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I: ....w... . ...... ..xu. v v.6...4...1..:.,:......67.13.47. ,. . 013.“. ~..’I. I up» . L. .. 4.151 3.11.5». .17}-.. ..I.. . r 4-» I a. ..vv..lvr.. I 4...... vlva. I . I: . .I. .8 . I ,. . I It“. 3.: ..I.... 4 ...I: .. .... ... 5.. .v .1... I III ... ... {-.II... 0.. III. . gen... . 4~I,....H1Iz~.4uag . 3...}...13 I. I. I ... I III .I .I-. I I. l .... .0 I 0 :0 .I. .v I. III... I .I.»? I. I I . I IIIJII ..Illll-II... III! I This is to certify that the dissertation entitled ESSAYS ON THE BIOECONOMIC ANALYSIS OF WILDLIFE AND LIVESTOCK DISEASE PROBLEMS presented by FANG XIE has been accepted towards fulfillment of the requirements for the Ph.D degree in Agricultural, Food and Resource Economics // ///1: ‘4.-.-n-.-o---u----.--—---- Major Professor’ 5 Signature -:’:?’(-'.x-/ II/ '3 “... '(J 4,9,. MSU is an Affinnative Action/Equal Opportunity Employer LIBRARY Michigan State University PLACE IN RETURN BOX to remove this checkout from your record. TO AVOID FINES return on or before date due. MAY BE RECALLED with earlier due date if requested. DATE DUE DATE DUE DATE DUE 5108 KthroleccatPres/CIRCIDateDueJndd ESSAYS ON THE BIOECONOMIC ANALYSIS OF WILDLIFE AND LIVESTOCK DISEASE PROBLEMS By Fang Xie A DISSERTATION Submitted to Michigan State University in partial fulfillment of the requirements for the degree of DOCTOR OF PHILOSOPHY Agricultural, Food and Resource Economics 2010 ABSTRACT ESSAYS ON THE BIOECONOMIC ANALYSIS OF WILDLIFE AND LIVESTOCK DISEASE PROBLEMS By Fang Xie The spread of infectious disease among both wild and domesticated animals has become a major problem worldwide. It not only becomes a global threat to human health, but also poses great economic losses to the society. Many infectious diseases are often unobservable without investments to discover disease status, and prior bioeconomic work focuses on imperfect controls in the sense that these investments have not been made -- generally because they are technologically not possible. Policies are also generally based on this situation. This dissertation focuses on analyzing two separate disease problems, Brucellosis and Bovine Tuberculosis (bTB), in order to better understand when it is worthwhile to invest in observability of disease status and how this can be used to improve management. This dissertation is divided into two essays. The first essay applies an econometric model of US cattle trade to forecast bTB transmission across cattle herds in the US. I first develop a gravity model of livestock trade and then link it to an epidemiological model of bTB transmission, with the goal being that this information could lead to improved disease surveillance and management. The findings suggest that targeting surveillance efforts towards high disease risks states could be more effective than targeting efforts only towards states known to be infected or targeting efforts equally across all states. The second essay is about brucellosis management in bison in Yellowstone National Park. It investigates the combination of vaccination and test-and-slaughter by applying an optimal control approach whereby the disease dynamics can be affected by both management actions and changes in human-environmental interactions. The results suggest that investments in observability (testing) are inferior because the manner in which vaccination performs is superior -—it not only prevents infections, but also helps contribute to the resistant population, in tum increase the overall healthy population. Yet, the differences are relatively small, and there might be transaction costs associated with implementing the oscillated controls. Therefore, the decision of choosing vaccination or test-and-slaughter would depend on whether the transaction costs of running the program actually offset differences of social net benefits. ACKNOWLEDGMENTS I would like to thank my major professor, Richard D. Horan, for his guidance and support during my master and PhD program. I am grateful to him for being a great inspiration in so many ways, for introducing me into the wonderful world of bioeconomic modeling with his intelligent and effective teaching, for continuing providing constructive comments and suggestions on my research. He cares about the career development of students and is willing to share his experience to provide guidance. I would like to express my appreciation to the members of my committee, Christopher Wolf, Scott M. Swinton and Jinhua Zhao for inspiring discussions and valuable comments that help improve my dissertation research. Other faculty members, including Frank Lupi, Robert Myers, Zhengfei Guan, Songqing J in, Jeffery Wooldridge and Scott Loveridge deserve my special thanks too. Thank you goes to Debbie Conway, Nancy Creed and Ann Robinson too for their help with administrative issues. Many thanks go to fellow graduate students. In particular, I would like to thank Nicole Mason, Shan Ma, Min Chen, F eng Wu, Chenguang Wang, Feng Song, Huilan Chen, Honglin Wang, Wei Zhang, Zhiying Xu, Lili Gao, Jacob Ricker-Gilbert, Tim Komarek, Adam Lovgren, Rie Muraoka, Christina Plerhoples, Alexandra Peralta, Joleen Hadrich, Nicole Olynk for their friendship and support. I would also like to thank all my dear friends who shared my laughter, excitement, depression and many other things together at East Lansing. It is because of you that my life in East Lansing was very enjoyable. I wish you good luck for the rest of the journey. Finally and most importantly, I would. like to express my gratitude to my family for their unwavering love and support. I am grateful to my parents, Hesong Xie and Silan Wu, for encouraging me to spread my wings, and for doing everything they could to help me pursue my dreams. I also thank my in-laws, Xiaolong Fang and Linhua Liu for treating me as their own daughter, supporting me and my husband unconditionally. I reserve my last words to express my gratitude and love to my dear husband, Yang Fang, for his love and faith in me, for his support, encouragement and understanding, and for sharing all the ups and downs with me throughout the process. TABLE OF CONTENTS LIST OF TABLES ........................................................................................................... viii LIST OF FIGURES ........................................................................................................... ix INTRODUCTION ............................................................................................................... I ESSAY I AN INTEGRATED MODEL OF ANIMAL TRADE AND DISEASE TRANSMISSION: PREDICTING BTB DISEAES RISKS IN CATTLE AND TARGETING SURVEILLANCE EFFORTS ...................................................................... 6 Introduction ................................................................................................................... 6 A Gravity Model of Trade ...................................................................... 8 Estimation of the Gravity Model ............................................................. l3 BTB Disease Dynamics for Cattle Movement .............................................. l9 Simulating Disease Dynamics and Risk ..................................................... 23 Comparison of Surveillance Policies ......................................................... 30 Conclusions ...................................................................................... 34 ESSAY 11 AN OPTIMAL CONTROL APPROACH TO BRUCELLOSIS CONTROL IN BISON IN YELLOWSTONE NATIONAL PARK ...................................................... 37 Introduction ...................................................................................... 37 History of Bison and Brucellosis in Yellowstone ........................................... 39 Brucellosis Management Options ............................................................ 40 The Epidemiological Model ................................................................... 42 Model Specification ........................................................................ 42 Impacts of Human Choices ................................................................ 44 The Bioeconomic Model .......... . ............................................................. 50 Numerical Example ............................................................................. 55 Partial Singular Solution for T when v is Constrained ................................ 56 Partial Singular Solution for v when T is Constrained ................................ 59 Comparison of social net benefits for the four policy scenarios ...................... 59 Sensitivity Analysis ............................................................................ 67 Optimistic Ecological Parameters ........................................................ 67 Pessimistic ecological parameters ........................................................ 68 Optimistic and Pessimistic Economic Parameters ..................................... 70 Increased Time Horizon ................................................................... 71 Conclusion ....................................................................................... 79 Appendix ......................................................................................... 83 (A)Double Singular Solution .............................................................. 83 (B) Partial singular solution when only v is constrained ............................... 85 vi (C)Partial singular solution when only T is constrained .............................. 86 REFERENCES ...................................................................................... 89 vii LIST OF TABLES Table 1.1 Maximum Likelihood Estimation of the Heckman Model ........................ 19 Table 1.2 Variables and Parameters in the Model for BTB across US ...................... 27 Table 2.1 Epidemiology and Economic Parameters ............................................ 61 Table 2.2 Comparison of Dynamic Outcomes and Social Net Benefits for Different Policy Scenarios ..................................................................................... 62 Table 2. 3 Comparison of Dynamic Outcomes and Social Net Benefits for Different Policy Scenarios (Optimistic and Pessimistic Ecological Parameters) ........................ 76 Table 2.4 Comparison of Dynamic Outcomes and Social Net Benefits for Different Policy Scenarios (Optimistic and Pessimistic Economic Parameters) ........................ 77 Table 2.5 Comparison of Dynamic Outcomes and Social Net Benefits for Different Policy Scenarios (Increased Time Horizon).................................. 78 viii LIST OF FIGURES Figure 1.1 The Monte Carlo Simulations for the BTB Disease Dynamics .................. 28 Figure 1.2 Predicted bTB New Infections From 2001 to 2010 ................................ 29 Figure 1.3 Probabilities under Different Surveillance Policies ................................ 33 Figure 2.1 Phase Plane of 8-1, Given Rdot=0 .................................................... 47 Figure 2.2 Bison Population Dynamics: The Partial Singular Solution for T when v=N.63 Figures 2.3a-b Bison Population Dynamics: The Partial Singular Solution for T when v=0 ..................................................................................................... 64 Figures 2.4a-b Bison Population Dynamics: The Partial Singular Solution for v when T=0 ..................................................................................................... 65 Figures 2.5a-b Bison Population Dynamics: The Partial Singular Solution for v when T=N .................................................................................................... 66 Figures 2.6a-b Bison Population Dynamics: The Partial Singular Solution for v when T=0(Optimistic Parameters) ....................................................................... 72 Figures 2.7a-b Bison Population Dynamics: The Partial Singular Solution for T when v=0 (Optimistic Parameters) ....................................................................... 73 Figures 2.8a-b Bison Population Dynamics: The Partial Singular Solution for v when T=O (Pessimistic Parameters) ....................................................................... 74 Figure 2.9 Bison Population Dynamics: The Partial Singular Solution for T when v=0 (Pessimistic Parameters) ....................................................................... 