Open AccessStochastic epidemic models with a backward bifurcation
Author(s) -
Linda J. S. Allen,
P. van den Driessche
Publication year - 2006
Publication title -
mathematical biosciences and engineering
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 0.451
H-Index - 45
eISSN - 1551-0018
pISSN - 1547-1063
DOI - 10.3934/mbe.2006.3.445
Subject(s) - stochastic modelling , mathematics , bistability , bimodality , population , markov chain , statistical physics , bifurcation , epidemic model , stochastic differential equation , population model , parameter space , nonlinear system , statistics , physics , demography , quantum mechanics , galaxy , sociology
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.