Cross-sectional study
We made use of data from an extensive field survey that was carried out in 19 1-ha sites located in forested areas in The Netherlands in 2013 and 2014. Data were collected on the density of questing I. ricinus (blanket dragging), vertebrate communities (camera and live trapping), and infection rates of tick-borne pathogens (qPCR detection). The sites, methodologies, and data have been described elsewhere as well as a series of detailed analyses [14, 22,23,24].
Host attribution
We arranged the encounter probabilities for all forest sites (n = 19) and vertebrate species (n = 32) into a matrix \(\mathbf{A}\in {\mathbb{R}}^{19\times 32}\) having 19 rows and 32 columns. We further arranged A. phagocytophilum DIN into a vector \(b\) matching the order of the forest sites along the rows of \(\mathbf{A}\). To attribute A. phagocytophilum DIN to an assemblage of 32 vertebrate species, we factored the matrix \(\mathbf{A}\) into two orthogonal matrices
$$\mathbf{U}=[{u}_{1},{u}_{2},\dots ,{u}_{19}]\in {\mathbb{R}}^{19\times 19} \mathrm{and} \mathbf{V}=[{v}_{1},{v}_{2},\dots ,{v}_{32}]\in {\mathbb{R}}^{32\times 32}$$
and a diagonal matrix \({\varvec{\Sigma}}\in {\mathbb{R}}^{19\times 32}\) with the singular values \({\sigma }_{1}\ge {\sigma }_{2}\dots \ge {\sigma }_{19}\ge 0\) and the remaining entries equal to zero. Theorem 2.5.2 [25] proves that \(\mathbf{A}=\mathbf{U}{\varvec{\Sigma}}{\mathbf{V}}^{T}\). The column vectors \({u}_{i}\) and \({v}_{i}\) are principal components, also known as singular vectors.
The A. phagocytophilum DIN increased at each forest site because of the first principal components by the amount (Theorem 5.5.1 [25])
$$\frac{{u}_{1}\cdot b}{{\sigma }_{1}}\mathbf{A}{v}_{1}\in {\mathbb{R}}^{19}.$$
(1)
We attributed Eq. (1) to roe deer because the highest contribution from the first principal component \({v}_{1}\) comes from roe deer. Next, we applied the theorem again to quantify the inputs from the lower principal components \({v}_{2},{v}_{3}\dots\),
$$y=\sum_{i=2}^{8}\frac{{u}_{i}\cdot b}{{\sigma }_{i}}\mathbf{A}{v}_{i}\in {\mathbb{R}}^{19}.$$
Lower components \({v}_{9}\dots {v}_{19}\) are ignored because the tail sum \(\sum_{i=9}^{19}{\sigma }_{i}^{2}\) is negligible (2.43%) compared to the whole cumulative sum \(\sum_{i=2}^{19}{\sigma }_{i}^{2}\). Next, we define
$$\begin{array}{cc}{y}^{+}& =\mathrm{max}(y,0),\\ & \end{array}$$
(2)
$$\begin{array}{cc}& \\ {y}^{-}& =\mathrm{max}(-y,0).\end{array}$$
(3)
Intuitively, we clipped the solution \(y\) into the positive part \({y}^{+}\) and the negative part \({y}^{-}\). We attributed Eq. (2) to fallow deer, red deer, and wild boar because the highest contributions from the second principal component \({v}_{2}\) and the third \({v}_{3}\) come from these species.
Larval tick burden on ungulates
Ungulates generally contribute relatively little as hosts for feeding I. ricinus larvae compared to rodents and birds [15]. Nevertheless, as important hosts for A. phagocytophilum, ungulates might feed a significant fraction of larvae that later become A. phagocytophilum-infected nymphs. Thus, differences in larval burden between ungulate species could contribute to differences in their importance as hosts contributing to A. phagocytophilum DIN. To explore this, we compiled data from published studies that collected I. ricinus larvae attached to individual ungulates (see Additional file 1: Table S1). We extracted species, the number of checked animals, and the number of I. ricinus larvae attached to the animals. We fit the negative binomial model (log-link) to the number of larval ticks using the number of animals and the species as predicting variables. We tested the significance of the species predicting variable by performing the likelihood ratio test.
Associations with other woodland species
We explored whether differences between species could be explained by associations of the different ungulate species with the availability of other host species. Here, two species might appear related because of the probability condition (the sum of rates over the host species must equal to one). To remove this potential bias, we performed the following analysis using the encounter rates instead of the encounter probability. For this, we calculated the Pearson correlation in the encounter rate for each ungulate species with each other woodland species. We fit the binomial model (logit link) to the frequency of positive and negative correlation values using the ungulate species as a predicting variable. We tested the significance of the predicting variable by performing the likelihood ratio test.
Absent species interaction, correlation values should be close to zero, and the deviation from the expected value zero should be symmetric. It is possible to calculate the probability of observing as many or more negative correlation values as actually observed in the vertebrate community,
$$\frac{1}{{2}^{-n}}\sum_{j=k}^{n}\left(\genfrac{}{}{0pt}{}{n}{j}\right).$$
(4)
This is a partial sum of binomial probability densities where a correlation value is negative with probability \(\frac{1}{2}\). The vertebrate community counts \(n\) members. The number of negative correlation values observed in the vertebrate community equals \(k\). All computations were implemented using the R language [26].
