1
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Barton N, Sachdeva H. Limits to selection on standing variation in an asexual population. Theor Popul Biol 2024; 157:129-137. [PMID: 38643838 DOI: 10.1016/j.tpb.2024.04.001] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [What about the content of this article? (0)] [Affiliation(s)] [Abstract] [Key Words] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 05/12/2023] [Revised: 01/23/2024] [Accepted: 04/08/2024] [Indexed: 04/23/2024]
Abstract
We consider how a population of N haploid individuals responds to directional selection on standing variation, with no new variation from recombination or mutation. Individuals have trait values z1,…,zN, which are drawn from a distribution ψ; the fitness of individual i is proportional to [Formula: see text] . For illustration, we consider the Laplace and Gaussian distributions, which are parametrised only by the variance V0, and show that for large N, there is a scaling limit which depends on a single parameter NV0. When selection is weak relative to drift (NV0≪1), the variance decreases exponentially at rate 1/N, and the expected ultimate gain in log fitness (scaled by V0), is just NV0, which is the same as Robertson's (1960) prediction for a sexual population. In contrast, when selection is strong relative to drift (NV0≫1), the ultimate gain can be found by approximating the establishment of alleles by a branching process in which each allele competes independently with the population mean and the fittest allele to establish is certain to fix. Then, if the probability of survival to time t∼1/V0 of an allele with value z is P(z), with mean P¯, the winning allele is the fittest of NP¯ survivors drawn from a distribution ψP/P¯. The expected ultimate change is ∼2log(1.15NV0) for a Gaussian distribution, and ∼-12log0.36NV0-log-log0.36NV0 for a Laplace distribution. This approach also predicts the variability of the process, and its dynamics; we show that in the strong selection regime, the expected genetic variance decreases as ∼t-3 at large times. We discuss how these results may be related to selection on standing variation that is spread along a linear chromosome.
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Affiliation(s)
- Nick Barton
- Institute of Science and Technology Austria, Am Campus 1, Klosterneuburg 3400, Austria.
| | - Himani Sachdeva
- Department of Mathematics, University of Vienna, Vienna 1090, Austria
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2
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Bauman N, Chigansky P, Klebaner F. An approximation of populations on a habitat with large carrying capacity. J Math Biol 2024; 88:44. [PMID: 38498209 PMCID: PMC10948565 DOI: 10.1007/s00285-024-02069-w] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [What about the content of this article? (0)] [Affiliation(s)] [Abstract] [Key Words] [MESH Headings] [Grants] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 03/27/2023] [Revised: 10/10/2023] [Accepted: 02/18/2024] [Indexed: 03/20/2024]
Abstract
We consider stochastic dynamics of a population which starts from a small colony on a habitat with large but limited carrying capacity. A common heuristics suggests that such population grows initially as a Galton-Watson branching process and then its size follows an almost deterministic path until reaching its maximum, sustainable by the habitat. In this paper we put forward an alternative and, in fact, more accurate approximation which suggests that the population size behaves as a special nonlinear transformation of the Galton-Watson process from the very beginning.
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Affiliation(s)
- Naor Bauman
- Department of Statistics, The Hebrew University of Jerusalem, Mount Scopus Campus, 9190501, Jerusalem, Israel
| | - Pavel Chigansky
- Department of Statistics, The Hebrew University of Jerusalem, Mount Scopus Campus, 9190501, Jerusalem, Israel.
| | - Fima Klebaner
- School of Mathematics, Monash University, Clayton, VIC, 3058, Australia
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3
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Olofsson P, Chipkin L, Daileda RC, Azevedo RBR. Mutational meltdown in asexual populations doomed to extinction. J Math Biol 2023; 87:88. [PMID: 37994999 DOI: 10.1007/s00285-023-02019-y] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [What about the content of this article? (0)] [Affiliation(s)] [Abstract] [Key Words] [MESH Headings] [Grants] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 02/10/2023] [Revised: 07/03/2023] [Accepted: 10/20/2023] [Indexed: 11/24/2023]
Abstract
Asexual populations are expected to accumulate deleterious mutations through a process known as Muller's ratchet. Lynch and colleagues proposed that the ratchet eventually results in a vicious cycle of mutation accumulation and population decline that drives populations to extinction. They called this phenomenon mutational meltdown. Here, we analyze mutational meltdown using a multi-type branching process model where, in the presence of mutation, populations are doomed to extinction. We analyse the change in size and composition of the population and the time of extinction under this model.