75 Introduction The spread of infectious disease among both wild and domesticated animals has become a major problem worldwide, with countless examples like Tuberculosis, Brucellosis, Avian influenza, Swine flu, and Foot-and-Mouth Disease (Daszak et al. 2000, World Research Institute, 2005, Bengis et al. 2002, National Academies 2005). As most newly emerging human infections of global importance are of animal origin (Taylor and et al. 2001, National Academies 2005), animal disease becomes a global threat to human health. Livestock disease poses an economic threat to not only to farmers, but also the entire livestock industry sector and even the national economy, as the experience of “Mad Cow” in the United Kingdom has shown (Delgado et al. 2003). In addition, a disease outbreak among wildlife may also impose costs on those who value wildlife products and/or services (Horan and Wolf 2005). Therefore, how to efficiently prevent and control both livestock and wildlife diseases become crucial to safeguard global animal and public health as well as related agricultural and trade issues. Human activities and environmental changes can affect patterns of disease transmission within and across species. For example, as cattle trade moves cattle herds from one region to another, it facilitates the spread of bovine Tuberculosis (bTB) infection from infected to non-infected herds and therefore transmits the disease from one state to another (Livingstone et al, 2006). Trade then expands the disease transmission to much larger scope, and makes it harder to manage disease risks. Another example is brucellosis in wildlife such as bison and elk. The human-environmental interactions that alter habitat and wildlife behavior, such as supplementary elk feeding or population control, might have a large impact on the disease transmission process (Hartup et al. 2000, Schmitt et al. 2002). Because economic forces help drive many infectious disease problems, we need to go beyond traditional solutions involving public and veterinary health and also focus on altering these economic incentives (Hennessy 2007, Fenichel and Horan 2007). Disease ecology models have been developed to understand disease transmission systems, and are crucial inputs in disease management decision models. However, the focus in disease ecology has been on disease dynamics in the absence of human impacts or a decision framework to guide disease management (e.g., McCallum and Dobson, 2002), and thus cannot offer guidance grounded in management decision models. Ignorance of human impacts on disease systems would involve allocating resources in imperfect ways to control the diseases, so that disease control is either overly costly or ineffective. Disease management would be straightforward if it were easy to identify and remove the infected animals, However, many infectious diseases are often unobservable without investments to discover disease status, as outwards signs of many diseases are rare and only appear in the final stages of infection (Williams et al. 2002; Lanfrachi et al. 2003). Prior bioeconomic work focuses on imperfect controls in the sense that these investments have not been made -- generally because they are technologically not possible (Horan and Wolf 2005). Policies are also generally based on this situation. Throughout this dissertation I focus on analyzing cases where investments can be made in observation (testing). In this dissertation, I analyze two separate disease problems, Brucellosis and Bovine Tuberculosis (bTB), in order to better understand when it is worthwhile to invest in observability of disease status and how this can be used to improve management. Bovine Tuberculosis (bTB) is a common and often deadly infectious mycobacterium disease that occurs in both animals and humans. Cattle and wildlife such as deer can get infected by bTB. Brucellosis is a bacterial disease that causes cattle, elk and bison to abort their calves. It is transmitted through sexual contact and direct contact with infected birthing materials, and it is one of the most infectious bacterial agents in cattle (Wyoming Brucellosis Coordination Team [WBCT], 2005). The only known focus of Brucellosis infection left in the nation is in the Greater Yellowstone Area (GYA). Both of these diseases could be transmitted to humans, and threaten human health. Meanwhile, these diseases have caused significant economic losses to livestock producers, and have proven difficult to eradicate. The first essay applies an economic gravity model of US cattle trade to forecast bTB transmission across cattle herds in the US. It is suspected that cattle movement across different farms and regions is one of the key factors of bTB transmission in the United States (Wolf et al. 2009). Prior attempts to model the epidemiology of bTB infection among cattle to predict disease transmission have not adequately captured the behavioral aspects of trade. A better understanding of livestock trade patterns would help in predicting disease transmission and the associated economic effects. In this essay, we develop a gravity model of livestock trade and link it to an epidemiological model of bTB transmission, with the goal being that this information could lead to improved disease surveillance and management. Although tests for bTB are non-selective, the improved predictions about disease risks help to figure out where to make these testing investments. The findings suggest that targeting surveillance efforts towards states that have high disease risks could be more effective than targeting efforts only towards states known to be infected, or targeting efforts equally across all states. The second essay is about brucellosis management in bison in Yellowstone National Park. It investigates the combination of vaccination and test-and-slaughter by applying an optimal control approach whereby the disease dynamics can be affected by both management actions and changes in human-environmental interactions. In the model, the wildlife host (bison) is valued for existence and recreation (viewing), which differs from most wildlife disease analyses. Previous studies on optimal wildlife disease management have mainly focused on solutions like population control, i.e., wildlife harvest (Horan and Wolf, 2005; Horan et al., 2008; Fenichel and Horan 2007). However, wildlife harvesting is often nonselective-- it fails to identify the harvested wildlife’s disease status prior to the harvest. This can limit the effectiveness of such a control. Moreover, non-selective harvesting is undesirable when dealing with a threatened population such as Yellowstone bison. The two control options—vaccination and test- and-slaughter -- work in different ways. The test-and-slaughter approach allows infected animals to be removed selectively, while vaccination is non-selective with respect to disease status. Though both controls reduce infection, they do so by different mechanisms. Test-and-slaughter reduces the force of infection on susceptible animals, while vaccination reduces the number of animals at risk of infection. The two controls are substitutes in disease control, as the marginal disease control benefits of each control are diminishing in theuse of the other control. The results suggest that investments in observability (testing) are inferior because the manner in which vaccination performs is superior —it not only prevents infections, but also helps contribute to the resistant population, in turn increase the overall healthy population. Yet, the differences are relatively small, and there might be transaction costs associated with implementing the oscillated controls, especially vaccination. Therefore, the decision of choosing vaccination or test-and-slaughter would depend on whether the transactioncosts of running the program actually offset differences of social net benefits. Essay I An Integrated Model of Animal Trade and Disease Transmission: Predicting bTB Risks in Cattle and Targeting Surveillance Efforts Introduction Tuberculosis (TB) is a common and often deadly infectious mycobacterium disease that occurs in both animals and humans. Bovine TB (bTB) in cattle has caused significant economic losses to livestock producers (with total indemnity costs exceeding $29 million in FY 2007) and has proven difficult to eradicate. Cattle movement across different farms and regions is one of the key factors of bTB transmission in the United States (Wolf et al. 2009). A better understanding of livestock trade patterns would help in predicting disease transmission and associated economic effects. Incorporating cattle trade effects into disease spread models could assist in targeting disease prevention and eradication efforts. The purpose of this paper is to link a gravity model of livestock trade with an epidemiological model of bTB transmission, with the goal being that this information could lead to improved disease surveillance and management. The gravity model has become the workhorse model to analyze trade patterns in international economics. Gravity models were originally inspired by Newton's Law of Gravitation in physics, which suggests that gravity depends positively on mass and negatively on distance. The basic idea is that larger places (in terms of population or economic size) attract people, ideas, and resources more than smaller places, and places closer together have a greater attraction. Gravity models represent one of the most empirically successful models in economics (Anderson and van Wincoop 2003), yielding sensible parameter estimates and explaining a large part of the variation in bilateral trade (Rose 2004). The theoretical foundation for these models has been developed by Anderson (1979), Anderson and van Wincoop (2003), and Eaton and Kortum (2002). Ecologists have recently applied non-behavioral forms of gravity models to invasive species problems to estimate long-distance dispersal of species between discrete points in heterogeneous landscapes, a problem similar in some respects to disease transmission. Bossenbroek et al. (2001, 2007) developed a production-constrained gravity model to forecast zebra mussel dispersal into inland lakes as a function of site characteristics, the relative locations of lakes, and the number and location of boats on which zebra mussels may hitch a ride. While their analyses showed the potential of this model, there was no behavioral model of boat movement. Rather, the estimates were based on lake characteristics and did not consider the explicit economic incentives of boat owners to travel from one lake to another. Prior attempts to model the epidemiology of bTB infection across cattle herds, in cases where trade is taken to be an important factor, have not adequately captured the behavioral aspects of trade. For instance, Barlow et al. (1997) applied both deterministic and stochastic models to investigate bTB dynamics within cattle herds in New Zealand (Barlow et al. 1997). The simulations suggested that bTB was unlikely to persist within a single herd, under present bTB testing policies, without an external source of reinfection. The most likely source of infection came from the movement of infected cattle into uninfected herds, and from surrounding wildlife species (Barlow et al. 1998). Cattle movements in these models take human behavior as exogenous and fixed, yet trade is a behavioral phenomenon that plays a key role in cattle movement and hence disease transmission. Each year, tens of millions of cattle are shipped into another state for feeding or breeding. Therefore, trade mechanisms must be integrated into epidemiology models to make transmission risk endogenous. That is, epidemiology and economics jointly determine bTB transmission patterns. In this paper, we develop a gravity model to capture the economic incentives for cattle movement in US, and we tie this to an epidemiology model to predict disease risks. In the gravity model, the cattle movement and disease risks are driven by production costs, transportation costs that are increasing in the distances between buyers and sellers, and costs caused by regulations imposed on regions that have lost TB-free disease status (Le, a state must have no findings of tuberculosis in any cattle or bison in the state for at least 5 years. Detection of tuberculosis in cattle or bison in two or more herds in the state within 48 months will result in revocation of accredited-free status.) This cattle movement model feeds into an epidemiology model, thereby incorporating an economic feedback affecting predicted disease transmission. As infections spread, predicted economic choices may be affected via an epidemiological feedback. In particular, new regulations may be introduced as infected herds become identified. Modeling these feedbacks improves our ability to predict trade and disease transmission patterns, and also allows us to analyze how the allocation of disease surveillance resources might be improved. A Gravity Model of Trade Our gravity model is based on the work of Eaton and Kortum (2002), who provide one of the few conceptual models that supports the use of a gravity model. We focus on beef cattle and assume they represent a homogenous good that is produced in each state. Beef cattle are generally bred and partially raised in one location and later moved to a final location for fattening before slaughter (Shields and Mathews, 2003). Trade in our model refers to this movement. Our unit of analysis for the gravity model is a US state, although there are many buyers and sellers in each state and movement can occur within or across states. We index sellers by i and buyers by j with the index referring to the state in which these producers reside. Sellers in state i produce cattle with an average input cost of ci. Within state i, there is heterogeneity in the efficiency of production. Denote this efficiency by the random parameter 2,- , so that effective unit production costs are c,- /z,' . Following Eaton and Kortum (2002), the efficiency parameter is taken to be random and is assumed to follow a F rechet distribution function1 —6 (I) F,(z)=e-z which is independent across states. The term ci /z,- represents the unit cost of production only, and does not reflect transportation and other trade related costs that would be relevant to the purchase decision. Trade costs are calculated as a multiplier on effective production costs. Specifically, the trade cost multiplier associated with moving cattle from state ito state j is denoted b ji 2 1, where b ji is a function that takes the following form The Frechet dlstr1but1on IS a spec1al case of the generallzed extreme value d1str1but10n, also called the Type II extreme value distribution. This is one of the common distribution models that can be applied when we generate N data sets from the same distribution, and create a new data set that includes the Viz maximum values from these N data sets. Eaton and Kortum (2002) assume Fi(z) = e- , with a higher Y ,- meaning a higher average realization for region 1'. In contrast, we treat 1’,- the same across states since the technology of cattle production does not vary substantially among cattle producers in the United States. (5‘. (2) bji(djia§)=er Idfi- This relation reflects two kinds of trade costs: (1) transportation costs, d it , and (ii) additional costs imposed by regulations due to bTB, erél . Transportation costs depend on the distance between buyers and sellers, d ji , measured here as the difference between geographic centers of the two states. Given the parameter p > 0, the specification in (2) ensures transportation costs are higher if the trading partners are farther apart. Disease- related regulatory costs only arise if the state has lost bTB-free status. We define 8,- to be a dummy variable for bTB-free disease status, with 5,- = 0 if the selling state is bTB-free and 6,- = 1 otherwise. The exponent y > 0 is a parameter. Hence, trade costs are higher when trade regulations are in place due to the presence of bTB within the state. After taking trade costs into account, the competitively-detennined price that buyers in state j must pay to buy one head produced in state i is (3) Pji = 01b ji / Zi However, buyers in state j would try to pay the lowest price across all sources, so the price they actually pay for cattle is: (4) pj =min{pj,-,i=1,...N} where N is the total number of states. Substituting the expression for p 1,. into (4) results in the following distribution for price p j,- : -0 t9 "C‘b" (5) Gji(P)=Pr[PjiSPI=1"Fi(Cibjj/p)=l-e ('1') p The distribution of prices at which state j will buy is N (9 _(D- (6) Gj(P)=1'1—II1-Gji(P)I=I-e 1” i=1 N 0 where (I)! = 2(cibfl) . i=1 The probability that a producer in state i sells an animal at the lowest price to a buyer in state j is (Cibji I49 (7) ¢ji=PR[pJ-,~_<.min{pjs,s¢i}]=In(l-st(p)dei(P))= (D- J s¢i The probability (bf,- represents the fraction of cattle that producers in state j buy from producers in state i: (8) ¢,-,- if: . WW - WW Xi “U N _. 