horand e wolf(2005) the economics of managing infectious wildlife disease

8/8/2019 Horand e Wolf(2005) the Economics of Managing Infectious Wildlife Disease

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THE ECONOMICS OF MANAGING INFECTIOUS

WILDLIFE DISEASE

RICHARD D. HORAN AND CHRISTOPHER A. WOLF

We use a two-state linear control model to examine the socially optimal management of disease in avaluable wildlife population when diseased animals cannot be harvested selectively. The two controlvariables are nonselective harvests and supplemental feeding of wildlife, where feeding increases bothin situ productivity and disease prevalence. We derive a double singular solution which depends on theinitial state and does not require bang-bang controls. The case of bovine tuberculosis among Michiganwhite-tailed deer is analyzed. In the base model, the disease is optimally maintained at low levels, withintermittent investments (via feeding) in deer productivity.

Key words: bovine tuberculosis, double singular solution, linear control, nonselective harvests, white-tailed deer.

The spread of infectious disease among andbetween wild and domesticated animals hasbecome a major problem worldwide, with ex-amples involving wildlife as the primary vectorof disease transmission almost too numerousto count (see the Journal of Wildlife Diseases).The economic costs of wildlife-to-livestock dis-ease transmission can be substantial. For in-

stance, a foot and mouth disease outbreak inthe U.S., which could easily spread through awildlife vector such as deer, could cost billionsof dollars the first year alone (USDA-APHIS2002). For this and other diseases, farmersincur losses when infected livestock die orbecome less productive due to the disease,because the demand for livestock products isdiminished, and because of strict regulationsimposed when a herd becomes infected (e.g.,depopulating a herd). But the costsare not lim-ited to infected farms, as trade sanctions are of-ten imposed on entire counties, states, or evencountries where the disease is present. Human

Richard D. Horan and Christopher A. Wolf are associate pro-fessors, Department of Agricultural Economics, Michigan StateUniversity.

The authors gratefully acknowledge funding provided by theEconomic Research Service-USDA cooperative agreement num-ber 43-3AEM-3-80105 through ERS’ Program of Research onthe Economics of Invasive Species Management (PREISM). Theviews expressed here are those of the authors and should not

be attributed to ERS or USDA. The authors thank StephenSwallow, three anonymous reviewers, Erwin Bulte, Eli Fenichel,Larry Karp, Ken Matthews, and Aart de Zeeuw for help-ful comments and suggestions. Any remaining errors are theauthors’.

health is also an important concern and is oftenthe primary justification for eradication pro-grams and controls.

A disease outbreak among wildlife may alsoimpose costs on those who value wildlife prod-ucts or services. These costs would arise if hunters place a premium on healthy wildlife,and if infected or even healthy populations in

close proximity to an outbreak are culled toprevent additional spread. The costs could begreater for threatened or endangered species,particularly those protected in parks not largeenoughtosupportaviablepopulation.Aspop-ulation members wander outside protected ar-eas, the risk of infection increases—both forwandering wildlife and for those in protectedareas. Conservation measures must thereforebe taken with disease control in mind (Simon-etti).

There has been relatively little research inthe area of the economics of disease con-trol among wildlife populations. Most ani-mal disease work has estimated the coststo farmers and consumers under alternativecontrol strategies, with little regard givento the wildlife dimension (e.g., Mahul andGohin; Kuchler and Hamm; McInerney; Ebel,Hornbaker, and Nelson; Dietrich, Amosson,and Crawford; Liu). An exception is Bicknell,Wilen, and Howitt, who developed a bioeco-nomic model to analyze a bovine tuberculosis

(TB) problem in New Zealand in which TB isspread by Australian brush-tailed possums todairy herds. Specifically, they explored optimaldisease control strategies (e.g., testing at the

Amer. J. Agr. Econ. 87(3) (August 2005): 537–551Copyright 2005 American Agricultural Economics Association

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538 August 2005 Amer. J. Agr. Econ.

farm level and hunting possums off the farm)from the perspective of a single farmer.

Ourwildlifediseasemodeldiffersfrompriorliterature in three important ways. First, weconsider the socially optimal management of an infected wildlife population as opposed to

controls taken by a single farmer. Wildlifedisease problems affect many people (e.g.,many landowners and hunters), and so an in-dividual farmer would tend to under-invest indisease control investments that provide ex-ternal benefits. Second, we assume infectedwildlife cannot be identified until after theyare killed and examined (Williams et al.), ren-dering it impossible to selectively harvest onlyinfected animals. Accordingly, any off-takeof infected animals is likely to be accompa-

nied by healthy animals that may be econom-ically valuable. Third, healthy but susceptiblewildlife such as deer and lions (which are in-fected with TB in South Africa) may hold con-siderable economic value. Thus, exterminatingwildlife as a way of eradicating a disease out-break, as is often proposed, might be a com-paratively costly approach.

The model is applied to the case of bovineTB among white-tailed deer in Michigan, theonly known area in North America wherebovine TB has become established in a wilddeer population. In the early to mid-1990s,signs of bovine TB started to re-emerge both inthe wild deer population and also among somefarms. Michigan lost bovine TB accredited-free status in June 2000 and was requiredto adopt a testing program for all Michigancattle, goats, bison, and captive cervids. Inaddition, other states could place movementrestrictions on Michigan livestock at their dis-cretion (MDA, USDA-APHIS 1999). On 1June 2004, Michigan received “split-state” sta-

tusforbovineTB,resultingintwodiseaseman-agement zones having separate requirementsfor animal movement, identification, and test-ing. This status came about because extensivetesting found the disease confined to the north-east corner of Michigan’s lower peninsula, andso regulatory costs are now primarily confinedto this area. Michigan agriculture is obviouslyconcernedabout disease-relatedcostsand sup-ports culling the deer population to eradicatethe disease. However, such extreme measures

could be very costly, particularly since deerhunting is arguably the highest-valued use of the land in the infected region.

