Multiple endemic states in agestructured $SIR$ epidemic models

1.
Dept. Mathematics, Università di Trento, Via Sommarive 14, 38123 Povo (TN)

2.
Department of Mathematics and Computer Science, University of Udine, via delle Scienze 206, I33100 Udine

Received:
01 June 2011
Accepted:
29 June 2018
Published:
01 July 2012


MSC :
Primary: 92D30; Secondary: 45G15, 65P30.


$SIR$ agestructured models are very often used as a basic model of
epidemic spread. Yet, their behaviour, under generic assumptions on contact
rates between different age classes, is not completely known, and, in the most
detailed analysis so far, Inaba (1990) was able to prove uniqueness of the endemic
equilibrium only under a rather restrictive condition.
Here, we show an
example in the form of a $3 \times 3$ contact matrix in which multiple
nontrivial steady states exist. This instance of nonuniqueness of positive
equilibria differs from most existing ones for epidemic models, since it arises
not from a backward transcritical bifurcation at the disease free equilibrium,
but through two saddlenode bifurcations of the positive equilibrium.
The dynamical behaviour of the model is analysed numerically around the range where
multiple endemic equilibria exist; many other features are shown to occur, from coexistence of multiple attractive periodic solutions, some with extremely long period, to quasiperiodic and chaotic attractors.
It is also shown that, if the
contact rates are in the form of a $2 \times 2$ WAIFW matrix, uniqueness of nontrivial steady states always holds, so that 3 is the
minimum dimension of the contact matrix to allow for multiple endemic equilibria.
Citation: Andrea Franceschetti, Andrea Pugliese, Dimitri Breda. Multiple endemic states in agestructured $SIR$ epidemic models[J]. Mathematical Biosciences and Engineering, 2012, 9(3): 577599. doi: 10.3934/mbe.2012.9.577

Abstract
$SIR$ agestructured models are very often used as a basic model of
epidemic spread. Yet, their behaviour, under generic assumptions on contact
rates between different age classes, is not completely known, and, in the most
detailed analysis so far, Inaba (1990) was able to prove uniqueness of the endemic
equilibrium only under a rather restrictive condition.
Here, we show an
example in the form of a $3 \times 3$ contact matrix in which multiple
nontrivial steady states exist. This instance of nonuniqueness of positive
equilibria differs from most existing ones for epidemic models, since it arises
not from a backward transcritical bifurcation at the disease free equilibrium,
but through two saddlenode bifurcations of the positive equilibrium.
The dynamical behaviour of the model is analysed numerically around the range where
multiple endemic equilibria exist; many other features are shown to occur, from coexistence of multiple attractive periodic solutions, some with extremely long period, to quasiperiodic and chaotic attractors.
It is also shown that, if the
contact rates are in the form of a $2 \times 2$ WAIFW matrix, uniqueness of nontrivial steady states always holds, so that 3 is the
minimum dimension of the contact matrix to allow for multiple endemic equilibria.
References