2(Ckbjk) 6 k=1 where x j is state j’s total cattle purchases and x ji is the number of cattle that state i supplies to state j. We can express x j, in the following form: Xj (Cibji (dji,5))—6 Zk=1(ckbjk(djk,5))— Equation (9) suggests that cattle movement, after controlling for size (total cattle purchases), depends negatively on both state i’s input costs and the trade costs between state i and state j, relative to the sum of an non-linear function of both input and trade costs across all states. So far we have focused on the supply side of cattle. Now consider the demand for cattle. Buyers in state j (feedlots) decide to buy x 1 cattle to fatten up and later sell to the slaughterhouse. Total purchases, x 1 , are chosen by maximizing expected profit a. (10) rgaxE[PR0FIT]=(Bj—tsj—Cj)K(w0xj) 1 —E[pj]xj 1 where 81 is the price the feedlot receives from the slaughterhouse, and t sj is the cost of transporting cattle to the slaughterhouse. These prices are based on weight (i.e., $/pound). The unit cost of production is C j, Production of beef by weight is a function of the weight of purchased animals, woxj, where wo is the average weight of cattle when sold to the feedlot. Specifically, production is given by K (wox j )aj , where 0< 01,- <1 and K are parameters. The final term in (10) is the expected costs paid for purchased feeder cattle, where E[ p j] is the expected price per head of cattle derived from the cattle price distribution in equation (6). The first order condition for determining cattle purchases is , a (11) K(Bj «,1- —Cj)w0al (21-ij = E[pj] The expected value for the price that buyers in state j pay is derived from (6): oo 9 oo 19 — <1>- — <1>- - 1 (12) E[pj]=jap9<1>je ” 1dp=j e p Jdp=¢j1/6F(1+5) o 0 12 where F is the Gamma function. We then combine equations (1 1) and ( 12) to solve for the derived demand for cattle: 1 EIPjI }aj—1 Ka j-woaj (B ,- -zsj - C j) (13) xj={ Plugging equation (13) into (9) yields -1/6 1 _ in = } <1) j Kajwoaj(Bj-tSj-Cj) 1 (I4) 1 a-—1 [F(1+g)l J (cibjiI-g 1 I —— I+————_— . (2" ._ '_ 'aj-l .6(aj_l) [KaJWO 1(31 ’5} CJ)] (I)! By considering the demand and supply of cattle trade, equation (14) shows that the cattle trade between state i and j depends on average input costs in both states, the trade cost variables {b fl }, including those not directly involving j, cattle prices sold at the slaughter house, as well as transportation costs to the slaughter house. Estimation of the Gravity Model To understand cattle movements our goal is to estimate equation (14). We begin by taking the natural log of both sides of (14): 1 aj—l 1 . [n le- = aj _] In(F(I +—6—))—91n(cibji)- ln(Kajw0aJ (Bj "tsj -Cj )) l —(l+m)ln¢j (15) The final term in (15) can be interpreted as a “multilateral resistance” index (Anderson and van Wincoop 2004), as it summarizes how the geographic barriers, disease barriers and input costs across states govern prices in each state. We could estimate (15) with nonlinear least squares, minimizing the sum of squared errors. However, it is difficult to get convergent estimates due to the complexity of the final term, which is a nonlinear function of 0. Anderson and van Wincoop (2004) suggest a technique that provides consistent estimates of (1 5). Consider a set of dummy variables that indicate whether a state is an importer or exporter of cattle when it trades with another state. Anderson and van Wincoop refer to this dummy as an outward region specific dummy. Specifically, let 0k be the outward region specific dummy for state k, with 0k = 1 if state k’s sales to another state are greater than its purchases from that state, and 0k = 0 if otherwise. Anderson and van Wincoop show that replacing the multilateral resistance index with the outward region specific dummy yields consistent estimates using ordinary least squares although this fixed-effects estimator is less efficient that the nonlinear least square estimator as the latter uses information on the full structure of the model. The gravity equation that we estimate then is lnxji = ,u-Hlnci -6p1n(djf§)—ay5 (16) ,9 .- N . 1 (a l)+l " ' lIn(Bj—tSj—Cj)+-—6—'—,——l—’20k “J- (“J“) k=1 where u is a constant: p = ln(F(1+ 1)) - ln(Kajw0aj ). l4 Data for the cattle movement variable x),- come from 2001 interstate livestock movement data from the USDA Economic Research Service (Shields and Mathews, 2003). The cattle movement flows are among the contiguous 48 US states (Hawaii and Alaska are not included).2 The outward region specific dummy 0,, was coded by comparing the cattle sales and purchases between state k and other states. Distance between state i and state j (d ji) was measured in kilometers between the center points of the two states. In 2001, only Michigan had lost TB-free status, so 5 =1 for the state of Michigan, and 0 for other states. The per-head cattle price received by the feedlot from the slaughterhouse is the steers and heifers price from USDA Agricultural Prices 2001. The transportation cost from feedlots to slaughterhouses was calculated by multiplying the average distance from the feedlot to a major slaughterhouse by the per mile cost of $0.09 per cattle head.3 The distance data is the average geographic distance from the center of each state to the 35 major slaughterhouses using ArcGIS 9.2.4 Finally, the average feedlot production cost of weight gain (C,) was calculated as average feed costs, assuming 57.4 bushels of corn 2 . . . . . . We s1mpltfy movement 1n the emp1r1cal example to feeder cattle and feed lots wh1ch are the vast majority of cattle movement. We understand that breeding stock also moves but we are unable to model the movement of both with available data. 3 As the distance is geographical distance, we cannot use real unit transportation cost. According to John D. Lawrence and Shane Ellis (http:ffiwuwv.extension.iastate.ed11’agd1n/livestock/htmlxb 1 ~35.html), the cost of the whole truck is $2.50 per loaded mile, and with the stocking density from http://www.ams.usda.mv/Al\'lSvl .0-"2ctfile’?dDocName;STELDE\-"3008268 From this we calculated a per- mile cost of $0.09 per head. 4 The average distances were calculated for three categories: 1. If there was slaughterhouse within the state, then we used the average of distances to the slaughterhouses in the states; 2. If there were no slaughterhouses within the state but there were slaughterhouses within 300 miles, then we used the average of those distances within 300 miles; 3. If there were no slaughterhouses within the state or within 300 miles in adjoining states, we used the average of all the distances to slaughterhouses from 300 to 1000 miles away. - 15 were consumed per head (Dartt and Schwab, 2001) and using state corn prices (USDA, 2002) The input costs for the feeder calf seller, C1, were proxied using feed costs--the most important expense in raising cattle--in dollars per head. The input costs for feeder calf sellers were constructed based on the feed proportion of a typical farm: 15 bushels of corn and 3.2 tons of hay are consumed per cattle head (Dartt, and Schwab, 2001). One complication arising from the log-linearized version of the gravity model is that there were many zero trade flow observations in the data set, and ln(0) is undefined. One option to address this issue is to drop all observations of zero trade flows from the sample. However, dropping zeros means disregarding potentially useful information, which could produce biased estimates of the coefficients.5 The zero trade flow could also occur due to non-observability. We then need to model the decisions that produce zeros, i.e., on whether or not trade occurs between certain states. This can be accomplished by modeling the decisions as a probit model, and the outcome of that decision determines whether we could observe the actual trade flow 5 . . . . . There are several approaches have been applled 1n the literature to solve th1s problem. One 15 the so called “Ad Hoc” approach. Although ln(O) is not defined, for a very small value of s, ln(0+s) is defined and can be used to approximate ln(0). Therefore, adding a small and positive number to all trade flows can be a sensible place to start, to see if including or excluding zeros appears to make much of a difference in the estimation. This “Ad Hoc” approach has been applied in the literature (e.g., Wang and Winters 1991, and Raballand 2003). However, there is no theoretical basis as the inserted value is arbitrary and doesn’t necessarily reflect the underlying expected value. The Tobit model has also been employed to deal with the zero flow problems (Anderson and Marcouiler, 2002; Carrillo and Li, 2002). The Tobit model describes a situation in which part of the observations on the dependent variable are censored and are represented instead by mapping them to specific values, generally zero (Linders and de Groot 2006). That means if the zero trade reflected either no trade or negative trade, then a Tobit model would provide consistent estimates. However, the underlying expected trade determined by the gravity model can never be negative (see equation 14). Therefore, the Tobit model is not the appropriate model to explain why there was zero trade flow. in the sample. This structure has been framed in the Heckman’s sample selection model (Heckman 1979, Emlinger 2008). First, a set ofcovariates (In C, , Ind 6 , ln(aj (Bj —tSj - Cj )) 10k ,xj) is used 1" ’ to determine the probability that two states engage in trade (i.e., that they are in the sample). The selection equation is defined as: Sjizlu-Blnci—Qplndfi-Byd— "(aj(Bj"’sj—Cj)) aj—l ('7) 6(011-—1)+1N +_—___ 20k + x - + p -- . ._ J 11 6(aj l) k=1 Estimation of (17) determines whether or not we observe cattle trade between two states in the sample, i.e., whether In x ji exists or can be dropped from subsequent estimations. Next, a second set of covariates determines the intensity of bilateral trade, subject to the existence of a trade relationship. The regression model for this relation is specified as: lnSEji=,u—6’lnc,-—6’plndji—l9y5-aj_ln(aj(BJ-—tsj-Cj)) 18 ( ) 6(aj—I)+1 N —— 0k+Xj+€jj (9(aj—l) k=1 where In x ji = In if) if Sji = 1, and In x 17: not observed (i.e., the observation is not used) if s 1., = 0. The error terms for the two equations are (,uji , 81',- ) ~bivariate normal 0 0 1 7- [a s a Ugsggyi- We first ran a Heckman two-step model (equations (17)-(18)) to test for sample selection. The inverse mills ratio was significant at 0.1%1 level, which suggested a sample selection problem in the data. Ignoring the zero trade flows would lead to this sample selection bias. Full information maximum likelihood estimation is more efficient than the two-step method (Maddala 1983; Davidson and MacKinnon 1993; Greene 2003), and so we applied Heckman maximum likelihood estimation to equations (I7)-(18). Table 1.1 shows the estimation results for the Heckman maximum likelihood model. With the parameters estimated below, we solved for p, y and 0. Note that the coefficient for the disease status ((5 ) was negative, which meant that when state i lost its bTB disease free status ((5 =1), the number of cattle that were sold to another states decreased. However, the dummy variable for disease status is not significantly different from zero (p = 0.491), suggesting that the impact of bTB disease status on cattle trade was not significant. This is'likely due to there being only one bTB nonvaccredited state in 2001 (Michigan). Compared to the economic benefits of cattle trade, the expected disease related costs for individual farms were very small. Therefore, cattle movement was not significantly affected by disease risk at the state level. In addition, distance (d ji) between the two trading states is significant and had a negative impact on cattle trade. Expected buyer profit, In (B j - t sj -— C j) , is also significant and has a positive impact on trade. Input costs (of) within the state of origin do not have significant impact on cattle trade. Table 1.1 Maximum Likelihood Estimation of the Heckman Model [95% Conf. Coef. Std. Err. P value Interval] Log input prices -1 .47 3.86 0.70 -9.07 6.07 Log distance -I .63 0.11 0 -1.83 -1 .38 TB disease free Status -2.32 3.38 0.49 -8.94 4.30 Log buyer’s expected net benefit 2.31 0.87 0.008 0.59 4.02 Constant 35.6 24.2 0.14 -11.8 83.0 46/1 -0.92 0.02 0's 3.34 0.1 I Inverse Mills ratio -3.09 0.15 BTB Disease Dynamics for Cattle Movement We use the estimation results from interstate trade in a simulation model to predict how interstate cattle trade generates disease risks across states. The bTB disease dynamics for cattle movement were modeled using a variation of Barlow et al.’s (1998) model. The model was comprised of difference equations that simulate changes in numbers of three herd categories in each of 48 states. In each state, farm-level herds are divided into one of three types: (I) herds that are healthy but susceptible (with the number in state i denoted Si), including accredited herds and those becoming reaccredited after coming off movement control; (2) herds that are infected with bTB and identified as such (with the number in state i denoted M,); and (3) herds that are infected with bTB but not detected (with the number in state i denoted I,-). We also assumed those infected and identified cattle herds will be under movement control due to government regulations. Cattle trade is assumed to occur only between herds that are either susceptible (S) or non-detected (I). The total number of herds in state i is given by Z,- = S,- + I,- + M,. 