Heterogeneous Biting • ramp.xds

In ramp.xds, the term heterogeneous biting is used to describe differences among population strata in the expected rate of exposure to infective mosquitoes. To be more rigorous, let $E$ denote the average daily entomological inoculation rate (dEIR) for a population with multiple strata, and $\xi_i$ the frailty term, then the dEIR for the $i^{th}$ stratum is: $E_i = \xi_i E.$ The implementation is part of a coherent model for blood feeding that serves at the interface between models of parasite/pathogen infections in humans (i.e. $\cal X$), and models of parasite infections in mosquitoes (i.e. $\cal YZ$). In this vignette, we introduce the concept of heterogeneous biting and its implementation in the blood feeding model using blood feeding search weights.

Heterogeneous blood feeding is a basic feature of malaria transmission, and an important aspect of Heterogeneous Transmission.

Definitions

Heterogeneous Biting is defined throughout the ramp.xds implementation and documentation as a difference in the relative biting rates for two strata that are otherwise identical. The implementation relies on two concepts:

blood feeding search weights or $\left\{\omega\right\}$
relative biting rates or $\left\{\xi\right\}$

The software deals mainly with the search weights because the denominators are changing, but it is useful to understand how search weights are related to relative biting rates.

Blood Feeding Search Weights

A flexible implementation is handled through the blood feeding model, which includes the the concepts of blood feeding search weights and availability. The search weights, $\left\{\omega\right\}$ , are a measure of how easy it is for mosquitoes to find and blood feed on a host.

We let $H_i$ denote the size of the $i^{th}$ population, where $H = \sum_i H_i.$ The total availability of humans for blood feeding is:

$W = \sum_i \omega_i H_i.$ Availability is used to compute the overall blood feeding rate for mosquitoes and the human fraction (human blood meals as a fraction of all blood meals). If we assign a biting weight to a stratum, then the fraction of bites received by that stratum is:

$\frac{\omega_i H_i}W.$ Example 1: For example, suppose that there are 200 people with a biting weight of 2.25 and 800 people with a biting weight of 1. In this model, the first stratum would get 36% of the total bites:

2.25*200/(2.25*(200) + 1*(800))
#> [1] 0.36

Relative Biting Rates

The relative biting rates are defined at the top.

The relative biting rates are constrained such that $\sum_i \xi_i \frac{H_i}H = 1$

Example 2:

For example, if 20% of the population gets bitten at a rate that is 80% higher than the population average, then the other 80% must get bitten (on average) at a rate that is 80% of the population average.

1.8*0.2+ 0.8*0.8
#> [1] 1

Relative biting rates are computed automatically from the blood feeding search weights, $\left\{\omega\right\},$ where

$\xi_i = \omega_i\frac{H}{W}.$

Example 3 computes the search weights from the biting weights

searchWts = c(2.25, 1)
Hi = c(200, 800)
H = sum(Hi) 
W = sum(searchWts*Hi)
xi = searchWts*H/W
xi 
#> [1] 1.8 0.8

sum(xi*Hi)/H
#> [1] 1

Implementation

In ramp.xds, the concept of human availability for blood feeding is also modified by time spent, and the algorithms were designed to deal with both changing denominators and search weights. In a patch, the available human population is a measure of the search weighted ambient population density. The infective bites in each patch are allocated among human population strata by taking a stratum’s availability divided by the total availability of all blood hosts. The blood feeding model outputs a vector of dEIR values for each stratum.

Relative biting rates are an important concept, and they are sometimes useful to compute, but they are not computed as part of the blood feeding model.