ORIGINAL PAPER

Bioeconomic management of invasive vector-borne diseases

Eli P. Fenichel • Richard D. Horan •

Graham J. Hickling

Received: 27 January 2009 / Accepted: 13 August 2009

� Springer Science+Business Media B.V. 2010

Abstract Invasive insects, arthropods, and other

invertebrates are of concern due to the role some play

in introducing and transmitting pathogens via a

pathogen–vector relationship. Indeed, vector-borne

diseases represent a significant portion of emerging

diseases. We compare and contrast three strategic

approaches to managing a vector-borne pathogen:

conventional strategies based on disease ecology

without regard to economic tradeoffs and cost-effec-

tive strategies based on a bioeconomic framework.

Conventional strategies entail managing the vector

population below a threshold value based on R0—the

basic reproductive ratio of the pathogen, which

measures a pathogen’s ability to invade uninfected

systems. This does not account for post-infection

dynamics, nor does it balance ecological and economic

tradeoffs. Thresholds take on a more profound role

under a bioeconomic paradigm: rather than unilaterally

determining vector control choices, thresholds inform

control choices and are influenced by them. Simulation

results show cost-effective strategies can lower overall

program costs and may be less sensitive to parameter

estimation.

Keywords Bioeconomics � Decision theory �Disease ecology � Host-density thresholds �Vector-borne pathogen system,

Introduction

Emerging infectious diseases (EIDs), whereby patho-

gens are newly evolved or are introduced into new

regions or species, are an increasingly important form

of species invasion that adversely affects the health of

people, plants, and wild and domestic animals. EIDs

can have substantial economic consequences (Daszak

et al. 2000). Vector-borne diseases in particular are

responsible for nearly a quarter of all EID events

(Jones et al. 2008). This fraction could increase due

to the globalizing effects of trade and travel that

inadvertently move insects and other vectors across

the globe, and due to climate change and other

processes that expand vector habitats both geograph-

ically and temporally (Dobson 2004).

E. P. Fenichel (&)

School of Life Science and ecoSERVICES group,

Arizona State University, Box 874501, Tempe,

AZ 85287-4501, USA

e-mail: eli.fenichel@asu.edu

R. D. Horan

Department of Agricultural, Food,

and Resource Economics, Agriculture Hall, Michigan

State University, East Lansing, MI 48824-1039, USA

e-mail: horan@msu.edu

G. J. Hickling

The Center for Wildlife Health/NIMBioS, The National

Institute For Mathematical and Biological Synthesis,

University of Tennessee, 1534 White Ave., Knoxville,

TN 37996-1527, USA

e-mail: ghickling@utk.edu

123

Biol Invasions

DOI 10.1007/s10530-010-9734-7

A vector-borne disease invasion process involves

the establishment of a pathogen–vector relationship

by one of three pathways (Juliano and Lounibos

2005; Lounibos 2002). First, the pathogens and

vectors may be introduced together or in close

sequence. This has been the case with malaria in

many places around the world (Tatem et al. 2006).

Second, an introduced ‘bridge’ vector may acquire a

native pathogen that previously had cycled only

among wild hosts, but which henceforth begins to

infect humans and domestic animals. For example,

the introduced Asian tiger mosquito Aedes albopictus

has carried the viruses causing dengue into urban

areas of Rio de Janeiro (Lourenco-de-Oliveira et al.

2004). Finally, the introduction of novel pathogens

may convert previously benign insects or other

animals into vectors of harmful disease. Purse et al.

(2008) propose this third pathway has contributed to

the expansion of bluetongue into northern Europe.

Ecological models have been developed to help

understand vector-borne disease systems, and these

models are crucial inputs into decision models for

pathogen management. These disease ecology mod-

els are often extended beyond their initial purpose of

understanding the system, so that the analysts may

offer management recommendations. However, gen-

erally such recommendations are not grounded in

decision theory. We refer to management based

solely on disease ecology as the ‘‘conventional’’

approach to disease management.

The purpose of this article is twofold. First, we show

conventional models of vector-borne pathogens are

limited in their ability to inform management of

already-infected systems (i.e., a system perturbed from

its pre-infection or disease-free equilibrium, as will be

the case with a typical emerging disease scenario).1

Second, we show how bioeconomic analysis that

integrates economic and vector-borne disease models

can be used to improve disease management recom-

mendations. Specifically, we compare and contrast

conventional and bioeconomic approaches for devis-

ing management strategies for vector-borne animal

diseases. Bioeconomic models have recently been

applied to directly-transmitted animal disease man-

agement problems (e.g., Bicknell et al. 1999; Horan

and Wolf 2005; Fenichel and Horan 2007a, b; Asano

et al. 2008) as well as human disease (Barrett and Hoel

2007) and plant disease (Gaff et al. 2007) problems.

Gersovitz and Hammer (2004, 2005) investigate

management of vector-borne human diseases and in

doing so they motivate the use of bioeconomic

analysis to guide management. However, none of

these articles compare bioeconomic and conventional

strategies. We fill a gap in the literature by explicitly

comparing conventional and bioeconomic approaches

to systems involving vectors and animal hosts.

We begin by describing some of the main elements

of conventional and bioeconomic models that influ-

ence our analysis and results. We then formally

review conventional vector-borne disease ecology

models, illustrate the role of human choices in these

models, and highlight the difference between host-

based and vector-based management. This discussion

also highlights the limitations of conventional

approaches and motivates bioeconomic analysis.

Finally, we introduce the bioeconomic model and

analytically and numerically compare the two

approaches. We find significant differences between

the approaches, both in terms of the recommended

strategies and the associated costs.

Background

Conventional strategies based on population- and

community-level disease management recommenda-

tions stem from disease ecology, which primarily

focuses on a pathogen’s ability to invade naıve (i.e.,

pre-disease) systems. Invasibility is typically quanti-

fied by calculating the basic reproductive ratio of the

pathogen (R0), i.e., the expected number of secondary

infections generated from a single infected individual

within an otherwise healthy but susceptible host

population. R0 is defined at the disease-free equilib-

rium (Diekmann et al. 1990), and so the pathogen

invades when R0 [ 1, and fails to invade when

R0 \ 1, implying a threshold effect (Roberts and

Heesterbeek 2003; Dobson 2004; Heffernan et al.

2005).

R0 is considered ‘‘the most pervasive and useful

concept in the mathematical epidemiology of

1 A reviewer points out that some may regard management of

a post-infected scenario as pest management, but the process of

biological invasion involves more than species introduction

(Williamson and Fitter 1996). Focusing on post-introduction

processes (i.e., establishment and spread) can help to prevent or

slow further invasion into surrounding areas.