19 The bTB disease dynamics for N = 48 states can be represented by the following difference equations, where the t subscripts are time-indices: S. M't 1” k Sj,tw (19) Sjrl+l-Sj9t=_1.- Z‘ngt—fl(z. )w 1.1"! q] kzl J” (20) Ij,1',=t+l"jt [(2];cm +fl(%— :(VljJ'gjljJ M' J” (21) Mj,t+I ’Mj,t =§j1j,t — q. I As herds transition from movement control status to susceptible status at the rate 1/ qj, the number of susceptible farms increases, and the number of farms under M-, Js ), qJ' movements controls decreases (so that the number of farms changing status is where q, is the average length of time a farm remains under movement controls. As new S j, I infections arising from interstate cattle trade ( 2 ‘Pk,’ ) and contact with other cattle k=1 herds within the same state ( “2131(0) 11-, t), the number of susceptible farms IS j, t decreased, while the number of infected farms is increased. These disease transmission terms are described in detail below. Finally, testing at the rate 4‘]- decreases the number of infected (non-detected) farms and increases the number of detected farms. Farms not under movement controls are periodically tested by the government or may be detected via slaughterhouse testing. Following Barlow et al. (1998), 5 j = a j + m/ rj is the rate 20 at which “I” herds become detected and go on movement controls. Here, a j is the rate of slaughterhouse detection, m is the test sensitivity per herd, and If is the regular herd testing interval in state j. For our baseline simulation, we assume that only states that lose their bTB free status will be under regular bTB testing, while slaughterhouse surveillance occurs at the same rate for all the states. Farms under movement control will not be allowed to trade with other cattle farms. In addition, if there are more than two cattle herds under movement control within 48 months in a state, the state will lose its bTB disease—free status, and hence decrease cattle sales because of the imposed trade restrictions by other states. Once infection occurs within a state, we assume the infection spreads S - t deterrninistically according to the intrastate transmission term ,B(-Z—J’—)w If” , where ,6 fat is the disease transmission parameter. The expression (flit—)0 is the susceptibility Z jJ function (Barlow, 1995), where o) is a spatial heterogeneity parameter.6 The use of a deterministic transmission function is a common approach in the disease ecology literature (e.g., Barlow et al. 1998) and will not affect whether a state becomes infected during the simulation period (though the deterministic nature of the function will affect the level of infection and hence the risks to other states). 6 Using the notation of footnote 5, “SM = ([1 - v]s / n)7 and g(N) = l. Dobson and Meagher investigate both g = l and g = N for the case of bison and find that g = 1 produces more reasonable results. These results are also in line with the view that g = 1 is often realistic for sexually-transmitted and indirectly-transmitted diseases (Barlow 1998). As cattle are behaviorally similar to bison, as both are herd species, we adopt g = 1. 2| However, assuming deterministic inter-state transmission will result in all states that purchase cattle from infected states becoming infected with probability one in each period. This is not realistic and it undermines the purpose of our simulation, which is to predict which states are most likely to become infected. We therefore adopt a stochastic simulation approach to modeling interstate transmission. This stochastic transmission is S . 1” represented by the term 2 ‘I’f t in (19) and (20). Specifically, we assume the k=l ’ probability of each farm getting infected via interstate cattle purchases follows a Bernoulli random process. Let 771-,- be the probability that one cattle farm in state j becomes infected after a purchase of one cow from state i. This probability is given by 771-; = p],- /Z,-, where (p is the prevalence rate of a typical infected farm. If the per farm cattle imports from state ito state j is n ji = x fl- / Z 1" then the probability of that one susceptible cattle farm does not become infected due to its trades with farms in state i is Zji = (l — 771-,- )"ji . The probability that the farm does not become infected due to N trades with farms in all states is TI 11-,- , so that the probability that the farm does i=1 N become infected is g =(1- Ill Zji)- Note that because this distribution will change i=1 over time as infection risks change and alter trade patterns, x,-,-, the cattle trade and bTB disease dynamics are linked together. If state 1' increase its cattle purchases from states that have bTB infection, individual farms’ disease risks (97) will increase accordingly, and therefore change the disease dynamics. 22 So far, we have described the probability that a single farm becomes infected. However, we are interested in outcomes at the state level. Define ‘I’ j as an infection dummy variable for one susceptible farm in state j: ‘I’j=l if the farm becomes infected via its cattle imports and \I’ j = 0 otherwise. This dummy variable is a Bernoulli random variable with ‘1’]: 1 occurring with probability 9'13 The total number of infections is determined by taking S,- random draws of the variable ‘I’ j (i.e., one draw for each susceptible farm in state j) and summing the values of those draws. The number of new S . J infections then equals 2 ‘1’ k=1 f , where ‘1’; represents the kth draw of the random SJ variable ‘I’j. Then this total number of new infection( 2 ‘I’ k=1 f ) will follow a binomial distribution with probability 9‘), and the number of trials is Sj. Simulating Disease Dynamics and Risk We now turn to a simulation of bovine TB disease dynamics across the 48 contiguous US states. The total number of cattle farms was taken from the 2002 Census of Agriculture (USDA 2002). Simulation results were derived using Matlab R2007b. The initial period is 2001, the period for which livestock movement data were available. In 2001, only Michigan had lost its bTB-free status (with eight infected herds), while there was one infected herd in both Oregon and Kansas. It is not clear how many herds were actually infected but not detected in 2001, but it is unlikely that all infected herds were detected. For simplicity, we assume the number of actual infected herds initially equals the number 23 of known infected herds. We also assume none of these herds is detected. Obviously, this assumption results in an underestimate of actual detected herds, and may or may not be an underestimate of infected but non-detected herds. The simulation parameters are shown in Table 1.2. Interstate cattle movements are predicted using the estimated gravity model (Table 1.1) and are used to simulate bTB dynamics. In this way, market factors, such as input costs, that change cattle trade patterns will also affect the bTB disease risks. States that have more than two cattle herds under movement controls will lose bTB disease free status. Accordingly, production costs were increased for both sellers and buyers (feedlots) in states that lost bTB-free status, as regular bTB testing is required for farms in those states. A testing cost of $10 per cattle head was assumed (Wolf, and Ferris, 2000), so that production costs become c,- / 21' +10 for sellers and C j + 10/ Wf for buyers, where Wf is the average weight of finished cattle (since buyers’ costs are expressed in terms of weight). Thus, the interstate cattle movements will respond to disease surveillance costs affecting cattle supply and demand in states where the disease has been detected. This creates an important feedback between the behavioral model and epidemiology model. The simulation takes initial values as given and proceeds over a 10 year-period according to equations (19)-(21). A Monte Carlo analysis, illustrated in Figure 1.1, is used to account for the randomness of new infections due to interstate cattle movement. First, the initial values of all state variables (S,I,M), which are assumed known, determine testing costs and trade restrictions, thereby influencing trade flows. The current state ' variables and predicted trade flows are used to calculate the number of new infections in each of the contiguous US. states, and the values of the state variables are updated. This 24 process is repeated for each of 10 years, with the output being 10 years of epidemiological and trade predictions. This output, however, is based on only 10 sets of random draws of new infections across the nation, and therefore only represents one possible outcome for the 10-year period. We therefore repeated the entire process N = 1000 times to derive 1000 possible outcomes for the 10-year period. This broader set of outcomes represents a distribution of results that we used to calculate the expected number of infections in each state over the 10-year period, along with the corresponding standard errors. The distributional results were used to calculate the probability that each state becomes infected (which we define as one or more infected herds) during the decade. We denote states for which the probability of infection is at least 0.2 as being “high-risk” states. The categorization of “high-risk” is subjective, reflecting both the likelihood of infection and the perceived costs that may result if infection spreads within the nation. Figure 1.2 indicates the three states that were initially infected in 2001 (grid patterns), along with the predicted high-risk states (solid shading) and states that were actually infected during this time period (solid shading and diagonal stripes). The high risk states extend from the central portion of the US. to the west, where many of the larger feedlots operate and where demand for cattle imports is the largest. The model’s performance can be judged by comparing predictions of infections with actual infections over the past decade. Overall, the model performs well. Seven of the predicted high-risk states (i.e., those not initially infected) have been detected with bTB infection since 2001: California, Colorado, Minnesota, Nebraska, Oklahoma, Texas, and South Dakota. In particular, the model predicts large infection probabilities for three 25 fourths of the states that have lost their bTB-free accreditation during the past decade (California, Texas, and Minnesota). The model failed to predict high infection probabilities for four states that did eventually become infected: Arizona, Kentucky, New Mexico, and Ohio, with New Mexico having lost its bTB-free accreditation. There are three reasons why the model may have failed to designate these states as high risk. First, these states may have simply been unlucky, as the model did predict positive (though small) probabilities of infection for each of these states, with the exception of Ohio. Second, some of these infections may have come from cattle trade with Mexico, which is not considered in the trade model but which is potentially important because bTB is endemic in cattle in many regions in Mexico. In particular, Arizona and New Mexico may have become infected due to Mexican trade, as approximately one million head of Mexican cattle enter the United States annually through ten ports of entry in Arizona, New Mexico, and Texas (Mitchell et al. 2001). In addition, many species regularly migrate across the US. and Mexican border (Lopez-Hoffman et al. 2010). Third, model predictions would likely improve if the model was updated during the simulation to reflect actual outbreaks as they occurred. We did not update the model in this manner because outbreak data only exists for detected herds (M), and not total infections (1+M). Updating the model with actual infections would have a bigger impact on the distribution of herd testing requirements over the landscape, possibly influencing trade flows. We explore the relation between testing requirements (surveillance) and risks in the next section, and we find that Kentucky and New Mexico are predicted to be two of a handful of states that might face greater risks when more states are required to adopt herd testing. 26 Finally, our model predicts three states remain at significant risk of infection: Idaho, Iowa, and Missouri. The fact that infections have not yet occurred in these states is not a negative indication of the model’s performance. Rather, this simply means there may be cause for increased vigilance in these states. We now turn to the notion of surveillance to explore how the simulation can be used to reduce infection risks. Table 1.2 Variables and Parameters in the Model for BTB across US Description Prevalence rate of a typical infected farm Average length from movement controls to susceptible (Month) Slaughterhouse detection rate Test sensitivity per herd Herd testing interval Initial infected herd Initial herds on movement control Parameter qj 1(2001) M(2001) Value or Calibration Method 0.02 9.6 0.01663 0.8 12 1,1, 8 for OR, KS, MI, and 0 for others 0 Source Assumption Barlow et al. (1998) Barlow et al. (1998) Barlow et al. (1998) Barlow et al. 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Sages I 2.0qu 29 csN 3 ZEN Each 2.2.8.3— 302 PH: 33:55A— ~.— 0.5»?