A final contribution of this article is amethodologicalone. Because our bioeconomicmodel involves two state variables, we adopt

a linear control model to make the problemtractable in two dimensions. In contrast to thebang-bang solutions that arise in conventionallinear control models, the solution is a nonlin-ear feedback law along a singular arc, wherethe particular arc pursued will vary accord-

ing to the initial state and where bang-bangcontrols are not required to initially move tothis arc.

A Model of Infectious Wildlife Disease

Consider a wildlife population that grows un-exploited on a fixed land area. The aggregatewildlife population, N , consists of two sub-populations: a healthy but susceptible stock, s, and an infected stock, z. In the absence of

exploitation or disease, the susceptible stockgrows according to the logistic growth func-tion, rs(1−N /k), where r is the intrinsic growthrate and k is the carrying capacity. The logisticfunction is not required for the general model,but the use of a specific functional form fa-cilitates understanding of the population dy-namics and the logistic is consistent with ournumerical model of Michigan white-taileddeer. Following Barlow (1991a), the density-dependent component of the logistic equation,

(1−N /k), depends on the aggregate popula-tion because susceptible and infected wildlifecompete for the same habitat.

Population growth is reduced as members of the susceptible stock become infected, whichoccurs when a susceptible animal comes intocontact with an infected animal. Disease trans-mission can be modeled in a number of ways.Historically, ecologists have used a mass actionor density-dependent transmission function,zs, where is the contact rate per infectious

deer. McCallum, Barlow, and Hone note, how-ever, that the mass action model often doesnot hold up empirically. A competing trans-mission function, which often fits data betterthan mass action for diseases such as cowpoxin bank voles and wood mice (Begon et al.1998, 1999) and brucellosis in Yellowstonebison (Dobson and Meagher), is frequency-dependent or density-independent transmis-sion. The frequency-dependent model, whichwe adopt, assumes that the z infected animalsmake on average z contacts in each time pe-riod, with s/N contacts being with susceptibleanimals so that total transmission is zs/N .1

1 McCallum, Barlow, and Honepropose a functionalform whichreduces to mass action at one extreme parameter value and


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Wildlife Management and Disease Controlfor Michigan White-Tailed Deer

Bovine tuberculosis among Michigan white-tailed deer is primarily concentrated in a four-county area in the northeastern part of the

lower peninsula, formally designated as deermanagement unit (DMU) 452 or less-formallyas the “core” (see Hickling). A few cases of infection have been found beyond this areabut the disease does not appear to be sus-tainable outside the core. This has led manyto speculate that unique, core-specific featuressuch as human–environment interactions—particularly feeding programs—have enabledthe disease to become endemic (Hickling).Indeed, prior to 1995, only eight cases of

bovine TB had ever been reported in wilddeer from North America (Schmitt et al.) andconventional wisdom held that the diseasewas not self-sustaining in wildlife populations(Hickling).

Several hunt clubs in the core sponsoredfeeding programs to increase deer density.Originating in the late 1800s, these clubs pur-chased large amounts of core area land onwhich its members could hunt. This land wasdesirable because it was easily accessible from

highways and, as it consisted of generally poorsoil for agronomic purposes, the land was in-expensive (Hickling). The historic density of deer in the area is estimated to have been 7–9deer/km2 (O’Brien et al.). The hunt clubs, de-siring greater density, began aggressive deerfeeding programs to encourage herd growth.These programs have been known to dumptractor–trailer loads of food in the woods andfringe areas, with the resulting massive foodpiles being visible from the air along with thetracks of many congregating deer. The result isthat deer density increased to an estimated 25deer/km2 by the mid-1990s. Of course, while in-creasing the carrying capacity of core deer gen-erates hunting benefits, while supplementalfeeding also generates external costs by lead-ing to increased disease transmission as deercongregate, and by supporting sick animals,thereby reducing disease-related mortality.

Denote f as feed provided by feeding pro-grams. Increased feed availability reduces thedensity-dependent component of growth by a

factor (1− f ), increases the disease transmis-sion coefficient by a factor (1 + f ), and de-creases mortality due to the disease by a factor(1− f ), where , , and are parameters. Theequations of motion for the deer populationand disease prevalence rate, (3) and (4), are

modified as follows to account for these im-pacts of feeding

˙ N = r N [1− ( N /k )(1− f )]

−(1− f ) N − h

(5)

= [(1+ f )− (1− f )](1 − ).(6)

With r > (as is widely believed), the diseasewould be endemic in the core if (1 + f ) >(1− f ).3 If >, thenthe disease will persistregardless of feeding or hunting choices (apartfrom wildlife eradication). In that case, someother effort to reduce disease transmission willbe required to eradicate the disease. But if < , then the disease would be eliminated bysetting f < [−]/[ + ] for some time.

A smaller f means the disease is eliminatedearlier but at an interim cost of lost deer pro-ductivity. Of course, it is important to considerwhether eradication is optimal.

Economic Specification

Hunters gain utility from the actual process of shooting deer or consuming meat and otherdeer products. The (constant) marginal utilityfrom harvesting healthy deer is denoted by p,which is not less than the (constant) marginalutility from harvesting infected deer, pz, thatis, p ≥ pz. For simplicity, we set pz = 0 sothat harvests of infected animals yield no ben-efits.4 The benefits from hunting are therefore phs/N = p(1− )h.

