History of Exposure

Estimating the annual EIR from a Prevalence Time Series

Author

David L Smith,…

Robust Estimation

We start with a time-series describing observed malaria prevalence, \(\hat x (t).\) To estimate malaria exposure, the daily EIR over time, \(E(t)\), we have developed a model-based estimation procedure. Each model includes:

  • A fully calibrated dynamic model for parasite infection in humans and immunity, \(\cal X\), including assumptions about drug taking;

  • An assumption about exposure while traveling, \(\delta(t),\) and the fraction of time spent traveling \(\xi\)

  • An assumption about exposure frailties by age, \(\omega(a).\)

  • A set of functions to describing the relation between the state space and prevalence, \(x,\) by various diagnostics: \[x = F_x \left(\cal X\right)\]

Decomposition

For any model, \(\cal X,\) we can fit functions describing the history of exposure, that is the product of three estimated quantities:

\[E(t) = \bar E \; T(t) \; S(t)\]

where:

  • \(\bar E\) - the mean entomological inoculation rate for the whole period.

  • \(T(t)\) - we use splines to describe a multiplicative factor to characterize inter-annual variability in transmission.

  • \(S(t)\) - we have developed functions that describe the seasonal pattern, including the phase and amplitude; we use the index of dispersal to characterize it.

In the simplest case, we solve:

\[\frac{d {\cal X}}{dt} = F_{\cal X}\left({\cal X} \; | \; E_d\left(t\right) (1-\xi) + \xi \delta\left(t\right) \right),\] to get predicted values of the parasite rate at the points in time when there are observations \[x(\hat t) = F_x \left({\cal X}\left(\hat t\right)\right),\] and for some objective function describing the goodness of fit, \[{\cal O}\left(x, \hat x \right).\]

The estimation procedure involves maximizing the GoF:

\[\max \left( {\cal O} \left(\hat x_{\hat t}, x\left(\hat t \;| \bar E, T, S \right) \right)\right)\]

Cohort Dynamics

Using that same model, we can translate \(E(t)\) into a model for exposure for a human cohort born on day \(d\) as it ages – using the transformation \(t=a+d:\)

\[E_d(a) = \omega(a) \; E(t-d)\]

Using functions from ramp.work and ramp.xds, we can then predict the PfPR using the xde_cohort functions to compute the PfPR at any age for a cohort born on day \(d\):

\[\frac{d {\cal X}}{da} = F_{\cal X}\left({\cal X} \; | \; E_d\left(a\right) (1-\xi) + \xi \delta\left(a\right) \right),\]

The algorithms depend on a model, \(\cal X,\) that includes information about care seeking and drug taking. Each model must supply The predictive algorithm can also be modified to consider effects of various study designs.