## Age structured models of mathematical epidemiology.

##### Abstract

We consider a mathematical model which describes the dynamics for the spread of a directly
transmitted disease in an isolated population with age structure, in an invariant habitat, where
all individuals have a finite life-span, that is, the maximum age is finite, hence the mortality
is unbounded. We assume that infected individuals do not recover permanently, meaning that
these diseases do not convey immunity (these could be: common cold, influenza, gonorrhoea)
and the infection can be transmitted horizontally as well as vertically from adult individuals to
their newborns. The model consists of a nonlinear and nonlocal system of equations of hyperbolic
type. Note that the above-mentioned model has been already analysed by many authors who
assumed a constant total population. With this assumption they considered the ratios of the
density and the stable age profile of the population, see [16, 31]. In this way they were able to
eliminate the unbounded death rate from the model, making it easier to analyse by means of
the semigroup techniques. In this work we do not make such an assumption except for the error
estimates in the asymptotic analysis of a singularly perturbed problem where we assume that the
net reproduction rate R ≤ 1.
For certain particular age-dependent constitutive forms of the force of infection term, solvability
of the above-mentioned age-structured epidemic model is proven. In the intercohort case, we
use the semigroup theory to prove that the problem is well-posed in a suitable population state
space of Lebesgue integrable vector valued functions and has a unique classical solution which
is positive, global in time and has the property of continuous dependence on the initial data.
Further, we prove, under additional regularity conditions (composed of specific assumptions and
compatibility conditions at the origin), that the solution is smooth. In the intracohort case,
we have to consider a suitable population state space of bounded vector valued functions on
which the (unbounded) population operator cannot generate a strongly continuous semigroup
which, therefore, is not suitable for semigroup techniques–any strongly continuous semigroup
on the space of bounded vector valued functions is uniformly continuous, see [6, Theorem 3.6].
Since, for a finite life-span of the population, the space of bounded vector valued functions is a
subspace densely and continuously embedded in the state space of Lebesgue integrable vector
valued functions, thus we can restrict the analysis of the intercohort case to the above-mentioned
space of bounded vector valued functions. We prove that this state space is invariant under the
action of the strongly continuous semigroup generated by the (unbounded) population operator
on the state space of Lebesgue integrable vector valued functions. Further, we prove the existence
and uniqueness of a mild solution to the problem.
In general, different time scales can be identified in age-structured epidemiological models. In
fact, if the disease is not terminal, the process of getting sick and recovering is much faster than a typical demographical process. In this work, we consider the case where recovering is much
faster than getting sick, giving birth and death. We consider a convenient approach that carries
out a preliminary theoretical analysis of the model and, in particular, identifies time scales of it.
Typically this allows separation of scales and aggregation of variables through asymptotic analysis
based on the Chapman-Enskog procedure, to arrive at reduced models which preserve essential
features of the original dynamics being at the same time easier to analyse.