Assume harvests occur according to theSchaefer harvest function (although in generalthis specification is not required), and that theunit cost of effort, c, is constant. Then, totalharvesting costs, restricted on the in situ stocks,are (c/q)h/N , where q is the catchability coef-

ficient. The unit cost of food is w.Finally, the costs of the disease, particularly

to farmers and related agribusinesses, mustalso be considered. Denote the variable eco-nomic damages caused by infected deer byD(z) (with D(0) = 0, D > 0, D ≥ 0). Thesevariable damages are due to infections in thecattle herd that result in lost stock, increasedtesting, and business interruption loss. Theimposition of trade restrictions and federallymandated testing requirements in response to

3 The disease would not be sustainable outside the core if 0 < 0, where 0 and 0 represent parameter values outside thecore. These parameters may differ from and due to human–environment interactions apart from feeding.

4 This assumption should not affect the qualitative nature of theresults, but it may affect the trajectories in the numerical exercise.


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Horan and Wolf Managing Infectious Wildlife Disease 541

the disease may also result in a significant lumpsum damage component. Such lump sum dam-ages are primarily policy-induced and, if largeenough, could affect the optimal plan. We be-gin by investigating an optimal plan withoutthese lump sum costs, as the solution is efficient

from Michigan’s point of view in the absenceof exogenous regulatory impositions. Later, inthe numerical section, we analyze the effect of the regulatory-based lump sum costs. We findthat these additional costs might sufficientlyalter Michigan’s optimal management plan sothat it is more in line with the federal govern-ment’s stated objectives.5

Optimality Conditions and the DoubleSingular Solution

Wildlife managers have two objectives whendealing with the disease: reduce the number of diseased animals and control the spread of thedisease. To accomplish these goals, managershave focused on harvest levels and the amountof food provided by feeding programs as theprimary choice variables (Hickling). Given thediscount rate , an economically optimal allo-cation of harvests andfeeding maximizessocialnet benefits, SNB, that is,

Maxh, f

SNB

=

∞0

[ p(1− )h − (c/q)(h/ N )

−w f − D( N )] e− t dt

(7)

subject to the equations of motion (5) and (6)and the feasibility conditions h ≥ 0 and 0 ≤ f ≤ f max.6 The current value Hamiltonian is

H = p(1− )h − (c/q)(h/ N )−w f − D( N ) + [r N [1− ( N /k )(1− f )]

−(1− f ) N − h]+ [([1+ f ]

−[1− f ])(1 − )]

(8)

where and are the co-state variables asso-ciated with N and , respectively.

5 Deer are also important causes of automobile accidents and

damage to agricultural crops (Rondeau, Rondeau and Conrad).We ignore these other damages in order to focus on the impacts of disease, but we note that these other damages could be important.

6 We do not place an upper bound on h. Rather, our primaryconcern is that f ≤min(1/, 1/ ). A value of f > 1/ would result ina negative mortality rate due to the disease, which is not possible.A value of f > 1/ would result in a negative density-dependencefactor, which also does not seem realistic.

The marginal impact of harvests on theHamiltonian is given by

∂ H /∂h = p(1− )− c/(qN ) − .(9)

If this expression is positive so that marginal

rents exceed the marginal user cost, then har-vests should be set at their maximum levels. If this expression is negative, then no harvestingshould occur. The singular solution is pursuedwhen marginal rents and the marginal usercost are equated. This is the standard condi-tion for linear control problems involving re-newable resources (e.g., Clark), except for twoimportant differences. First, marginal rents arereduced by p because not all harvested ani-mals are valued (as some are infected). Sec-

ond, because harvests of N are nonselective,the marginal user cost of N can be positive ornegative, that is, the sign of is ambiguous. Itis easy to show that the following relation musthold: = (∂SNB/∂ s)(1− ) + (∂SNB/∂z),where ∂SNB/∂z < 0 and ∂SNB/∂ s > 0 or <0depending on whether increases in s only endup fueling the growth of a larger infected pop-ulation z. If = 1, then all additions to N only add to the infected stock so that < 0,and vice versa when = 0. There must exist a

set of values for the state variables such that = 0, other sets such that > 0, and still oth-ers such that < 0. A potential nonconvex-ity therefore emerges, with the possibility of multiple optimality candidates (Rondeau; seealso Tahvonen and Salo; Huffaker and Wilen;Maler, Xepapadeas, and de Zeeuw). The po-tential for nonconvexities does not arise whenharvests can be made selectively.

Now consider the marginal impacts of feed-ing on the Hamiltonian

∂ H /∂ f = −w + [r ( N 2/k ) + N ]

+[ + ](1− ).

(10)

Feeding can be thought of as an investmentin the productivity of the resource, althoughit has the unwanted side-effect of increasingdisease prevalence. As we show below, the so-lution has similarities to, but also importantdifferences from, Clark, Clarke, and Munro’sproblem of investing in harvesting capital. The

singular solution should be followed wheneverthe unit cost of feeding equals the in situ netmarginal value of feeding on the two state vari-ables. The in situ net marginal value is thedifference between the marginal benefits of feeding on the overall stock (which includes


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increased productivity and decreased mortal-ity, and which may be negative when < 0)and the marginal costs of feeding in terms of anincreased proportion of infected animals (dueto increased transmission and decreased mor-tality among the infected stock). If the

marginal in situ values exceed the unit cost,then feeding should proceed at the maximumrate. If the unit cost exceeds the in situ value,then feeding should optimally cease.

It must be the case that < 0, for disease isnever beneficial. Equation (10) therefore im-plies that > 0 must hold along a singular path,and so nonconvexities can only emerge alonga nonsingular feeding path. Clearly f = f max

is not optimal if < 0 as feeding would onlycreate costs in this case. The only nonsingular

feeding path consistent with < 0 is f = 0, but < 0 would not be sustainable in this case be-cause → 0 as long as f = 0 is maintained (seealso footnote 10).

The necessary arbitrage conditions for anoptimal solution are given by:

= − ∂ H /∂ N = − ch/(qN 2)

+ D − [r − 2r ( N /k )(1− f )

−(1− f )]

(11)

= − ∂ H /∂ = + ph + D N

+(1− f ) N − [(1+ f )−(1− f )](1 − 2).

