at the endemic level can be approximated by a multivariate normal distribution for large population size. The result of this theorem may be used for estimating parameters of the model by applying the methods given in [177,178]. However, as noted in [177,178], in order to justify their method a local limit theorem will be required.…”

Section: Discussionmentioning

confidence: 99%

“…The result of this theorem may be used for estimating parameters of the model by applying the methods given in [177,178]. However, as noted in [177,178], in order to justify their method a local limit theorem will be required. I leave this problem for future studies.…”

Section: Discussionmentioning

confidence: 99%

“…However, for certain diseases such as gonorrhea and chlamydia, a gamma distribution may be a more realistic model of the infectious period [48]. The model studied in this work could incorporate gamma distributed infectious periods using a similar construction to that employed in [177]. Alternatively, the model could be generalised by incorporating a general infectious period along the lines of [153].…”

Section: Discussionmentioning

confidence: 99%

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A model for the spread of an SIS epidemic in a human population

Shausan¹

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The aim of this work is to understand how infectious diseases spread through human populations.Attention is given to those diseases which follow the Susceptible-Infective-Susceptible (SIS) pattern.When modelling diseases spread in a human population, it is important to consider the social and spatial structure of the population. Humans usually live in groups such as work places, households, towns and cities. However, an individual's membership of a particular group is not fixed. Rather, it changes over time. This structure determines two paths for a disease to spread through the population.Disease is spread between individuals in the same group by contact between infected and susceptible individuals, and is spread from one group to another by the migration of infected individuals. This type of population structure can be modelled by a metapopulation network. I develop a continuoustime Markov chain (CTMC) model that describes the spread of an SIS epidemic in a metapopulation network.I establish an ordinary differential equations (ODE) and a Gaussian diffusion analogue of the stochastic process by applying, respectively, the theory of differential equation approximations for Markov chains, and the theory of density dependent Markov chains. I use the ODE model to derive analytic expressions for various epidemiological quantities of interest. In particular, I obtain expressions for two threshold quantities; the basic reproduction number, and a quantity called T 0 which is greater than the basic reproduction number. If the basic reproduction number is above 1, then the disease persists and if the basic reproduction number is below 1, then the disease-free equilibrium (DFE) is locally attractive. However, if T 0 is less than or equal to 1, then the DFE is globally attractive. Using the theory of cooperative differential equations and the theory of asymptotically autonomous differential equations, I show the existence and global stability of a unique endemic equilibrium (EE) and the global stability of the DFE in terms of the basic reproduction number, provided that the migration rates of susceptible and infected individuals are equal. Numerical examples indicate that a unique stable EE exists when the condition on the migration rates is relaxed. The approximating Gaussian diffusion shows that the distribution of the population at the endemic level has an approximate multivariate normal distribution whose mean is centered at the endemic equilibrium of the ODE model. iThe results of this study can serve as a basic framework on how to formulate and analyse a more realistic stochastic model for the spread of an SIS epidemic in a metapopulation which accounts for births, deaths, age, risk, and level of infectivities.Assuming that the model presented here accurately describes the spread of an SIS epidemic in a metapopulation, another question which I address is how to control the spread of the disease. Since most control strategies such as vaccination, treatment and public awareness require a high cost for their imp...

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On parameter estimation in population models II: Multi-dimensional processes and transient dynamics

Cited by 39 publications

References 19 publications

Limit theorems for discrete-time metapopulation models

Limit theorems for discrete-time metapopulation models

Modelling population processes with random initial conditions

A model for the spread of an SIS epidemic in a human population

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