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

Abstract

A new deterministic model for the transmission dynamics of two strains of in-

uenza is designed and used to qualitatively assess the role of cross-immunity on the transmission process. It is shown that incomplete cross-immunity could in- duce 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 im- munity against infection with the other strain, it is shown that the disease-free equilibrium of the model is globally-asymptotically stable whenever the repro- duction 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.