In `exDE,`

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
`exDE`

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 `exDE,`

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.