Journal of Mathematical Biology

, Volume 76, Issue 4, pp 911–944 | Cite as

Birth/birth-death processes and their computable transition probabilities with biological applications

  • Lam Si Tung HoEmail author
  • Jason Xu
  • Forrest W. Crawford
  • Vladimir N. Minin
  • Marc A. Suchard


Birth-death processes track the size of a univariate population, but many biological systems involve interaction between populations, necessitating models for two or more populations simultaneously. A lack of efficient methods for evaluating finite-time transition probabilities of bivariate processes, however, has restricted statistical inference in these models. Researchers rely on computationally expensive methods such as matrix exponentiation or Monte Carlo approximation, restricting likelihood-based inference to small systems, or indirect methods such as approximate Bayesian computation. In this paper, we introduce the birth/birth-death process, a tractable bivariate extension of the birth-death process, where rates are allowed to be nonlinear. We develop an efficient algorithm to calculate its transition probabilities using a continued fraction representation of their Laplace transforms. Next, we identify several exemplary models arising in molecular epidemiology, macro-parasite evolution, and infectious disease modeling that fall within this class, and demonstrate advantages of our proposed method over existing approaches to inference in these models. Notably, the ubiquitous stochastic susceptible-infectious-removed (SIR) model falls within this class, and we emphasize that computable transition probabilities newly enable direct inference of parameters in the SIR model. We also propose a very fast method for approximating the transition probabilities under the SIR model via a novel branching process simplification, and compare it to the continued fraction representation method with application to the 17th century plague in Eyam. Although the two methods produce similar maximum a posteriori estimates, the branching process approximation fails to capture the correlation structure in the joint posterior distribution.


Stochastic models Birth-death process Infectious disease SIR model Transition probabilities 

Mathematics Subject Classification

60J27 92D30 62F15 

Supplementary material


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Copyright information

© Springer-Verlag GmbH Germany 2017

Authors and Affiliations

  • Lam Si Tung Ho
    • 1
    Email author
  • Jason Xu
    • 2
  • Forrest W. Crawford
    • 3
  • Vladimir N. Minin
    • 4
  • Marc A. Suchard
    • 5
  1. 1.Department of BiostatisticsUniversity of California, Los AngelesLos AngelesUSA
  2. 2.Department of BiomathematicsUniversity of California, Los AngelesLos AngelesUSA
  3. 3.Department of BiostatisticsYale UniversityNew HavenUSA
  4. 4.Departments of Statistics and BiologyUniversity of WashingtonSeattleUSA
  5. 5.Departments of Biomathematics, Biostatistics and Human GeneticsUniversity of California, Los AngelesLos AngelesUSA

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