An Age Structured S.I. Epidemic Problem in a Heterogeneous Environment

摘要

In this work we analyze an S.I. epidemic model with age dependence and a heterogeneous spatial structure. Our main motivation rests on a tentative qualitative comparison in Naulin between continuous models and matrix population models, i.e., discrete models, with age and space structures of the S.E.I.R. form, used in modelling the propagation of rabies in red fox populations (Vulpes vulpes) in Suppo et al. [19]; see also Naulin [17]. A derivation of continuous S.E.I.R.models from the corresponding matrix models is found in Naulin [18], along the lines of Cushing [7] and Gurtin [10]. Then, the first question to address is a global (in time) existence problem. While this does not pose any problem for matrix population models because they are set in an explicit form, X(t + 1) = P(t, X(t))X(t) where P(t,X) is the reproduction matrix (Caswell [5]), this is far from being obvious for continuous models with age dependence because they are both non linear and non local. A further feature is that population fluxes are age dependent and may contain a transport term, as it is the case for territorial populations where juveniles must find their own territory to settle down and reproduce ([17]), [19]. This yields new age dependent mathematical problems much more structured than the one usullly considered (see the books of Webb [20], Iannelli [11] and Anita [3]).