Modeling seasonal behavior changes and disease transmission with application to chronic wasting disease
Posted Dec 31 2013 2:39pm
Modeling seasonal behavior changes and disease transmission with application to chronic wasting disease
Tamer Orabya, Corresponding author contact information E-mail the corresponding author, Olga Vasilyevab, Daniel Krewskia, c, Frithjof Lutscherb a McLaughlin Centre for Population Health Risk Assessment, University of Ottawa, Ottawa, Ontario, Canada b Department of Mathematics and Statistics, University of Ottawa, Ottawa, Ontario, Canada c Department of Epidemiology and Community Medicine, Faculty of Medicine, University of Ottawa, Ottawa, Ontario, Canada
• A new model is built to study spread chronic wasting disease in free-ranging deer.
• The model employs two modes of transmission based on seasonal behavior.
• Birth and change in seasonal home range are impulsive.
• The basic reproduction number and stability of disease-free equilibrium are studied.
• Under certain conditions, culling can eradicate the disease.
Behavior and habitat of wildlife animals change seasonally according to environmental conditions. Mathematical models need to represent this seasonality to be able to make realistic predictions about the future of a population and the effectiveness of human interventions. Managing and modeling disease in wild animal populations requires particular care in that disease transmission dynamics is a critical consideration in the etiology of both human and animal diseases, with different transmission paradigms requiring different disease risk management strategies. Since transmission of infectious diseases among wildlife depends strongly on social behavior, mechanisms of disease transmission could also change seasonally. A specific consideration in this regard confronted by modellers is whether the contact rate between individuals is density-dependent or frequency-dependent. We argue that seasonal behavior changes could lead to a seasonal shift between density and frequency dependence. This hypothesis is explored in the case of chronic wasting disease (CWD), a fatal disease that affects deer, elk and moose in many areas of North America. Specifically, we introduce a strategic CWD risk model based on direct disease transmission that accounts for the seasonal change in the transmission dynamics and habitats occupied, guided by information derived from cervid ecology. The model is composed of summer and winter susceptible-infected (SI) equations, with frequency-dependent and density-dependent transmission dynamics, respectively. The model includes impulsive birth events with density-dependent birth rate. We determine the basic reproduction number as a weighted average of two seasonal reproduction numbers. We parameterize the model from data derived from the scientific literature on CWD and deer ecology, and conduct global and local sensitivity analyses of the basic reproduction number. We explore the effectiveness of different culling strategies for the management of CWD: although summer culling seems to be an effective disease eradication strategy, the total culling rate is limited by the requirement to preserve the herd.
Most modeling studies of human and wildlife disease assume that the mechanism of individual contacts and therefore the functional dependence of the force of infection remains unchanged, even if parameters may vary seasonally. Instead, we argue that seasonal changes in behavior can lead to a more fundamental change in the disease transmission mechanism (see also Potapov et al., 2013), so that the functional dependence of the force of infection changes seasonally. In particular, roaming and aggregation behavior in wildlife populations could lead to a shift from DD to FD disease transmission. We developed and analyzed a simple, strategic model for such a shift, and applied it to CWD in deer. In principle, these same considerations could be applied to modeling of childhood diseases that show outbreak patterns highly correlated with school terms. Such an approach could give a more mechanistic underpinning of the contact rate, which is often formulated as a periodically forced function. An interesting future topic is to compare the predictions of a multi-season model to those of a temporally constant model where disease transmission is modeled by some suitably interpolated transmission term.
The simplicity of our model allows for an elegant reduction to a pair of impulsive equations and an explicit expression for R0. While culling is a viable control strategy in pure DD models, it is not in pure FD models (Lloyd-Smith et al., 2005). We found that culling can be a useful control strategy in our two-season model, but culling rates need to be chosen carefully to ensure survival of the herd(see also Choisy and Rohani, 2006). According to our analysis, the contact rate during the summer season has greater influence on R0 than the contact rate during the winter. Previous authors had argued otherwise (Habib et al., 2011). Accordingly, if culling were equally costly during the summer and winter, we argue that harvesting efforts should be concentrated in the summer.However, since herds tend to be spread out over larger areas during the summer, this might not be feasible. Our analysis also shows that increasing the length of the summer season, as predicted under some global change scenarios, would increase R0 and make disease eradication more difficult.