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Affiliation(s)
- Peter Olofsson
- Department of Mathematics, Trinity University, San Antonio, TX, 78212, USA
- Department of Mathematics, Physics and Chemical Engineering, Jönköping University, 551 11, Jönköping, Sweden
| | - Logan Chipkin
- Department of Biology and Biochemistry, University of Houston, Houston, TX, 77204, USA
| | - Ryan C Daileda
- Department of Mathematics, Trinity University, San Antonio, TX, 78212, USA
| | - Ricardo B R Azevedo
- Department of Biology and Biochemistry, University of Houston, Houston, TX, 77204, USA.
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4
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Li A, Kibby D, Foo J. A comparison of mutation and amplification-driven resistance mechanisms and their impacts on tumor recurrence. J Math Biol 2023; 87:59. [PMID: 37707631 DOI: 10.1007/s00285-023-01992-8] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [What about the content of this article? (0)] [Affiliation(s)] [Abstract] [Key Words] [MESH Headings] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 06/01/2023] [Revised: 08/15/2023] [Accepted: 08/19/2023] [Indexed: 09/15/2023]
Abstract
Tumor recurrence, driven by the evolution of drug resistance is a major barrier to therapeutic success in cancer. Tumor drug resistance is often caused by genetic alterations such as point mutation, which refers to the modification of a single genomic base pair, or gene amplification, which refers to the duplication of a region of DNA that contains a gene. These mechanisms typically confer varying degrees of resistance, and they tend to occur at vastly different frequencies. Here we investigate the dependence of tumor recurrence dynamics on these mechanisms of resistance, using stochastic multi-type branching process models. We derive tumor extinction probabilities and deterministic estimates for the tumor recurrence time, defined as the time when an initially drug sensitive tumor surpasses its original size after developing resistance. For models of amplification-driven and mutation-driven resistance, we prove law of large numbers results regarding the convergence of the stochastic recurrence times to their mean. Additionally, we prove sufficient and necessary conditions for a tumor to escape extinction under the gene amplification model, discuss behavior under biologically relevant parameters, and compare the recurrence time and tumor composition in the mutation and amplification models both analytically and using simulations. In comparing these mechanisms, we find that the ratio between recurrence times driven by amplification versus mutation depends linearly on the number of amplification events required to acquire the same degree of resistance as a mutation event, and we find that the relative frequency of amplification and mutation events plays a key role in determining the mechanism under which recurrence is more rapid for any specific system. In the amplification-driven resistance model, we also observe that increasing drug concentration leads to a stronger initial reduction in tumor burden, but that the eventual recurrent tumor population is less heterogeneous, more aggressive and harbors higher levels of drug-resistance.
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Affiliation(s)
- Aaron Li
- School of Mathematics, University of Minnesota, Minneapolis, MN, USA
| | | | - Jasmine Foo
- School of Mathematics, University of Minnesota, Minneapolis, MN, USA.
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5
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Ren M, Zhang G. Some properties of branching processes with random control functions and affected by viral infectivity in random environments. Adv Contin Discret Model 2023; 2023:26. [PMID: 37216170 PMCID: PMC10184110 DOI: 10.1186/s13662-023-03775-3] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [What about the content of this article? (0)] [Affiliation(s)] [Abstract] [Key Words] [Grants] [Track Full Text] [Figures] [Subscribe] [Scholar Register] [Received: 07/04/2022] [Accepted: 05/09/2023] [Indexed: 05/24/2023]
Abstract
In this paper, a model of branching processes with random control functions and affected by viral infectivity in independent and identically distributed random environments is established, and the Markov property of the model and a sufficient condition for the model to be certainly extinct under some conditions are discussed. Then, the limit properties of the model are studied. Under the normalization factor {Sn:n∈N}, the normalization processes {Wˆn:n∈N} are studied, and the sufficient conditions of {Wˆn:n∈N} a.s., L1 and L2 convergence are given; A sufficient condition and a necessary condition for convergence to a nondegenerate at zero random variable are obtained. Under the normalization factor {In:n∈N}, the normalization processes {W¯n:n∈N} are studied, and the sufficient conditions of {W¯n:n∈N} a.s., and L1 convergence are obtained.