— Comparison of Surveillance Policies We now use the simulation model to examine how surveillance resources can be allocated among different states to improve disease management. Broadly speaking, there are three ways to allocate surveillance resources: towards states where infections have been detected, towards states that are at highest risk of infection, or to all states regardless of actual infections or risks. Accordingly, we compare three different govemment-directed surveillance scenarios that may be implemented in addition to standard slaughterhouse surveillance: (a) Status quo testing, where testing is only applied 1 in states that have already lost their bTB-free status; (b) Non-targeted testing, based on periodic testing in all states; and (c) Targeted testing, based on periodic testing in only the highest-risk states. To make valid comparisons of these scenarios, each scenario is assumed to employ the same level of surveillance efforts, as described below. , (a) Status Quo. This scenario, which is also the baseline scenario we analyzed in the previous section, is based on the current US. bTB approach of focusing surveillance efforts on states where infections have been detected. We assume only states that have lost their bTB-free status-~with more than two herds detected with infection--will test herds at an average testing interval of once every twelve months. (b) Non-targeted testing. For this scenario, surveillance resources are allocated to all states, i.e., farms in all states are under regular bTB testing. The testing interval is calculated based on the assumption that the expected cost of surveillance efforts is the same as what was expended over the ten year interval under the status quo. Specifically, we find that testing all US. herds an average of once every 26 months yields the same total surveillance efforts as Scenario (a). The available surveillance resources cannot 30 sustain annual testing in the present case, due to testing occurring in more states than in the status quo. The'simulation results are presented in Figure 3. Predicted infection probabilities increase in all high-risk states relative to the status quo. Moreover, infection probabilities increase in six states that were previously in the low risk category. Tennessee and Illinois become high risk states in this scenario. We also find predicted infection probabilities increase to about 0.15 in Kentucky and New Mexico (and also Alabama), which we previously indicated did actually become infected within the past decade. Predicted infection probabilities are larger in this scenario, relative to the status quo, because the new distribution of surveillance costs reduces the relative cost of trading with infected states, altering trade flows in favor of higher risk trades.7 (c) Targeted testing. For this scenario, surveillance is targeted at the highest risks states. Specifically, states with a very high probability (0.5) of having more than one infected farm will be tested for bTB every 8 months. As with the non-targeted scenario, the testing frequency is calculated according to the expected surveillance costs under the status quo. Here, testing occurs more frequently than in the status quo. This is because the targeted approach is more effective than the status quo, which ultimately requires testing in more states due to there being more infected states under the status quo. Indeed, by focusing the surveillance efforts in states having larger disease risks, the 7 There are two other things to note about these results. First, the reduction in total bTB infections in our simulation alters who pays for bTB surveillance, as we have assumed cattle farms incur testing costs. With a mandatory testing program in all states, even cattle farms that have a very small disease risk will pay testing costs in our model. Some might view this as unfair, though all farms benefit from reduced risks. Second, we have not considered administrative costs in our model, and these might depend on the number of states in which testing occurs. Total surveillance costs might be higher than the status quo if the administrative costs of testing in all states are higher than those associated with administering tests in fewer states. 31 predicted probability of infection falls in all high-risk states except South Dakota, and in all previously defined low-risk states except Illinois and Indiana (Figure 3). Targeted bTB testing reduces overall infection risks because it requires actively searching for infections in places where infections are most likely to occur. Accordingly, targeted surveillance generally addresses disease problems earlier. There are some unintended consequences of targeted surveillance, however. Relative to status quo surveillance, targeted surveillance reduces the demand for cattle imports by the highest risk states, which results in more cattle purchases by lower risk states. With increased imports into the low risk states comes increased risk. 32 I osc mafia. .695 mBSm xmcasoq 0:3 33% .525 33% xmtéwE I> II? \I I} \I I} 2... 5.2 ...2 >¥ Z. 1: ._< X... Dw v.0 m2 OS 2.2 n: <_ , 00 <0 _ if 3 .3. J n. J LJ . . fir .1 C M m, . I1 1 w. I m. T1 11 "W1 P.o l I M. F I l - Nd I I 1 -- 1 - 2 2:69th .1 I 1 - ed w $693.82 I , . . . e m 80 .855 D , MI I - md m I 06 No we ,_ _ iIIIIIIIIIIIIIIIlliIlIIltlIl II III IIIIIIIIIIIIIIIIII lLr @ o 3.2-o.— oo=a=_o?_=m Ens—ob:— uo—E: 85.3395 MA «...—arm 33 Conclusions In this paper, we did not find evidence that producers’ market transactions were significantly affected by social or private bTB risks. This is probably due to at least of couple of reasons. First, there exists a very small disease risk for individual farms. Compared to the economic benefits of cattle trade, the expected disease related costs for individual farms are very small. Therefore, cattle movements are not significantly affected by the state of origin’s disease risk. Second, the metric for disease risk in this paper is the bTB-free status of the state of origin. Once a state loses bTB free status, trade restrictions are imposed, and additional costs are incurred for interstate cattle trade. Disease-free status is a very good indicator of disease risk as well as economic cost that affect trade decisions. However, in 2001, only Michigan had bTB free status, so the disease status variable does not exhibit enough variation with which to explain cattle trade. More variation in the variable representing disease risk would be more useful, but data are not available at this. ' Despite the fact that disease risk did not significantly affect cattle movement, interstate cattle movement can affect the transmission and spread of bTB. As we showed in the simulation, states with greater cattle imports, especially from infected states, have a much higher probability of becoming infected. Therefore, more emphasis on bTB surveillance in these states could be justified. The bTB disease predictions from our simulation are consistent with the actual detected infections, which suggest that the gravity model of cattle trade could be useful in predicting bTB transmission in the United States. Of course, cattle trade with Mexico and Canada should be considered in a more complete analysis. In addition, our analysis was 34 performed at a broad scale — the state level. A more useful approach for predicting infection risks and guiding policy choices would involve using a smaller unit of analysis such as the county level. However, current state level cattle trade data sets do not allow for this. Given the concern over bTB, and other animal diseases, and the role that animal movement plays in disease spread, it would seem that improved data collection would be the first step to proactively managing disease risks. Finally, the purpose of comparing different surveillance policy scenarios in this essay is to provide insights into how surveillance resources can be allocated among different states to improve disease management, but not to suggest that the targeted testing approach we considered is economically efficient. Though the particular targeted testing approach that we considered was the most cost-effective method of reducing overall infection risks, at least among the three scenarios we compared, there is no guarantee that this approach is the most efficient surveillance strategy relative to all possible options. Indeed, only a few options were compared in this essay. Moreover, farmers’ cattle movement decisions remain decentralized within the model, and do not reflect the costs of disease outbreaks to the states’ economies. A truly efficient solution would take all external costs into account, and would involve a combination of movement and surveillance decisions that will vary by state to reflect the costs and benefits of implementing controls in each state. For instance, efficient surveillance might involve testing not only in high risk states (which is what we modeled for the targeted surveillance scenario), but also in low risk states - though likely would involve less- frequent testing in low risk states. Indeed, as shown in Figure 1.3, testing in high risk states will likely affect disease risks in some low risk states due to the increased cattle 35 purchase incentives for those states. Therefore, completely ignoring the low risk states would not be optimal. 36 Essay II An Optimal Control Approach to Brucellosis Control in Bison in Yellowstone National Park Introduction The threat of wildlife disease is rising due to increased international trade and human expansions. Risks posed by infectious diseases are significant and escalating (Millennium Ecosystem Assessment, 2005) and pathogen introductions may achieve a status similar to invasive species, the second most important cause of extinction (Daszak Cunningham, and Hyatt, 2000). Some wildlife species serve as natural hosts for diseases that affect livestock and humans (e.g., brucellosis, bovine Tuberculosis, Foot and Mouth Disease), and can incur substantial economic costs. Moreover, wildlife diseases are considered as an important driver of biodiversity loss (World Resources Institute, 2005). Therefore, wildlife disease management has become a challenge and an opportunity to integrate biological and human dimensions for improved wildlife management. Indeed, interest in wildlife diseases is growing among researchers from a wide range of disciplines, including conservation biology, public health, ecology, biology and of course, economics. There are three basic forms of management strategies for wildlife disease: prevention of introduction of disease, control of existing disease or eradication. Once a disease has been introduced, managers often focus on eradication. Eradication may involve extreme population controls to reduce wildlife densities and infectious contacts, particularly when vaccination is not an option. These controls are generally. nonselective with respect to an animal’s disease status, as infected wildlife is often not observable prior to harvest and postmortem (Lanfranchi et al. 2003). Eradication controls may be 37 acceptable when the infected species is a nuisance, but it may imply large costs for valuable wildlife such as bison, deer and elk. These wildlife usually hold significant existence value or use values such as tourism and hunting. Population management may be particularly costly for threatened and endangered species. Yet doing nothing may also put the population at risk. Brucellosis disease control policies in wildlife are still being formulated and implemented on the basis of sociopolitical considerations, rather than on the preponderance of scientific evidence (Bienen and Tabor, 2006), i.e., the ecosystem and its interactions with human management choices. Optimal wildlife disease management therefore may not involve eradication; but should consider the trade off between benefit of the wildlife’s existence and economic costs that the disease imposes on society, such as on livestock sectors or human health. Moreover, the method of control is an important choice. Previous studies on optimal wildlife disease, management have mainly focused on solutions like population control, i.e., wildlife harvest or reducing supplemental feeding of wildlife(Horan and Wolf, 2005; Horan et al., 2008; Fenichel and Horan 2007). Disease control approaches that have been applied extensively in livestock disease management, such as vaccination and test-and-slaughter have only received limited attention in wildlife disease-management. Vaccination has been used to manage some diseases, mainly non- valued wildlife, such as rabbits (Calvete, 2006). There have been concerns about vaccinating threatened wildlife, as the vaccines usually require extensive safety trials and involve with some potential risks to the wildlife (Cleaveland et al., 2002). Test and slaughter is only feasible for wildlife when a good test can be administered with fast results in the field. Moreover, capturing wild animals safely may be costly. 38 In the case of bison and brucellosis, safe vaccinations and accurate field tests have recently been developed, making these tools feasible options. These tools work in very different ways—vaccination reduces the number of animals at risk, while test-and- removal reduces the force of infection (i.e., the risk to susceptible animals). In this paper, we will investigate the combination of vaccination and test-and-slaughter in a bioeconomic framework, when the wildlife host (bison) generates both existence and ecotourism values. The combination of the two approaches could improve the efficiency of disease control in Greater Yellowstone Area (GYA). and make it more acceptable to the public than test-and-slaughter only approach. Cheville et al. (1998) also compared the pros and cons of the two approaches. However, none of the previous research about brucellosis management in wildlife has studied these approaches in an optimal fashion. History of Bison and Brucellosis in Yellowstone As symbol of the American west, bison are an essential component of Yellowstone National Park (YNP) because of their contribution to the biological, ecological,cultural, and aesthetic purposes of the park. Starting with a herd of 23 original bison within YNP at the beginning of the 20th century, the wild bison increased and intermixed with captive bison brought to the park in 1902. Today, nearly 4,000 wild bison roam Yellowstone, delighting visitors with a glimpse of American history (Yellowstone National Park, 2009). The population has increased steadily since the 19605, with some fluctuations in recent years due to human management actions. Brucellosis is a bacterial disease that causes bison, cattle and elk to abort their calves. It is transmitted through sexual contact and direct contact with infected birthing materials, and it is one of the most infectious bacterial agents in cattle (Wyoming 39 Brucellosis Coordination Team [WBCT], 2005). Brucellosis has caused devastating losses to US. farmers over the last century. The USDA and animal industry embarked on a plan to eradicate brucellosis in the United States in the 19305, and this effort required 70 years and an estimated $3.5 billion in state, federal, and private funds (WBCT, 2005). The only known focus of Brucella abortus infection left in the nation is in the GYA. Brucellosis is thought to have been transmitted to different bison herds in the national parks of the Greater Yellowstone Area in the early 19003 (Godfroid, 2002). It was first diagnosed serologically in the park in 1917 from two bison that aborted their fetuses (Mohler, 1917). About 50 percent of the park’s bison test positive for exposure to the brucella organism. Testing positive for exposure (seropositive) means the animal has been exposed to the bacteria at some time in its life; it doesn’t mean the infection, is active or contagious. Tests reveal that less than half of seropositive female bison were shown to have the. infection (Yellowstone National Park website, 2010). 