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).

Finally, while the effort levels derived from (4) would prevent an infection from occurring, prevention is not the purpose of such analysis in the disease ecology literature. Indeed, prior literature does not specifically address disease prevention, even though significant investments in prevention are shown to be efficient in related contexts such as invasive species (e.g., see Leung et al. 2002).

Optimal Bioeconomic Management

We now consider management from a bioeconomic perspective, considering both ex ante disease prevention and ex post disease control. We also explicitly consider multiple controls. For simplicity, we redefine our variables of interest. Following Horan and Wolf (2005), we work with the variables [N.sub.i], [[theta],sub.i] = [I.sub.i]/[N.sub.i], and [h.sub.i] : [[gamma].sub.i][N.sub.i] (written in vector form as N, [theta], and h), so that the relevant equations of motion become:

[[??].sub.i] = [N.sub.i][g.sub.i] ([N.sub.i]) - [[alpha].sub.i] [[theta].sub.i] [N.sub.i] - [h.subi.i]

= [F.sub.i]([N.sub.i]) - [[alpha].sub.i] [[theta].sub.i] [N.sub.i] - [h.subi.i] (5)

[[??].sub.i] = [[theta].sub.i] (1 - [[theta].sub.i])[[[beta].sub.i] [N.sub.i] + [[beta].sub.ji] [N.subi.j] [[theta].sub.j] / [[theta].sub.i] - [[alpha].sub.i] (6)

We also redefine x as the scalar x to represent biosecurity to reduce cross-host infectious contacts (i.e., [[beta].sub.ij](x) < 0 i [not equal to] j). Biosecurity costs are given by cx, where c is the unit cost.

[R.sub.0] is of limited use for risk management because it is calculated for a pre-infection equilibrium, and it does not reflect risks to individual hosts. Instead, we focus on host-density thresholds, which reflect host-specific pathogen risks at each point in time: the pathogen will not spread within host i when [N.sub.i](t) is less than the host-density threshold [[??].sub.i](t), which is the value of [N.sub.i](t) that solves [[??].sub.i](t) = 0.

Suppose both hosts are pathogen-free ([[theta].sub.L] = [[theta].sub.W] = 0) prior to t = T. At t = T, the pathogen is introduced such that [[theta].sub.i](T) = [[eta].sub.i] > 0, where [[eta].sub.i] is small (i = L, W). (4) The post-T threshold [[??].sub.i](t [greater than or equal to] T) = [[??].sub.i,t>T] [[theta](t), [N.sub.j](t), [h.sub.i](t), x(t)] depends on current controls and the current states [theta] and [N.sub.j]. The pre-T threshold [[??].sub.i](t < T) = [[??].sub.i,t[less than or equal to]T] ([h.sub.i](t), x(t)) is also endogenous but depends only on current controls. A newly introduced pathogen will only spread in host i if [N.sub.i](t) > [[??].sub.i](T).

Pre-Infection Case

Suppose both hosts are pathogen-free and that T is unknown. Social net benefits in each period t < T are G = [[summation].sub.i=L, W] [B.sub.i][h.sub.i]- cx. The present value of net benefits from time T onward is denoted V(N(T), [theta](T)), which reflects ex post optimal management of the populations and the pathogen (see the Post-infection section below). (5) Note that V is contingent on ex ante management of the livestock and wildlife systems, as decisions prior to T affect he state variables arising at T, thereby impacting the ability to control the disease ex post. Ex ante management may also affect the likelihood of a pathogen introduction.

Let E be the expectations operator reflecting the uncertainty of time T, and let r be the discount rate. Ex ante efficient management is then defined as the solution to:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (7)

subject to [N.sub.i](0) (i = L, W) and the equation of motion (5). Following Reed and Heras (1992), the probability that a pathogen is introduced at any time t is given by:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (8)

with [[psi].sub.x] < 0 and [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (potential contacts are reduced by biosecurity but increased by greater density). Problem (7) is rewritten in a manner similar to Reed and Heras (1992):

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (9)

subject to [N.sub.i](0) (i = L, W) and the equations of motion (5) and:

[??] = [psi] (N, x), y(0) = 0. ((10)

Hence, both the probability of pathogen introduction and the ability of an introduced pathogen to establish are endogenous.

