Aging & Cohorts
Galerkin Methods
Let \(\cal X\) denote the state space defining a model family for human malaria infections. We can think of any model family for the dynamical component defining human infections and immunity as a set of PDEs with the following form:
\[\frac{\partial {{\cal X}(a,t)}}{\partial a} + \frac{\partial {{\cal X}(a,t)}}{\partial t} = F_{\cal X}\left({\cal X}(a,t)\; | \; h(a,t) \right)\]
where \(a\) denotes age, and \(t\) denotes time, and \(h(a,t)\) is the daily FoI, which we treat as an input.
The boundary condition (i.e. the state of a birth cohort) could depend on the states. For example, we might want to consider maternal protection in newborns as a function of the immune status of their mothers:
\[{\cal X}(0, t) = B_{\cal X} \left( {\cal X} \left( t \right) \; | \; h \left( 0, t \right) \right).\]
Solving equations over age and time is cumbersome, so many analyses simply integrate over age, and focus on the temporal dynamics. We get solutions \({\cal X}(t|h)\) by solving:
\[\frac{\partial {\cal X(t)}}{d t} = F_{\cal X}\left({\cal X(t)}| h (t) \right)\]
To understand the patterns wrt age, we could ignore time and focus on dynamics of a cohort born on day, \(d\). We redefine the FoI \(h_d(a)\) defined such that \(a + d = t\). We set initial conditions for a cohort, and we get solutions \({\cal X}(a|h)\) by solving:
\[\frac{d {\cal X\left(a\right)}}{d a} = F_{\cal X}\left({\cal X(a)}, h \right)\]
Computational Galerkin Methods
We seek algorithms that allow us to approximate the full dynamics simultaneously dealing with age and time. The idea is to stratify the population into \(n + 1\) discrete age classes by defining a mesh:
\[\left\{0, \ell_1, \ell_2, ..., \ell_n, \infty\right\}\]
which defines a set of age classes, either:
The youngest cohort (\(i=1\)), \(a\in[0, \ell_1)\), or
The oldest cohort (\(i=n+1\)), \([\ell_n, \infty)\), or
for all the other cohorts (\(i \in 2,\ldots n\)), \(a \in[\ell_{i-1}, \ell_i)\)
For each age class, we have a set of state spaces:
\[{\cal X}_i(t)\] The dynamics within an age class are defined by \(F_{\cal X}\), but they are linked by the aging operator, \(\cal A\), such that:
\[\frac{\partial {\cal X_i(t)}}{d t} = F_{\cal X}\left({\cal X(t)}, h \right) + {\cal A} \left({\cal X}, \vec \ell\right)\]
To evaluate the resulting system as Ideally, the variables for the \(i^{th}\) age stratum would closely approximate the system that average for :
\[{\cal X_i}(h(t)) \approx \int_{\ell_{i - 1}}^{\ell_i} {\cal X(a|h)}\; d a\]