6. Dynamic Perspectives on the Control of Animal Disease: Merging Epidemiology and Economics

Note R(t) differs from R _t or R ₀, which were mentioned in the introduction and are described in greater detail below. Though these measures differ, the use of this similar notation is conventional. Furthermore, in some cases, the R in SIR stands for removed.

Vaccines are more common for human diseases than animal diseases. When vaccines are available for animals, they are not always used. Vaccines may not be given to livestock because healthy, vaccinated animals may test positive for disease, thereby resulting in trade restrictions that lower the value of the animals in trade (USDA-APHIS 2002). Wildlife populations are seldom vaccinated because vaccines are unavailable or ineffective for many diseases (Smith and Cheeseman 2002), or vaccination of wildlife may be controversial – particularly for threatened and endangered species. This concern has arisen, at least in part, because a study population of Serengeti wild dogs became extinct following interventions that used vaccination (Burrows 1992; Burrows et al. 1994).

As indicated in the introduction, the basic population-based epidemiology model (6.1) and (6.2) could be expanded to include ecologically defined compartments like resistant and exposed or carrier subpopulations (Hethcote 2000), or human-defined compartments such as subpopulations under quarantine or in reserve areas. The model can also be expanded to incorporate populations in additional locations, with animal movement between locations (Fulford et al. 2002; Rowthorn et al. 2009). Wild animal movement could depend on ecological and human factors. For instance, humans can manage population densities and construct habitat corridors that encourage movement to areas of greater resource abundance, and of reduced competition and predation (e.g., Gichohi 2003; Kaiser 2001; Ewing 2005). Often, though not always (e.g., see Sanchirico and Wilen 1999; Horan et al. 2005), spatial metapopulation models define the state variables as the number of “patches” of habitat, or the number of farms, in each disease state (cf. Levins 1969) (as opposed to animal densities). “Patch-based” models are simply based on a coarser unit of analysis, achieved by rescaling the system (6.1) and (6.2), to focus on changing disease patterns across the landscape. These models are well suited for farm sector analysis, where authorities are primarily concerned about the number of infected farms (Barlow et al. 1998). However, as population-based and patch-based models are related, the insights for both types of models are similar.

Subscripts denote partial derivatives.

By the LeChatelier principle, selective harvests would reduce disease management costs, resulting in greater social welfare, but selective harvests are not always feasible. Harvests may be selective with respect to disease status for diseases exhibiting easily observable outward signs, or after diagnostic testing to elicit an animal’s health status. Diagnostic testing has been proposed for some disease problems where field tests are quick and accurate. For instance, diagnostic testing has been proposed for Devil Facial Tumor Disease (DFTD) in Tasmanian devils (STTD 2009; Platt 2009), brucellosis in bison (Bienen and Tabor 2006), and chronic wasting disease in mule deer (Watry et al. 2004; Wolfe et al. 2004).

More generally, R ₀ is derived as the dominant eigenvalue of the linearized epidemiological system at the disease-free equilibrium.

For problems where vaccination and immunity is possible, S will not equal N in the predisease equilibrium. Rather, S will equal (1 − υ)N, where υ is the proportion of the population that is immune due to vaccination. (Note that υ is not a control, but a state variable. We use υ to proxy for the immune population, which we previously defined as R, since R = υN. Here, we have used υ to avoid confusion between R and R ₀.) In that case, the R ₀ < 1 criterion could be satisfied by manipulating either N or υ. The basic insights are not affected by this complication: the approach is still to manage R ₀ by affecting the pre-infection state of the world, rather than through current-period control choices.

Fenichel et al. (2010) find the opposite result for the case of multiple hosts: host-density thresholds predicted by R ₀ may be insufficient to eradicate a pathogen from an already-infected system.

For instance, if we assume harvest benefits take the form B(hN) − ch, where B is willingness to pay and costs correspond to Schaefer-type costs (Clark 2005) since h is a rate, then ∂h/∂N > 0 and ∂h/∂I > 0.

Suppose harvests are selective and that SNB takes the form B(S, I, h _S ) − c(h _I I, I). Then the current-value Hamiltonian is \( H = B(S,I,{h_S}) - c({h_I}I,I) + \lambda [F(N) - \delta N - \alpha I - {h_S}[N - I] - {h_I}I] + \mu [\tau (N,I) - (\delta + \alpha )I - {h_I}I] \).