# American Institute of Mathematical Sciences

2006, 3(3): 445-458. doi: 10.3934/mbe.2006.3.445

## Stochastic epidemic models with a backward bifurcation

 1 Department of Mathematics and Statistics, Texas Tech University, Lubbock, TX 79409-1042, United States 2 Department of Mathematics and Statistics, University of Victoria, Victoria B.C., Canada V8W 3P4

Received  March 2006 Revised  April 2006 Published  May 2006

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 $N\geq 1000$, 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.
Citation: Linda J. S. Allen, P. van den Driessche. Stochastic epidemic models with a backward bifurcation. Mathematical Biosciences & Engineering, 2006, 3 (3) : 445-458. doi: 10.3934/mbe.2006.3.445
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