E. P. Fenichel et al.

123

infectious diseases’’ due to its perceived role in

guiding disease management (Roberts and Heester-

beek 2003), with ‘‘threshold behavior [being] the

most important and useful aspect of the R0 concept’’

(Heffernan et al. 2005). Using R0 (or variations

thereof; see Heesterbeek and Roberts 1995), manag-

ers have estimated the minimum constant level of

harvesting, vaccination, or other efforts required to

prevent a disease outbreak or to eradicate a previ-

ously-established disease. The conventional manage-

ment approach is to eradicate a disease by engaging

in management efforts (e.g., vector control or vacci-

nation) to satisfy a threshold criterion based on R0. In

the case of vector-borne pathogens, R0 has been used

to compute thresholds for malaria control (Hagmann

et al. 2003) and other vector-borne pathogen control

programs (e.g., Lord et al. 1996).

The conventional approach described above is

based on the pre-determined, fixed performance

objective of disease eradication, and also on a fixed

performance criterion. Thresholds based on R0 depend

on exogenously-fixed ecological parameters in a pre-

disease equilibrium. Such strategies limit the ability of

managers to consider economic and epidemiological

tradeoffs when choosing the level of disease control

(i.e., eradication or not) or the level of control effort.

Indeed, Heffernan et al. (2005) note that economic

tradeoffs are often ignored. Such an approach leaves

managers with little flexibility to actively manage

disease risks and relegates the role of ‘‘economics’’ to

an accounting exercise. Specifically, costs are com-

puted only after disease control strategies have

already been devised from R0-based analyses, and

the lowest-cost approach is then selected from this

small group of strategies (e.g., Wilkinson et al. 2004;

Smith et al. 2007; Baly et al. 2007). We demonstrate

below that strategies chosen in this manner are

unlikely to be least cost over all possible strategies.

Truly cost-effective management involves choos-

ing the strategy that is least cost overall (or,

equivalently, yields maximum net benefits overall).

This means economic costs and epidemiological

impacts must be evaluated simultaneously, which

requires the application of economics in its most

fundamental role as a decision science to evaluate

economic and epidemiological tradeoffs (e.g., the

marginal benefit of an intervention in terms of

reduced disease prevalence versus the marginal cost

of the intervention).

Bioeconomic models (Clark 2005) incorporate

information on epidemiological processes, and how

these are influenced by human actions, into an

economic decision framework (e.g., Bicknell et al.

1999; Gersovitz and Hammer 2004; Fenichel and

Horan 2007a). No exogenous performance metrics

are imposed in this framework. Rather, the system

dynamics are determined endogenously by selecting a

management strategy to minimize the costs of the

disease and its control. A bioeconomic approach

recognizes that the economic and epidemiological

systems are jointly-determined (i.e., that human

choices affect disease outcomes, which in turn

influence human choices), and that there are eco-

nomic and epidemiological tradeoffs that require

quantification. Knowledge of these joint interactions

and tradeoffs is then used to make cost-effective

disease management decisions.

Epidemiological model

Vector-transmitted diseases represent a special case

of multiple-host disease and involve complex species

and pathogen interactions. We make no attempt to

analyze the most general of all epidemiological

models that captures every conceivable interaction

or management response. Adding more interactions

and management options would significantly increase

the complexity of the analysis while not affecting our

primary conclusion that there are some fundamental

differences between conventional and bioeconomic

management strategies. We illustrate these differ-

ences with the simplest model possible—a model that

involves a small number of interactions and only a

few management opportunities.

Consider a pathogen that is transmitted only

between the host and a vector—and not horizontally

between conspecifics. Specifically, consider an inter-

acting insect vector population and an animal (e.g.,

wildlife or livestock) host population on a fixed area,

in which individuals of each population are either

susceptible, S, or infected, I, and there is no recovery

from disease—a simple SI model.2 The vector

2 Implicitly, we have assumed vaccination is not an option,

which is often the case for emerging diseases. While the SIframework is a special case of more complex models involving

recovered, immune, or exposed population compartments (e.g.,

Bioeconomic management of vector-borne diseases

123

population is indexed by i, so that the density of the

susceptible vector population is Si, the density of the

infected vector population is Ii, and the aggregate

density of vectors is Ni = Si ? Ii. Analogously, the

host population is indexed by w. We also define

hj = Ij/Nj to be the prevalence of infection in

population j = {i, w}.

Insect vector populations generally have short

generational times relative to vertebrate hosts.

Following the convention set forth in models of

vector populations, we model a disease outbreak

within the time span of the vector population so that

the aggregate host population is only affected by

disease mortality and any host population manage-

ment made in response to the disease outbreak (e.g.,

Lord et al. 1996; Wonham et al. 2004; Gersovitz and

Hammer 2005).3 Host reproduction and natural

mortality are not modeled, as these are assumed to

occur on a longer time span. Sub-populations fluctu-

ate due to feedbacks related to disease mortality and

pathogen transmission. Given these assumptions, the

system dynamics are

_Sw ¼ �biwIiSw=Nw � hwSw ð1Þ_Iw ¼ biwIiSw=Nw � awIw � hwIw ð2Þ_Si ¼ ðri � DiNiÞNi � bwiSiIw=Nw � diSi � hiSi ð3Þ_Ii ¼ bwiSiIw=Nw � diIi � hiIi ð4Þ

Here, bjk is the horizontal transmission rate at which

population j infects population k. We follow convention and

model transmission to be frequency-dependent, since

insect and other common vectors often make a finite

number of contacts (bites) during their lifetime (Dob-

son 2004). The disease-induced mortality rate for

animal hosts is aw, whereas the vector is only a carrier

and does not experience disease-related mortality. The

vector population’s recruitment rate is ri, Di is the

density-dependent reduction in the net recruitment

rate, and di is the natural mortality rate. Finally,

hj [ [0,1] is the harvest rate of population j. In the case

of an insect vector, harvests can be thought of as

mortality due to insecticide application, or vector

control. It is generally not possible to distinguish

between infected and susceptible individuals when

implementing harvest efforts. Therefore, we follow

convention (e.g., Heesterbeek and Roberts 1995) and

model effort as non-differentially applied to suscepti-

ble and infected individuals (i.e, non-selective har-

vesting). We focus on harvests as the primary control to

reduce population densities and therefore to limit the

number of infectious contacts. A more general model

might also include treatment and vaccination as

options (at least for valued livestock or wildlife,

though infected livestock and wildlife are often culled

instead of treated). In conventional analysis the harvest

rate hj is generally fixed. In contrast, in bioeconomic

analyses hj generally varies over time to address

changing ecological and economic conditions.