Problem (9) is deterministic, with the discount factor modified by the probability the system has remained pathogen-free until time t (i.e., the survival probability), [e.sup.-y]. This leads to the conditional current value Hamiltonian (Reed and Heras 1992):

[bar.H] = [summation over (i)] [B.sub.i] [h.sub.i] - cx + [[psi] V + [summation over (i)] [[lambda].sub.i] [[??].sub.i] + [rho] [??], (11)

where [[lambda].sub.i] is the shadow value of an extra unit of he ith stock at time t, conditional on no disease roving occurred up until that time, and [rho] is the co-state variable for y.

The optimality conditions for (11), assuming singular solution for [h.sub.i], are:

[partial derivative][bar.H] / [partial derivative][h.sub.i] = [B.sub.i] - [[lambda].sub.i] = 0 (12)

[partial derivative][bar.H] / [partial derivative]x = -c + [[psi].sub.x] [V + [rho]] [less than or equal to] 0;

(-c + [[psi].sub.x] [V + [rho]])x = 0 (13)

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (14)

[??] = [r + [psi]] [rho] + [summation over (i=L,W)] [B.sub.i][h.sub.i] - cx + [psi] V.

Condition (12) ensures the marginal benefits of [h.sub.i] equal the marginal user cost of host i. To interpret conditions (13) and (14), it is useful to first use condition (15) to derive:

[rho](t) = - [[integral].sup.[infinity].sub.t] [G + [psi] V] [e.sup.-r(s-t)-y(s)] ds. (16)

This is the negative of the expected present value of net benefits along the ex ante optimal path. Ex ante net benefits must not be less than ex post net benefits (i.e., -[rho] [greater than or equal to] V); otherwise, society would be better off to introduce the pathogen on purpose. In the special case in which ex ante management prior to T eliminates the risk that an introduced pathogen will spread (i.e., [N.sub.i](T) < [[??].sub.i](T) for i = L, W), then -[rho] = V; the planner is indifferent between the ex ante and expost cases because the pathogen fails to establish, and the management problem at time T is unchanged from the ex ante case.

Condition (13) says x should be set such that the marginal cost of biosecurity equals the marginal intertemporal welfare savings stemming from reduced risk of invasion. Biosecurity should not occur absent any risk of pathogen introduction ([[psi].sub.x] = 0) or risk of spread after an introduction ([rho] + V = 0). In other words biosecurity should not be used to eliminate risk because the marginal benefits of x would vanish.

Taking the time-derivative of (12), setting this equal to (14), and using (12) to substitute for [[lambda].sub.i] yields the following golden rule condition for managing host i:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (17)

Condition (17) equates the discount rate with the own rate of return to holding the resource in situ. Absent any risk of pathogen introduction ([MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]), the own rate of return equals the marginal productivity of the stock in reproduction plus a marginal cost savings term owing to the fact that a larger stock reduces unit harvesting costs (at least in the wildlife sector--this term likely vanishes for livestock). This is the standard outcome in models that do not account for pathogen risks. Denote the value of [N.sub.i](t) that satisfies the optimality conditions in this no-risk case as [N.sup.NR.sub.i](t), and assume [N.sup.NR.sub.i](t) > [[??].sub.i](t [less than or equal to] T) for at least one host. Otherwise, we are left with the uninteresting case in which there is no risk of invasion even when [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (i.e., an introduced pathogen cannot establish).

The last two terms in condition (17) are relevant if introduction risks are positive ([MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]). These terms are the expected intertemporal costs of a pathogen invasion arising from a larger host i density at the margin. If an invaded pathogen is expected to spread, then [rho] + V < 0 and also the ex post marginal value of the host is reduced ([V.sub.Ni] < [B.sub.i]). Accordingly, the last two terms are negative, which implies a reduction in host i's density relative to the no-risk case, ceteris paribus: [N.sup.*.sub.i][(t) < [N.sup.NR.sub.i](t), where [N.sup.*.sub.i](t) solves problem (9). It is not optimal, however, to reduce the host density to eliminate post-introduction risks of spread (i.e., [N.sup.*.sub.i](T) < [[??].sub.i](T) for both i = L, W). In that case, [rho] + V = 0 and [V.sub.Ni] = [B.sub.i], so that the last two terms vanish, indicating the marginal benefits of reducing risk vanish. The solution is then [N.sup.*.sub.i](t) = [N.sup.NR.sub.i](t), which exceeds the host-density threshold for at least one host and contradicts the condition for eliminating risk. Eliminating risk is therefore not optimal because the marginal costs of reducing risk exceed the marginal benefits.