(12)

Conditions (11) and (12) reflect intertem-poral changes in optimal marginal resourcevalues.

The remainder of this section is devoted toderiving the conditions that characterize thedouble singular path—that is, the solution pathinvolving singular controls for both harvest-ing and feeding, so that conditions (9) and

(10) both vanish. As we show in the numeri-cal section to follow, partial singular solutions,in which the solution is singular for only onecontrol variable, will also be part of the over-all solution. Partial singular solutions in thiscontext arise as part of a blocked interval, a

period of time during which one of the con-trols is “blocked” or constrained from follow-ing the double singular path (Arrow, Clark).Blocked intervals will be shown to introducesome interesting complexities into the model.However, to understand howthe solution tran-

sitions from the double singular solution to ablocked interval, it is first necessary to under-stand the unconstrained solution, for “the opti-mal path must always lie as close as possible tothe [double] singular path” (Clark, p. 56). In-deed, it will be helpful to first graph the doublesingular solution in the context of a numericalexample before examining the complete solu-tion of double and partial singular paths. Wetherefore delay our discussion of partial singu-lar solutions until the numerical section below.

Differentiating condition (9) with respect totime, substituting the right-hand side (RHS) of condition (12) in for , and using (9) to substi-tute for the co-state variable , we have

= r −2r N

k (1− f ) − (1− f )

+c/(qN 2)[r N [1− ( N /k )(1− f )] − (1− f ) N ]− D

p(1− )− c/(qN )

− p[(1+ f ) − (1− f )](1− )

p(1− )− c/(qN ).

(13)

Equation (13) is a variant of the conventional“golden rule” for renewable resource manage-ment: the rate of return for holding the healthystock in situ equals the marginal productiv-ity of the stock, plus net marginal stock ef-fects (i.e., the marginal cost savings that accrueas harvests come from a larger stock minusthe marginal damages, normalized by marginaluser cost), minus the (normalized) value of

foregone revenues as some of the remaininghealthy in situ stock will become infected andresult in a larger proportion of infected deer infuture harvests.

Equation (13) must hold at all times alongthe double singular path (or even a partial sin-gular path involving only the harvest; see be-low). In conventional autonomous renewableresource models involving a single state vari-able, the singular solution is a point, N ∗, be-cause the golden rule is only a function of the

stock and can be solved for a unique value of N . In contrast, condition (13) is a (linear) func-tion of one of the control variables, f . If wesolve equation (13) for f as a function of thecurrent state variables, N and , the result is anonlinear feedback law (Bryson and Ho). This


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feedback rule implies that the double singularsolution will be a path.

Now differentiate condition (10) with re-spect to time, substitute the right-hand side(RHS) of conditions (11) and (12) for and ,respectively, and use conditions (9) and (10) to

substitute for and (the procedure is anal-ogous to the one described for equation (13),and is outlined by Conrad and Clark, and oth-ers). The result is a golden rule expression formanaging disease prevalence

= −{[−2 pr 2 (1− f ) N 3/k 2

− c(+ )](1− )

+ h[cr /k − 2 pr ( N /k )

+ p ](1− ) + N [ D(+ )

+ rp+ p[+ ]

− p(2+ + f )](1− )

+ (r N 2/k )[2 pr − p

− p[ + ] − p[ − 2] f ]

× (1− ) + (r N /k )[(r N /k )c

× (1− f )− cr + c (1− f )]}

× {− pr ( N 2/k )(1− )

+ (w + c) + Ncr /k − N p(1− )}−1.

(14)

Equation (14), which depends linearly on bothcontrols, h and f , must also hold along the dou-ble singular path. If we plug the feedback lawfor f into this expression, it is possible to solveequation (14) for a feedback law for the har-vest, h(N , ).

The feedback laws h(N , ) and f (N , ) canbe plugged into the differential equations (5)

and (6) to (numerically) solve for the doublesingular path, given the initial states, N 0 and0, and assuming that the feedback laws sat-isfyfeasibility conditions at these initial states.7

Note that the double singular path—the solu-tion to the equations of motion—depends onthe values of the initial state variables. Hence,different singular arcs will be defined for dif-ferentinitial states, and so the system can beginon a singular arc (an interior solution) without

7 That equations (9) and (10) both vanish when the feedbackrules arefollowed, for anystate variable combination such that thenonnegativity constraints are satisfied, is verified by setting equa-tions (9) and (10) equal to zero and noticing that the coefficientmatrix for the vector [ ] for this system is not singular—thus, aunique value of both and satisfy the singular conditions for allrelevant combinations of N and .

the need for bang-bang controls to move tothe arc. In contrast, more conventional singu-lar arcs are independent of the initial state—they are defined by a single curve (or a point),with bang-bang controls required to move tothe curve if it does not pass through the initial

state.8

Numerical Example

We now examine the optimal solution numeri-cally because the feedback rules and the differ-entialequationsthatdefinethesolutionaretoocomplex to analyze analytically. The data usedto parameterize the model are described in theAppendix. While we have made every effort tocalibrate the model realistically, research onthe Michigan bovine TB problem is still evolv-ing and at a fairly early stage so knowledge of many parameters is somewhat limited. The fol-lowinganalysisisthereforebestviewedasanu-merical example rather than a true reflectionof reality. Nonetheless, the results shed light onthe economics of wildlife disease managementin general and specifically on TB in Michigandeer.

