The HTC Matrix
Infectious Days Spent
To understand parasite dispersal by humans, we need the HTC matrix
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This follows the notation in [1]
We define the following terms:
\(D\) is the HTC for each population stratum, a vector of length \(N_h\)
\(b\) is the probability of infection per infectious bite for each population stratum, a vector of length \(N_h\)
Time at risk \((\Psi)\) is a \(N_p \times N_h\) matrix:
\(H\) is human population density, a vector of length \(N_h\)
\(w\) is a vector of biting search weights of length \(N_h\)
Availability \((W)\) is a vector of length \(N_p\): \[W = \Psi \cdot w H\]
The biting distribution matrix: \[\beta = \left<w \right> \cdot \Psi^T \cdot \left< \frac{1}{W} \right>\]
The EIR is \[\beta \cdot fqZ = \left<w \right> \cdot \Psi^T \cdot \left< \frac{fqZ}{W} \right>\]
HTC
The HTC matrix \([D]\) is a \(N_p \times N_p\) matrix that describes how humans allocate time at risk among the patches, from [1]:
\[[D] = \left< W \right> \cdot \beta^T \cdot \left< bDH \right> \cdot \beta\]
or equivalently:
\[[D] = \Psi \cdot \left<wDHb\right> \cdot \beta\]
We can understand \([D]\) as the product of two matrices:
\(\beta\) is the fraction of the bites in each patch that are received by a single member of each stratum:
and days spent fully infectious in each patch (weighted) by each stratum:
\[\Psi \cdot \left<wDHb \right>\]
So it measures the number of fully infectious days spent by hosts in each patch from an infective bite in each patch:
\[[D] = \Psi \cdot \left<wDHb\right> \cdot \beta\]