An Introduction to Stochastic Processes with Applications to Biology

Allen, Linda J. S.

doi:10.1201/b12537

Cited by 553 publications

(680 citation statements)

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“…wheref ðXðtÞÞ is the drift vector, DðXðtÞÞ is the diffusion matrix, and dõðtÞ is a vector of independent Wiener processes (Allen, 2003). The stochastic differential model can be explicitly expressed by definingf ðXðtÞÞ ¼ EðDXðtÞÞ, where, for example,…”

Section: Article In Pressmentioning

confidence: 99%

A comparative evaluation of modelling strategies for the effect of treatment and host interactions on the spread of drug resistance

Alexander

Dietrich

Hua

et al. 2009

Journal of Theoretical Biology

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The evolutionary responses of infectious pathogens often have ruinous consequences for the control of disease spread in the population. Drug resistance is a well-documented instance that is generally driven by the selective pressure of drugs on both the replication of the pathogen within hosts and its transmission between hosts. Management of drug resistance therefore requires the development of treatment strategies that can impede the emergence and spread of resistance in the population. This study evaluates various treatment strategies for influenza infection as a case study by comparing the long-term epidemiological outcomes predicted by deterministic and stochastic versions of a homogeneously mixing (mean-field) model and those predicted by a heterogeneous model that incorporates spatial pair-wise correlation. We discuss the importance of three major parameters in our evaluation: the basic reproduction number, the population level of treatment, and the degree of clustering as a key parameter determining the structure of heterogeneous interactions. The results show that, as a common feature in all models, high treatment levels during the early stages of disease outset can result in large resistant outbreaks, with the possibility of a second wave of infection appearing in the pair-approximation model. Our simulations demonstrate that, if the basic reproduction number exceeds a threshold value, the population-wide spread of the resistant pathogen emerges more rapidly in the pair-approximation model with significantly lower treatment levels than in the homogeneous models. We tested an antiviral strategy that delays the onset of aggressive treatment for a certain amount of time after the onset of the outbreak. The findings indicate that the overall disease incidence is reduced as the degree of clustering increases, and a longer delay should be considered for implementing the large-scale treatment.

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Section: Article In Pressmentioning

confidence: 99%

A comparative evaluation of modelling strategies for the effect of treatment and host interactions on the spread of drug resistance

Alexander

Dietrich

Hua

et al. 2009

Journal of Theoretical Biology

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“…We can, in principle, use these transition probabilities to construct master equations and generating functions to formulate ODEs for the moments (Allen, 2010). However, as is usual, master equations are difficult to analyse and the formulated ODEs suffer from the identification of suitably reliable moment closure conditions; see Chalub andSouza (2011), Isham (1991), Krishnarajah et al (2005) and references therein for examples in epidemiological modelling.…”

Section: A Stochastic Modelmentioning

confidence: 99%

Modelling the outbreak of infectious disease following mutation from a non-transmissible strain

Chen

Ward

Xie

2019

Theoretical Population Biology

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In-host mutation of a cross-species infectious disease to a form that is transmissible between humans has resulted with devastating global pandemics in the past. We use simple mathematical models to describe this process with the aim to better understand the emergence of an epidemic resulting from such a mutation and the extent of measures that are needed to control it. The feared outbreak of a human-human transmissible form of avian influenza leading to a global epidemic is the paradigm for this study. We extend the SIR approach to derive a deterministic and a stochastic formulation to describe the evolution of two classes of susceptible and infected states and a removed state, leading to a system of ordinary differential equations and a stochastic equivalent based on a Markov process. For the deterministic model, the contrasting timescale of the mutation process and disease infectiousness is exploited in two limits using asymptotic analysis in order to determine, in terms of the model parameters, necessary conditions for an epidemic to take place and timescales for the onset of the epidemic, the size and duration of the epidemic and the maximum level of the infected individuals at one time. Furthermore, the basic reproduction number R is determined from asymptotic analysis of a distinguished limit. Comparisons between the deterministic and stochastic model demonstrate that stochasticity has little effect on most aspects of an epidemic, but does have significant impact on its onset particularly for smaller populations and lower mutation rates for representatively large populations. The deterministic model is extended to investigate a range of quarantine and vaccination programmes, whereby in the two asymptotic limits analysed, quantitative estimates on the outcomes and effectiveness of these control measures are established.

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“…The conventional SEIR (susceptible-exposed-infected-removed) compartment models, both deterministic and stochastic, suppose that all individuals have an equally small chance (probability) of being infected [4]. In this section, we describe an alternative model structure where infection can only occur along certain predefined paths [6] which link subsets of individuals.…”

Section: Small World Network Modelmentioning

confidence: 99%

“…Models, however, are only as good as their underlying assumptions allow. For example, standard epidemiological models [4] such as susceptible-infected-removed (SIR) equations assume a constant infection and recovery rate across all individuals at any given time. That is, each infected individual has an equal chance of infecting any susceptible in the population regardless of its social structure.…”

Section: Introductionmentioning

confidence: 99%

Parameter inference in small world network disease models with approximate Bayesian Computational methods

Walker

Allingham

Lee

et al. 2010

Physica A: Statistical Mechanics and its Applications

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a b s t r a c tSmall world network models have been effective in capturing the variable behaviour of reported case data of the SARS coronavirus outbreak in Hong Kong during 2003. Simulations of these models have previously been realized using informed ''guesses'' of the proposed model parameters and tested for consistency with the reported data by surrogate analysis. In this paper we attempt to provide statistically rigorous parameter distributions using Approximate Bayesian Computation sampling methods. We find that such sampling schemes are a useful framework for fitting parameters of stochastic small world network models where simulation of the system is straightforward but expressing a likelihood is cumbersome.

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An Introduction to Stochastic Processes with Applications to Biology

Cited by 553 publications

References 0 publications

A comparative evaluation of modelling strategies for the effect of treatment and host interactions on the spread of drug resistance

A comparative evaluation of modelling strategies for the effect of treatment and host interactions on the spread of drug resistance

Modelling the outbreak of infectious disease following mutation from a non-transmissible strain

Parameter inference in small world network disease models with approximate Bayesian Computational methods

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