Our analysis of the dynamic system (1)–(4)

proceeds based on the assumption that disease control

efforts are implemented after the pathogen has been

introduced and detected. Thus, the vector and host

populations initially exhibit some level of infection

(i.e., Ij(0) [ 0 for j = w, i), and hosts have experi-

enced some level of disease-induced mortality so that

Nw(0) is less than the disease-free host population

equilibrium, denoted Kw. Management in this setting

is an intervention during the invasion and establish-

ment process. We allow for the possibility that the

pathogen is eradicated, in which case management

prevents the pathogen from establishing.

The conventional threshold-based approach

Roberts and Heesterbeek (2003) analyze threshold-

based management in the context of vector-borne

diseases, with the objective being to determine how

much control effort is required to eradicate the

pathogen from an infected system—or, alternatively,

to prevent the pathogen from entering in the first

place. Specifically, the level of control they advocate

is based on a metric related to R0, called the type

reproduction number. The type reproduction number,

denoted T, represents the expected number of infec-

tions caused (directly or by a chain of infections

Footnote 2 continued

SIS, SIR or SEIR models), the basic insights developed for our

SI model—that the current state of infection matters and

focusing on tradeoffs as opposed to eradication leads to qual-

itatively different results—are generally applicable to these

more complex models. Also note that the vector may also

transmit pathogen to sink hosts, causing damages, but these do

not affect the basic disease dynamics between the vector and

host populations that we model here (Chaves and Hernandez

2004).3 See Song et al. (2002) for an approach to combining

infection dynamics on multiple time scales.

E. P. Fenichel et al.

123

occurring through host and vector populations) when

one infected individual is introduced into a com-

pletely susceptible population, i.e., a pathogen-free

equilibrium.

A pathogen is prevented from invading a multiple-

host system through a particular host population (or

group of populations) when measures are taken to

reduce the type reproduction number, T, below one

(Roberts and Heesterbeek 2003). Roberts and Heest-

erbeek (2003) argue that such calculations can guide

the minimal effort levels needed to eradicate a

pathogen from an infected system. This approach

ignores the current state of infection and important

system dynamics, however, because the basis for

these calculations, T, is predicated on the assumption

of a pathogen-free equilibrium. Even so, such

recommendations are convention, and references to

R0 and T are ubiquitous in the disease management

literature (e.g., Roberts 1996; Wobeser 2002;

Hagmann et al. 2003). Below we show that such

recommendations are not cost-effective and are

potentially misleading.

Host management

Consider the possibility of managing the pathogen

problem by reducing the density of the animal host

population, Nw. From Eq. 2 note that the number of

new host infections depends on the ratio Sw=Nw.

Denote initial values of the host state variables by S0w,

I0w, and N0

w. After non-selective harvests are admin-

istered at the rate hw, the values of these state

variables are denoted by S0w ¼ ð1� hwÞS0w, I0w ¼

ð1� hwÞI0w, and N 0w ¼ ð1� hwÞN0

w. Host transmission

after population management therefore depends

on the ratio S0w=N 0w ¼ ð1� hwÞS0w= ð1� hwÞN0

w

� �¼

S0w=N0

w. Hence, management of the host animal

population has no effect on disease transmission (all

else being equal). Analogously, from Eq. 4, the

number of newly-infected insect vectors depends on

the ratio Iw=Nw, and this ratio is also unaffected by

harvests hw. Reducing an animal host population

imposes costs on society but has no effect on

pathogen transmission. Therefore, it is not econom-

ically optimal to invest resources in this method of

control.

If harvests of animal hosts were somewhat selec-

tive (i.e., harvesting was biased towards removing

infected animals), then it is likely that harvests of

host animals would reduce Nw proportionately more

than Sw. In that case, the ratio Sw/Nw increases and

there would be more new infections among host

animals. The reason is that infectious insects, which

make a fixed number of bites during their lifetime,

‘‘waste’’ fewer infectious bites when a larger propor-

tion of the total population is susceptible, so that the

probability of biting an already-infected host

declines. This is consistent with Wonham et al.’s

(2004) finding that killing hosts worsened a West

Nile Virus problem.

Vector management

Now consider management of the vector population.

The numbers of new animal and insect infections in

Eqs. 2 and 4 depend on the absolute number of

infected or susceptible insects, and not the propor-

tions. Hence, insect population controls can reduce

pathogen transmission.

Conventional analyses determine the required

level of control efforts by first calculating a threshold

value of Ni that prevents pathogen invasion. Specif-

ically, this vector-density threshold, which is derived

from R0 (or variants of R0 or T), is denoted NTi. NTi is

the largest value of Ni that prevents pathogen

invasion when a single infected vector is introduced

into an otherwise susceptible population in equilib-

rium (McCallum et al. 2001; Holt et al. 2003).

Threshold NTi is generally derived without consider-

ation of human-ecosystem interactions. NTi is defined

according to the relation

limSi!Ni;Nw!Kw;

hj!0 j¼i;w

_IijIi¼1;Iw¼I�w¼ bwiNiI

�w=Kw � di ¼ 0

! NTi ¼awdi

biwbwi

Kw ð5Þ

where I�w ¼ biw=aw is the solution to limSw!Nw

_Iw ¼ 0

when Ii ¼ 1. The disease-free equilibrium of the

animal host, Kw, is determined from biological

processes occurring outside the presently-modeled

time span, and would be larger than values of Nw

during an outbreak.

Conventional thresholds are fundamentally linked

to R0 : Ni \ NTi if and only if R0 \ 1 (Roberts and

Heesterbeek 2003; Heesterbeek and Roberts 2007).

The correct policy interpretation of NTi is that a

pathogen cannot invade an uninfected population

Bioeconomic management of vector-borne diseases

123

when the vector density is maintained below the

threshold (Gubbins et al. 2008). Yet this result is

often applied to infected populations as well, as it is

conventionally understood that maintaining vector

densities below NTi in already-infected populations

will reduce infections and ultimately eradicate the

pathogen. This is not necessarily true. Maintaining

Ni\NTi will be too weak a requirement for disease

reduction in some instances (and, once benefits and

costs are considered, will be overly strong in others).

The reason is that vector control is not explicitly

included in derivations of NTi, and that the current

infection levels matter. Observe that condition (4)

indicates _Ii\0 when

hi [bwihw

hi� ðbwihw þ diÞ ð6Þ

If we were to impose the requirement that hi \ 1,

then the following condition could be derived from

condition (6) (see ‘‘Appendix’’):

Si\ NTi þawKw

biwbwi

� �Ii

Iw

Nw

Kw

biw

awð7Þ

The right hand side (RHS) of (7) is less than NTi when

NTi\ð1=bwiÞðIi=IwÞNw

1� ðIi=IwÞðNw=KwÞðbiw=awÞand ðIi=IwÞðNw=KwÞðbiw=awÞ\1:

ð8Þ

When condition (8) is not satisfied, then Si\NTi

may not reduce Ii. In this case, the condition Ni\NTi

may not be sufficient to reduce Ii. When condition (8)

is satisfied, the condition Si\NTi ensures Ii is

reduced, but this may be stronger than necessary.