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Affiliation(s)
- Min Ren
- College of Mathematics and Statistics, Suzhou University, Suzhou, 234000 China
| | - Guanghui Zhang
- College of Mathematics and Statistics, Suzhou University, Suzhou, 234000 China
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6
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Gunnarsson EB, Leder K, Foo J. Exact site frequency spectra of neutrally evolving tumors: A transition between power laws reveals a signature of cell viability. Theor Popul Biol 2021; 142:67-90. [PMID: 34560155 DOI: 10.1016/j.tpb.2021.09.004] [Citation(s) in RCA: 4] [Impact Index Per Article: 1.3] [Reference Citation Analysis] [What about the content of this article? (0)] [Affiliation(s)] [Abstract] [Key Words] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 02/23/2021] [Revised: 08/24/2021] [Accepted: 09/11/2021] [Indexed: 11/28/2022]
Abstract
The site frequency spectrum (SFS) is a popular summary statistic of genomic data. While the SFS of a constant-sized population undergoing neutral mutations has been extensively studied in population genetics, the rapidly growing amount of cancer genomic data has attracted interest in the spectrum of an exponentially growing population. Recent theoretical results have generally dealt with special or limiting cases, such as considering only cells with an infinite line of descent, assuming deterministic tumor growth, or taking large-time or large-population limits. In this work, we derive exact expressions for the expected SFS of a cell population that evolves according to a stochastic branching process, first for cells with an infinite line of descent and then for the total population, evaluated either at a fixed time (fixed-time spectrum) or at the stochastic time at which the population reaches a certain size (fixed-size spectrum). We find that while the rate of mutation scales the SFS of the total population linearly, the rates of cell birth and cell death change the shape of the spectrum at the small-frequency end, inducing a transition between a 1/j2 power-law spectrum and a 1/j spectrum as cell viability decreases. We show that this insight can in principle be used to estimate the ratio between the rate of cell death and cell birth, as well as the mutation rate, using the site frequency spectrum alone. Although the discussion is framed in terms of tumor dynamics, our results apply to any exponentially growing population of individuals undergoing neutral mutations.
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Affiliation(s)
- Einar Bjarki Gunnarsson
- Department of Industrial and Systems Engineering, University of Minnesota, Twin Cities, MN 55455, USA.
| | - Kevin Leder
- Department of Industrial and Systems Engineering, University of Minnesota, Twin Cities, MN 55455, USA.
| | - Jasmine Foo
- School of Mathematics, University of Minnesota, Twin Cities, MN 55455, USA.
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7
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Levesque J, Maybury DW, Shaw RD. A model of COVID-19 propagation based on a gamma subordinated negative binomial branching process. J Theor Biol 2021; 512:110536. [PMID: 33186594 PMCID: PMC7654309 DOI: 10.1016/j.jtbi.2020.110536] [Citation(s) in RCA: 5] [Impact Index Per Article: 1.7] [Reference Citation Analysis] [What about the content of this article? (0)] [Affiliation(s)] [Abstract] [Key Words] [Track Full Text] [Download PDF] [Figures] [Journal Information] [Subscribe] [Scholar Register] [Received: 09/03/2020] [Revised: 10/19/2020] [Accepted: 11/02/2020] [Indexed: 12/16/2022]
Abstract
We build a parsimonious Crump-Mode-Jagers continuous time branching process of COVID-19 propagation based on a negative binomial process subordinated by a gamma subordinator. By focusing on the stochastic nature of the process in small populations, our model provides decision making insight into mitigation strategies as an outbreak begins. Our model accommodates contact tracing and isolation, allowing for comparisons between different types of intervention. We emphasize a physical interpretation of the disease propagation throughout which affords analytical results for comparison to simulations. Our model provides a basis for decision makers to understand the likely trade-offs and consequences between alternative outbreak mitigation strategies particularly in office environments and confined work-spaces. Combining the asymptotic limit of our model with Bayesian hierarchical techniques, we provide US county level inferences for the reproduction number from cumulative case count data over July and August of this year.
