Aging & Cohorts

Galerkin Methods

Let $\mathcal{X}$ 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:

$$\frac{\partial \mathcal{X}(a,t)}{\partial a}+\frac{\partial \mathcal{X}(a,t)}{\partial t}=F_{\mathcal{X}}(\mathcal{X}(a,t)|h(a,t))$$

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:

$$\mathcal{X}(0,t)=B_{\mathcal{X}}(\mathcal{X}\left(t\right)|h(0,t)).$$

Solving equations over age and time is cumbersome, so many analyses simply integrate over age, and focus on the temporal dynamics. We get solutions $\mathcal{X}(t|h)$ by solving:

$$\frac{\partial \mathcal{X}\mathcal{(}\mathcal{t}\mathcal{)}}{dt}=F_{\mathcal{X}}(\mathcal{X}\mathcal{(}\mathcal{t}\mathcal{)}|h(t))$$

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 $\mathcal{X}(a|h)$ by solving:

$$\frac{d\mathcal{X}\left(\mathcal{a}\right)}{da}=F_{\mathcal{X}}(\mathcal{X}\mathcal{(}\mathcal{a}\mathcal{)},h)$$

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:

$$\{0,{\ell }_1,{\ell }_2,...,{\ell }_n,\mathrm{\infty }\}$$

which defines a set of age classes, either:

• The youngest cohort ($i=1$), $a\in [0,{\ell }_1)$, or

• The oldest cohort ($i=n+1$), $[{\ell }_n,\mathrm{\infty })$, or

• for all the other cohorts ($i\in 2,\dots n$), $a\in [{\ell }_{i-1},{\ell }_i)$

For each age class, we have a set of state spaces:

$${\mathcal{X}}_i(t)$$ The dynamics within an age class are defined by $F_{\mathcal{X}}$, but they are linked by the aging operator, $\mathcal{A}$, such that:

$$\frac{\partial {\mathcal{X}}_{\mathcal{i}}\mathcal{(}\mathcal{t}\mathcal{)}}{dt}=F_{\mathcal{X}}(\mathcal{X}\mathcal{(}\mathcal{t}\mathcal{)},h)+\mathcal{A}(\mathcal{X},\overrightarrow{\ell })$$

To evaluate the resulting system as Ideally, the variables for the $i^{th}$ age stratum would closely approximate the system that average for :

$${\mathcal{X}}_{\mathcal{i}}(h(t))\approx {\int }_{{\ell }_{i-1}}^{{\ell }_i}\mathcal{X}\mathcal{(}\mathcal{a}|\mathcal{h}\mathcal{)}da$$

Example