Because Ni C Si, it follows that Ni\NTi is more than

sufficient to reduce Ii when condition (8) is satisfied.

But if maintaining Ni\NTi is extremely costly

relative to the benefits, then society may do better

by investing in less control.

Threshold NTi is a special case of a more general

threshold, ~Ni, such that _Ii\0 when Ni\ ~Ni given

current infection levels and current harvest rates. As

with NTi, threshold ~Ni is derived by setting Eq. 4

equal to zero, but now the pre-infection steady state

assumptions are not imposed at the outset. Specifi-

cally, substitute the relations Si ¼ Ni � Ii and hw ¼Iw=Nw into Eq. 4 to yield _Ii ¼ bwi½Ni � Ii�hw�diIi � hiIi. Then set _Ii ¼ 0 and solve for the following

critical value of Ni:

~Niðhi; Ii; hwÞ ¼ðbwihw þ di þ hiÞIi

bwihwð9Þ

The critical value ~Ni is a vector-density threshold for

Ni, which is used to re-write Eq. 4

_Ii ¼ bwihw Ni � ~Ni hi;Ii; hw

� �� �ð40Þ

Eq. 40 indicates that infection will spread (i.e., _Ii [ 0)

when Ni [ ~Ni and it will decline (i.e., _Ii\0) when

Ni\ ~Ni.

In contrast to NTi, the threshold ~Ni depends on the

current disease prevalence level in the host and the

current density of infected insects. For instance, when

Ii [ 0 and in the limit as hw ! 0, then the threshold

approaches infinity: infection will never spread.

When hw [ 0 and Ii ¼ 0, then ~Ni ¼ 0 and infection

will always spread. Another difference from NTi is

that threshold ~Ni depends on vector controls, hi. Thus

the threshold is an ‘endogenous’ component of the

modeled system. This feature also highlights a

limitation of threshold-based management. Because

the threshold depends on human choices about vector

control, the policy interpretation of ~Niðhi; Ii; hwÞbecomes muddled: invasions are prevented when Ni

is reduced below ~Niðhi; Ii; hwÞ, but greater vector

control reduces the need to control vectors because

o ~Ni=ohi [ 0. This confusion is often sidestepped by

imposing additional constraints that are often ad hoc

(e.g., imposing time-invariant policies and pre-

disease equilibria; see Heesterbeek and Roberts

1995) and fails to address economic and ecological

tradeoffs. In what follows, we present an alternative

management approach that explicitly accounts for the

endogeneity of the thresholds and hence the system

dynamics.

A bioeconomic approach to management

The objective of a bioeconomic model is to choose

the management variables (vector control mortality

rates in the current case) to maximize some economic

criterion that a resource manager might care about,

such as social welfare. To do this, one considers how

the management choices affect disease dynamics, and

also how the intertemporal changes in the state

variables from the disease ecology model affect

economic welfare.

E. P. Fenichel et al.

123

Suppose the economic criterion of social welfare is

defined as the present value of economic net benefits

that people derive from the host species and vector.

These net benefits are assumed to accrue over two

distinct ‘‘phases’’: an outbreak phase during a time

interval of t [ [0,s), and a post-outbreak phase begin-

ning at s. Specifically, suppose we are dealing with a

vector that flourishes during warm months, but dies out

naturally once the temperature cools (see Wonham

et al. 2004 for a similar set-up). Vector control costs

therefore only accrue during the outbreak phase.

Denote these control costs by cyi=Ni, where yi ¼hiNi is the total harvest mortality of vectors and c is a

cost parameter. A larger value of c indicates control

efforts are either more costly or less effective. The

management choice variable is yi, and Eqs. 3 and 4 can

be adjusted accordingly. This formulation is used to

highlight the fact that marginal vector control costs

become quite large as the overall vector population

falls. Because the outbreak time frame is short, we do

not discount costs. Assuming vector controls are the

only economic activity during the outbreak phase,

social welfare during the outbreak phase is

Zs

0

�cyi

Ni

� �dt: ð10Þ

All other economic activity, such as livestock sales

or recreational wildlife harvests, occurs during the

post-outbreak phase. Any infected hosts remaining at

time s are assumed to yield no direct economic value

(e.g., infected livestock are unproductive and not

marketable), and these animals do not contribute to

disease transmission once the vector dies out at time s(e.g., infected livestock are often culled to prevent

future spread, and other infected animals may

experience disease mortality). We assume that the

disease may re-emerge cyclically, but that this re-

emergence is strongly influenced by environmental

conditions and independent of the state of the world

at the time when the vector dies out. Given these

assumptions, social welfare after the outbreak phase

depends only on the number of healthy animals

remaining, Sw(s). Denote social welfare in the post-

outbreak phase by V[Sw(s)], which represents the

present value of future net benefits resulting from the

healthy animal stock. This present value may include

economic impacts from pathogen re-emergence in

future periods.

Given the formulation outlined above, total social

welfare over both phases is given by

W ¼Zs

0

�cyi

Ni

� �dt þ V Sw sð Þ½ � ð11Þ

The social welfare measure in (11) does not explicitly

incorporate any values associated with infected host

animals. However, there is an implicit opportunity

cost associated with infected animals: a larger

number of infected animals during the outbreak

phase implies a smaller value of Sw(s) and hence a

smaller present value V[Sw(s)]. This opportunity cost

is considered during the maximization of (11).

Indeed, this opportunity cost is the only reason to

invest in disease control in this model.

An economically efficient, or cost-effective, strat-

egy is one in which vector harvest levels are chosen

for each point in time during the outbreak phase in

order to maximize social welfare, W, subject to the

constraints implied by the dynamic Eqs. 1–3, 40, and

the initial conditions. There is no need to consider

vector control choices beyond time period s since any

future re-emergence of the disease is independent of

the state of the world at time s, and since the present

value function V accounts for the net benefits of

future management. Mathematically, this problem is

written as

MaxyiðtÞ

W s:t:ð1Þ � ð4Þ; and Sjð0Þ ¼ Sj0; Ijð0Þ ¼ Ij0

for j ¼ i;w ð12Þ

This bioeconomic problem can be solved as an

optimal control problem (Clark 2005). The solution

to problem (12) is characterized by a set of conditions

(see ‘‘Appendix’’) that require the manager to balance

economic and ecological tradeoffs. For instance, if V

is small, then disease-related opportunity costs will

be small and hence the incentives to reduce disease

prevalence will be small. Disease control incentives

will also be small when c is large, as the marginal

costs of disease control will be large in this case.