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Affiliation(s)
- Jérôme Levesque
- Public Services and Procurement Canada, 270 Albert Street, Ottawa, ON K1P 6N7, Canada,Public Health Agency of Canada, 130 Colonnade Road, Ottawa, ON K1A 0K9, Canada
| | - David W. Maybury
- Public Services and Procurement Canada, 270 Albert Street, Ottawa, ON K1P 6N7, Canada,Public Health Agency of Canada, 130 Colonnade Road, Ottawa, ON K1A 0K9, Canada,Corresponding author
| | - R.H.A. David Shaw
- Public Services and Procurement Canada, 270 Albert Street, Ottawa, ON K1P 6N7, Canada
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8
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Ledesma-Terrón M, Peralta-Cañadas N, Míguez DG. FGF2 modulates simultaneously the mode, the rate of division and the growth fraction in cultures of radial glia. Development 2020; 147:147/14/dev189712. [PMID: 32709691 PMCID: PMC7390635 DOI: 10.1242/dev.189712] [Citation(s) in RCA: 2] [Impact Index Per Article: 0.5] [Reference Citation Analysis] [What about the content of this article? (0)] [Affiliation(s)] [Abstract] [Key Words] [Track Full Text] [Figures] [Journal Information] [Subscribe] [Scholar Register] [Received: 03/06/2020] [Accepted: 06/18/2020] [Indexed: 01/16/2023]
Abstract
Radial glial progenitors in the mammalian developing neocortex have been shown to follow a deterministic differentiation program restricted to an asymmetric-only mode of division. This feature seems incompatible with their well-known ability to increase in number when cultured in vitro, driven by fibroblast growth factor 2 and other mitogenic signals. The changes in their differentiation dynamics that allow this transition from in vivo asymmetric-only division mode to an in vitro self-renewing culture have not been fully characterized. Here, we combine experiments of radial glia cultures with numerical models and a branching process theoretical formalism to show that fibroblast growth factor 2 has a triple effect by simultaneously increasing the growth fraction, promoting symmetric divisions and shortening the length of the cell cycle. These combined effects partner to establish and sustain a pool of rapidly proliferating radial glial progenitors in vitro. We also show that, in conditions of variable proliferation dynamics, the branching process tool outperforms other commonly used methods based on thymidine analogs, such as BrdU and EdU, in terms of accuracy and reliability. Highlighted Article: When mode and/or rate of division are changing, a branching process, rather than a thymidine analog method, provides temporal resolution, it is more robust and does not interfere with cell homeostasis.
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Affiliation(s)
- Mario Ledesma-Terrón
- Departamento de Física de la Materia Condensada, Instituto de Física de la Materia Condensada, IFIMAC, Instituto Nicolas Cabrera, INC, Centro de Biología Molecular Severo Ochoa, CBMSO, Universidad Autónoma de Madrid, Madrid 28012, Spain
| | - Nuria Peralta-Cañadas
- Departamento de Física de la Materia Condensada, Instituto de Física de la Materia Condensada, IFIMAC, Instituto Nicolas Cabrera, INC, Centro de Biología Molecular Severo Ochoa, CBMSO, Universidad Autónoma de Madrid, Madrid 28012, Spain
| | - David G Míguez
- Departamento de Física de la Materia Condensada, Instituto de Física de la Materia Condensada, IFIMAC, Instituto Nicolas Cabrera, INC, Centro de Biología Molecular Severo Ochoa, CBMSO, Universidad Autónoma de Madrid, Madrid 28012, Spain
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Abstract
In this paper we consider Susceptible \documentclass[12pt]{minimal}
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\begin{document}$$\rightarrow $$\end{document}→ Infectious \documentclass[12pt]{minimal}
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\begin{document}$$\rightarrow $$\end{document}→ Recovered (SIR) epidemics on random graphs with clustering. To incorporate group structure of the underlying social network, we use a generalized version of the configuration model in which each node is a member of a specified number of triangles. SIR epidemics on this type of graph have earlier been investigated under the assumption of homogeneous infectivity and also under the assumption of Poisson transmission and recovery rates. We extend known results from literature by relaxing the assumption of homogeneous infectivity both in individual infectivity and between different kinds of neighbours. An important special case of the epidemic model analysed in this paper is epidemics in continuous time with arbitrary infectious period distribution. We use branching process approximations of the spread of the disease to provide expressions for the basic reproduction number \documentclass[12pt]{minimal}
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\begin{document}$$R_0$$\end{document}R0, the probability of a major outbreak and the expected final size. In addition, the impact of random vaccination with a perfect vaccine on the final outcome of the epidemic is investigated. We find that, for this particular model, \documentclass[12pt]{minimal}
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\begin{document}$$R_0$$\end{document}R0 equals the perfect vaccine-associated reproduction number. Generalizations to groups larger than three are discussed briefly.