An optimal strategy manages risk differentially by host, incorporating population controls and biosecurity. Biosecurity may be more efficient at the margin, particularly when it is a well-targeted approach to reducing cross-host transmission, and when it is applied in a highly managed setting (e.g., livestock production). Biosecurity may in turn reduce the planner's incentives to apply population controls, particularly to wild hosts.

Post-Infection Case

Suppose the pathogen has been introduced and is able to establish (or it established some time ago, but was only recently discovered and management has just begun). Redefining the current time period as t = 0, the bioeconomic problem is now given by:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (18)

subject to the equations of motion (5) and (6), where W is as defined in (9), and [tau] is the time period in which the pathogen is eradicated. The Hamiltonian for this problem is:

[H.sub.v] = [summation over (i=L,W)] [B.sub.i]([N.sub.i], [[theta].sub.i])[h.sub.i] - cx

+ [summation over (i=L,W)] [[lambda].sub.i][[??].sub.i], + [[mu].sub.i] [[theta].sub.i], (19)

where [[mu].sub.i] is the co-state variable for [[theta].sub.i]. The optimality conditions for (19), assuming a singular solution for [h.sub.i] and an interior solution for x, are:

[partial derivative][H.sub.V]/[partial derivative][h.sub.i] = [B.sub.i] - [[lambda].sub.i] = 0 (20)

[partial derivative][H.sub.V]/[partial derivative]x = -c + [summation over (i=L,W)] [[mu].sub.i] [partial derivative][[??].sub.i]/[partial derivative]x = 0 (21)

[[lambda].sub.i] = r[[lambda].sub.i] - [B.sun.iN][h.sub.i]

- [summation over (i=L,W)] [[[lambda].sub.i] [partial derivative][[??].sub.i]/[partial derivative][N.sub.i] + [[mu].sub.i] [partial derivative][[??].sub.i]/[partial derivative][N.sub.i]] (22)

[[??].sub.i] = r[[mu].sub.i] - [B.sub.i[theta]]

- [summation over (i=L,W)] [[[lambda].sub.i] [partial derivative][[??].sub.i]/[partial derivative][[theta].sub.i] + [[mu].sub.i] [partial derivative][[??].sub.i]/[partial derivative][[theta].sub.i]], (23)

plus the equations of motion (5) and (6) and transversality conditions for the terminal time and stocks (not presented here due to space limitations, though they imply [[lambda].sub.i]([tau]) = [partial derivative]W/[partial derivative][N.sub.i]([tau]) and [[theta].sub.i]([tau]) = 0). Conditions (20)-(23) have been interpreted elsewhere (e.g., Fenichel and Horan [in press]), so we do not do so here. These conditions differ from those of the ex ante case, indicating that efficient threshold and population management prior to T generally differ from efficient post-T threshold and population management. The result is that it can be costly to return the system to an uninfected state (Horan and Wolf 2005; Fenichel and Horan 2007).

We now explore the question of whether to eradicate the pathogen. Taking the time derivative of (20) and using the resulting expression for [[??].sub.i] in (22), we obtain the following golden rule condition for population management:

r = [F'.sub.i]([N.sub.i]) - [[alpha].sub.i] [[theta].sub.i] + [[B.sub.iN][[F.sub.i] - [[alpha].sub.i] [[theta].sub.i] N]

+ [[mu].sub.i][[beta].sub.ii](1 - [[theta].sub.i])[[theta].sub.i] + [[mu].sub.j] [[beta].sub.ij] [[theta].sub.i] (1 - [[theta].sub.j])]/ [[B.sub.i]. (24)

At first glance, all disease terms appear to vanish as [[theta].sub.i], [[theta].sub.j] [right arrow] 0, implying [N.sub.i] [right arrow] [N.sup.NR.sub.i] as [[theta].sub.i], [[theta].sub.j] [right arrow] 0. However, because [N.sup.NR.sub.i] > [[??].sub.i,t>T] (as has been assumed), eradication cannot occur and hence can never be optimal. The conclusion that eradication cannot be optimal is incorrect, though, for the disease terms do not vanish as [[theta].sub.i], [[theta].sub.j] [right arrow] 0, as we show below.