The solution to problem (12) is a path of vector

harvest levels, denoted y�i ðtÞ, that determines optimal

paths for the state variables, S�j ðtÞ and I�j ðtÞ (and

hence N�j ðtÞ, h�j ðtÞ, and h�i ðtÞ) for j = i,w, and also the

vector-density threshold, ~NiðtÞ. The vector-density

threshold is not an explicit element of the optimiza-

tion problem. Rather, it is implicit: the choices of

Bioeconomic management of vector-borne diseases

123

y�i ðtÞ drive changes in the state variables and hence~NiðtÞ. However, as ~Ni is an implicit component of

Eq. 40, the choices of y�i ðtÞ are optimally made to

account for the impacts on this ecological metric and

the feedbacks that they imply. Moreover, the choices

of y�i ðtÞ define the optimal time-varying vector-

density threshold as ~N�i ðtÞ ¼ ~Niðy�i ðtÞ; I�i ðtÞ; h�wðtÞÞ.

Human choices in each period cause the threshold to

change endogenously over time. The bioeconomic

model accounts for this, and consequently will result

in the overall least-cost strategy.

Illustrative example

We illustrate the conventional and bioeconomic

management approaches with a simple numerical

example that is representative of a range of impor-

tant vector-host diseases (see Wonham et al. 2004,

Lord et al. 1996, and Purse et al. 2008 for specific

examples). We consider one animal host population

and one vector population. The epidemiological

processes and economic relations are as described

above, with V[Sw(s)] = vSw(s), where v is a param-

eter. Parameter values for the simulation are shown

in Table 1. As indicated earlier, initial values of the

state variables are based on the assumption that

disease control efforts are implemented only after

the pathogen has already been introduced and

detected.

Conventional management

First, consider management based on the threshold

NTi = 364, derived using Eq. 5. Specifically, we

examine implementing vector controls so as to hold

the vector population at a constant level NiðtÞ ¼�Ni\NTi for all t \ s. We find the social welfare

associated with such a strategy is greatest when �Ni is

just less than NTi, i.e., �Ni ¼ 363. Social welfare will be

lower if the vector population is held constant at any

lower level, as the added costs of vector control exceed

the additional benefits that would follow from having

fewer infected hosts and a larger value of Sw(s). Social

welfare will actually achieve its maximum value,

under a constant-vector policy, at a value of �Ni ¼ 421.

We refer to this as the optimal constant vector (OCV)

strategy, but such a strategy violates the conventional

management requirement that the vector population be

maintained below the threshold NTi.

These results are illustrated in Fig. 1, which shows

the social welfare values associated with different

values of �Ni. The social welfare associated with

maintaining �Ni ¼ 363 is negative, yielding W =

-$3,233. Social welfare is negative because the

control costs incurred during the infection period

exceed the present value of the remaining hosts.

Managing further below �Ni ¼ 363 only leads to

greater economic loses, as described above. Social

welfare associated with the OCV strategy of �Ni ¼421 is also negative, yielding W = -$2,498. We

Table 1 Parameters and

values used in the model

All rates are days-1. Other

units are as documented

Parameter Interpretation Value

biw Transmission rate from vectors to hosts 0.0144

aw Disease induced mortality rate of hosts 0.0143

bwi Transmission rate from hosts to vectors 0.0792

ri Maximal vector recruitment rate 0.0075

di Vector mortality rate 0.029

Di Density-dependence term for vector growth, N�1i

� �0

v Constant marginal value of a uninfected host at time T ($) 30

c The cost of vector control ($) 10,000

Nw(0) Host population when pathogen is detected (after some

disease induced mortality has occurred)

988

hw(0) Host prevalence when pathogen is detected 0.1

Ni(0) Vector population when pathogen is detected 1,419

hi(0) Vector prevalence when pathogen is detected 0.1

Kw Equilibrium host population density prior to disease 1,000

s Time horizon (days) 100

E. P. Fenichel et al.

123

show below that social welfare is non-negative under

a cost-effective strategy, highlighting the inefficien-

cies of a constant �Ni strategy.

Figure 2a indicates infections in the animal host

initially increase when maintaining �Ni ¼ 363, but then

begin to diminish. The initial increase in infections

indicates that the pre-disease vector-density threshold

is not stringent enough given the state of the world

when management is initiated. Holding the vector

population further below the threshold NTi will yield a

more rapid decline in host prevalence, but at rapidly

increasing cost (Fig. 1). Vector control would have to

be extreme to lead to eradication within the time

horizon of the problem (s = 100 days).

Bioeconomic management

We now develop an alternative management strategy

based on the bioeconomic model presented in (12).

As described above, and in contrast to the conven-

tional management results, the optimal vector

controls, state variables, and vector-density threshold

are each allowed to vary over time in the bioeconomic

model. In particular, the optimality conditions (see

‘‘Appendix’’) suggest, and we verify numerically, that

the optimal course of action when 0 \ s � ? is to

exert vector control effort such that 0 \ yi(0)

\ Ni(0) and yi(t) = 0, for all t [ (0, s].4 Specifically,

conditional on the parameter values in Table 1,

y�i ð0Þ ¼ 1; 163, so as to kill 82% of the initial vector

population. Following this initial pulse of vector

control, no vector control should take place. This

management strategy yields W = $6,519 (Fig. 3

s = 100). Any other initial pulse of vector control

would result in lower economic returns (Fig. 3).

Social welfare would also be reduced by applying

additional vector controls after the initial period. This is

because future costs are not discounted, and marginal

harvesting costs (c/Ni) will rise as the vector population

is reduced.5 It is therefore optimal to apply significant

controls only in the initial period to reduce the vector

population and hence prevalence in all future periods.

This result illustrates the simplest time-varying control

Fig. 1 Net benefits from maintaining a given density of vector

population, �Ni for three different terminal time periods,

s = {50, 100, 250} days. The horizontal dotted line is zero

net benefits. The vertical dashed line is the conventional

recommendation where R0 ? e = 1, and e is small

Fig. 2 Host prevalence (a) and population density (b) over a

100 day infectious period for a conventional pre-disease

equilibrium threshold management that maintains Ni ¼ �Ni ¼363, and for cost-effective management

4 If there are capacity constraints that only permit a

yi(0) \ yi*(0), then subsequent vector control pulses may be

optimal within the interval t [ (0, s]. We also note that for this

Footnote 4 continued

problem if s is large enough, then it is not optimal to engage in

any management resulting in W = 0.5 Rising marginal costs are also a factor in determining the

initial pulse harvest. The costs associated with a pulse harvest

are cln(Ni/[Ni - yi]) (see Clark 2005). This implies that costs

increase as a larger proportion of the vector population is

initially harvested.

