A systematic review of mechanistic models used to study avian influenza virus transmission and control

Table 4 Estimates of spatial kernel parameters used in between-farm transmission models of highly pathogenic avian influenza.

Spatial kernel \(K\left({d}_{ij}\right)\)	Reference	Parameters
Pareto distribution \(\left\{\begin{array}{c}1, 0\le {d}_{ij}<{x}_{min}\\ {\left(\frac{{x}_{min}}{{d}_{ij}}\right)}^{\alpha +1}, {d}_{ij}\ge {x}_{min}\end{array}\right.\)	[46, 47]	\({x}_{min}=0.1 \mathrm{km}\)	\(\alpha =-0.358 \left(-0.666,-0.159\right)\)
	[46, 47]	\({x}_{min}=0.1 \mathrm{km}\)	\(\alpha =0.0136\) \(\left(-0.122, 0.143\right)\)
Power-law \(1-{e}^{-{\left(\frac{\delta }{{d}_{ij}}\right)}^{\rho }}\)	[54]	\(\delta =1.529\times {10}^{-5} \mathrm{km}\)	\(\rho =2.2\)
	[55]	\(\delta =9.334\times {10}^{-5} \mathrm{km}\)	\(\rho =2.5\)
	[50]	\(\delta =5.899\times {10}^{-5} \mathrm{km}\)	\(\rho =2.009\)
Logistic expression \(\frac{1}{1+{\left(\frac{{d}_{ij}}{{r}_{0}}\right)}^{\alpha }}\)	[71]; [74]	\({r}_{0}=1.9 \mathrm{km}\) \(\left(1.1, 2.9\right)\)	\(\alpha =2.1 \left(1.8, 2.4\right)\)
	[67]	\({r}_{0}=3.160 \mathrm{km}\) \(\left(1.770, 4.549\right)\)	\(\alpha =2.192 \left(1.894, 2.490\right)\)
	[56]	\({r}_{0}=7.02 \mathrm{km}\) \(\left(3.07, 16.16\right)\)	\(\alpha =2.46 \left(1.80, 4.38\right)\)
	[59]	\({r}_{0}=3.4 \mathrm{km}\) \(\left(1.001, 10.0\right)\)	\(\alpha =1.4\) \(\left(1.001, 5.0\right)\)

For estimated parameter values, the mean/median and 95% confidence/credible interval (when reported) are indicated. \({d}_{ij}\) is the distance between an infectious farm \(i\) and a susceptible farm \(j\). In the Pareto distribution, \({x}_{min}\) is the minimum possible value of the function and \(\alpha \ge -1\) determines the shape of the kernel: \(\alpha =-1\) corresponds to distance-independent transmission, with increasing values of \(\alpha\) increasing local transmission and diminishing long-range transmission. In the Power-law function, the distance-scaling parameter \(\delta\) is the distance at which the relative risk of transmission is \(1-\frac{1}{e}\) (or about 0.63), and \(\rho\) determines whether the decrease from probability 1 to 0 is gradual (small \(\rho\)) or step like (large \(\rho\)). In the logistic expression, the half-kernel distance \({r}_{0}\) corresponds to the distance at which the relative risk of transmission is 0.5, and the shape parameter \(\alpha\) determines whether the decrease from probability 1 to 0 is gradual (small \(\alpha\)) or step like (large \(\alpha\)).

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