Economics and ecology of managing emerging infectious animal diseases.

Emerging infectious diseases (EIDs) greatly concern wildlife conservationists and livestock producers, with pathogen transmission between wildlife and livestock of particular concern (Dobson and Foufopolous 2001). (1) The disease ecology literature mostly focuses on single host-pathogen interactions, but has recently turned to EIDs infecting multiple-hosts. This literature generally derives post-infection management advice (and does not specifically address prevention) from models evaluated at pre-disease equilibria free of pathogen risks (Roberts and Heesterbeek 2003: Dobson 2004). The economic aspects of this problem, however, have not previously been addressed.

We adopt a bioeconomic approach to explore multiple-host disease management problems. Prior recommendations are shown to be "revealed preferences" for some underlying economic objective. We then determine how socially efficient management may deviate from these recommendations. We consider efficient management for both uninfected systems that are at risk of infection, and previously infected systems.

Our analysis is motivated by wildlife-livestock-pathogen interactions. Livestock models have addressed disease dynamics within herds (Barlow et al. 1997) and between herds (Barlow et al. 1998), but they have generally ignored infectious interactions with wildlife (Chi et al. 2002; Scantlebury et al. 2004). Similarly, wildlife models often do not address interactions with livestock, even when livestock impacts motivate the analysis (Barlow 1996; Dobson and Meagher 1996). Bicknell, Wilen, and Howitt (1999) do consider livestock and wildlife interactions, but theirs is not a true multiple-host model because they do not model disease dynamics within both populations.

An Ecological Approach

For both wild and domestic systems, EID management recommendations have generally been derived from epidemiological models that focus primarily on a pathogen's ability to expand within existing host-pathogen systems or invade new systems. This ability is calculated as the basic reproductive rate of the pathogen ([R.sub.0]), or the expected number of secondary infections generated from a single infected individual within an otherwise healthy host population. The disease invades or spreads when [R.sub.0] > 1 and fails to invade or diminishes in prevalence when [R.sub.0] < 1 (Dobson 2004; Roberts and Heesterbeek 2003).

For a single-host and a single control, the [R.sub.0] = 1 relation can be used to derive a host-density threshold: either the disease cannot invade or disease prevalence diminishes when the host-density is held below the level that makes [R.sub.0] = 1. For multiple populations or controls, the threshold is a multi-dimensional concept (Roberts and Heesterbeek 2003).

While analyses involving [R.sub.0] and related host-density thresholds yield insight, they have limited applicability for management for three reasons: (a) [R.sub.0] depends on the densities of susceptible animals in a disease-free equilibrium, yet the [R.sub.0] = 1 criterion has been used to guide post-infection management efforts; (b) the [R.sub.0] = 1 criterion generally ignores the endogeneity of disease management choices and the economic and ecological tradeoffs these choices imply; and (c) the analyses are based on disease eradication goals, which may not be economically efficient.

Consider two animal host populations in which infection follows the conventional susceptible-infected (SI) model (e.g., Heesterbeck and Roberts 1995). (2) Denote host i = W as wildlife and host i = L as livestock. Host i's aggregate density is [N.sub.i] = [S.sub.i] + [I.sub.i], where [S.sub.i] and [I.sub.i] are the densities of susceptible and infected animals within host i. Host-pathogen dynamics are defined by:

[[??].sub.i] = [S.sub.i][g.sub.i]([N.sub.i]) - [S.sub.i] [summation over (i)] [[beta].sub.ji] [I.sub.j] - [[gamma].sub.i][S.sub.i] (1)

[[??].sub.i] = [I.sub.i][g.sub.i] ([N.sub.i]) + [S.sub.i] [summation over (i)] [[beta].sub.ji] [I.sub.j] - [[alpha].sub.i] [I.sub.i] - [[gamma].sub.i] [I.sub.i] (2)

where [g.sub.i] represents density-dependent, average net natural growth, excluding disease mortality ([g.sub.i](0) = 0, [g.sup.''.sub.i] [less than or equal to] 0). The specification assumes all offspring of infected animals are also infected, which simplifies the algebra but has no bearing on the qualitative results (see Fenichel and Horan (in press) for a more general specification). The parameter [[beta].sub.ji] is the per capita rate of pathogen transmission from host j to host i; [[alpha].sub.i] is the additional mortality rate due to the disease; and [[gamma].sub.i] is the harvest rate from the aggregate population (i.e., [[gamma].sub.i] = [h.sub.i]/[N.sub.i], where [h.sub.i] is the aggregate harvest).

