IDEAS home Printed from https://ideas.repec.org/a/eee/spapps/v126y2016i6p1839-1883.html
   My bibliography  Save this article

Branching within branching: A model for host–parasite co-evolution

Author

Listed:
  • Alsmeyer, Gerold
  • Gröttrup, Sören

Abstract

We consider a discrete-time host–parasite model for a population of cells which are colonized by proliferating parasites. The cell population grows like an ordinary Galton–Watson process, but in reflection of real biological settings the multiplication mechanisms of cells and parasites are allowed to obey some dependence structure. More precisely, the number of offspring produced by a mother cell determines the reproduction law of a parasite living in this cell and also the way the parasite offspring is shared into the daughter cells. In this article, we provide a formal introduction of this branching-within-branching model and then focus on the property of parasite extinction. We establish equivalent conditions for almost sure extinction of parasites and find a strong relation of this event to the behavior of parasite multiplication along a randomly chosen cell line through the cell tree, which forms a branching process in random environment. We then focus on asymptotic results for relevant processes in the case when parasites survive. In particular, limit theorems for the processes of contaminated cells and of parasites are established by using martingale theory and the technique of size-biasing. The results for both processes are of Kesten–Stigum type by including equivalent integrability conditions for the martingale limits to be positive with positive probability. The case when these conditions fail is also studied. For the process of contaminated cells, we show that a proper Heyde–Seneta norming exists such that the limit is nondegenerate.

Suggested Citation

  • Alsmeyer, Gerold & Gröttrup, Sören, 2016. "Branching within branching: A model for host–parasite co-evolution," Stochastic Processes and their Applications, Elsevier, vol. 126(6), pages 1839-1883.
    • Handle: RePEc:eee:spapps:v:126:y:2016:i:6:p:1839-1883
    DOI: 10.1016/j.spa.2015.12.007
    as

    Download full text from publisher

    File URL: http://www.sciencedirect.com/science/article/pii/S0304414915003245
    Download Restriction: Full text for ScienceDirect subscribers only
    File URL: https://libkey.io/10.1016/j.spa.2015.12.007?utm_source=ideas
    LibKey link: if access is restricted and if your library uses this service, LibKey will redirect you to where you can use your library subscription to access this item
    ---><---

    As the access to this document is restricted, you may want to search for a different version of it.

    References listed on IDEAS

    as
    1. Coffey, John & Tanny, David, 1984. "A necessary and sufficient condition for noncertain extinction of a branching process in a random environment (BPRE)," Stochastic Processes and their Applications, Elsevier, vol. 16(2), pages 189-197, February.
    2. Kuhlbusch, Dirk, 2004. "On weighted branching processes in random environment," Stochastic Processes and their Applications, Elsevier, vol. 109(1), pages 113-144, January.
    3. Biggins, J. D. & D'Souza, J. C., 1993. "The supercritical Galton-Watson process in varying environments--Seneta-Heyde norming," Stochastic Processes and their Applications, Elsevier, vol. 48(2), pages 237-249, November.
    4. Yuh Nung Jan & Lily Yeh Jan, 1998. "Asymmetric cell division," Nature, Nature, vol. 392(6678), pages 775-778, April.
    5. Tanny, David, 1988. "A necessary and sufficient condition for a branching process in a random environment to grow like the product of its means," Stochastic Processes and their Applications, Elsevier, vol. 28(1), pages 123-139, April.
    6. Delmas, Jean-François & Marsalle, Laurence, 2010. "Detection of cellular aging in a Galton-Watson process," Stochastic Processes and their Applications, Elsevier, vol. 120(12), pages 2495-2519, December.
    7. Afanasyev, V.I. & Geiger, J. & Kersting, G. & Vatutin, V.A., 2005. "Functional limit theorems for strongly subcritical branching processes in random environment," Stochastic Processes and their Applications, Elsevier, vol. 115(10), pages 1658-1676, October.
    8. Bansaye, Vincent, 2009. "Surviving particles for subcritical branching processes in random environment," Stochastic Processes and their Applications, Elsevier, vol. 119(8), pages 2436-2464, August.
    Full references (including those not matched with items on IDEAS)

    Citations

    Citations are extracted by the CitEc Project, subscribe to its RSS feed for this item.
    as


    Cited by:

    1. Marguet, Aline & Smadi, Charline, 2024. "Spread of parasites affecting death and division rates in a cell population," Stochastic Processes and their Applications, Elsevier, vol. 168(C).

