Table 2 Best models for prediction of density of nymphs (DON), nymphal infection prevalence (NIP), and density of infected nymphs (DIN)

Eq. no.ResponseEquationTypeYearTrend
1DONt+1$${ = 1} . 2 8\times {\text{rodent density }} - 8. 7 5\times I_{\text{year = 2014}} + 1 1. 7 7\times I_{\text{treatment = control}}$$LM↑***
2NIPt+1 B. afzelii$$= - 3.25 + 0. 0 2\times {\text{rodent density}}$$GLM, binomial↑***
3DINt+1 B. afzelii$$= 5.33 + 0. 7 5\times {\text{rodent density}}$$LM↑***
4NIPt+1 N. mikurensis$$= - 2.85 + 0. 0 3\times {\text{rodent density}}$$GLM, binomial↑***
5DINt+1 N. mikurensis$$= 6. 8 4+ 1.52 \times {\text{rodent density }}$$LM↑***
6NIPt+1 B. miyamotoi$$= - 3.33 + 0. 0 3\times {\text{rodent density }} \times I_{\text{year = 2014}}$$GLM, binomial2013
2014↑**
7DINt+1 B. miyamotoi$$= 4.88 + 0. 3 4\times {\text{rodent density}}$$LM↑*
8NIPt+1 B. microti$$= - 2. 9 3- 0.02 \times {\text{rodent density}}$$GLM, binomial↓***
9DINt+1 B. microti$$= 2.64 - 0.04 \times \left( {\text{rodent density}} \right){}_{{}}^{2} + 1.62 \times {\text{rodent density}}$$LM↑↓*
10NIPt+1 B. garinii$$= - 4.28 - 0. 0 4\times {\text{rodent density }} + 0. 8 3\times I_{\text{year = 2014}}$$GLM, binomial↓***
11DINt+1 B. garinii$$= 3.00 \left( {null} \right)$$LM
12NIPt+1 R. helvetica$$= - 3.52 + 0.03 \times {\text{rodent density}} \times I_{\text{year = 2014}}$$GLM, binomial2013
2014↑***
13DINt+1 R. helvetica$$= 3.21 + 0. 7 0\times {\text{rodent density}}$$LM↑*
14NIPt+1 S. ixodetis$$= - 1.04 - 0. 0 1\times {\text{rodent density}}$$GLM, binomial↓***
15DINt+1 S. ixodetis$$= 24.94 - 0.12 \times \left( {\text{rodent density}} \right){}_{{}}^{2} + 5.88 \times {\text{rodent density}}$$LM↑↓***
1. Notes: Only significant interactions are shown in the equations; full equations can be found in Additional file 4: Table S6. Arrows indicate whether an effect of rodent density was positive, negative or none. Two arrows, one going up and one going down indicate non-linear association (parabola). Asterisks denote significance of an effect (*P ≤ 0.05, **P ≤ 0.01, ***P ≤ 0.001)