The numerical solution, arrived at usingthe software Mathematica 5.0 (WolframRe-search), is presented in figure 1 for the case

of = 0.1 (we explore the sensitivity of thischoice of below). Although not presented,an interior but unstable steady state arises atthe point (N = 5,561, = 0.013), just northeastof the point d. This steady state is an unsta-ble focus, which means that it is only optimalto be at this point if the system begins at thispoint. Otherwise, it is optimal to spiral awayfrom this point (see Clark), which explains thecyclical nature of paths in its vicinity. Indeed,we find an interior cycle, involving a portion of

path 3 along with paths 4 and 5, is optimal.The optimal solution begins at the initialstate values N 0 and 0, represented by pointa in figure 1. From this point, the doublesingular path 1 is followed. This path spiralsaway from the interior equilibrium, initiallyincreasing and N . The result that feedingshould be initially encouraged runs contrary toMichigan’s current policy approach of banning

8 Linearcontrol problems having nonlinear feedback laws as the

solution seem to be more common in noneconomicapplications, atleast for the class of autonomous problems. For instance, Brysonand Ho provide a famous example of optimal thrust programsfor rockets. Swallow provides the only example of which we areaware in the resource economics literature. He derives a nonlinearfeedback rule for the harvest of a resource stock; however, hissingular solution lies along a single curve, with bang-bang controlsrequired when the system is initially off the curve.


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Figure 1. Solution of the benchmark numerical example ( N 0 = 13,298; 0 = 0.023; r = 0.5703;k = 14,049; = 0.339; = 0.356; = 0.00008; = 2.64 × 10−6; = 0.3623; p = 1270.8;c / q= 231,192; w= 36.53; = 5491; = 0.1)

feeding (although it is consistent with the vol-untary actions of the hunt clubs in recentyears). Feeding represents an investment instockproductivity, initially increasing the stockwhile enabling large harvests along the singu-

lar path 1. The disease prevalence rate goesup along the singular path, but the increaseddamages are offset by the rewards of largernear-term harvests.

Feeding and also prevalence rates con-tinue to grow along the path 1. Eventually f (N , ) = f max = 10,000, represented by theboundary f = f max in figure 1. This bound-ary creates a blocked interval that preventsthe state variables from following the dou-ble singular path (Arrow; Clark, p. 56).9 Thefarsighted planner knows the boundary is ap-proaching. Therefore, at some point prior tothe singular path becoming constrained at the f = f max boundary, the planner will set f equalto an extremal value and abandon the my-opic double singular path, as is required by the“premature switching principle” (Clark, p. 57).Arrow (p. 16) points out that actually findingthe optimal path “is a process of trial and er-ror,” first finding the eligible intervals and thennumerically finding when it becomes optimalto jump from one interval to another.

9 The continuation of the singular path 1 is not illustrated, butafter the boundary it changes direction and moves northwest: withincreasing prevalence, damages are reduced by reducing the stocksince prevalence is increasing.

The first interval consists of path 1, and thesecond interval begins by setting f = f ma x priorto having reached the f = f ma x boundary. De-fine T as the time at which this extremal valueof f is chosen and the double singular path 1 is

abandoned. At T ,itbecomesoptimaltopursuethe partial singular solution for N conditionalon f = f ma x—at least provided that f = f ma x

will be maintained for more than an instant.Suppose this was the case. Then, the result-ing partial singular solution is characterized byequation (13), holding f fixed at its constrainedvalue. Specifically, equation (13) can be solvedfor N (, f ma x), a unique deer population giventhe value of at time T and also given f = f ma x. This value is optimally approached alonga most rapid approach path (MRAP). How-ever, N (, f ma x)liesatpoint c infigure1,whichis to the left of the f = 0 curve. At this point,it is actually optimal to set f = 0 and movetowards the f = 0 curve along a MRAP thatalso involves zero harvests (so that N mightincrease), resulting in damages accruing alongthis path without any offsetting benefits. Sincewe know that f = 0 is optimal immediately af-terthecullat T ,itisbettertoset f = f ma x foraninstant at point b, and then immediately jumpto point c, where f = 0 is optimal. Path 2 in

figure 1 illustrates this jump.The instantaneous move from the f = f ma x

boundary to the f = 0 boundary may at firstglance seem excessive, but it makes sense fromthe perspective of the premature switching


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principle. For instance, suppose we had not jumped prematurely at point b but rather hadfollowed the double singular solution to the f = f ma x boundary, and then set f = f ma x andmoved northwesterly along the boundary. Inthis case, the f = f ma x constraint would not

remain binding indefinitely, for it eventuallyleads to very high prevalence rates and lowstocks. Rather, at some point, it would becomeoptimal to follow another double singular so-lution: moving northwesterly along the f = f ma x boundary, a double singular path (not il-lustrated) does eventually emerge from the f = f ma x boundary and moves towards the f = 0 boundary. The f = 0 boundary wouldthen be followed, reducing prevalence and al-lowing the stock to recover. This long pro-

cess would eventually lead to point c, butthe jump along path 2 gets there sooner.The premature switching principle says thatif we foresee constraints on the horizon, thenan early adjustment will produce larger netbenefits. Our numerical simulations confirmthis.

We now move on to define the third inter-val in our solution. Once on the f = 0 bound-ary, we must check to see if it becomes optimalto pursue a double singular solution given thecurrent stock and prevalence levels. At pointc, the unconstrained double singular solutionwould move to the left of the f = 0 curve,which implies f < 0 and hence is infeasible.Hence, it is optimal to maintain zero feeding.Given that f is constrained exogenously in thisfashion, equation (13) can be solved for N (,0), with moving exogenously through time.Specifically, N (, 0) is derived from the fol-lowing modified form of equation (13) with f = 0

= r −2r N

k −

+c/(qN 2)(r N [1− N /k ]− N )− D

p(1− )− c/(qN )

−p( − )(1− ) p(1− ) − c/(qN )

.

