Economics and ecology of managing emerging infectious animal diseases.

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.

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