14.4 Heterogeneity & Stratification
Human populations are heterogeneous. Some kinds of heterogeneity affect how we understand malaria and what we should do, including who to target. To deal with heterogeneity in models, we will often need to segment a human population into sub-populations, or strata. When we talk about stratification, we mean it the narrow sense of segmenting a human population (i.e. not subdividing landscapes spatially3), because the model predictions made by creating strata that are more homogeneous should be more accurate. The guiding principle is that our analytics will should strive to be more accurate, and that we should thus identify and remove those sources of heterogeneity that would affect policy advice, whether it affects estimating the impact of interventions in the past or projecting those impacts into the future. We acknowledge that models are approximations, and that our approximations don’t have to be perfect. The goal is to find ways of propagating uncertainty that are good enough for the task at hand.
In malaria epidemiology, some kinds of endogenous heterogeneity could be built into the epidemiological state space. Other kinds of heterogeneity, including consistent differences in exposure, differences in care seeking and drug taking, and differences created by malaria control (e.g. net ownership or vaccination), usually require stratification. The decision about how to strike the right balance depends on the model and the purpose of a study.
The framework and supporting software offer a toolbox for stratification. It is designed to stratify populations in a principled way, so that we can understand how the heterogeneity affects transmission or outcomes that we care about, but we can also combine effects. We want to stratify populations by applying rules that split populations when the differences are large enough. (If we started with complex models, we might choose to join populations if the differences were small.) By so doing, we can compare the behaviors of models that differ from each other in only one way. If the differences are not too large, or if the differences in dynamical behaviors we care about are not too large, we might decide not to split the strata, and use the average. Because of the way models are encoded, it’s easy to build models that split the strata in multiple, independent ways.
14.4.1 Strata in the Ross model
As a simple example, consider a simple Ross-style model for infection with exposure and recovery (described in Section 10.1.5):
\[\frac{dX}{dt} = h (H-X)-r X\]
If exposure is heterogeneous, we could split this population into two strata and add subscripts (i.e., indexed by \(i \in \left\{1,2, \ldots \right\}\)):
\[\frac{dX_i}{dt} = k_i h (H_i -X_i)-r X_i\]
We hold the average FoI constant by constraining the values of \(k_i\):
\[\frac{\sum_i k_i H_i}{H} = 1\] Stratification is important if the differences are large. With two strata, it would not make sense to stratify if \(k_1 \approx k_2\), but if \(k_2 \gg k_1\) then it might change our expectations, or it might change what we recommend.
14.4.2 Frailties
We will introduce segmentation first through models of [Heterogeneous Exposure] to malaria, where we consider various sources of frailty – proportional differences in the average hazard rate for infection (\(k_i\), in the example above). These differences in exposure can arise because of age, house type, risky behaviors, other factors. Frailty that is attributable to location (e.g. proximity of home to aquatic habitats) can be dealt with by sub-dividing space into patches, a topic that is taken up in [Space] below and Spatial Dynamics. Depending on the size of the patches, some differences in average rates of exposure due to location can persist, and these could be dealt with by generic stratification into high vs. low exposure strata.
Some of the heterogeneous traits that we care about change dynamically, so we will also need to consider population flows among strata, which change the sizes of the strata. We would like to deal with these flows in a principled way. Bed net ownership and use are among the human behaviors that matters most for programs. In some cases, we will want to understand dynamic changes in bed net ownership, the patterns of use among those who own a net, personal protection, and community effects. Later, we show how to construct an example that describes all of these aspects of bednets.
Segmentation is what we need to build models of pharmaceutical interventions with waning effectiveness, such as mass vaccination. Among the most important factors in malaria is age. We have defined algorithms to model [Aging] and other demographic change, the loss of bednets, waning protection or changing housing quality.
14.4.3 Age
Immunity to malaria develops with age and exposure. The development of immunity is probably changing throughout life, so it makes sense to think of malaria epidemiology as ontogeny.
For systems described generically by the state space, \(\mathscr X\), the dynamics we care about have the form:
\[\frac{\partial {\mathscr X}(a,t)}{\partial a} + \frac{\partial {\mathscr X}(a,t)}{\partial t}\]
We might want to deal with malaria differently if we are studying malaria in cohorts. In a population where the FoI over time is \(h(t)\), we might want to follow a birth cohort, so we define \(h_d(a) = h(t-a)\) for all \(t>d\). We can then solve:
\[\frac{d{\mathscr X}}{d a} \] which produces states in cohorts as they age, \({\mathscr X}(a|h).\)
When we simulate malaria transmission dynamics in populations for policy, we will want to put a mesh on age and segment the population. The dynamics are define for age strata, where the FoI is defined differently for each age stratum:
\[\frac{d{\mathscr X}_a}{d t}\]
which produces age-dependent states over time, \({\mathscr X}_a(t|h).\)
Our algorithms should guarantee that the epidemiological states over time provide an accurate match for the epidemiological states over age.
In a broader sense, stratification is also about subdividing landscapes into a set of spatial domains that share relevant features in order to tailor interventions to context. That is a topic we take up in a separate book, ( Robust Analytics for Malaria Policy. ).↩︎