    Most related items

    These are the items that most often cite the same works as this one and are cited by the same works as this one.
    1. Vincent Bansaye, 2019. "Ancestral Lineages and Limit Theorems for Branching Markov Chains in Varying Environment," Journal of Theoretical Probability, Springer, vol. 32(1), pages 249-281, March.
    2. Kuhlbusch, Dirk, 2004. "On weighted branching processes in random environment," Stochastic Processes and their Applications, Elsevier, vol. 109(1), pages 113-144, January.
    3. Huang, Chunmao & Liu, Quansheng, 2012. "Moments, moderate and large deviations for a branching process in a random environment," Stochastic Processes and their Applications, Elsevier, vol. 122(2), pages 522-545.
    4. Gao, Zhiqiang & Liu, Quansheng, 2016. "Exact convergence rates in central limit theorems for a branching random walk with a random environment in time," Stochastic Processes and their Applications, Elsevier, vol. 126(9), pages 2634-2664.
    5. Huang, Chunmao & Liu, Quansheng, 2024. "Limit theorems for a branching random walk in a random or varying environment," Stochastic Processes and their Applications, Elsevier, vol. 172(C).
    6. Bitseki Penda, S. Valère, 2023. "Moderate deviation principles for kernel estimator of invariant density in bifurcating Markov chains," Stochastic Processes and their Applications, Elsevier, vol. 158(C), pages 282-314.
    7. Hoffmann, Marc & Olivier, Adélaïde, 2016. "Nonparametric estimation of the division rate of an age dependent branching process," Stochastic Processes and their Applications, Elsevier, vol. 126(5), pages 1433-1471.
    8. Biggins, J. D. & Cohn, H. & Nerman, O., 1999. "Multi-type branching in varying environment," Stochastic Processes and their Applications, Elsevier, vol. 83(2), pages 357-400, October.
    9. S. Valère Bitseki Penda & Adélaïde Olivier, 2017. "Autoregressive functions estimation in nonlinear bifurcating autoregressive models," Statistical Inference for Stochastic Processes, Springer, vol. 20(2), pages 179-210, July.
    10. Bernard Bercu & Vassili Blandin, 2015. "Limit theorems for bifurcating integer-valued autoregressive processes," Statistical Inference for Stochastic Processes, Springer, vol. 18(1), pages 33-67, April.
    11. Böinghoff, Christian & Kersting, Götz, 2010. "Upper large deviations of branching processes in a random environment--Offspring distributions with geometrically bounded tails," Stochastic Processes and their Applications, Elsevier, vol. 120(10), pages 2064-2077, September.
    12. Li, Yingqiu & Liu, Quansheng & Peng, Xuelian, 2019. "Harmonic moments, large and moderate deviation principles for Mandelbrot’s cascade in a random environment," Statistics & Probability Letters, Elsevier, vol. 147(C), pages 57-65.
    13. Böinghoff, Christian, 2014. "Limit theorems for strongly and intermediately supercritical branching processes in random environment with linear fractional offspring distributions," Stochastic Processes and their Applications, Elsevier, vol. 124(11), pages 3553-3577.
    14. Grama, Ion & Liu, Quansheng & Miqueu, Eric, 2017. "Berry–Esseen’s bound and Cramér’s large deviation expansion for a supercritical branching process in a random environment," Stochastic Processes and their Applications, Elsevier, vol. 127(4), pages 1255-1281.
    15. Shieh, Narn-Rueih & Yu, Jinghu, 2004. "Dimensions of supercritical branching processes in varying environments," Statistics & Probability Letters, Elsevier, vol. 70(4), pages 299-308, December.
    16. Afanasyev, V. I., 2001. "On the maximum of a subcritical branching process in a random environment," Stochastic Processes and their Applications, Elsevier, vol. 93(1), pages 87-107, May.
    17. Bansaye, Vincent, 2009. "Surviving particles for subcritical branching processes in random environment," Stochastic Processes and their Applications, Elsevier, vol. 119(8), pages 2436-2464, August.
    18. de Saporta, Benoîte & Gégout-Petit, Anne & Marsalle, Laurence, 2012. "Asymmetry tests for bifurcating auto-regressive processes with missing data," Statistics & Probability Letters, Elsevier, vol. 82(7), pages 1439-1444.
    19. Najme Khorasani & Mehdi Sadeghi & Abbas Nowzari-Dalini, 2020. "A computational model of stem cell molecular mechanism to maintain tissue homeostasis," PLOS ONE, Public Library of Science, vol. 15(7), pages 1-25, July.
    20. Li, Zenghu & Xu, Wei, 2018. "Asymptotic results for exponential functionals of Lévy processes," Stochastic Processes and their Applications, Elsevier, vol. 128(1), pages 108-131.

    Corrections

    All material on this site has been provided by the respective publishers and authors. You can help correct errors and omissions. When requesting a correction, please mention this item's handle: RePEc:eee:spapps:v:126:y:2016:i:6:p:1839-1883. See general information about how to correct material in RePEc.

    If you have authored this item and are not yet registered with RePEc, we encourage you to do it here. This allows to link your profile to this item. It also allows you to accept potential citations to this item that we are uncertain about.

    If CitEc recognized a bibliographic reference but did not link an item in RePEc to it, you can help with this form .

    If you know of missing items citing this one, you can help us creating those links by adding the relevant references in the same way as above, for each refering item. If you are a registered author of this item, you may also want to check the "citations" tab in your RePEc Author Service profile, as there may be some citations waiting for confirmation.

    For technical questions regarding this item, or to correct its authors, title, abstract, bibliographic or download information, contact: Catherine Liu (email available below). General contact details of provider: http://www.elsevier.com/wps/find/journaldescription.cws_home/505572/description#description .

    Please note that corrections may take a couple of weeks to filter through the various RePEc services.

    IDEAS is a RePEc service. RePEc uses bibliographic data supplied by the respective publishers.