Chronic Wasting Disease: Transmission Mechanisms and the Possibility of Harvest Management

We develop a model of CWD management by nonselective deer harvest, currently the most feasible approach available for managing CWD in wild populations. We use the model to explore the effects of 6 common harvest strategies on disease prevalence and to identify potential optimal harvest policies for reducing disease prevalence without population collapse. The model includes 4 deer categories (juveniles, adult females, younger adult males, older adult males) that may be harvested at different rates, a food-based carrying capacity, which influences juvenile survival but not adult reproduction or survival, and seasonal force of infection terms for each deer category under differing frequency-dependent transmission dynamics resulting from environmental and direct contact mechanisms. Numerical experiments show that the interval of transmission coefficients β where the disease can be controlled is generally narrow and efficiency of a harvest policy to reduce disease prevalence depends crucially on the details of the disease transmission mechanism, in particular on the intensity of disease transmission to juveniles and the potential differences in the behavior of older and younger males that influence contact rates. Optimal harvest policy to minimize disease prevalence for each of the assumed transmission mechanisms is shown to depend on harvest intensity. Across mechanisms, a harvest that focuses on antlered deer, without distinguishing between age classes reduces disease prevalence most consistently, whereas distinguishing between young and older antlered deer produces higher uncertainty in the harvest effects on disease prevalence. Our results show that, despite uncertainties, a modelling approach can determine classes of harvest strategy that are most likely to be effective in combatting CWD.

For multiple infected compartments the increase is    C n . The rate of passing through each compartment is greater, but the mean duration of the disease remains the same, see  We assume that the main way of natural deer population regulation is densitydependent juvenile mortality, which is usually related with predators and food limitation 4 during most critical season, which is winter for Alberta. We used two models of density dependence, and both of them are based on the ratio of required R F and available A F food for deer population. The required food is estimated with the help of daily food for each deer category (see Table 1). Then where C is a constant proportional to the duration of food-limiting period and the area occupied by the population. The available food is estimated from the same expression , which assumed to be known: The first way of characterizing food limitation is through the value that we call starvation index V, [15]. If there is excessive food, i.e., the population is way below winter carrying capacity, then F A >F R and V=0. If F A is much less than F R and starvation rates are high, V approaches 1. At a food-based equilibrium (at carrying capacity) V takes some value V 0 between 0 and 1, corresponding to partial food limitation. We assume that juvenile mortality is related with V as: We assume that maximum per year fawn survival corresponds to V=0, the minimum one to V=1, and the equilibrium survival corresponds to 0 V .
The second model of density dependent mortality is a generalization of expression given in [17]. It uses different expression for V in (A13), . This model has been parameterized by the same data: maximum survival corresponds to V=0 and equilibrium one to V=1. Note that the density independent part vj m 0 for (A12) and (A14) should coincide, but the density dependent coefficients vj m 1 are different.
Our modeling results do not show any influence of the chosen type of densitydependence in the model on disease prevalence and hence on the effect of harvest disease management. Choice of the density-dependence, however, may be critical for the study of the effect of harvest on deer density.

Density-dependence visualization (Fig. A1)
The goal is to visualize the behaviour of deer mortality and recruitment as density change. In general this is a complicated task, because the required food depends not only on density, but on proportions of deer of each categories. For this reason, we assume the 6 population healthy ( 0  x I ) and the proportions D S x Sx Then the required food is where 0 D is the density of population used for model parameterizing. Therefore, Substituting this for the expression for V and using (A13), we obtain Fig. A1a.
If in addition we assume that buck:doe ratio >0.05 and juveniles are not harvested, then at where Sj m is density dependent juvenile mortality (A13), then per year recruitment

Disease transmission
In [6] we develop the mathematical expressions for the force of infection terms for 7 transmission mechanisms and use four of them, which have been considered most plausible. They give the following force of infection terms for the 5 combination of age classes used in our harvest senarios: The two terms in square brackets in (A15)-(A17) correspond to disease transmission within sexually segregated groups (summer and early autumn) and within mixed groups (rut, winter and early spring  and R  are the corresponding transmission coefficients for within-group and rut transmission. For rut transmission, we assume that if there are enough older males, they alone mate with females, but if the density of the older males drops below one older male 8 per more than ten females, that is the ratio , then younger males also take part in mating, which is described by L in (A18). ). Therefore, the pre-harvested deer density is , and hence the proportion of the pre-hunted deer of the category x removed by harvest is

Disease-related mortality: multiple compartments
The reason for introducing multiple compartments for infected individuals is explained in Section 2.5: to reduce variability in infected deer survival times. This is illustrated in Fig.   A2. Mathematical explanations are given in Appendix B. Here we discuss the effects that we observed comparing the results for different numbers of compartments C n .
We fitted transmission coefficients for models with 25 , 5 , 1  C n to the same Alberta data for the initial stage of the disease. The results are shown in Table A3. An interesting effect is observed: the greater is C n , the smaller is the fitted transmission coefficient. This may be the result of slower decrease in infected deer density in the first 1-1.5 years of the disease development. The total number of the secondary infections may be the same, but they are created in a shorter period. Therefore, the same initial disease dynamics can be reproduced with less intensive transmission.  Fig. 2 shows that the disease prevalence at the same harvest level is higher at smaller C n . At the same time, optimal harvest policy remains practically the same.

Experiments with Healthier older male class: influence on harvest policies
In some publications on CWD, see [4,5], there are data that can be interpreted such that male disease prevalence grows with deer age reaching its maximum around 5 years old, and then decreasing. If disease transmission rates and mortality of adult male deer do not depend on age, as in our model, this can not happen: older males have higher disease prevalence just because of longer exposure to the infection. However, the older male class may have less contact rate because of less participation in bachelor groups, or higher mortality e.g. because of poorer condition in the beginning of the winter due to starvation and exhaustion during rut. In either of cases their prevalence may be less or not higher than that of young males, and there appears one more reasonable harvest strategy: harvest of young males only. We did a series of model calculations for the original model and the cases of twice less contact rate and twice greater mortality rate for the older male class. We searched for optimal harvest strategy with three independent harvest rates: antlerless, young males and older males. The optimal policies were less stable than in Figs. 2-3, and in most stable cases were close to that for the original model.
However, the results of comparing the seven harvest policies were more interesting. For the transmission mechanisms TM3 (environmental) and TM5 (higher male susceptibility) 12 they are shown in Fig. A5. One can see that in case of healthier old male class (panels cf) the results of harvesting young males only give slightly higher than harvest all males, but close to them. If the older male class is indeed healthier, then harvesting only young males could be an alternative management policy provided there is a goal to preserve older males.
Another interesting effect related with the hypothesis of healthier older male class is that fitting the model with twice less contact rate to data gives much higher value of rut     (Fig. 3), twice less contacts for older males (c,d) and twice greater mortality rate for older males (e,f). In panels c-f harvest of young adults is closer by the effect to harvest of all adults.   Fig. 4). b) Proportions of removed females and juveniles vs proportion of removed males for this policy. c) The disease prevalence under the mixed policy for each of TMs: for TM3, TM5 and TM6 the disease can be eradicated, but this requires greater harvest effort.  We consider the cases 1  n , 2, 3, 4, 6, 8, 12, 24, which corresponds to the mean duration of staying within each compartment 24, 12, 8, 6, 4, 3, 2 and 1 month.
In The greater is n, the closer is the survival distribution to a step-like function.