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Tahir D, Glémin S, Lascoux M, Kaj I. Modeling a trait-dependent diversification process coupled with molecular evolution on a random species tree. J Theor Biol 2018; 461:189-203. [PMID: 30340056 DOI: 10.1016/j.jtbi.2018.10.032] [Citation(s) in RCA: 3] [Impact Index Per Article: 0.5] [Reference Citation Analysis] [What about the content of this article? (0)] [Affiliation(s)] [Abstract] [Key Words] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 03/24/2018] [Revised: 10/10/2018] [Accepted: 10/12/2018] [Indexed: 12/23/2022]
Abstract
Understanding the evolution of binary traits, which affects the birth and survival of species and also the rate of molecular evolution, remains challenging. In this work, we present a probabilistic modeling framework for binary trait, random species trees, in which the number of species and their traits are represented by an asymmetric, two-type, continuous time Markov branching process. The model involves a number of different parameters describing both character and molecular evolution on the so-called 'reduced' tree, consisting of only extant species at the time of observation. We expand our model by considering the impact of binary traits on dN/dS, the normalized ratio of nonsynonymous to synonymous substitutions. We also develop mechanisms which enable us to understand the substitution rates on a phylogenetic tree with regards to the observed traits. The properties obtained from the model are illustrated with a phylogeny of outcrossing and selfing plant species, which allows us to investigate not only the branching tree rates, but also the molecular rates and the intensity of selection.
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Affiliation(s)
- Daniah Tahir
- Department of Mathematics, Uppsala University, Box 480, Uppsala SE-751 06, Sweden.
| | - Sylvain Glémin
- Department of Plant Ecology and Evolution, Uppsala University, Norbyvägen 18D, Uppsala SE-752 36, Sweden; UMR 5554 ISEM (Université de Montpellier-CNRS-IRD-EPHE), Montpellier Cedex 5 FR-34095, France.
| | - Martin Lascoux
- Department of Plant Ecology and Evolution, Uppsala University, Norbyvägen 18D, Uppsala SE-752 36, Sweden.
| | - Ingemar Kaj
- Department of Mathematics, Uppsala University, Box 480, Uppsala SE-751 06, Sweden.
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11
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Voinson M, Alvergne A, Billiard S, Smadi C. Stochastic dynamics of an epidemic with recurrent spillovers from an endemic reservoir. J Theor Biol 2018; 457:37-50. [PMID: 30121292 PMCID: PMC7094102 DOI: 10.1016/j.jtbi.2018.08.017] [Citation(s) in RCA: 2] [Impact Index Per Article: 0.3] [Reference Citation Analysis] [What about the content of this article? (0)] [Affiliation(s)] [Abstract] [Key Words] [Track Full Text] [Download PDF] [Figures] [Journal Information] [Subscribe] [Scholar Register] [Received: 11/02/2017] [Revised: 07/12/2018] [Accepted: 08/12/2018] [Indexed: 12/16/2022]
Abstract
Stochastic SIR with reservoir to describe the dynamics of pathogens in a host. Branching process approximations are provided. Recurrent spillover cause multiple outbreaks even for a pathogen barely contagious. Spillover and direct transmission have similar importance for pathogens dynamics.
Most emerging human infectious diseases have an animal origin. While zoonotic diseases originate from a reservoir, most theoretical studies have principally focused on single-host processes, either exclusively humans or exclusively animals, without considering the importance of animal to human transmission (i.e. spillover transmission) for understanding the dynamics of emerging infectious diseases. Here we aim to investigate the importance of spillover transmission for explaining the number and the size of outbreaks. We propose a simple continuous time stochastic Susceptible-Infected-Recovered model with a recurrent infection of an incidental host from a reservoir (e.g. humans by a zoonotic species), considering two modes of transmission, (1) animal-to-human and (2) human-to-human. The model assumes that (i) epidemiological processes are faster than other processes such as demographics or pathogen evolution and that (ii) an epidemic occurs until there are no susceptible individuals left. The results show that during an epidemic, even when the pathogens are barely contagious, multiple outbreaks are observed due to spillover transmission. Overall, the findings demonstrate that the only consideration of direct transmission between individuals is not sufficient to explain the dynamics of zoonotic pathogens in an incidental host.