There is a long-standing discussion about whether DD or FD is a more appropriate modeling assumption in a given situation (Begon et al., 2002; Lloyd-Smith et al., 2005). For wildlife diseases, FD is sometimes favored (McCallum et al., 2001; Begon et al., 1999), but deciding between the two alternatives based on data fitting is often difficult, and, in the case of CWD, remains unclear (Wasserbergetal.,2009). We speculate that some of the confusion may arise by pooling data from different seasons when different transmission mechanisms may be operating. In practice, transmission may be neither ‘purely’ DD nor ‘purely’ FD. Some authors have addressed this problem by employing various interpolations between DD and FD (Almberg et al., 2011; Habib et al., 2011). In practice, model selection criteria are then required to decide whether the improved fit to data warrants the inclusion of an additional parameter. Our modeling approach also works for such interpolated forms of disease transmission; however, an explicit solution necessary for model reduction is not available. The relative size of the habitat that the herd occupies in different seasons would then affect R0 and all other model characteristics.
We are currently extending this work to include the rut season explicitly, where social behavior changes again, so that disease transmission might change, and where harvesting is not allowed. At that point, gender and potentially age structure should also be introduced into the population since males, females and fawns engage in social contact in very different ways; see Al-arydah et al. (2012) for an age and gender-structured model of CWD. Such a model is too complex to yield explicit solutions, so that the analysis has to proceed numerically. Our results here can inspire simulation studies of the properties of such a model, and our weighted average formula for R0 can provide guidance for R0 in a more complex model.
So far,we considered only one of the three potential transmission pathways, namely direct transmission. The extension of our model to include vertical transmission is straight forward, and all the analytical results can be extended (see Appendix D). The basic reproduction number increases as the probability of vertical transmission increases. Since there are no reliable estimates of the vertical transmission probability, we did not include it in our sensitivity analysis.
The inclusion of environmental transmission into our model is a lot more delicate and is beyond the scope of this work. A number of recent empirical and theoretical studies point to the importance of environmental transmission of CWD in addition to, or instead of, direct contact transmission (Almberg et al., 2011; Miller et al., 2004; Wasserberg et al., 2009; Smith et al., 2011; Johnson et al., 2006). To justify the absence of an environmental compartment in many models, it is typically argued that since the rate of degradation of the environmentally available CWD agent is faster than the prevalence growth rate, this compartment will be proportional to the number of infected individuals and hence can be incorporated into direct transmission (Potapovetal., 2012). A more thorough investigation into the conditions under which indirect transmission can be modeled as direct transmission was recently given by Breban (2013).
Environmental transmission is relatively easily explicitly incorporated into the disease model when the herd remains in the same location. One needs to add an ‘environmental’ compartment and define appropriate deposition and uptake functions for the CWD agent (prions) (Almberg et al., 2011; Vasilyeva et al., submitted for publication). Model formulation is more challenging when a herd migrates between seasons. Since environmental prions are not expected to decay within a single season, one needs to keep track of prions in the winter and summer areas separately, there by introducing an additional compartment to the model. If summer and winter areas overlap, the modeling process becomes even more difficult. It is also unclear to what degree environmental prions are available for uptake under snow cover. Based on our results without environmental transmission, we speculate that if the herd is much more aggregated during the winter, then the prion concentration is much higher in the winter season, and that R0 would be more sensitive to (some) winter parameters than summer parameters.
We believe that by splitting the year into different seasons where different behavioral mechanisms such as aggregation and reproduction operate, our model can capture important aspects of disease etiology not embodied in current models, thereby facilitating investigation of questions related to optimal timing of disease control,as well as other issues that have a seasonal dimension.