1-2% of non- feeding-ground elk are seropositive; elk at feeding grounds have a much higher rate of about-37%-because dense concentrations of animals create conditions favorable to disease transmission. (Cheville et al., 1998). Brucellosis Management Options There have been numerous approaches proposed to controlling or eliminating from the GYA. The interagency bison management for YNP proposed eight alternatives in order to maintain a wild, free-ranging population of bison and address the risk of brucellosis transmission to protect the economic interest and viability of the livestock industry. 40 Those alternatives involve approaches like bison hunting, testing and slaughtering seropositive bison, quarantine of seronegative bison and vaccination..0pinions differ as to the likelihood of successful outcomes of the various alternatives. For example, rigorous bison hunting is not favored by the public due to bison’s special value to Americans. Among these approaches, vaccination and test-and-slaughter are common strategies in brucellosis control in cattle, and a combination of these two actions could potentially eradicate the disease in the GYA over time (Cheville et al. 1998). The two vaccines, $19 and strain RB 51 (RB 51), are currently available for use against bovine brucellosis caused by Brucella abortus. Neither vaccine is 100% effective, but RB 51 has the advantage of not inducing antibodies that are detected on standard surveillance tests (Roffe and Olsen, 2002). R351 has been used extensively on cattle herds, and is the only available vaccine used on commercial bison herds. Although there is no consensus on the efficacy of RB 51 on bison herds, many studies suggest efficiency for RB 51(Roffe and Olsen, 2002). Unlike vaccination for livestock, which usually uses with parentreal delivery (vaccine injected directly into the body), vaccination for bison uses remote delivery—using charge or compressed gas-powered guns delivering a biodegradable bullet containing the vaccine. However, disease control will be very slow by only vaccinating the bison, and it would be expensive if we are trying to vaccinate all the free-ranging bison (Cheville et al., 1998). Test-and-slaughter as a disease control method involves serology testing for Brucella antibodies, and then culling seropositive individuals (Bienen and Tabor, 2006). Diagnostic testing and slaughter of test positive herd is one of the common procedures in livestock disease control, especially at the slaughter surveillance. A three-year capture 41 and test program for all bison was proposed in the interagency bison management plan (IBMP). Those testing negative would be released in the park, and seropositives would be shipped to slaughter. The Epidemiological Model Model Specification We begin with an epidemiological model of population and disease dynamics for bison in the YNP. The model is based on Dobson and Meagher’s (1996) single-species susceptible-infected-recovered (SIR) model, modified to include vaccination and test- and-slaughter management choices. The bison population (N) consists of three sub-populations: susceptible, S, infected, I, and resistant, R. The population S changes over time according to (1) s =[S+771(1—C’)+R][a—¢N]-mS-flIS/N+ch—va/N The first term of the right-hand-side (RHS) represents the natural reproduction of susceptible bison, accounting for the impact of density-dependent competition. The birth rate is a, 4 is the proportion of infected females that produce infected offspring, 77 is the reduction of fecundity in infected animals, and (p represents the magnitude of the density- dependent competition effect. The second term represents natural mortality, with the natural mortality rate given by m. The third term represents the number of bison infected by contacting with other infectious bison. Following Dobson and Meagher, the disease transmission is of the form ,BIS / N , with ,6 representing the rate of infectious contact per infectious animal, and S /N is the proportion of contacts with susceptible individuals. The fourth term represents the number of newly-susceptible bison that were previously 42 recovered and immune, but which have lost their resistance to brucellosis. The rate of the lost resistance is o. The last term is the number of bison that become resistant due to vaccination, where v is the number of bison in YNP that are vaccinated at time t, and K is the effectiveness of the vaccine. Animals are vaccinated non-selectively so that S/N is the proportion of vaccinated animals that are converted from susceptible (S) to resistant (R). Also note that vaccination is non-selective in the sense that it simultaneously affects multiple states: it affects S as well as R, as we will show below. The change in the infected stock of the bison is (2) 1'=n41(a-¢N)-m1+asI/N—wI—ar—T1/N The first RHS term of equation (2) represents the reproduction of infected bison. The second term is natural mortality. The third term represents the number of bison being infected. The fourth term reflects disease-related mortality, where a) is virulence (disease mortality rate). The fifth term is the number of infected bison that recover from brucellosis, where 5 is the recovery rate. The final term represents the reduction in I due to test-and-slaughter, where T is the number of bison in YNP captured for brucellosis testing, and then the infected bison will be removed after identification. Brucellosis disease is not detectable before testing, so animals are tested non-selectively, and I/N are the proportion of animals that are actually infected. However, even though testing is economically non-selective (i.e., you have to pay to test animals that may or may not be infected), this control has a selective ecological effect in that it only affects the infected population. The change in the resistant stock of bison is (3) R=61—mR—aR+vxS/N 43 The first RHS term represents the number of bison that recover from infection. The second and third RHS terms reflect the decrease in the number of resistant bison due to mortality and loss of resistance. The final term is the number of bison that become resistant due to vaccination. Impacts of Human Choices Among the two bison management controls, test-and-slaughter is a selective control that only affects the infected population, while vaccination is also a non-selective control affecting the susceptible and resistant population. These controls affect disease management in different ways. To see how, note that the objective of disease management is to attain I = 7741(a - (0N) — m] + ,BSI / N — col - 51 — TI / N <0. The control Treduces I directly. The controliv, however, has no immediate impact onI . Vaccination reduces future values of S and increases future values of R. Together, these impacts have a neutral effect on future values of N. The only indirect effect on I therefore comes from a reduction in future values of S and hence transmission. Hence, vaccination indirectly reducesI . These effects can be shown graphically, though it is difficult to visualize in three dimensions. We can show the impacts in two dimensions if we make assumptions about one of the state variables. Specifically, we analyze how the controls affect the dynamics in S-I space, assuming R = 0. These effects are shown in Figures la-lc. The S = 0 isocline has an intercept at the S axis and an intercept at the I axis. S > 0 for combinations of S and 1 below the isocline and S < Owhen above. The isoclineI = 0 is represented by the upward-sloping line, it can be thought as a threshold boundary for 44 changes in the disease state: I > 0 for combinations of S and 1 to the right of the isocline andI < 0 for the combinations to the left of the isocline. Figure 2.1a represents the case where both v and Tare zero. The two isoclinesintersect, and imply a stable interior equilibrium in which both the susceptible and infected population would exist. Figure 2.1b illustrates the case when v=0, T>0. The disease threshold, which depends directly on T, shifts to the right as T is increased. As T is ecologically selective, the isocline is unaffected by T. The overall effect of T is to increase the range of the state space over which infection levels are decreasing, and results in a smaller equilibrium value of I and a larger equilibrium value of S. The resistant population also decreases in equilibrium, as it moves in the same direction as the infected population when v = 0. Relative to Figure 1.13, the increase in v and Tshifts both isoclines. The S = 0 isocline shifts to the left since vaccination reduces the number of susceptible animals. TheI = 0 isocline rotates downward. Though we have already indicated the I = 0 isocline does not depend on vaccination, this isocline does depend on the equilibrium value of R since we are evaluating the isoclines along the locus of points for which R = 0. An increase in vaccination increases the equilibrium value of R. Other things equal, this increases N and results in more reproduction, reduced transmission, and reduced effectiveness of T. The overall effect is to reduce the locus of equilibrium values of 1. Therefore, relative to Figure 2. lb, the effect of vaccination is to increase the range of the state space over which infection levels are decreasing, and results in smaller equilibrium values of both I and S. Though both controls reduce 1, they do so for different reasons. Test-and- slaughter reduces the force of infection on susceptible animals, while vaccination reduces 45 the number of animals at risk of infection. The two controls are substitutes in disease control, as the marginal disease control benefits of each control are diminishing in the use of the other control. Alternatively, the marginal costs (in terms of increased infection) of reducing either control are decreased as use of the other'control is increased. Hence, there may not be incentives to use both controls, and a solution where both controls are . utilized may not be optimal. The analysis of Figure 2.1 considers time-invariant changes in the controls, so as to understand the impacts of the controls on disease management. In the following sections, we construct a bioeconomic model to explore economically optimal management. The time-variant control strategies derived under this approach are chosen by considering economic the economic and ecological tradeoffs. 46 Figure 2.1 Phase Plane of S-I, Given Rdot==0 I 3000 _ 2500 1“ ’ ’ " ’ ’ I K r r r r ..- ..- ‘_. 2000 - r - - - .— .. .. ._ ._ . 1500 — . _ . ' ’ . _ ‘ _ _ _ J' o 1‘ ': d5/ ‘1’: 0 ,v'” dI/dt=0 500 ~ I In 1 1 1 L 1 L L 1 1 1 1 1 1 1 ' 1 1 1 1 4 4 S 1000 2000 3000 4000 5000 Figure 2.1a T=0, v=0 47 Figure 2.1 Phase Plane of S-I, Given Rd0t=0 I 3000‘ / / / / / / / / / / / / ZSOOi/i/ //////// / I / / x / / / / / / / / 20001 F I l I / / / / / / / / isooj- , , , . dS/dFO I\ 10001—\ ‘ 500. auaa=0 ‘1000‘ 42000L I A 130001 4000 II 1 ‘5000 Figure 2.1b T>0,v=0 48 S Figure 2.1 Phase Plane of S-I, Given Rdot=0 3000 I 25001////////////// I 2000 I 1500 dS/dt=0 1 000 500 dI/dt=0 q-v"'L.--1-T-1..1-T-1.--1..1 1 1 1 1 1 1 14 1 1 1 S 1000 2000 3000 4000 5000 Figure 210 D0, v>0 49 The Bioeconomic Model Suppose the manger of YNP wants to control the bison population and disease prevalence rates in a manner that maximizes the discounted net economic benefits to society. These net benefits include the existence and eco-tourism values of bison less the damage costs associated with infection to the livestock sector, as well as costs of implementing vaccination and test-and-slaughter. Ecotourism and existence values, denoted U(S+R) (with U '>0, U ”<0), depend only on healthy animals. We assume U(N) take the form of a ln(S + R), where a is a parameter. The YNP manger’s choices include vaccination (v) and test-and-slaughter (I), i.e., he or she chooses to vaccinate or test-and-slaughter bison, and 0 S v 5 M0 5 T S N . Each choice will incur a unit cost c. and cr respectively. The cost of the disease, particularly to farmers, must also be considered. Denote the economic damage caused by infected bison by D(I), which we assume to take form of 01 . Given the discount rate p, an economically optimal allocation of vaccination and test-and-slaughter maximizes social net benefits, SNB, that is °° c c (4) if? SNB = j [a ln(S + R)-7V‘i v"; T — 61]ep'dt subject to three equations of motions (1), (2) and (3), and the feasibility conditionsO S v,T S N , and given S(0),I(0),R(0). The current value Hamiltonian is (5) H = aln(S + R)'%V'%T'61 + 25$ + 1,1' + ARR where 15,111,)“ are the co-state variables associated with S, I, R respectively. Note that the Hamiltonian does not model the constraints explicitly, though the constraints are 50 considered implicitly. The optimality condition for the choice of vaccination is >0, if] vt=N 6H cv S S' , 111 (6) E=-W‘}t5Kfi+ARK-N— =0, lff V =V(S,I,R) <09 {fl v*=0 The middle of expression (6) is the linear coefficient of vaccination in the Hamiltonian. If this expression is positive so that marginal value of vaccination AR exceed the total marginal cost of vaccination (CV/KS + 2.5 ), which includes current period costs and the intertemporal cost that stems from a smaller S when susceptible animals become resistant, then vacc1nat10n should be set at 1ts max1mum level, 1.e. v = N . If this express1on 13 negative, then no vaccination would occur. The singular solution, denoted by the feedback rule v(S,1, R), is pursued when marginal vaccination costs and intertemporal marginal net benefit of vaccination are equated, so that cv / 16 = AR — 2.5 = 4V (S, I , R), where 1,. (S, I , R) is the net gain from making a susceptible bison resistant. When 1,, >0, i.e., it R > 2.5 , this means that resistant stocks are more valuable than the susceptible ones. It is the necessary but not sufficient condition for vaccination to occur. On the other hand, vaccination should never occur when AV < 0. Similarly, the optimality condition for the choice of test-and-slaughter is 6H 1 >0, iff T =N (7) _—=-EL_,t,- = , iff T*=T(S,I,R) 6T N N , 1 <0, if T =0 The middle of expression (7) is the linear coefficient of test-and-slaughter in the Hamiltonian. If this expression is positive, then testing should be set at its maximum level, 51 i.e. T * =N. If this expression is negative, then T=0 is Optimal. The singular solution, denoted by the feedback rule T (S, I, R), should be followed when the express is zero. In that we could solve for A, = —c.,./I , which implies marginal test-and—slaughter costs equals its intertemporal marginal net benefit. Note that ,1, <0, as greater disease prevalence reduce welfare. The necessary arbitrage conditions for an optimal solution are given by . aH ._ (8) II] =pfitj——aj—,wherej—S,I,R These equations may be expressed in the form of the following “golden rule” equafions: (9) p=§§+ ii—LflJr’I—REK +iS_+___C_Y___ 55 45 55 4s 65 is 15(S+R) at ,1 as ,1 ate ,1 19 (IO) ,0=—+ i—+-—R——}+_l.