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Affiliation(s)
- Marina Voinson
- University of Lille, CNRS, UMR 8198 - Evo-Eco-Paleo, Lille F-59000, France.
| | - Alexandra Alvergne
- School of Anthropology and Museum Ethnography, University of Oxford, Oxford, OX2 6PE, UK
| | - Sylvain Billiard
- University of Lille, CNRS, UMR 8198 - Evo-Eco-Paleo, Lille F-59000, France
| | - Charline Smadi
- IRSTEA UR LISC, Laboratoire d'ingénierie pour les Systèmes Complexes, 9 avenue Blaise-Pascal CS 20085, Aubière 63178, France; Complex Systems Institute of Paris Ile-de-France, 113 rue Nationale, Paris, France
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12
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Moore S, Mörters P, Rogers T. A Re-entrant Phase Transition in the Survival of Secondary Infections on Networks. J Stat Phys 2018; 171:1122-1135. [PMID: 31007280 PMCID: PMC6445506 DOI: 10.1007/s10955-018-2050-9] [Citation(s) in RCA: 2] [Impact Index Per Article: 0.3] [Reference Citation Analysis] [What about the content of this article? (0)] [Affiliation(s)] [Abstract] [Key Words] [Grants] [Track Full Text] [Figures] [Subscribe] [Scholar Register] [Received: 02/07/2018] [Accepted: 04/20/2018] [Indexed: 06/09/2023]
Abstract
We study the dynamics of secondary infections on networks, in which only the individuals currently carrying a certain primary infection are susceptible to the secondary infection. In the limit of large sparse networks, the model is mapped to a branching process spreading in a random time-sensitive environment, determined by the dynamics of the underlying primary infection. When both epidemics follow the Susceptible-Infective-Recovered model, we show that in order to survive, it is necessary for the secondary infection to evolve on a timescale that is closely matched to that of the primary infection on which it depends.
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Affiliation(s)
- Sam Moore
- University of Bath, Claverton Down, Bath, BA2 7AY UK
| | - Peter Mörters
- Mathematisches Institut, Universität zu Köln, Weyertal 86-90, 50931 Köln, Germany
| | - Tim Rogers
- University of Bath, Claverton Down, Bath, BA2 7AY UK
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13
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Abstract
Real time, or quantitative, PCR typically starts from a very low concentration of initial DNA strands. During iterations the numbers increase, first essentially by doubling, later predominantly in a linear way. Observation of the number of DNA molecules in the experiment becomes possible only when it is substantially larger than initial numbers, and then possibly affected by the randomness in individual replication. Can the initial copy number still be determined? This is a classical problem and, indeed, a concrete special case of the general problem of determining the number of ancestors, mutants or invaders, of a population observed only later. We approach it through a generalised version of the branching process model introduced in Jagers and Klebaner (J Theor Biol 224(3):299–304, 2003. doi:10.1016/S0022-5193(03)00166-8), and based on Michaelis–Menten type enzyme kinetical considerations from Schnell and Mendoza (J Theor Biol 184(4):433–440, 1997). A crucial role is played by the Michaelis–Menten constant being large, as compared to initial copy numbers. In a strange way, determination of the initial number turns out to be completely possible if the initial rate v is one, i.e all DNA strands replicate, but only partly so when \documentclass[12pt]{minimal}
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\begin{document}$$v<1$$\end{document}v<1, and thus the initial rate or probability of succesful replication is lower than one. Then, the starting molecule number becomes hidden behind a “veil of uncertainty”. This is a special case, of a hitherto unobserved general phenomenon in population growth processes, which will be adressed elsewhere.
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Affiliation(s)
- Pavel Chigansky
- Department of Statistics, The Hebrew University, Mount Scopus, 91905, Jerusalem, Israel
| | - Peter Jagers
- Mathematical Sciences, Chalmers University of Technology and University of Gothenburg, 412 96, Gothenburg, Sweden.
| | - Fima C Klebaner
- School of Mathematical Sciences, Monash University, Monash, VIC, 3800, Australia
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14
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Abstract
We consider a class of Sevastyanov branching processes with non-homogeneous Poisson immigration. These processes relax the assumption required by the Bellman-Harris process which imposes the lifespan and offspring of each individual to be independent. They find applications in studies of the dynamics of cell populations. In this paper, we focus on the subcritical case and examine asymptotic properties of the process. We establish limit theorems, which generalize classical results due to Sevastyanov and others. Our key findings include novel LLN and CLT which emerge from the non-homogeneity of the immigration process.
