The time evolution of the influenza A virus is linked to a nonfixed landscape driven by interactions between hosts and competing influenza strains. Herd-immunity, cross-immunity, and age-structure are among the factors that have been shown to support strain coexistence and/or disease oscillations. In this study, we put two influenza strains under various levels of (interference) competition. We establish that cross-immunity and host isolation lead to periodic epidemic outbreaks (sustained oscillations) in this multistrain system. We compute the isolation reproductive number for each strain ($\Re_i$) independently, as well as for the full system ($\Re_q$), and show that when $\Re_q < 1$, both strains die out. Subthreshold coexistence driven by cross-immunity is possible even when the isolation reproductive number of one strain is below 1. Conditions that guarantee a winning type or coexistence are established in general. Oscillatory coexistence is established via Hopf bifurcation theory and confirmed via numerical simulations.

MSC codes

  1. 92D30
  2. 92D25
  3. 34C25
  4. 34C60


  1. influenza
  2. multiple strains
  3. cross-immunity
  4. isolation
  5. stability
  6. bifurcation
  7. oscillations
  8. coexistence

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Information & Authors


Published In

cover image SIAM Journal on Applied Mathematics
SIAM Journal on Applied Mathematics
Pages: 964 - 982
ISSN (online): 1095-712X


Published online: 31 July 2006

MSC codes

  1. 92D30
  2. 92D25
  3. 34C25
  4. 34C60


  1. influenza
  2. multiple strains
  3. cross-immunity
  4. isolation
  5. stability
  6. bifurcation
  7. oscillations
  8. coexistence



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