(15)

The solution N (, 0) is a partial singular path

for N , which is essentially a nonautonomoussingular path given the exogenous movementof (e.g., see Clark or Conrad and Clark) andis optimally approached along a most rapidapproach path (MRAP). This singular path,which we label as path 3 in figure 1, coin-

cides with the f = 0 curve in the presentmodel.10

Disease prevalence diminishes while wild-life stocks increase along the partial singularpath 3. Continuation along this path wouldeventually lead to an outcome with a disease-

free wildlife stock, after which time feedingcould be reintroduced without creating anydisease problems. But that outcome is not pur-sued. When point d is attained on path 3, f = 0 is no longer a binding constraint butrather the solution to the feedback function f (N , ) along a double singular path, labeled4, that moves northeasterly away from the f = 0 boundary. At this point, small amountsof supplemental feeding cansignificantly boostproductivity while adding little to disease

prevalence, as is reflected by the relatively flatslope of path 4 in the vicinity of d. The op-portunity cost of waiting for the disease to dieout therefore becomes too high relative to thegains that can be made from re-investing indeer productivity at d. So, the planner aban-dons the partial singular path 3 for the doublesingular path 4, which is the fourth solutioninterval.

Path 4 eventually turns around and headsback to the f = 0 boundary, as the marginalcosts of feeding in terms of increased dis-ease prevalence approach the marginal ben-efits of feeding on increased deer productivity.But knowing that this boundary is imminent,the premature switching principle requires theplanner to optimally cull the stock some timeprior to reaching the boundary, jumping to apoint such as e by way of path 5, the fifth inter-val in the solution. Point e lies on path 3 andso, once at e, the remaining portion of path 3 ispursued and the cycle 3-4-5-3 repeats. The dis-ease is never eradicated because the deer are

highly valuable and feeding intermittently be-comes a good investment to boost productivityof the stock.

Once the eligible intervals have been iden-tified, the solution algorithm proceeds as fol-lows. Starting at point d (which is easilycalculated numerically as N = 5,379, =

0.009) and moving along path 4, determine the

10 Note that > 0 everywhere along path 3, andso nonconvexities

do not emerge. Nonconvexities could emerge on the f = 0 curvewhere p(1− ) < c/(qN ), or 1− c/( pqN ) < , but this only occursat very large values of . Such an outcome could optimally ariseas part of a solution for a problem involving much larger initialprevalence rates than the one in this example. But even in suchinstances would diminish as f = 0 is maintained, and eventually would becomepositive. The remainderof thesolutionwouldthenconverge with the one being described for the present example.


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stock-prevalence rate combination on thispath for which switching from path 4 to path 5(and then continuing in the equilibrium cyclein perpetuity) results in maximum discountednet benefits. Once the equilibrium cycle is cal-culated, it is possible to start at point a and

determine the stock-prevalence rate combina-tion for which switching from path 1 to path 2(and then moving on path 3 and into the equi-librium cycle) results in maximum discountednet benefits. The solution is (1) starting at pointa, follow path 1, which involves increasing sup-plemental feeding and harvesting rates, untilN = 14,123 and = 0.05; ( 2) an instantaneouscull is required to bring the deer populationto N = 2,970; ( 3) follow path 3, which involveszero feeding and increasing harvests, until N =

5,379 and = 0.009; (4) follow path 4, whichinvolves increasing supplemental feeding andharvesting rates, until N = 7,681 and = 0.018;( 5) cull to a population of N = 4,874, and con-tinue the equilibrium cycle.

In many respects, the optimal path is simi-lar to that of Clark, Clarke, and Munro, whoanalyze irreversible investments in harvestingcapacity for renewable resources. They find itis optimal to temporarily over-capitalize (rel-ative to the steady state) prior to a stock-depletion phase. The reason is that the largercapital levels allow more harvesting earlyon, which generate greater near-term bene-fits prior to advancing to the steady state.Somewhat analogously in our model, we findthat initial and intermittent future investmentsin resource productivity create opportunitiesfor near-term gains. But, an important differ-ence is that a steady state is not optimal inour model. Unlike Clark, Clarke, and Munro,investment in our model (via feeding) pro-duces adverse effects on resource dynamics:

along with the productivity enhancing invest-ments comes the unwanted side-effect of thedisease, and sustained investment (feeding)would only lead to increasing disease preva-lence. If allowed to continue unabated, thisincreasing prevalence eventually causes dam-ages to swamp benefits. Therefore, intermit-tent dis-investment in the disease is warranted.

Incorporating Fixed Damage Costs

Consider now the imposition of trade restric-tions and federally mandated testing require-ments in response to the disease that resultin a significant lump sum damage component.Wolf and Ferris estimated $4 million annually

in such costs under the current split-state TB-free status. Such costs do not affect the op-timal interior cycle that we just described, asthe marginal incentives are unaffected. But, of course, the lump sum costs never vanish underan interior strategy, and so these costs could

affect whether the interior cycle is in fact op-timal over a strategy to eradicate the diseaseand hence eliminate the lump sum costs.

Given the magnitude of the lump sum dam-age costs, we find the optimal eradication strat-egy is to immediately stop feeding and imme-diately cull to the f = 0 curve. Once on thatcurve, the solution is of the partially singulartype as harvesting is set at levels to achieve thestock levels that solve equation (15). The f =0 curve is followed until = 0, at which point

feeding can resume. It can easily be verifiedthat the singular solution in this case involvesequation (9) being satisfied as a strict equal-ity and equation (10) as a strict inequality, sothat feeding should be set at itsmaximum level.Equation (13) then uniquely determines thesingular stock, N ∗=0, f = f max, which should beapproached along a most rapid approach path.This implies zero harvests until the stock hasincreasedto the steady-state value N ∗=0, f = f max,

which equals 30,942 deer (compared to N 0 =

13,298). Simulated discounted net benefits un-der this scenario appear to be about $0.5 mil-lion larger under the eradication strategy.11

Hence, the eradication strategy appears opti-mal in the presence of lump sum damage costs.Because these costs are exogenously imposed,primarily as a result of federal policies, andbecause the federal government has clearlystated eradication as a policy goal (USDA-APHIS 1999), the results suggest that federalpolicies ensure that eradication is an incentivecompatible strategy for Michigan to follow.