Bioeconomic management of vector-borne diseases

123

strategy. It is possible that vector control efforts would

optimally be spread out over additional periods in

models where vector control costs are nonlinear in yi,

the future is discounted, or when the system dynamics

involve more complex relations. But the most basic

result—that control efforts should optimally vary over

time in response to the state of the world—will be

unaffected.

The $6,519 in net benefits under the cost-effective

strategy is $9,017 greater than the best constant vector

density management program, and $9,752 greater than

the conventional management program described

above (Table 2). The differences in net benefits arise

because the cost-effective plan yields significantly

lower control costs during the outbreak phase, and

greater benefits during the post-outbreak phase. Con-

sider the outbreak phase. The cost-effective plan

initially reduces the vector population density to 256,

well below the conventional management threshold

NiT . This initial reduction is chosen because of the

dynamic effects it has on the endogenously determined

threshold (Eq. 5). At the instant of control, ~NiðtÞ rises

to 83,765 vectors, but then immediately falls to 542

vectors (Eq. 5). This new threshold is much larger than

the conventional threshold NiT , and it is more than

twice the vector population that remains after the

initial cull. Management has shifted the threshold to

ensure that the infected vector population will con-

tinue to decline even if no further controls are

implemented. In contrast, under conventional man-

agement, the vector population is maintained only

slightly below its threshold value. In that case, controls

must continue to be implemented to keep the infected

vector population in check. Moreover, these controls

are implemented at higher costs after the initial period

due to the larger marginal control costs (i.e., c/Ni)

associated with maintaining a reduced vector popula-

tion in each period.

The value of V also is also larger under cost-

effective and smaller under the conventional man-

agement. This is because the large initial cull in the

cost-effective scenario yields a smaller number of

infected vectors and infected hosts in almost every

subsequent period (Fig. 2a). With less disease

pressure over the most of the management interval,

the total density of surviving hosts is greater. The

result is a greater density of susceptible hosts at time

s and a larger V, under bioeconomic management

(Fig. 2b).

Sensitivity analysis

We explore the sensitivity of the model in two steps.

First, we consider the influence of the length of the

time horizon s, and then we consider other parameters

in the model. The value of s is particularly important,

as it may vary by location for climatic reasons, and

over time as influenced by climate change (Purse et al.

2008). These analyses can be viewed as alternative

climate scenarios. The time horizon does not influence

conventional management recommendations (noted

by the vertical dashed line in Fig. 1), though a shorter

time span s does result in larger net benefits. The time

horizon does influence the OCV strategy (i.e., the

vector density associated with the maximum on each

curve in Fig. 1) and the net benefits under this strategy.

Specifically, a larger vector population is allowed

under shorter time spans in the OCV strategy, as this

reduces control costs and also the vectors have less

time to infect the host population when s is small. The

sensitivity analysis associated with the OCV strategy

clearly shows how conventional threshold-based man-

agement may be too stringent in some cases (i.e.,

s = 50, as the maximum lies to the right of the vertical

dashed line) and not stringent enough in others (i.e.,

s = 250, as the maximum lies to the left of the vertical

dashed line).

The magnitude of s also influences the optimal

bioeconomic strategy. Figure 3 illustrates the net

benefits accruing from the optimally chosen yi(0)

conditional on three different values of s. On the one

Fig. 3 Social welfare arising under cost-effective manage-

ment for different initial pulses (with zero harvests afterward),

for three different terminal time periods, s = {50, 100, 250}.

The x-axis differs from that in Fig. 1. Zero social welfare is

indicated with a dotted line

E. P. Fenichel et al.

123

Ta

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Bioeconomic management of vector-borne diseases

123

hand, longer infectious periods imply larger damage

costs (i.e., reduced V) since there is greater time for

infection to spread. This feature increases the net

benefits of vector controls. On the other hand, longer

infectious periods imply the pathogen has more time

to rebound, within both the vector and host popula-

tions, after the initial cull. This feature reduces the

incentives to invest in vector control. The net effect

depends on the initial value of s and the magnitude of

the change in s. When s is small (e.g., s = 50), the

optimal response to a small-to-moderate increase in s(say, to s = 100) is a larger initial pulse, i.e., a larger

value of y�i ð0Þ (Fig. 3). The net economic effect of

increasing s from 50 to 100 is a reduction in social

welfare, as control costs increase and V is diminished.

However, if s is increased by a large value (e.g.,

s = 250), then it may be optimal to not control the

vector and forgo animal-related benefits (note: the

global maximum of the s = 250 curve is at

y�i ð0Þ ¼ 0).6 It is not optimal to wait to initiate vector

controls because this delay results in more infected

hosts, reducing Sw(s) and hence V. Overall, our

results suggests that if an exogenous force such as

climate change increased s, then this too could

impose costs on society by reducing W, though it is

possible to adapt the management response to offset

larger potential losses.

We also perform sensitivity analysis for other

relevant epidemiological and economic parameters

for the case of s = 100 (Table 2). The results for

changes in the economic parameters are intuitive:

smaller control costs or larger economic benefits of

preserving healthy animals encourage a larger initial

vector harvest, under both the bioeconomic and OCV

strategies. In contrast, the conventional strategy does

not change in response to changes in economic

parameters, as these parameters are not considered in

calculating the R0-based threshold. In all cases, social

welfare is increased when control costs are reduced or

the benefits of Sw are increased. These sensitivity

results illustrate one of the short-comings of conven-

tional management. A move towards the OCV

strategy is an improvement from the conventional

approach, but does not take full advantage of

economics as decision science. The bioeconomic

management strategy is relatively insensitive to

changes in economic parameters (as measured by

the elasticity of the optimal initial cull with respect to

the parameter in question; Table 2). Therefore,

precise estimation of these parameters may not be

essential to develop a near-optimal bioeconomic

strategy.

Now consider the sensitivity results for changes in

the ecological parameters. Changes in the parameters

bwi and dwi have virtually no impact on management or

welfare in either the bioeconomic or OCV scenarios.

The reason is that these parameters affect biological

processes that are of second-order importance to the

host population (i.e., they do not directly affect

transmission to the host) once an outbreak occurs.

These parameters do, however, significantly impact

management and welfare levels associated with the

conventional strategy. The reason is that these param-

eters affect the calculation of NTi (Eq. 5), on which

conventional management is based. This result high-

lights the importance of basing management strategies

on the dynamics associated with current infection

levels as opposed to the initial point of infection.

Increases in ri and biw result in ‘‘speeding up’’

those processes having direct and adverse effects on

the host population. Specifically, a larger ri results in

the vector population rebounding more quickly after

a cull, and a larger biw results in more transmission to

animal hosts. Speeding up these processes is akin to

having a slightly larger time horizon, s, as a larger

number of adverse pathogen interactions can occur

during the actual time horizon. It is therefore not

surprising that, for both the bioeconomic and OCV

strategies, increases in each of these variables has

qualitatively similar impacts as a slight increase in s.

