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

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 ii drives out Strain jj 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 ii dominating (but not driving out Strain jj), whenever R0i>R0j>1R0i>R0j>1.