Condition (24) differs from standard golden rule conditions. In a first-best problem with selective harvests, another set of first-order conditions would allow us to eliminate [[mu].sub.i] and [[mu].sub.j] in condition (24), so that this condition could be solved for the optimal current states. The optimal strategy in that case would be either to eradicate infected hosts as quickly as possible, provided marginal harvesting costs are not too great as [I.sub.i] [right arrow] 0, or to move as quickly as possible to an equilibrium outcome in which [[theta].sup.*.sub.i] > 0. With nonselective harvests, the golden rule conditions do not define a unique optimal state that can be attained quickly. Rather, an optimal strategy is second-best and involves slower adjustment, owing to the fact that the controls are not well-targeted, and so changing prevalence is difficult and more costly (Horan and Wolf 2005).

Condition (24) (for i and j) can be used to solve for [[mu].sub.i] and [[mu].sub.j]. Then recognizing from equation of motion (6) that [[theta].sub.j]/[[theta].sub.i] [right arrow] 1 must hold as [[theta].sub.i], [[theta].sub.j] [right arrow] 0, we can derive:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (25)

where [[LAMBDA].sub.i] = [F'.sub.i]([N.sub.i]) + [B.sub.iN] [F.sub.i]/[B.sub.i] - r. The expression in (25) is finite, indicating that the incentives to manage the hosts in response to the disease do not vanish as [[theta].sub.i], [[theta].sub.j] [right arrow] 0. The pathogen will optimally be eradicated if: (a) eradication is also optimal in the first-best case, and (b) adjustment is not too slow. Otherwise, the disease optimally remains endemic. Slow adjustment increases the costs of control and hence eradication.

The optimality of eradication may depend on the ability to target transmission risks, as efficiency would be improved and adjustment sped up as better-targeted controls are implemented (Fenichel and Horan (in press)). In a wildlife-livestock system, livestock are often easier to target selectively by disease status (due to diagnostic testing), and biosecurity may be effective at targeting cross-host infections. Eradication may be optimal if livestock are responsible for sustaining the disease in wildlife. Conversely, if wildlife continually infects livestock, then well-targeted controls in the livestock sector may mitigate livestock-sector damages but not optimally lead to eradication. All else equal, the more effective are livestock controls at reducing livestock damages, the fewer incentives the social planner will have to reduce prevalence in wildlife. Finally, note that the risk of re-infection after eradication will reduce disease control incentives overall.

Discussion

Disease ecologists have made tremendous advances in understanding dynamic host-pathogen relationships. However, we find that useful ecological metrics such as [R.sub.0] cannot be directly applied to guide policy. Policy derived from such metrics is myopic and overly constrained. The implicit goal of such policy is to eliminate all pathogen risks--resulting in inefficiencies. Pathogen risks are endogenous and should be managed (differentially by host), but eliminating these risks is too costly; the marginal benefits of reducing risk tend toward zero, while the marginal costs are increasing at low-risk levels.

Even if it were optimal to manage risk at low levels prior to infection, it may not be optimal to eradicate an already-invaded pathogen. Indeed, returning to the uninfected state may be too costly because the economic and ecological tradeoffs are fundamentally altered after an invasion. Risk of re-infection further reduces incentives for eradication.

References

Barlow, N.D. 1996. "The Ecology of Wildlife Disease Control: Simple Models Revisited." Journal of Applied Ecology 33:303-14.

Barlow, N.D., J.M. Kean, N.P. Caldwell, and T.J. Ryan. 1998. "Modelling the Regional Dynamics and Management of Bovine Tuberculosis in New Zealand Cattle Herds." Preventive Veterinary Medicine 36:25-38.

Barlow, N.D., J.M. Kean, G. Hickling, P.G. Livingstone, and A.B. Robson. 1997. "A Simulation Model for the Spread of Bovine Tuberculosis within New Zealand Cattle Herds." Preventive Veterinary Medicine 32:57-75.

Bicknell, K.B., J.E. Wilen, and R.E. Howitt. 1999. "Public Policy and Private Incentives for Livestock Disease Control." Australian Journal of Agricultural and Resource Economics 43:501-21.

Chi, J., A Weersink, J.A. VanLeeuwen, and G.P. Keefe. 2002. "The Economics of Controlling Infectious Diseases on Dairy Farms." Canadian Journal of Agricultural Economics 50:237-56.

Dobson, A. 2004. "Population Dynamics of Pathogens with Multiple Hosts Species." The American Naturalist 164:s64-s78.

