Economics and ecology of managing emerging infectious animal diseases.

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