Ecological relations may depend on human choices, denoted by the vector x: [g.sub.i]([N.sub.i], x), [[beta].sub.ji](x), and [[alpha].sub.i](x). Elements of x affecting wildlife might be human-environmental interactions, such as supplemental feeding and habitat alterations. Elements of x affecting livestock might include feed and biosecurity choices.

We follow the convention of wildlife disease models (e.g., Heesterbeek and Roberts 1995) and assume that harvests are nonselective with respect to health status, which is often unobservable prior to post-mortem testing (Lanfranchi et al. 2003). Accordingly, for a given harvest level h, only a proportion (I/N) is infected. We also model livestock harvests as nonselective. This assumption is too strong when diagnostic testing is available, although testing is not always performed (e.g., due to costs or nonreporting of suspect animals) and is subject to error. Modeling nonselective livestock harvesting captures the essence of imperfect testing and greatly simplifies the analysis (Bicknell, Wilen, and Howitt (1999) model livestock testing error). Relaxing this assumption results in shifting optimal disease controls towards livestock producers.

Revealed Preferences of the Disease Ecology Literature

We begin by analyzing disease control as it is generally advocated in the disease ecology literature. Conceptual models in this area generally focus on the post-infection case and a single control, such as harvests (implicitly holding other controls fixed) (Roberts and Heesterbeek 2003; Heesterbeek and Roberts 1995; Dobson 2004). We focus on harvests, though the results for other controls are analogous. The standard objective is to eradicate a pathogen from host populations. The basic strategy analyzed is a constant effort policy, [[gamma].sub.i](t) = [[bar.[gamma]].sub.i] [for all]t, such that the following condition is satisfied:

[R.sub.0]([[bar.N].sub.W]([[bar.[gamma]].sub.W]|[I.sub.W], [I.sub.L] = 0), [[bar.N].sub.L] ([[bar.[gamma]].sub.L] | [I.sub.W], [I.sub.L] = 0)) = 1 - [epsilon]. (3)

Here, [epsilon] is an arbitrarily small parameter, and [[bar.N].sub.i]([[bar.[gamma]].sub.i]|[[I.sub.W], [I.sub.L] = 0) is the steady state value of [N.sub.i] that corresponds to [[bar.[gamma]].sub.i] when [I.sub.W], [I.sub.L] = 0 (Heesterbeek and Roberts 1995). [R.sub.0] is calculated as the dominant eigenvalue of the "next-generation" matrix (Dobson 2004), each element of which represents the expected number of secondary infections in host j that would arise from an initial infection within host i, assuming the populations are at a preinfection equilibrium. Hence, [R.sub.0] depends on pre-infection equilibrium population densities, which in turn depend on pre-infection equilibrium harvest rates. (3)

Condition (3) defines a frontier of harvest rates as opposed to a pair of specific rates (and if additional controls such as vaccination were available, then the frontier would encompass additional dimensions). As such, it does not indicate how to target efforts differentially across host types: any point on the frontier can be chosen to achieve the stated objectives.

However, once chosen, a particular choice could be viewed as a revealed preference--that is, it is "as if" the planner chose the effort levels to maximize some economic net benefit function. Suppose the net benefits associated with managing hosts i and j are separable and given by [B.sub.i]([N.sub.i], [I.sub.i])/[[gamma].sub.i] [N.sub.i], with [B.sub.iN] [greater than or equal to] 0, [B.sub.iI] < 0. The implicit optimization problem takes the following form, owing to the focus on the pre-infection steady state with no consideration given to the current state of the world:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (4)

subject to condition (3). The objective function NB represents the time-invariant net benefits associated with the effort choices and steady state stock levels. It is straightforward to incorporate multiple controls, as represented by x, into this framework.

The solution to (4) is myopic, due to the constant effort rates and the focus on a pre-infection steady state. Myopic solutions ignore economic and ecological trade-offs arising en route to a steady state. This is particularly true if the pathogen is initially present, the situation problem (4) is supposed to address. Another concern is that the solution to problem (4) is constrained to eliminate pathogen risks, though such an outcome may not be optimal (see Horan and Wolf 2005; Fenichel and Horan 2007).