Dobson, A., and J. Foufopoulos. 2001. "Emerging Infectious Pathogens of Wildlife." Philosophical Transactions of the Royal Society of London B 356:1001-12.

Dobson, A.P., and M. Meagher. 1996. "The Population Dynamics of Brucellosis in the Yellowstone National Park." Ecology 77:1026-36.

Fenichel, E.P, and R.D. Horan. 2007. "Jointly-Determined Ecological Thresholds and Economic Tradeoffs in Wildlife Disease Management." Natural Resource Modeling 20:511-547.

--. 2007. "Gender-Based Harvesting in Wildlife Disease Management." American Journal of Agricultural Economics 89: 904-920.

Heesterbeek, J.A.P., and M.G. Roberts. 1995. "Mathematical Models for Microparasites of Wildlife." In B.T. Grenfell and A.P. Dobson, eds. Ecology of Infectious Diseases in Natural Populations. New York: Cambridge University Press.

Horan, R.D., and C.A. Wolf. 2005. "The Economics of Managing Infectious Wildlife Disease." American Journal of Agricultural Economics 87:537-51.

Lanfranchi, P., E. Ferroglio, G. Poglayen, and V. Guberti. 2003. "Wildlife Vaccination, Conservation and Public Health." Veterinary Research Communications 27:567-74.

Leung, B., D.M. Lodge, D. Finoff, J.F. Shogren, M.A. Lewis, and G. Lamberti. 2002. "An Ounce of Prevention or a Pound of Cure: Bioeconomic Risk Analysis of Invasive Species." Proceedings of the Royal Society of London B 269:2407-13.

Reed, W.J., and H.E. Heras. 1992. "The Conservation and Exploitation of Vulnerable Resources." Bulletin of Mathematical Biology 54:185-207.

Roberts, M.G., and J.A.P. Heesterbeek. 2003. "A New Method for Estimating the Effort Required to Control an Infectious Disease." Proceedings of the Royal Society of London B 270:1359-64.

Scantelbury, M., M.R. Hutchings, D.J. Allcroft, and S. Harris. 2004. "Risk of Disease from Wildlife Reservoirs: Badgers, Cattle, and Bovine Tuberculosis." Journal of Dairy Science 87:330-39.

United States Department of Agriculture, Animal and Plant Health Inspection Service (USDA-APHIS). 2002. Foot-and-Mouth Disease Vaccine Factsheet. Washington, DC: U.S. Department of Agriculture, APHIS Veterinary Services, March.

(1) Dobson and Foufopoulos (2001) define EIDs as infectious diseases that are increasing in prevalence, spatial range, or number of host types. Most are not newly evolved but occur historically in only a few populations and are exotic to recently invaded populations.

(2) There is no recovered population in SI models, implying that vaccination is not an option. This is the case for many EIDs because vaccines: (a) must be developed for particular disease strains and so may be ineffective against new outbreaks: (b) often only protect against clinical signs of the disease and not the disease itself, making it harder to detect an actual outbreak; and (c) can cause inoculated animals to test positive for the disease, risking sanctions by trading partners (e.g., USDA-APHIS 2002).

(3) The [R.sub.0] = 1 criterion is often discussed for unmanaged populations. In this case the result that invasion cannot occur when host density combinations result in [R.sub.0] < 1 should not be interpreted as a policy prescription because harvest mortality is not explicit in the model.

(4) The pathogen is likely to be introduced first into a single host population, but disease transmission in our model is such that the pathogen would be introduced into the second host population at the next instant. Because of this, we simplify matters and assume both hosts are initially infected, so as to not worry about which is infected first.

(5) Infected animals are unobservable, so detection and response likely occur after T. For simplicity we ignore this delay, but note that delay in switching from prevention to control: (a) will depend on monitoring effort, which will also be endogenously determined, and (b) will likely increase the incentives to invest in prevention.

Richard D. Horan is Associate Professor, Department of Agricultural Economics, and Eli P. Fenichel is Research Assistant, Department of Fisheries and Wildlife, both at Michigan State University.

Funding was provided by the Economic Research Service-USDA cooperative agreement number 58-7000-6-0084 through ERS's Program of Research on the Economics of Invasive Species Management (PREISM), and by NRI, USDA, CSREES, grant #2006-55204-17459. The views expressed here are the authors'.

This article was presented in a principal paper session at the AAEA annual meeting (Portland, OR, July 2007). The articles in these sessions are not subjected to the journal's standard refereeing process.