Sensitivity Analysis

We conclude our numerical example with asensitivity analysis (of the model with no fixedcosts)designedtoshedlightonhowcertainkeyparameters influence the optimal interior solu-tion. First, consider figure 2, which illustrates

11 It is difficult to report a comparison of social net benefits, aseradication is an asymptotic process that does not occur in finite

time in the simulation. Our estimate of a $0.5 million advantagefor the eradication strategy arises for eradication time periods forwhich the combination of prevalenceratesand stock levelssuggestabout 25–50 infected deer. While not extremely close to zero, theprevalence rate seemed sufficiently low and the time intervals suf-ficiently long to believe the disease would be wiped out in reality.


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Figure 2. Solution of the numerical example when = 0.05 (with all other parameter valuesas in figure 1)

the solution when = 0.05. There are severalimportant differences between figures 2 and 1.First, the double singular path 1 is negativelysloped in figure 2 but positively sloped in fig-ure 1. This difference arises because a smallerdiscount rate results in a more equal weightingof the near-term benefits that accrue from in-vesting in insitu deer productivity (via feeding)and the costs that stem from the greater associ-ated disease prevalence. Accordingly, feedingoccurs at lower levels under smaller discountrates and hence provides a smaller productiv-ity boost. Moreover, the jump to the f = 0curve occurs at a greater distance from the f = f max curve (when N = 13,298 and = 0.028).Another difference is that the double singularpath 4 in figure 2 has shifted down and exhibitsa shorter upward path relative to its analogueinfigure1(withpath4startingat N =6,163, =0.006, and path 5 starting at N = 7,681, =

0.018). The shorter upward path results fromthe same sort of tradeoff influencing sin-gular path 1. The downward shift resultsbecause the smaller discount rate reduces theopportunity cost of waiting for smaller diseaseprevalence levels relative to the benefits of feeding-induced productivity enhancements.In sum, the equilibrium cycle is shorter andoccurs at lower disease prevalence rates forsmaller discount rates.

Solutions analogous to that in figure 2 could

also arise for larger marginal damages, feed-ing costs, or disease mortality rate. For in-stance, upon increasing from 0.356 to 0.373(so that / increases from 1.05 to 1.1), hold-ing all other parameters from the basic model

constant, the solution looks almost identicalto figure 2 but for different reasons.12 Withlargerdisease mortality, infected deer die morerapidly. This means the productivity boost tothe aggregate deer stock is not as great andthe stock declines more rapidly for a given har-vest level—hence the negatively sloped dou-ble singular path 1 and the short upwardportion of the double singular path 4. Greaterdisease-related mortality also decreases theopportunity cost of waiting for reduced diseaseprevalence. Hence, the downward shift in sin-gular path 4. Similar results occur for largermarginal damages, which reduces the oppor-tunity cost of waiting for reduced prevalence,and for larger feeding costs, which reduces themarginal benefits of productivity investmentvia feeding and hence reduces the cost of wait-ing for reduced prevalence.

If feeding costs, marginal damages, or /

is increased enough, we find that the oppor-tunity cost of eradication of the disease be-comes optimal in the long run. This process ispresented in figure 3 for the case of increasedfeeding costs. After an initial productivity in-vestment, there is a jump to the partial singu-lar path 3 along the f = 0 curve. This partialsingular path is optimally followed until =0 and N = N ∗=0, f =0, as there is no double

singular path that moves easterly out of the

12 The solution arising under a mass action transmission functionis also qualitatively similar to figure 2. This is because density-dependent disease transmission implies larger opportunity costsassociated with holding the deer stock in situ, so that there areincentives to start depleting the stock sooner and to maintain it atlower levels.


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Figure 3. Solution of the numerical example when feeding costs are increased tenfold (with allother parameter values as in figure 1)

f = 0 curve (coinciding with the fact that thereis no interior focus point in this case). Once = 0, feeding is set at its maximum level,provided the feeding costs are not too great(otherwise, the system remains at N ∗=0, f =0 =

5,921 deer). Equation (14) then uniquely de-

termines the singular stock, N ∗

=0, f = f max ,whichshould be approached along a most rapid ap-proach path 4. This implies zero harvests un-til the stock has increased to the steady-statevalue N ∗=0, f = f max = 30,942 deer.

Discussion and Conclusions

This article represents a first step in under-standing the economics of disease controlin wildlife populations. A general model of

wildlife growth and disease transmission wasused to illustrate limitations of harvestingstrategies when harvests cannotbe made selec-tively from the diseased population. Strategiesto address disease prevalence must thereforefocus on more than just the harvest, and canbe particularly effective if they address diseasetransmission and mortality.

The state of Michigan announced a goal of eradicating bovine tuberculosis in Michigandeer populations by 2010. To that end, the wild

white-tailed deer population in the core dis-ease area was to be decreased through hunt-ing programs that sold increased licenses. Inaddition, the practice of legally feeding deer inthe infected area was ended and the practice of baiting was temporarily ended. Both of these

disease control instruments were examined inour model. With deer being a highly valuedresource, we find that eradicating the diseaseis not likely to be economically optimal—atleastwhenMichigandoesnotbearfixedpolicy-related costs due to the exogenous impositionof trade restrictions and federally mandatedtesting requirements. It takes too long for thedisease to dissipate naturally once supplemen-tal feeding is halted, which is not surprisingconsidering that it took sixty-two years to pre-viously eliminate the disease in cattle herds un-der much more controlled conditions. It is alsotoo difficult and costly to kill all the deer in theinfected area, as policy-makers in Michigan arecurrently discovering. Instead, it is optimal forthe disease to remain endemic in the area atvery low levels, with intermittent investments

(via supplemental feeding) in in situ deer pro-ductivity.