The conventional vector control strategy is unaf-

fected by changes in ri (as this parameter does not

affect NTi), but it is affected by changes in biw, as this

parameter does affect NTi.

Finally, increases in aw results in fewer infected

hosts, which reduces both transmission of the patho-

gen back to the vector and also the number of

‘‘wasted’’ vector bites. The former is beneficial and

reduces the need for vector controls, though this effect

is of secondary importance relative to vector-to-host

transmission. The latter result indicates increased

vector-to-host transmission, which is costly and

6 For a sufficiently large s it will be important to incorporate

discounting and net natural growth within the host population,

neither of which are currently included in our model.

We hypothesize that including these factors will result in a

time-varying positive level of vector control.

E. P. Fenichel et al.

123

increases the incentives for vector controls.7 We find

the latter effect dominates under the bioeconomic

strategy, so that a larger aw results in more vector

controls being economically optimal (though the

response is small). In contrast, the first effect dominates

for the OCV strategy. The reason is that an increase in

aw reduces host prevalence, hw, which in turn increases

the current vector-density threshold ~Ni defined in

expression (9). A larger threshold means the number

of infected vectors can be kept under control at higher

vector densities. Hence, control costs are reduced by

reducing vector controls. The net effect is a reduction

in social welfare as the second effect (i.e., reduced

wasted bites) reduces V, but investing in more vector

controls to prevent these losses would be even more

costly than easing up on controls and allowing some

losses to occur. Finally, vector controls are also eased

up under the conventional strategy, as increases in aw

increase the conventional vector density threshold NTi.

Sensitivity analysis with respect to aw indicates

another short-coming of basing management on the

point of invasion once the invasion has occurred.

Discussion

Programs to manage vector-borne pathogens will

likely expand in coming decades, motivated by the

link between wildlife, domestic animal, and human

health (Cleaveland et al. 2001) and global environ-

mental changes that lead to disease emergence (Daszak

et al. 2000; Jones et al. 2008). Historically a concern in

tropical climates (e.g., malaria), vector-borne patho-

gens are now of increasing concern in temperate

climates (e.g., West Nile virus in North America,

bluetongue in Europe)—either due to the increasing

number of biological invasions, often with human help,

or due to the potential effects of climate change (Purse

et al. 2008). It is imperative to prioritize management

actions within such programs to make the best use of

scarce resources. We have presented an approach that

bases disease management decisions on both biolog-

ical and economic decision science.

Prior bioeconomic models of vector-borne disease

management are limited and have not compared

bioeconomically-derived strategies to strategies

based only on disease ecology –conventional recom-

mendations. One goal of this manuscript has been to

compare and contrast conventional and bioeconomic

approaches to deriving vector-borne disease manage-

ment recommendations. Our results indicate there are

potentially large gains to be made from explicitly

considering economics at the outset of vector-borne

disease management planning.

An interesting feature of the problem is that the

bioeconomically optimal solution we present is less

complicated and less costly to implement than the

conventional approach. Moreover, the bioeconomic

management program is less sensitive to the estima-

tion of biological parameters. This means that bioeco-

nomic management may be more robust to ecological

and economic uncertainty than the conventional

approach. While this is not a global feature of

bioeconomic models, it illustrates the point that both

bioeconomic and conventional ecological approaches

can involve complicated analysis and complex and

costly implementation. Generally, analysis of natural

resource management problems is less costly than

implementation of the management programs. Finding

the best management program that accounts for

economic and ecological processes, transactions costs,

and institutional constraints requires interdisciplinary

teams willing to explore new approaches. Such novel

approaches will become increasingly common as

interdisciplinary teams complement each other’s

strengths and bring new and innovative tools to

ecological problems.

The use of ‘‘ecological’’ thresholds is increasing in

natural resources management (Martin et al. 2009).

Much of this work has come out of disease manage-

ment (e.g., R0 based thresholds) and invasive species

biology (invasion thresholds). A related literature

address pest control with accounting based thresholds

related to ‘‘economic’’ injury levels (Koonce et al.

1993). Though occasionally mentioned (Walker et al.

2004), the rapidly growing literature on thresholds

largely neglects the multi-dimensional nature of the

determinants of these thresholds and how management

actions shape both the system relative to the threshold

and the threshold itself. Bioeconomic management

provides a framework for jointly considering the effect

of management action relative to the threshold and the

7 Increases in aw effectively result in selective culling of

infected host animals. As indicated in footnote 4, selective

culling of infected hosts can in some circumstances be

detrimental because this increases the probability that an

infected vector randomly bites a susceptible (as opposed to

infected) host.

Bioeconomic management of vector-borne diseases

123

effects of management actions on the value of future

thresholds. State-dependent management that affects

the system relative to a threshold and the future value

of the threshold itself can lead to lower management

cost and more effective management. We have

illustrated this with a simple vector-borne pathogen

model.

Acknowledgments The authors gratefully acknowledge

funding provided by the Economic Research Service-USDA

cooperative agreement number 58-7000-6-0084 through ERS’

Program of Research on the Economics of Invasive Species

Management (PREISM), and by NRI, USDA, CSREES,

grant #2006-55204-17459. This work was conducted as part of

the SPIDER working group at the National Institute for

Mathematical and Biological Synthesis (NIMBioS), sponsored

by the National Science Foundation and the U.S. Department of

Agriculture through NSF Award #EF-0932858, with additional

support from the University of Tennessee, Knoxville. The views

expressed here are the authors and should not be attributed to

ERS, USDA, or NIMBioS.

Appendix

(A) Derivation of condition (7)

Upon imposing the requirement that hi \ 1, condition

(6) indicates that

1 [ bwihw=hi � ðbwihw þ diÞ¼ bwihwð1=hi � 1Þ � di: ðA1Þ

Condition (A1) can be re-written as

di þ 1 [ bwihwð1=hi � 1Þ ¼ bwihwðNi=Ii � Ii=IiÞ¼ bwihwðSi=IiÞ: (A2)

Write hw ¼ Iw=Nw and solve for Si:

Si\ di þ 1½ � IiNw

bwiIw¼ di þ 1½ � awKw

biwbwi

Ii

Iw

Nw

Kw

biw

aw; ðA3Þ

where the last equality simply comes from multiply-

ing both the numerator and the denominator by

awKwbiw. Upon applying the distributive property,

condition (A3) becomes

Si\awdiKw

biwbwi

þ awKw

biwbwi

� �Ii

Iw

Nw

Kw

biw

aw

¼ NTi þawKw

biwbwi

� �Ii

Iw

Nw

Kw

biw

aw; ðA4Þ

which is condition (7).

