Stochastic epidemic models with a backward bifurcation - Linda J. S. Allen; P. van den Driessche (Mathematical Biosciences and Engineering)
Two new stochastic epidemic models, a continuous-time Markov chainmodel and a stochastic differential equation model, are formulated. These arebased on a deterministic model that includes vaccination and is applicableto pertussis. For some parameter values, the deterministic model exhibitsa backward bifurcation if the vaccine is imperfect. Thusa region of bistability exists in a subset of parameter space.The dynamics of the stochastic epidemic models are investigated in this regionof bistability, and compared with those of the deterministic model. In this region the probabilitydistribution associated with the infective population exhibits bimodality withone mode at the disease-free equilibrium and the other at the larger endemicequilibrium. For population sizes $N≥ 1000$, the deterministic and stochastic models agree,butfor small population sizes the stochastic models indicate that the backward bifurcation may have little effect on the disease dynamics.