__ 61 A, a1 2., a1 ,1, ,1, (11) _6R+_/1§__6_S_+,{12[_ (33+ 0: p725} 1,. 6)? Zara 2,, m Equation (9) equates the rate of return for holding the healthy stock in situ to its opportunity cost (p). The first right-hand-side (RHS) term is the marginal productivity of the susceptible stock, minus the rate at which susceptible bison become resistant. The second RHS term (in brackets) is the impact of a larger susceptible population on the other two populations. A larger susceptible population creates more opportunities for infectious contacts, reducing the rate of return to holding susceptible animals. Meanwhile, a larger susceptible population leads to more resistant bison, for a particular vaccination rate, increasing the rate of return. The remaining RHS terms represent, 52 respectively, the additional marginal benefits of investing in a susceptible stock (i.e., the capital gain and marginal cost savings from having a larger susceptible stock. As in equation (9), the left-hand side (LHS) in equation (10) is the discount rate, which represents the rate of return elsewhere in the economy. The RHS represents the rate of return for controlling the disease. The first RHS term is the marginal productivity of infected stock minus the rate of bison being removed. The second RHS term (in brackets) is the impact of a increased infected population on the other two populations: more infected creates higher risk of infection, so susceptible stock might be decreasing accordingly; Yet, more bison become resistant due to natural immunization. The last two RHS terms represent, respectively, the capital loss from having larger infected population and the marginal damages to the livestock industry or the public resulting from greater infection population. Equation (1 l) is the adjoint condition associated with resistant bison. It equates the opportunity cost (p) with the rate of return from holding resistant bison stock. The first RHS term is the marginal productivity of the resistant stock. The second RHS term (in brackets) is the impact of increased resistant bison stock on the other two populations: More bison become susceptible if they lose their natural immunity; the infected bison population will decrease as there are less infectious contacts. The remaining RHS terms represent, respectively, the additional marginal benefits of investing in a resistant population and the additional benefits of investing in a larger susceptible stock. The optimal solutions for control variables in a linear control problem are feedback rules, with the optimal values at each point in time depending on current state (Conrad and Clark 1987). At any point in time, a control may take a constrained value or 53 its singular value. A double singular solution arises when both controls are singular, i.e., when both conditions (6) and (7) vanish. In this case, we have found that a double singular solution of vaccination and Test-and-slaughter does not exist (See Appendix). A partial singular solution arises when one control is singular (either condition 6 or 7 vanishes) and the other is constrained or “blocked” from taking on its singular value. For instance, if v(S, I , R) >N, then v=N would be optimal. The singular solution may suggest infeasible values because the singular feedback rule for a particular variable is derived under conditions when the constraints on that variable are not binding, the potential combinations of possibilities render analytical analysis intractable. (see the Appendix for derivations of the partial singular solutions that arise in the numerical example). As the decision of when to pursue singular or blocked solutions is inherently numerical (Arrow 1968), the potential combinations of singular and constrained controls must be analyzed numerically. We therefore will examine the problem numerically. Numerical Example The data used to parameterize the model are provided in Table 2.1. The ecological parameters come from existing ecological literature such as Dobson and Meagher (l 996). While we have made every effort to calibrate the economic parameters realistically, research on this problem is still evolving. Therefore, the following analysis is best viewed as a numerical example rather than a case study. A discrete-time approximation was used to solve problem (4) numerically. The discrete-time model is specified identically to the continuous problem (4), except the cost functions in discrete-time model for vaccination and test-and-slaughter are 54 cv In[N /(N - v)] and cT In[N /(N - T )] respectively (Conrad and Clark 1987). The results from those specifications are known to approximate those of linear, continuous-time models as in problem (4) (Clark 1976). The discrete-time problem was solved in Excel using the Solver module. The optimization was run for a 100-year period. The initial values for the three subpopulation are assumed to be S = 1,400, I = 1,400, and R = 1,200. As shown in the appendix, the double singular solution of vaccination and test- and-slaughter does not exist, which suggests that a solution where both controls are utilized may not be optimal. This result is consistent with our earlier speculations in the ecological analysis. Therefore, in the numerical example, we will focus on the partial singular solutions, i.e., when either v or T is constrained. Given our specification for costs, we realize setting v or Tto the upper bound of N implies infinite costs and will therefore not be efficient. However, it is worth examining these cases simply to show how the use of one control affects the incentives to use the other. Partial Singular Solution for T when v is Constrained The partial singular solution in this case involves constraining v=0 or v=N, and also setting condition (7) equal to zero to derive the optimality condition. Our results suggest that when v is constrained at N, i.e., all the bison population is subject to vaccination, then engaging in test-and-slaughter is never optimal (i.e., test-and-slaughter equal to zero across the 100 years). The reason is that Treduces the force of infection on the susceptible population, but this has little value when vaccination is applied at the maximum rate so that few animals remain at risk. As you can see from Figure 2.2, with the constant force vaccination among the whole population, most of the bison become 55 resistant, while the susceptible population is largely decreased. Because the structure of the model requires animals to first transition from resistant to susceptible before vaccination can have impact, some animals might still remain at risk when all animals are vaccinated. As the at-risk population is reduced by continuous vaccination, the number of infected bison approaches to zero after 15 years. A steady state of S ‘ = 595, I. = 0, and R‘ = 3,405 is attained after about 20 years. Test-and-slaughter becomes optimal when v=0, as there are benefits to reducing the force of infection in this case. Figures 3a and 3b indicate the solution involves oscillations that dampen over time. The system begins with a significant and growing number of infected animals, which spurs significant but tempered early investments in test-and-slaughter. T is large enough to substantially reduce, but not eliminate the force of infection. The most obvious reason is that the required level of Twould be high, implying significant costs. A less obvious reason is that infected animals also contribute to reproduction, with many offspring contributing to the susceptible population. Infected animals also contribute to the stock of resistant animals, which are valuable. Moderate levels of T continue until I is significantly diminished, around year 3. Once this happens, the incentives for Tare reduced and T falls temporarily. Notice, however, that T begins to increase again around year 6, even though I is still small. The reason is that S has now grown so large that many animals are at risk of infection. Eventually, S is so large that it becomes too costly to protect them all, and I is allowed to grow while S declines. As S declines, the value of protection falls and T is reduced until 1 becomes sufficiently large. At this point, the entire process begins again, though with progressively smaller cycles until the system seems to approach a steady 56 state towards the end of the time horizon. The approximate steady state is S‘ e 1600, 1‘ = 250, and R‘ = 1000. The eigenvalues we calculated for test-and-slaughter suggest that the steady state is a saddle. The oscillations arise for two reasons. First, oscillations arise because the uncontrolled system naturally oscillates. The system continues to oscillate even when T is applied because T is an imperfect control that cannot fully dampen the oscillations -- at least initially. Second, there are a few instances where a constraint on T becomes binding and T =0 -- this also contributes to oscillations in the baseline scenario. Eventually T is maintained at positive, interior values and is able to dampen the oscillations so that the system moves along a smooth approach path to a saddle point. Partial Singular Solution for v when T is Constrained The partial singular solution for v in this case involves constraining T=0 or T=N, and also setting condition (6) equal to zero to derive the optimality condition. When T is constrained at 0, i.e., there is no test-and-slaughter available, it is optimal to vaccinate in order to reduce the at-risk population (susceptible). Figures 4a and 4b indicate that solution involves oscillations over time. The system begins with a significant and growing number of infected and susceptible bison, which spurs significant early investments in vaccination. v is large enough to substantially reduce, but not eliminate the at-risk population (susceptible). One reason is that the required level for v would be high, which implies significant costs. It is also due to that fact that the structure of the model requires bison to first transition to susceptible before vaccination can have impact. Resistant bison increase rapidly as both susceptible and infected animals become resistant either by natural immunity or vaccination. With significantly diminished S, 57 around year 2, the incentives for v are reduced and v falls temporally. Notice, however, that v begins to increase again around year 4, even though S is still small. The reason is that there is still a significant number of I that might cause more susceptible bison to be infected. Moderate levels of v continue and decline slowly until S has grown significantly, around year 28. The incentives for v are increased again and v increases accordingly. S falls again, the value of protection falls and v is reduced until S becomes sufficiently large. At this point, the entire process begins again, though with progressively smaller cycles. The system shows similar patterns on and on: the fluctuation of vaccination number leads to the change in susceptible and resistant population, while the oscillation of susceptible and resistant population also affects the optimal vaccination choices. The infected population, however, is unaffected, and remains at a steady level of about 2 head. The oscillations occur in the optimal solution because it is difficult to control an interacting system of three state variables by using just one control—vaccination in this case. Vaccination directly affects two populations, S and R, which makes it impossible to separately maneuver S and R along any particular trajectories. For instance, the use of v to move S along a particular path will have spillover effects on R. Matters are further complicated by the uncontrolled third population, I, which interacts with the other two. The result of this imperfect control over the system is that managers must over apply or under apply v to move all three variables in the right direction. Indeed, the optimal vaccination control hits upper and lower bounds along the optimal path, becoming constrained. So even though the steady state is a saddle (as indicated by the eigenvalues for the linearized system), the saddle path to a steady state must be periodically 58 abandoned as the controls become constrained. Once off the saddle path, the system tends to veer off course until interior values of the controls again become optimal, so as to put the system back on the saddle path. The result is oscillatory behavior that is akin to chattering. Chattering occurs when there is no interior optimal control. Imperfect control also results in increased control costs. Control costs are further increased by the fact that vaccination is not selective with respect to disease status, so that Some non- susceptible animals may be vaccinated by mistake. These wasted vaccinations