2.6. Model structure and explanatory variables

In our model, Inf was a two-level explanatory factor (infected versus uninfected), describing the individual state of infection at the time of release. As LPAIV infections are of short duration (often less than 8 days) [31], we analysed individual movements based on GPS data collected within a week after each release. If a bird was recaptured in the 7 days following its release, the time scale was reset and the 7 days thereafter were considered according to the updated state of infection. To avoid potential handling effects on movement due to capture, we reran the analyses excluding the first (12 h) period following release.

Time after last release (T.aft.Rel) was included in models as an explanatory continuous covariate. For uninfected individuals, we expected almost no variation in movement metrics with time after release. By contrast, we expected movement metrics for infected birds to change with time following recovery from infection (visualized in figure 1). As a result, we included the interaction Inf*T.aft.Rel to account for these two alternative trends in the movement metric variation.

Theoretical predictions of the influence of infection on movement metrics. If infection affects spatial behaviour, infected (blue) and uninfected (red) birds should behave differently at the time of release. We postulate that, at this time, movement metrics for infected birds should be lower than for uninfected birds, which would be revealed as different intercepts of the regression of the movement metrics against time for uninfected (β₀) and infected birds (β₀+β_Inf). As infected birds recover with time, their movement metrics will approach and eventually meet the values for uninfected birds. This happens when the slope of the regression of the movement metrics against time for infected individuals (β_{T.aft.Rel*inf}) reaches the slope for uninfected birds (β_T.aft.Rel), which is expected to be null.

We also included three other explanatory covariates to control for extra sources of variation in movement metrics. First, the number of GPS locations is a function of the settings of the GPS units and the specific localities of the birds, and represents a technical bias influencing, for example, the minimum convex polygon (MCP) area, speed and travel distance of the birds. Therefore, the number of GPS locations (Reloc) was included as an explanatory covariate to account for varying sample size. For similar reasons, we excluded the last (incomplete) day/night of the total period for which each individual was followed. Second, a first inspection of the data revealed apparent differences in behaviour strategies among the mallards. They showed two main behaviours (Behtrap): ‘trap-dependent’ (i.e. returning to the trap on a daily basis; trapd) and ‘trap-independent’ (i.e. not returning to the trap; ntrap; figure 2). Third, mallard movements were expected to differ substantially between day (D) and night (N). Day/night was assigned based on solar angle of −6 degrees (which marks the end of civil twilight), i.e. when the sun was below this angle it was considered to be night and when above this angle it was considered day. As these two factors, trap behaviour and day/night movements (Behtrap and DN, respectively), are potentially important sources of variation in movement metrics, they were included by default as explanatory variables in the analyses.

Examples of different types of movement behaviour in autumn-staging mallards for (a) ‘trap-dependent’ (i.e. returning to the trap) and (b) ‘trap-independent’ (i.e. not returning to the trap) individuals. T marks the location of the duck trap.

The models were, therefore, as follows:

where

— β₀ is the intercept, or reference, for the metrics recorded on uninfected trap-dependent mallards during the day. Trap-dependent observations are used as a reference as they were the most frequent in the data.

— β_Inf is the regression coefficient quantifying the effect of infection on the movement metrics compared with the reference β₀, i.e. the metrics recorded on uninfected individuals.

— β_T.aft.Rel is the slope of the regression of the movement metrics against time after release in uninfected, trap-dependent mallards during the day.

— β_Reloc is the slope of the regression of the movement metrics against the number of relocations.

— β_Behtrap is the regression coefficient quantifying the difference between trap-independent mallards and the reference, i.e. trap-dependent individuals, on the movement metrics.

— β_DN is the regression coefficient quantifying the difference observed between night and the reference, i.e. day, on the movement metrics.

— β_{Inf*T.aft.Rel} is the regression coefficient quantifying the interaction between time after release and infection on the movement metrics. It is the difference in the slope of the regression of the movement metric against time after release between infected individuals and the reference, i.e. uninfected individuals.

— ε are the normal errors, with ε∼N(0,1).

— γ is the individual random term, accounting for variance heterogeneity among clustered observations recorded within a same individual.