Box 3. Using mechanistic models to measure the impact of predators on parasite transmission.
Individuals have been advocating for a relatively long time that ecologists and evolutionary biologists need to confront mechanistic models of ecological and evolutionary processes with data (e.g., Hilborn & Mangel 1997; Turchin 2003). Over the last 25 years, we as a field have taken that advice to heart particularly with regards to the study of disease ecology and evolution (e.g., Dwyer et al. 1997; Duffyet al. 2011; Elderd 2019). This advice also holds true for thinking about how predation affects disease spread and incidence. While there are multiple mechanisms that we outline throughout the paper and potential pathways for predators to spread parasites, whether or not these mechanisms are important for determining the rate and the extent of parasite transmission will often come down to a series of statistical tests using standard statistical (e.g., linear regression) or mechanistic models. Here we advocate for a more mechanistic approach. As a heuristic, consider the following model of disease transmission where the prey becomes infected with a lethal parasite (Elderd & Dwyer 2020),
, (1) , (2) . (3)
Here, S represents susceptible individuals, E represents exposed individuals, and P represents the parasites in the system. is the transmission rate and is the rate at which infected individuals are converted into parasites. In a field experiment, we can control the number of susceptibles and the number of parasites. If we conduct an experiment with a known number of parasites, we know P at time 0 or P(0). If the same experiment is conducted over a period of time from time 0 to time T, we can integrate equation 1, so that:
. (4)
Now, consider that we have added a predator to our experiment. The predator could either increase or decrease the spread of the disease. We could then add a term to equation 4 that changes the transmission rate based on predator addition. The modified equation would read:
. (5)
Here the sign and the magnitude of dictates the effect of the predator on the system as a linear function of predator density, D. If = 0, the predator has no effect. If < 0, the predator essentially follows the healthy herd hypothesis by decreasing disease transmission. If > 0, the predator spreads the disease and increases disease transmission. Thus, this simple addition to a standard mass-action model of disease transmission can serve as a first pass on whether or not a predator will have an impact on transmission. Note that the model can be easily modified to consider specific aspects of the predator. That is, while here is a linear function of simply predator density, it can take on a variety of forms such as linear functions that account for direct and indirect effects of predator presence or various non-linear models whose terms differ based on experimental treatments.
To confront the model with data, we can take a number of approaches; here we advocate two. The first approach is using standard information theory (Burnham & Anderson 2002; Gotelli & Ellison 2004) such as the Akaike Information Criterion (AIC), whereby we have multiple models that we directly compare. Since our solved model (eqn. 5) only has two parameters to estimate, we can directly compare a model that just estimates the transmission rate to a model that estimates both the transmission rate and the effect of the predator via on transmission. The same approach can also be analyzed using a Bayesian framework and the appropriate model comparison metrics as the Watanabe Akaike Information Criterion (WAIC) (Hobbs & Hooten 2015); this is the second approach that we advocate. The advantage of the Bayesian framework is not only the ability to compare multiple models but also the ability to derive a probability distribution associated with each of the transmission parameters estimated (Elderd 2018).