Multiple endemic states in age-structured $SIR$ epidemic models - Andrea Franceschetti; Andrea Pugliese; Dimitri Breda (Mathematical Biosciences and Engineering)
$SIR$ age-structured models are very often used as a basic model ofepidemic spread. Yet, their behaviour, under generic assumptions on contactrates between different age classes, is not completely known, and, in the mostdetailed analysis so far, Inaba (1990) was able to prove uniqueness of the endemicequilibrium only under a rather restrictive condition. &nbsp Here, we show anexample in the form of a $3 × 3$ contact matrix in which multiplenon-trivial steady states exist. This instance of non-uniqueness of positiveequilibria differs from most existing ones for epidemic models, since it arisesnot from a backward transcritical bifurcation at the disease free equilibrium,but through two saddle-node bifurcations of the positive equilibrium.The dynamical behaviour of the model is analysed numerically around the range wheremultiple endemic equilibria exist; many other features are shown to occur, from coexistence of multiple attractive periodic solutions, some with extremely long period, to quasi-periodic and chaotic attractors. &nbsp It is also shown that, if thecontact rates are in the form of a $2 × 2$ WAIFW matrix, uniqueness of non-trivial steady states always holds, so that 3 is theminimum dimension of the contact matrix to allow for multiple endemic equilibria.