Stability and Persistence in a Model for Bluetongue Dynamics

A model for the time evolution of bluetongue, a viral disease in sheep and cattle that is spread by midges as vectors, is formulated as a delay differential equation system of six equations. Midges are assumed to have a pre-adult stage of constant duration, and a general incubation period for bluetongue. A linear stability analysis leads to identification of a basic reproduction number that determines if the disease introduced at a low level dies out, or is uniformly weakly persistent in the midges. Stronger conditions sufficient for global stability of the disease free equilibrium are derived. The control reproduction numbers, which guide control strategies for midges, cattle or sheep, are determined in the special case in which the incubation period for midges is exponentially distributed. The possibility of backward bifurcation is briefly discussed as is an equilibrium situation in which the disease wipes out sheep populations that are introduced in small numbers.