The Relative Contribution of Direct and Environmental Transmission Routes in Stochastic Avian Flu Epidemic Recurrence: An Approximate Analysis

  • May Anne Mata
  • Priscilla Greenwood
  • Rebecca Tyson
Special Issue: Mathematical Epidemiology


We present an analysis of an avian flu model that yields insight into the roles of different transmission routes in the recurrence of avian influenza epidemics. Recent modelling work suggests that the outbreak periodicity of the disease is mainly determined by the environmental transmission rate. This conclusion, however, is based on a modelling study that only considers a weak between-host transmission rate. We develop an approximate model for stochastic avian flu epidemics, which allows us to determine the relative contribution of environmental and direct transmission routes to the periodicity and intensity of outbreaks over the full range of plausible parameter values for transmission. Our approximate model reveals that epidemic recurrence is chiefly governed by the product of a rotation and a slowly varying standard Ornstein–Uhlenbeck process (i.e. mean-reverting process). The intrinsic frequency of the damped deterministic version of the system predicts the dominant period of outbreaks. We show that the typical periodicity of major avian flu outbreaks can be explained in terms of either or both types of transmission and that the typical amplitude of epidemics is highly sensitive to the direct transmission rate.


Avian influenza Disease transmission Recurrent epidemics Host pathogen model Stochastic SIR Sustained oscillations 



We gratefully acknowledge the BRAES institute at UBCO for supporting this work.

Author’s contributions MAM formulated the research problem, developed the methods of analysis, simulated and analysed the model, and drafted the manuscript; PEG and RCT evaluated the results and helped draft the manuscript. All authors gave final approval for publication.

Compliance with ethical standards

Conflict of interest

The authors declare that they have no conflict of interest.

Supplementary material


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Copyright information

© Society for Mathematical Biology 2018

Authors and Affiliations

  1. 1.Barber School of Arts and SciencesUniversity of British Columbia OkanaganKelownaCanada
  2. 2.Department of MathematicsUniversity of British ColumbiaVancouverCanada

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