Competitive exclusion in a vector–host epidemic model with distributed delay†

A multi-strain model of a vector-borne disease with distributed delay in the vector and the host is investigated. It is shown that if the reproduction number of the model ℛ(0) < 1, the unique disease-free equilibrium is globally asymptotically stable. Without loss of generality, strain one is assumed to have the largest reproduction number. In this case, the dominance equilibrium of strain one is shown to be locally stable. The basic reproduction number for a strain i (ℛ(0)(i)) is written as a product of the reproduction number of the vector (ℛ(v)(i)) and the reproduction number of the host (ℛ(i)(h)), i.e. ℛ(i)(0) = ℛ(i)(h)ℛ(v)(i). The competitive exclusion principle is derived under the somewhat stronger condition that if strain one maximizes both the reproduction number of the host ℛ(h)(i) < ℛ(h)(1), i ≠ 1 and the reproduction number of the vector ℛ(v)(i) < ℛ(v)(1), i ≠ 1, strain one dominance equilibrium is globally asymptotically stable.