ramp.demog

Demography

Demographic functions supported include:

• Aging - set up and solve models that stratify a population by age

• Age specific mortality - to develop models for age-specific mortality using data describing human populations by age

• Births - set up and solve models with a births; in age-structured models, into the class that includes newborns

• Migration - defined here to include a change of residence, including seasonal labor migration and some forms of nomadism

Orthogonal Stratification

To support development of models for malaria policy, we have also developed a set of utilities to stratify a population along other traits. To set up and solve models with dynamical strata, we have implemented a method we call principled stratification that defines a subset of all possible models. In a nutshell, we create one matrix for each process. For example:

1. A matrix describing residence and time spent;
2. A matrix describing migration among patches;
3. A matrix describing aging among age classes

In the set up, each stratum belongs to a residence and time spent matrix. In the next step, we develop a migration matrix, where every resident of a patch has the same probability of migrating to every other patch. Next, we split each stratum by age; every daughter stratum inherits the features of its parent stratum. The total number of strata is the product of the number of substrata for each trait. In this case, migration does not split so we have the product of the number of strata along a time spent vector.

Setup thus outputs a generalized linear matrix dHdt describing demography and dynamic transitions among strata. We also have a matrix dAdt that describes aging (for tracking variables).

We note that it is possible to formulate and run a model with any matrix, Hmatrix, so long as it has the right shape.

For example, suppose we have a residence vector $1, 1, 2, 3$ on $3$ patches with a migration matrix $M$ and the residence matrix $J_r$, and we have two age classes, $1,2$ with an aging matrix $A$. There are a total of $8$ strata, but $M$ is a $3 \times 3$ matrix and $A$ is a $2 \times 2$ matrix. We formulate the membership matrix for aging, $J_a$ and full matrix by taking:

$$J_a^T \cdot A \cdot J_a + J_r^T \cdot M \cdot J_ r$$

At some point, ramp.xds made the decision to compute the derivative dHdt for human population density. In compartment models, a population is sub-divided into $N$ mutually exclusive and collectively exhaustive states (MECE), but since the state variables describe densities and they sum up to $H,$ only $N-1$ of the state variables need to be computed. We also adopted the convention that the compartment that would be omitted is the state with newborns.