Stochastic epidemic models with a backward bifurcation

Linda J.S. Allen, P. Van Den Driessche

Research output: Contribution to journalArticlepeer-review

27 Scopus citations


Two new stochastic epidemic models, a continuous-time Markov chain model and a stochastic differential equation model, are formulated. These are based on a deterministic model that includes vaccination and is applicable to pertussis. For some parameter values, the deterministic model exhibits a backward bifurcation if the vaccine is imperfect. Thus a region of bistability exists in a subset of parameter space. The dynamics of the stochastic epidemic models are investigated in this region of bistability, and compared with those of the deterministic model. In this region the probability distribution associated with the infective population exhibits bimodality with one mode at the disease-free equilibrium and the other at the larger endemic equilibrium. For population sizes N ≥ 1000, the deterministic and stochastic models agree, but for small population sizes the stochastic models indicate that the backward bifurcation may have little effect on the disease dynamics.

Original languageEnglish
Pages (from-to)445-458
Number of pages14
JournalMathematical Biosciences and Engineering
Issue number3
StatePublished - Jul 2006


  • Bistability
  • Continuous-time Markov chain
  • Epidemic model
  • Stochastic differential equations
  • Vaccination


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