IDEAS home Printed from https://ideas.repec.org/a/gam/jijerp/v7y2010i3p1186-1204d7528.html
   My bibliography  Save this article

Branching Processes: Their Role in Epidemiology

Author

Listed:
  • Christine Jacob

    (National Agricultural Research Institute, UR341, Department of Applied Mathematics and Informatics, F-78352 Jouy-en-Josas, France)

Abstract

Branching processes are stochastic individual-based processes leading consequently to a bottom-up approach. In addition, since the state variables are random integer variables (representing population sizes), the extinction occurs at random finite time on the extinction set, thus leading to fine and realistic predictions. Starting from the simplest and well-known single-type Bienaymé-Galton-Watson branching process that was used by several authors for approximating the beginning of an epidemic, we then present a general branching model with age and population dependent individual transitions. However contrary to the classical Bienaymé-Galton-Watson or asymptotically Bienaymé-Galton-Watson setting, where the asymptotic behavior of the process, as time tends to infinity, is well understood, the asymptotic behavior of this general process is a new question. Here we give some solutions for dealing with this problem depending on whether the initial population size is large or small, and whether the disease is rare or non-rare when the initial population size is large.

Suggested Citation

  • Christine Jacob, 2010. "Branching Processes: Their Role in Epidemiology," IJERPH, MDPI, vol. 7(3), pages 1-19, March.
    • Handle: RePEc:gam:jijerp:v:7:y:2010:i:3:p:1186-1204:d:7528
    as

    Download full text from publisher

    File URL: https://www.mdpi.com/1660-4601/7/3/1186/pdf
    Download Restriction: no
    File URL: https://www.mdpi.com/1660-4601/7/3/1186/
    Download Restriction: no
    ---><---

    References listed on IDEAS

    as
    1. Klebaner, Fima C., 1993. "Population-dependent branching processes with a threshold," Stochastic Processes and their Applications, Elsevier, vol. 46(1), pages 115-127, May.
    2. Christine Jacob & Pierre Magal, 2007. "Influence of Routine Slaughtering on the Evolution of BSE: Example of British and French Slaughterings," Risk Analysis, John Wiley & Sons, vol. 27(5), pages 1151-1167, October.
    3. J. A. P. Heesterbeek & K. Dietz, 1996. "The concept of Ro in epidemic theory," Statistica Neerlandica, Netherlands Society for Statistics and Operations Research, vol. 50(1), pages 89-110, March.
    4. Klebaner, F. C. & Nerman, O., 1994. "Autoregressive approximation in branching processes with a threshold," Stochastic Processes and their Applications, Elsevier, vol. 51(1), pages 1-7, June.
    Full references (including those not matched with items on IDEAS)

    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. Klebaner, F. C. & Liptser, R., 1999. "Moderate deviations for randomly perturbed dynamical systems," Stochastic Processes and their Applications, Elsevier, vol. 80(2), pages 157-176, April.
    2. Ramanan, Kavita & Zeitouni, Ofer, 1999. "The quasi-stationary distribution for small random perturbations of certain one-dimensional maps," Stochastic Processes and their Applications, Elsevier, vol. 84(1), pages 25-51, November.
    3. Högnäs, Göran, 1997. "On the quasi-stationary distribution of a stochastic Ricker model," Stochastic Processes and their Applications, Elsevier, vol. 70(2), pages 243-263, October.
    4. Eugenio Valdano & Davide Colombi & Chiara Poletto & Vittoria Colizza, 2023. "Epidemic graph diagrams as analytics for epidemic control in the data-rich era," Nature Communications, Nature, vol. 14(1), pages 1-11, December.
    5. Antoine Djogbenou & Christian Gouriéroux & Joann Jasiak & Paul Rilstone, 2022. "An Econometric Panel Data Model of the COVID-19 Pandemic," Journal of Statistical and Econometric Methods, SCIENPRESS Ltd, vol. 11(1), pages 1-3.
    6. András Schubert & Wolfgang Glänzel & Gábor Schubert, 2022. "Eponyms in science: famed or framed?," Scientometrics, Springer;Akadémiai Kiadó, vol. 127(3), pages 1199-1207, March.
    7. Carnehl, Christoph & Fukuda, Satoshi & Kos, Nenad, 2023. "Epidemics with behavior," Journal of Economic Theory, Elsevier, vol. 207(C).
    8. Imelda Trejo & Nicolas W Hengartner, 2022. "A modified Susceptible-Infected-Recovered model for observed under-reported incidence data," PLOS ONE, Public Library of Science, vol. 17(2), pages 1-23, February.
    9. Lin William Cong & Ke Tang & Bing Wang & Jingyuan Wang, 2021. "An AI-assisted Economic Model of Endogenous Mobility and Infectious Diseases: The Case of COVID-19 in the United States," Papers 2109.10009, arXiv.org.
    10. Gong, Jiangyue & Gujjula, Krishna Reddy & Ntaimo, Lewis, 2023. "An integrated chance constraints approach for optimal vaccination strategies under uncertainty for COVID-19," Socio-Economic Planning Sciences, Elsevier, vol. 87(PA).
    11. Rezapour, Shabnam & Baghaian, Atefe & Naderi, Nazanin & Sarmiento, Juan P., 2023. "Infection transmission and prevention in metropolises with heterogeneous and dynamic populations," European Journal of Operational Research, Elsevier, vol. 304(1), pages 113-138.
    12. Williams, Noah, 2004. "Small noise asymptotics for a stochastic growth model," Journal of Economic Theory, Elsevier, vol. 119(2), pages 271-298, December.
    13. Ioannis Kioutsioukis & Nikolaos I. Stilianakis, 2021. "On the Transmission Dynamics of SARS-CoV-2 in a Temperate Climate," IJERPH, MDPI, vol. 18(4), pages 1-17, February.
    14. Besjana MEMA & Sabina TOSUNI, 2024. "Implementation of mathematics model in public health: Albanian case study," Smart Cities and Regional Development (SCRD) Journal, Smart-EDU Hub, Faculty of Public Administration, National University of Political Studies & Public Administration, vol. 8(3), pages 41-54, April.
    15. Florin Avram & Rim Adenane & Lasko Basnarkov, 2024. "Some Probabilistic Interpretations Related to the Next-Generation Matrix Theory: A Review with Examples," Mathematics, MDPI, vol. 12(15), pages 1-16, August.
    16. Fabricius, Gabriel & Maltz, Alberto, 2020. "Exploring the threshold of epidemic spreading for a stochastic SIR model with local and global contacts," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 540(C).
    17. Chang, Joseph T. & Kaplan, Edward H., 2023. "Modeling local coronavirus outbreaks," European Journal of Operational Research, Elsevier, vol. 304(1), pages 57-68.

    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:gam:jijerp:v:7:y:2010:i:3:p:1186-1204:d:7528. 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: MDPI Indexing Manager (email available below). General contact details of provider: https://www.mdpi.com .

    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.