Of course, an endemic disease is not alwaysoptimal. If marginal damages, feeding costs ordiseasemortalityarelargeenough,wefindthatit may be optimal to delay feeding-inducedproductivity enhancements in favor of diseaseeradication. Perhaps moreimportantly, we findthat eradicating the disease may be optimalwhen Michigan bears exogenously imposed,fixed policy-related costs. Specifically, the re-

sults suggest that the restrictions imposed bystates and the federal government ensure thatdisease eradication is an incentive compatiblestrategy for Michigan to follow.

Although the model was applied to the spe-cific case of bovine TB in deer herds, the model


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and results are likely to be applicable to otherwildlife disease problems—even those prob-lems where supplemental feeding is not anissue. Supplemental feeding decisions in ourmodel represent the easiest method of affect-ing disease transmission for the Michigan case,

and control of disease transmissions wouldlikely be a part of any wildlife disease manage-ment strategy. For other diseases, alternativeenvironmental variables could be manipulatedin ways that reduce disease transmission, andit is reasonable to believe that such actionsmight result in tradeoffs in in situ productivity(e.g., if contact is somehow reduced then fertil-ity might also be expected to decline). Hence,the current model provides a foundation foranalyzing a range of wildlife disease problems.

Finally, two caveats are worth mentioning.First, the disease was assumed to be unsus-tainable beyond the core area. This appearsreasonable for the Michigan bovine TB prob-lem, but it may not be the case for some otherdiseases. Rather, it might be possible for someotherdiseasestospreadamongadditionalpop-ulations. Such a situation might imply greatermarginaldamagesduetothediseaseandhencemore incentives to contain the disease. Ad-ditional tradeoffs may also arise for spatiallydifferentiated populations that possibly inter-act through migratory processes. A spatiallyexplicit analysis would be required in suchinstancesto fully assess the implications of spa-tial disease transmission. A second caveat isthat livestock sector management responses(e.g., biosecurity investments) were not in-cluded in our model. The economics of jointlymanaging valuable wildlife and livestock (andeven human) populations to control zoonoticdiseases is left to future research.

[Received January 2004;accepted September 2004.]

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Appendix

The model is calibrated using parameters obtained

from a variety of sources. The initial core-area deerpopulation, N 0, was estimated to be 13,298 in Spring2002 (Hill). Since 1995, core prevalence rates havefluctuated between 2.2% and 4.8%, averaging 2.3%during the period 1998–2000 (O’Brien et al.). Weadopt a value of 0 = 0.023, as it is believed thishas been fairly constant over the past few years(Hickling, O’Brien et al.).

Core carrying capacity is taken to be the up-per bound of estimates from the 1960s prior to ex-tensive hunt club feeding activities. This value is 9deer/km2 (Miller et al., O’Brien et al.), which im-

plies k = 14,049 for the 1,561 km

2

core area. Re-centpopulations, which benefit fromextensivefeed-ing, have peaked around 19–23 deer/km2 (O’Brienet al., Hickling). Since these populations havebeen subject to significant exploitation, we use aslightly higher effective carrying capacity value of 27 deer/km2, whichtranslates to k/(1− f )= 42,147.Miller et al. report approximately 8,212 kg/km2 of fruits, vegetables, and grains being fed to deer in thecore area (probably an underestimate). We adoptthis value and set f = 8,212, solving for = 0.00008.Note that, in ouranalysis, we set themaximum valueof f equal to f max = 10,000. This choice is somewhat

arbitrary but it has little bearing on our qualitativegraphical results.

For disease transmission, we derive (1 + f ) =0.346 using reported rates of infected contact by sexalong with survival rates from the time of contactto that of infection (Miller and Corso), and also re-ported deer sex ratios (McCarty and Miller). Milleret al.’s results on the relation between increasedfeeding and increased transmission are used to cal-ibrate = 2.64 × 10−6. Given this value and f =8,212, we can solve for = 0.339.

The intrinsic growth rate for white-tailed deeris taken to be r = 0.5703 (Rondeau and Conrad).Then, using results from Hill on nonhunting mor-tality, we adopt an effective disease-related mortal-ity of (1− f ) = 0.2. This rate does not imply that20% of all infected deer die as a direct result of thedisease, as few deer actually die from tuberculosis.Rather, the deer are weakened by their infectionand ultimately die from something else. More in-formation is needed to calibrate and . The un-

sustainable nature of the disease outside the coresuggests that > . We assume / = 1.05, as thisvalue produces reasonable results relative to histor-ical changes in disease prevalence when recent deerpopulations, disease prevalence rates, harvests, andfeeding choices are plugged into the model. Usingour previously defined values, we can solve for =0.356 and = 0.3623.

The price per harvested deer is p = $1,270.80,which is derived from various estimates of con-sumer’s surplus, hunting effort, and expendituresprovided by Boyle, Roach, and Waddington; Fraw-

ley; and U.S. DOI-FWS. Scaled harvesting costs, c/q,aretakenfromRondeauandConradtobe$231,192.This cost might be smaller in the Michigan case, butwe find the results are insensitive to even fairly sig-nificant changes in this value (Rondeau and Con-rad report a similar finding). The price of feed is setat w = 36.53, which is imputed from Miller et al.’sfeed density rates for the core along with anecdo-tal evidence about feed expenditures. Finally, usingthe results of Wolf and Ferris (given “split-state”status), we take variable damages in the core areato be about $1.6 million per year and, using a lin-ear damage function of the form z, we derive =

5,491. Fixed damage costs are taken to be $4 millionper year.


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horand e wolf(2005) the economics of managing infectious wildlife disease

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