Compartmental Models in Epidemiology - The Influence of Age: Age-structured Models

The Influence of Age: Age-structured Models

Maybe "the most specific parameter of biological system is the age" (M. Iannelli), and, especially for some infectious diseases, it has a deep influence on the dynamics of its spreading in a population. Many of the parameters we have seen may depend on age, and especially the contact rate, which summarizes the 'infectious effectiveness' of contacts between susceptible and infectious subjects. This effectiveness has, thus, to take into account both the age of the infectious and the age of the susceptible. Epidemic models modeling the age structure of a population are very complex. In fact, they are infinite dimensional model since we have to deal with density through the ages of the epidemic classes (to limit ourselves to the susceptible-infectious-removed scheme) such that:

(where is the maximum admissible age)and their dynamics is not described, as one might think, by "simple" partial differential equations, but by integro-differential equations:

where:

is the force of infection, which, of course, will depend, though the contact kernel on the interactions between the ages.

Complexity is added by the initial conditions for newborns (i.e. for a=0), that are straightforward for infectious and removed:

but that are nonlocal for the density of susceptible newborns:

where are the fertilities of the adults.

Moreover, defining now the density of the total population one obtains:

In the simplest case of equal fertilities in the three epidemic classes, we have that in oder to have demographic equilibrium the following necessary and sufficient condition linking the fertility with the mortality must hold:

and the demographic equilibrium is

automatically ensuring the existence of the disease-free solution:

A basic reproduction number can be calculated as the spectral radius of an appropriate functional operator.