Subcritical endemic steady states in mathematical models for animal infections with incomplete immunity

Many classical mathematical models for animal infections assume that all in fected animals transmit the infection at the same rate, all are equally sus ceptible, and the course of the infection is the same in all animals. Howev er for some infections there is evidence that seropositives may still trans mit the infection, albeit at a lower rate. Animals can also experience more than one episode of the infection although those who have already experien ced it have a partial immune resistance. Animals who experience a second or subsequent period of infection may not necessarily exhibit clinical sympto ms. The main example discussed is bovine respiratory syncytial virus (BRSV) amongst cattle. We consider simple models with vaccination and homogeneous and proportional mixing between seropositives and seronegatives. We derive an expression for the basic reproduction number, R-o, and perform an equil ibrium and stability analysis. We find that it may be possible for there to be two endemic equilibria (one stable and one unstable) for R-o < 1 and in this case at R-o = 1 there is a backwards bifurcation of an unstable endem ic equilibrium from the infection-free equilibrium, Then the implications f or control strategies are considered. Finally applications to Aujesky's dis ease (pseudorabies virus) in pigs are discussed. (C) 2000 Elsevier Science Inc. All rights reserved.