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15
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Abstract
Identifying the critical domain size necessary for a population to persist is an important question in ecology. Both demographic and environmental stochasticity impact a population's ability to persist. Here we explore ways of including this variability. We study populations with distinct dispersal and sedentary stages, which have traditionally been modelled using a deterministic integrodifference equation (IDE) framework. Individual-based models (IBMs) are the most intuitive stochastic analogues to IDEs but yield few analytic insights. We explore two alternate approaches; one is a scaling up to the population level using the Central Limit Theorem, and the other a variation on both Galton-Watson branching processes and branching processes in random environments. These branching process models closely approximate the IBM and yield insight into the factors determining the critical domain size for a given population subject to stochasticity.
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Affiliation(s)
- Jody R Reimer
- Mathematical and Statistical Sciences, 632 Central Academic Building, University of Alberta, Edmonton, AB, T6G 2G1, Canada
| | - Michael B Bonsall
- Mathematical Ecology Research Group, Department of Zoology, University of Oxford, Tinbergen Building, South Parks Road, Oxford, OX1 3PS, UK.
| | - Philip K Maini
- Wolfson Centre for Mathematical Biology, Mathematical Institute, University of Oxford, Andrew Wiles Building, Radcliffe Observatory Quarter, Woodstock Road, Oxford, OX2 6GG, UK
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16
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Abstract
For proliferating cells subject to both division and death, how can one estimate the average generation number of the living population without continuous observation or a division-diluting dye? In this paper we provide a method for cell systems such that at each division there is an unlikely, heritable one-way label change that has no impact other than to serve as a distinguishing marker. If the probability of label change per cell generation can be determined and the proportion of labeled cells at a given time point can be measured, we establish that the average generation number of living cells can be estimated. Crucially, the estimator does not depend on knowledge of the statistics of cell cycle, death rates or total cell numbers. We explore the estimator's features through comparison with physiologically parameterized stochastic simulations and extrapolations from published data, using it to suggest new experimental designs.
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Affiliation(s)
- Tom S Weber
- Hamilton Institute, Maynooth University, Maynooth, Ireland
| | - Leïla Perié
- Division of Immunology, Netherlands Cancer Institute, Amsterdam, The Netherlands
- Department of Theoretical Biology and Bioinformatics, Utrecht University, Utrecht, The Netherlands
- Institut Curie, PSL Research University, CNRS UMR168, Paris, France
| | - Ken R Duffy
- Hamilton Institute, Maynooth University, Maynooth, Ireland.
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Drakos NE, Wahl LM. Extinction probabilities and stationary distributions of mobile genetic elements in prokaryotes: The birth-death-diversification model. Theor Popul Biol 2015; 106:22-31. [PMID: 26383090 DOI: 10.1016/j.tpb.2015.09.001] [Citation(s) in RCA: 2] [Impact Index Per Article: 0.2] [Reference Citation Analysis] [What about the content of this article? (0)] [Abstract] [Key Words] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 10/29/2014] [Revised: 08/27/2015] [Accepted: 09/02/2015] [Indexed: 11/23/2022]
Abstract
Theoretical approaches are essential to our understanding of the complex dynamics of mobile genetic elements (MGEs) within genomes. Recently, the birth-death-diversification model was developed to describe the dynamics of mobile promoters (MPs), a particular class of MGEs in prokaryotes. A unique feature of this model is that genetic diversification of elements was included. To explore the implications of diversification on the longterm fate of MGE lineages, in this contribution we analyze the extinction probabilities, extinction times and equilibrium solutions of the birth-death-diversification model. We find that diversification increases both the survival and growth rate of MGE families, but the strength of this effect depends on the rate of horizontal gene transfer (HGT). We also find that the distribution of MGE families per genome is not necessarily monotonically decreasing, as observed for MPs, but may have a peak in the distribution that is related to the HGT rate. For MPs specifically, we find that new families have a high extinction probability, and predict that the number of MPs is increasing, albeit at a very slow rate. Additionally, we develop an extension of the birth-death-diversification model which allows MGEs in different regions of the genome, for example coding and non-coding, to be described by different rates. This extension may offer a potential explanation as to why the majority of MPs are located in non-promoter regions of the genome.