(B) Optimal vector management

Problem (12) can be formulated as a linear control

problem (i.e., the problem is linear in the control

variable yi). The Hamiltonian for the problem is:

H ¼ �cyi

Niþ kw

_Sw þ lw_Iw þ ki

_Si þ li_Ii; ðB1Þ

where kj is the co-state variable associated with the

susceptible stock of population j, and lj is the

co-state variable associated with the infected stock of

population j (j = w,i).The marginal value of yi on the

Hamiltonian is:

oH

oyi¼ � c

Ni� ki

Si

Ni� li

Ii

NiðB2Þ

The marginal value in (B2) vanishes along a singular

path. When the value is positive (negative), then hi

should be set at its maximum (minimum) value.

Generally, the system will not be on the singular path

initially, which means an extremal control must be

used to move the system as quickly as possible to the

singular path (if one exists) (Clark 2005). As there is

no upper bound on yi, harvests can only occur at a

maximum rate for an instant. Harvests can persist at

yi = 0 or at the singular level for a longer time.

Hence, except for an initial jump, we have

yi ¼ _yi ¼ 0; or oH=oyi ¼ 0: ðB3Þ

The remaining necessary conditions for problem (12)

are:

_kw ¼ �oH

oSwðB4Þ

_lw ¼ �oH

oIwðB5Þ

_ki ¼ �oH

oSiðB6Þ

_li ¼ �oH

oIiðB7Þ

The transversality conditions are:

kwðsÞ ¼oV

oSw[ 0 ðB8Þ

lwðsÞ ¼ 0 ðB9ÞkiðsÞ ¼ 0 ðB10Þ

E. P. Fenichel et al.

123

liðsÞ ¼ 0 ðB11Þ

Conditions (B10) and (B11), along with (B2), indi-

cates that hi(s) = 0. Taking the time derivative of the

Hamiltonian yields:

oH

ot¼ oH

oyi_yi þ

X

j¼w;i

oH

oSj

_Sj þoH

oIj

_Ij þoH

okj

_kj þoH

olj

_lj

" #

oH

ot¼ 0þ

X

j¼w;i

oH

oSj

_Sj þoH

oIj

_Ij þ _Sj_kj þ _Ij _lj

� �

oH

ot¼ 0þ

X

j¼w;i

�oH

oSj

_Sj þoH

oIj

_Ij þ _Sj �oH

oSj

� �

þ _Ij �oH

oIj

� ��¼ 0: (B12)

The second row of (B12) comes from condition (B3),

while the third row comes from conditions

(B4)–(B11). Hence, H is constant after possibly an

initial pulse harvest. Define this constant value of H

by v, i.e., H = v. Using the results H = v and

hi(s) = 0, along with the transversality conditions

(B8)–(B11), we have

HðsÞ ¼ 0þ kwðsÞ _SwðsÞ ¼ �oV

oSwðsÞbiwIiðsÞ

SwðsÞNwðsÞ

¼ v\0; ðB13Þ

which means H(t) \ 0 for all t [ [0, s).

As H is a measure of economic welfare, the result

H(s) \ 0 means that extending the time horizon to sreduces welfare. It would be optimal to diminish the

time horizon to some value s0\s, such that

Hðs0Þ ¼ 0, as setting Hðs0Þ ¼ 0 means there is no

value to extending the time horizon any further

(Clark 2005). However, this is not possible since s is

biologically-determined and is therefore exogenously

fixed. The implication is that it is optimal to take all

actions as soon as possible: an impulse control at time

t = 0. The intuition is as follows. First, there is no

incentive to delay management because problem (12)

is a linear control problem with no discounting. If

discounting occurred at a very high rate, then future

costs (and benefits) would be worth less and so one

would want to wait to invest in vector controls. With

no discounting, costs today and in the future are

valued equally, so there is no penalty from investing

early. But there is a penalty to investing later, as the

only stock of concern, Sw, can only decrease when the

disease problem gets worse. So it is optimal for all

vector reduction to take place immediately. The cost

associated with an impulse control used for the initial

cull is c ln½Ni0=ðNi0 � yið0ÞÞ� (Clark 2005).To deter-

mine the optimal initial harvest, rewrite the problem

as

Maxyið0Þ

�c ln Ni0

Ni0�yið0Þ

h iþ VðSwðsÞÞ ðB14Þ

subject to the equations of motion and the initial

conditions as before, except that now we have the

following initial conditions for the vector popula-

tions: Sið0þÞ ¼ Si0ð1� yið0Þ=Ni0Þ and Iið0þÞ ¼Ii0ð1� yið0Þ=Ni0Þ. The objective function in (B14)

now does not depend on time, and the equations of

motion no longer depend on human choices. We

solve the equations of motion (1)–(4) as functions of

the stock levels after the initial cull, and then rewrite

problem (B14) as

Maxyið0Þ�c ln

Ni0

Ni0 � yið0Þ

� �þ VðSwðs; Sw0; Iw0; Si0

½1� yið0Þ=Ni0�; Ii0½1� yið0Þ=Ni0�ÞÞ; ðB15Þ

or, more simply, as

Maxhið0Þ�c ln

1

ð1� hið0ÞÞ

� �þ VðSwðs; Sw0; Iw0; Si0

½1� hið0Þ�; Ii0½1� hið0Þ�ÞÞ ðB16Þ

The optimality condition for this problem is:

c

ð1� hið0ÞÞ¼ oV Sw sð Þð Þ

oSw� oSw

oSið0þÞSi0 �

oSw

oIið0þÞIi0

� �

ðB17Þ

The left hand side of (B17) is the marginal cost of

vector controls. Marginal costs approach infinity as

the harvest rate hið0Þ approaches unity; hence it is not

optimal to eradicate the vector population assuming

the marginal benefits of eradication are finite. The

right hand side is the marginal net benefit of vector

controls in time period 0. Society benefits from a

larger value of SwðsÞ. This value is increased at the

margin as Iið0þÞ is reduced (i.e., �oSw=oIið0þÞ[ 0),

and it may increase or decrease as Sið0þÞ is reduced

(i.e., �oSw=oSið0þÞ�\0).

It is worth noting that the impacts of the initial cull

will depend on the time horizon s. The smaller is s, the

larger the impact of a given initial cull on SwðsÞ, as the

vector population will have less time to rebound and

infect the host population. In contrast, a larger s gives

Bioeconomic management of vector-borne diseases

123

infected vectors more time to rebound and infect hosts

after the initial cull. Our numerical sensitivity analysis

indicates that the less control is required when s is

small, as having only few controls can yield effective

protection of the host population in this case while

keeping costs low. The optimal level of vector controls

is initially increasing in s, as vector control efforts are

initially substituted for reduced protection effective-

ness as s is increased. If s is too large, however, the

level of control is reduced in response to reduced

effectiveness; more hosts become infected while

society saves from reduced control costs.

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