Cross-immunity-induced backward bifurcation for a model of transmission dynamics of two strains of influenza

S. M. Garba, M. A. Safi, A. B. Gumel

Research output: Contribution to journalArticle

13 Scopus citations


A new deterministic model for the transmission dynamics of two strains of influenza is designed and used to qualitatively assess the role of cross-immunity on the transmission process. It is shown that incomplete cross-immunity could induce the phenomenon of backward bifurcation when the associated reproduction number is less than unity. The model undergoes competitive exclusion (where Strain i drives out Strain j to extinction whenever R0i>1>R0j;i,j=1,2,i≠j). For the case where infection with one strain confers complete immunity against infection with the other strain, it is shown that the disease-free equilibrium of the model is globally-asymptotically stable whenever the reproduction number is less than unity. In the absence of cross-immunity, the model can have a continuum of co-existence endemic equilibria (which is shown to be globally-asymptotically stable for a special case). When infection with one strain confers incomplete immunity against the other, numerical simulations of the model show that the two strains co-exist, with Strain i dominating (but not driving out Strain j), whenever R0i>R0j>1.

Original languageEnglish (US)
Pages (from-to)1384-1403
Number of pages20
JournalNonlinear Analysis: Real World Applications
Issue number3
StatePublished - Jun 1 2013



  • Co-existence
  • Cross-immunity
  • Equilibria
  • Multiple strains
  • Stability

ASJC Scopus subject areas

  • Analysis
  • Engineering(all)
  • Economics, Econometrics and Finance(all)
  • Computational Mathematics
  • Applied Mathematics

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