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Xu J, Guttorp P, Kato-Maeda M, Minin VN. Likelihood-based inference for discretely observed birth-death-shift processes, with applications to evolution of mobile genetic elements. Biometrics 2015; 71:1009-21. [PMID: 26148963 DOI: 10.1111/biom.12352] [Citation(s) in RCA: 9] [Impact Index Per Article: 1.0] [Reference Citation Analysis] [What about the content of this article? (0)] [Affiliation(s)] [Abstract] [Key Words] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 02/01/2015] [Revised: 05/01/2015] [Accepted: 05/01/2015] [Indexed: 11/28/2022]
Abstract
Continuous-time birth-death-shift (BDS) processes are frequently used in stochastic modeling, with many applications in ecology and epidemiology. In particular, such processes can model evolutionary dynamics of transposable elements-important genetic markers in molecular epidemiology. Estimation of the effects of individual covariates on the birth, death, and shift rates of the process can be accomplished by analyzing patient data, but inferring these rates in a discretely and unevenly observed setting presents computational challenges. We propose a multi-type branching process approximation to BDS processes and develop a corresponding expectation maximization algorithm, where we use spectral techniques to reduce calculation of expected sufficient statistics to low-dimensional integration. These techniques yield an efficient and robust optimization routine for inferring the rates of the BDS process, and apply broadly to multi-type branching processes whose rates can depend on many covariates. After rigorously testing our methodology in simulation studies, we apply our method to study intrapatient time evolution of IS6110 transposable element, a genetic marker frequently used during estimation of epidemiological clusters of Mycobacterium tuberculosis infections.
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Affiliation(s)
- Jason Xu
- Department of Statistics, University of Washington, Seattle, WA, U.S.A
| | - Peter Guttorp
- Department of Statistics, University of Washington, Seattle, WA, U.S.A
| | - Midori Kato-Maeda
- School of Medicine, University of California, San Francisco, CA, U.S.A
| | - Vladimir N Minin
- Department of Statistics, University of Washington, Seattle, WA, U.S.A.,Department of Biology, University of Washington, Seattle, WA, U.S.A
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Abstract
We study the spread of stochastic SIR (Susceptible \documentclass[12pt]{minimal}
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\begin{document}$$\rightarrow $$\end{document}→ Recovered) epidemics in two types of structured populations, both consisting of schools and households. In each of the types, every individual is part of one school and one household. In the independent partition model, the partitions of the population into schools and households are independent of each other. This model corresponds to the well-studied household-workplace model. In the hierarchical model which we introduce here, members of the same household are also members of the same school. We introduce computable branching process approximations for both types of populations and use these to compare the probabilities of a large outbreak. The branching process approximation in the hierarchical model is novel and of independent interest. We prove by a coupling argument that if all households and schools have the same size, an epidemic spreads easier (in the sense that the number of individuals infected is stochastically larger) in the independent partition model. We also show by example that this result does not necessarily hold if households and/or schools do not all have the same size.
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Lopes FM. Epidemics on a weighted network with tunable degree-degree correlation. Math Biosci 2014; 253:40-9. [PMID: 24721555 DOI: 10.1016/j.mbs.2014.03.013] [Citation(s) in RCA: 2] [Impact Index Per Article: 0.2] [Reference Citation Analysis] [What about the content of this article? (0)] [Affiliation(s)] [Abstract] [Key Words] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 10/14/2013] [Revised: 03/23/2014] [Accepted: 03/24/2014] [Indexed: 11/30/2022]
Abstract
We propose a weighted version of the standard configuration model which allows for a tunable degree-degree correlation. A social network is modeled by a weighted graph generated by this model, where the edge weights indicate the intensity or type of contact between the individuals. An inhomogeneous Reed-Frost epidemic model is then defined on the network, where the inhomogeneity refers to different disease transmission probabilities related to the edge weights. By tuning the model we study the impact of different correlation patterns on the network and epidemics therein. Our results suggest that the basic reproduction number R0 of the epidemic increases (decreases) when the degree-degree correlation coefficient ρ increases (decreases). Furthermore, we show that such effect can be amplified or mitigated depending on the relation between degree and weight distributions as well as the choice of the disease transmission probabilities. In addition, for a more general model allowing additional heterogeneity in the disease transmission probabilities we show that ρ can have the opposite effect on R0.
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