do not affect disease dynamics, but they do affect the costs of control. Now how about the partial singular solution when T =N? Our results suggest that it is only optimal to invest in vaccination in the first period. The reason is that there is risk from the infected population in the first year, but no infected bison are left afterward because of the imposed constraint that the entire population is tested for infection. Meanwhile, resistant bison initially increase rapidly as both susceptible and infected animals become resistant either by natural immunity or vaccination. However, the resistant population again becomes susceptible after I has been eliminated and vaccination has ceased (See Figure 2.5a, 2.5b). The population dynamics reach a steady state after year 70, with s‘ = 4,000, 1‘ = 0, and R‘ =0. Comparison of social net benefits for the four policy scenarios Table 2.2 compares the present value of social net benefits for the four different policy scenarios: (a) Partial Singular Solution for T(Given v=0); (b) Partial Singular Solution for T (Given v=N); (0) Partial Singular Solution for v (Given T=0); ((1) Partial Singular Solution for v (Given T=N). As discussed in section 5, setting v or T to the upper 59 bound of N implies infinite costs and therefore will not be efficient. We include scenarios (b) and (d) simply to show how the use of one control affects the incentives to use the other. As you can see from Table 2.2, scenario (c) outperforms scenario (a) due to the way in which the controls operate. Vaccination prevents infections from arising, which protects the susceptible population. At the same time, vaccination helps contribute to the resistant population. Transitioning bison from susceptible to resistant status has a neutral effect on existence value, whereas protection from infection enhances the overall healthy population to increase these values. Test-and-slaughter reduces the infected population (increasing existence values), but has a secondary effect of reducing new recruits to the resistant population (decreasing existence values). As a result, the partialsingular solution for vaccination when constraining T at zero achieves the highest social net benefit. On the other hand, the differences of present net social benefits between scenario (a) and scenario (c) are comparatively small (26 million USD). We have not modeled transactions costs associated with implementing controls, and these could be relevant and possibly larger for the vaccination scenario due to the oscillations associated with the controls in that scenario. Policy makers might prefer the control that has fewer oscillations since it is easier to implement. Therefore, whether vaccination is preferred would depend on whether the transaction costs of running the program actually offset its larger value of social net benefits relative to test-and-slaughter. 60 Table 2.1 Epidemiology and Economic Parameters Value Parameter Source (if applicable) c ,(vaccination cost) 29,080 Montana department of livestock (2004)i cz(test-and-slaughter cost) 905,000 Bienen and Taylor (2006)ii a (parameter for existence Assumptioniii and Yellowstone National Park 5,874,477 . value) Websue 6 (parameter for economic . iv damage of infected bison ) 427 Assumpt1on x (rate of lost resistance 0 01 Assum tion for bison) ' p ( . t' Assume same effectiveness as in cattle, KffvaIcCIna 1011 0.7 Wyoming Dept. of Administration and e ec 1veness) Information (2004) ’6 (b1son mfecttous 2 Dobson and Meagher (1996) contact rate) a (bison birth rate) 0.26 Dobson and Meagher (1996) 355150“ natural mortal1ty 0.1 Dobson and Meagher (1996) Q (proportion of infected female bison that produce 0.9 Dobson and Meagher (1996) infected offspring) 1] (reduction of fecundity 0.5 Dobson and Meagher (1996) in infected bison) (p (density-dependent 4x10'5 Dobson and Meagher (1996) competition effect) 8 (bison recovery rate) 0.5 Dobson and Meagher (1996) a (bison virulence rate) 0.005 Dobson, and Meagher (I996) X(0) (initial bison stock) 4,000 Yellowstone National Park website 61 .5505 2e 28» cm as 05 SB 2:3 omega 5 om_3..o£o 438 .: 32m.» 08% >3on a o :8 83. 8.: .9: =85 2353. o N o 8N .8 Sam Bees 8.... am. am 82 .5 :85 05.2.0015 88m 3.» oz mo> mo> >3on Mo 00:05me Em: m..o....s. .o:_m> E885 2.5.2. 2.332 m .... 2.51.5 Desmoz :3 £23. 82 seam Sun 529 ans 3:9 Eu. .328 at. 8:9 a ..8 notiom a .8.— =o==_om h .8.— =e_~=_om & .8.— :eEEom Easietaia Esauaaafaie ......»efataie 3.5523513 Stanwoh 333% mote-Sum zo=om «=0.—eta .8.— manom— .oz 732% can 85330 352.3 he near—2.590 N.N «Bah 62 Figure 2.2 Bison Population Dynamics: The Partial Singular Solution for T when v=N Bison Population 4000 .3.-. 2.... ___ .--.--.,._-._,..--_____ "l 1 3500 J] 3000 —+—-—-—- 1' 1 2500 j 2000 ‘i 1500 'fi % 1 ‘1. l 1000 4K A I, i 500 1 x i 0 r r. .r. 7 .374”, 1 1 1 1 r 1'; v1 1 T l'T'v'T'f'l‘l'TTTv’TlT‘xT‘IWT‘l‘YT‘I' rrr 71‘1‘1'1'Th'r ; r r 11".”: r‘rr. 1'r71‘1‘r-Trf'7‘1". T‘l‘t‘T'T‘Y‘l‘.TT l1"T'T‘I“.":"‘."-’" 15 9 1317 2125 29 333741 4549 5357 61 6569 73 7781 85 8993 97101 Year — — Susceptible -----— Infected Resistant Figure 2.2 63 Figures 2.3a-b Bison Population Dynamics: The Partial Singular Solution for Twhen v=0 4000 3500 3000 2600 2000 :3” 1500 g 1000 m 500 1. . 0 ma flRna—fi-mmrfirnn 15 913172125293337414549535761653973778185899397101 Year [ - - Susceptible —— hfected Resistant] Figure 2.3a 5 flfi'l‘yi“ 1,? >3 “'31:” ‘2." ;’§:. 3ij TQN‘ 1:". ) w‘r‘“h:r ~_'., \ ‘3 M w) , , L117. 3 (a 7" ' .1 'a . 'V.‘. c - ;. 3 —. .J A, 5 ii; 3 . 3t 1 6 11 16 21 26 31 36 41 46 51 56 61 66 71 76 81 86 91 96 101 Year Figure 2.3b Figures 2.4a-b Bison Population Dynamics: The Partial Singular Solution for v when T=0 2500 \ 1500 I‘,\._ ‘\..‘ \ \quMI \¢ v \/\‘, 1000"' * "" Bison Population O , f'f’fi‘TT‘l‘f'T'TT‘TT WT T'TTT 15 913172125293337414549535761656973778185899397101 Your Infected Resistant i L — — Susceptible Figure 2.4a 5000 <5." .. 1., v . m .. we .. 36’3‘373'1. ' A"Yfif-AJ'T-szi9&333‘5153 7%.. mm ’. . u . 4000 if tyr- 17915.1: {fir-33:, . . . , .1 .; V241” {fiwéa ‘w'v- Ik , ”J ‘ 413:... "fir ',~- .. -‘-- we... -’ a J . _ . .. "... . D . a. ._u;_(. r.-'-. w . '_ .1) .-. ‘ ‘ . b . . ._ ,- ?“4~5.392m x a». no?“ .1726:- a .. . . {W 3000 2000 1000 .P ' ,. rd .' '\ ‘ ‘ " .... . . ,.- l", - a # of Vaccination 17131925313743495561677379859197 Year Figure 2.4b 65 Figures 2.5a-b Bison Population Dynamics: The Partial Singular Solution for v when T =N Bison Population 4500 .... ---.-..---..-.---,_...,_-__._ ..-- . ..-.-- - ---._.,-._--_-__~_.. .. _,-.__. .---.. -._._ --_..._ _. --.- .--......_.. ...._.,:,_.“,_,l 4000 ‘W—fl'v—WW— 3500 74 3000 ,4 2500 ~ 2000 «--—- 1 1500 II ' / 1000 § / 500 1 o 7 T—Tj—Tfiflj~TT-T'T”ITT7-V1""T'TYTF—IT ‘lT-T'TT'T'T‘T T' 'VI'T! I h "'IT'I I 2-7 7'11”] I l—rl—ll-1I'-rl-F|-rl IY'ZT"T11 T“. 7": T"! I .T [1 ."I I l-PI'rI'TI'l 15 9 13 1721 252933 3741 4549 535761 6569 737781 8589 9397101 l K _..L _1L--- .1 --- Year Infected Resistant 1] [ — — Susceptible Figure 2.5a : 3 6 DE 0 0 G > '6 fi . 17131925313743495561677379859197 Year Figur62.5b 66 Sensitivity Analysis Sensitivity analyses are usually used to examine how changes in one or more parameters influence the results. As there are many parameters in this model, I did not perform a sensitivity analysis for each parameter one at a time. Rather, I performed sensitivity analyses based on groups of ecological and economic parameters to get insights on how these groups of parameters impact the simulation results. For each sensitivity analysis scenario, we examine (a) the partial singular solution for Twhen v = O, and (b) the partial singular solution for v when T = O. The other two partial singular solutions (when T=N or when v = N) are not examined because they imply infinite costs given the cost specification. Specifically, I analyzed the following five scenarios: Optimistic Ecological Parameters For this scenario, we simultaneously changed all key epidemiological parameters so that they are "better". Specifically, the transmission rate ( ,6 ), mortality rate (m), loss of resistance rate (x), and virulence rate (a) are each reduced by 33%, while the birth rate (a), recovery rate (6) and vaccination effectiveness rate (K) are increased by 33%. The results for the two partial singular solutions are presented in Table 2.3. The vaccination strategy remains Optimal in this scenario, though the gains fromevaccinating relative to pursuing the test-and-slaughter strategy are significantly reduced relative to the baseline scenario. For both management strategies, the optimistic ecological parameters result in a larger long-run healthy population (susceptible or resistant bison) and a smaller infected population. The larger birth rate in the current scenario results in more susceptible bison being born, while the smaller transmission rate results in fewer infected 67 bison. In turn, these effects result in disease control being easier and hence less costly, and so it is optimal to maintain infections at a lower level; As you can see from Figure 2.6a and Figure 2.6b, the disease is easier to control with vaccination and these fluctuations are reduced -- more time is spent with no vaccination. For test-and-slaughter, the system fluctuates even more when no controls are applied (See F igure2.7a and F igure2.7b). In particular, the larger birth rate gives rise to much larger increases in S in the uncontrolled system, putting more animals at risk. The result is there are incentives for a much larger impulse of T initially, which pushes I to a lower level than in the baseline case. The result is less risk for a longer period of time and hence smaller values of T during the interim. The lower values of I also result in fewer fluctuations in the early part of the time horizon. Another pulse is needed around period 41, which is again much larger than the second pulse required under the baseline parameters. Again, this reduces risks and significantly dampens the oscillations. Note, however, that T is maintained at larger levels, relative to the baseline case, throughout the remainder of the time horizon. The reason is that S is larger in this scenario and so there are more animals at risk and hence more incentives to keep the infection at bay. Pessimistic Ecological Parameters For this scenario, which is opposite of the optimistic ecological parameters scenario, we simultaneously change all key epidemiological parameters so that they are "worse". Specifically, transmission rate (,6 ), mortality rate (m), loss of resistant rate (x), and virulence rate (or) are each increased by 33%, while the birth rate (a), recovery rate (5) and vaccination effectiveness rate (K) are reduced by 33%. 68 Table 2.3 also compares the two different policy scenarios with the pessimistic ecological parameters. Similar to the baseline and optimistic ecological parameters setting, the vaccination strategy outperforms the test-and-slaughter strategy. The difference between the social net benefits remains small, though the difference is larger than in the baseline scenario. For each management scenario, the pessimistic ecological parameters result in smaller healthy populations but a larger infected population. The low birth rate and high mortality rate in the current scenario results in fewer susceptible bison, while the larger transmission rate results in more infected bison. In turn, these effects result in disease control being more difficult and hence more costly, and so less is done to control the infected bison population. Under pessimistic ecological parameters, the system still exhibits oscillations even when no controls are applied. However, the oscillations dampen much faster than in the baseline scenario. The only big oscillation occurs initially due to the initial values S(O), [(0), and R(O) being much larger than can be supported naturally under pessimistic parameters. Specifically, the value of S falls quickly fall due to mortality and resource competition, but not before generating one large "baby boom" that causes one small rebound of susceptible animals (a single oscillatory peak). There is also an initial peak for infections and resistant animals: these values initially rise due to the large initial values of S(O) and 1(0) generating a lot of infected animals initially, but they subsequently decline due to high mortality and resource competition. The fast dampening of the uncontrolled system makes it possible to optimally manage the system without oscillating controls (atlleast, afler one initial oscillation) — for both vaccination and test-and- slaughter (Figure 2.8a,b and Figure 2.9). 69 Optimistic Economic Parameters and Pessimistic Economic Parameters In the Optimistic (pessimistic) economic parameters scenario, we simultaneously change all key economic parameters so that they are "better". Specifically, parameters for economic damages (6), vaccination costs (0 1), and test-and-slaughter costs (Q) are reduced (increased) by 50%, while the existence value parameter (a) is increased (decreased) by 50%. I chose a 50% change in parameters, as opposed to a 33% change (as in the ecological parameter scenarios), because the results were less responsive to changes in economic variables. This lack of responsiveness is also why economic parameter results are presented together in Table 2.4. Long run population levels for the two disease control strategies are quite similar to the baseline case. This suggests that the simulation results are robust to the economic parameters. As expected, there are slightly more infected animals in the pessimistic case, and slightly fewer infected animals in the optimistic case. The lack of responsiveness is partially due to the fact that control costs and damage costs move in the same direction. Larger control costs reduce the incentives for disease control, while larger damage costs increase these incentives. The opposite is true for smaller control costs and damage costs. The vaccination strategy remains optimal in both scenarios. The gains from vaccinating relative to pursuing the test-and-slaughter strategy are small in the optimistic case, but large in the pessimistic case. In both cases, social net benefits are quite different from the baseline scenario due to the very different economic parameters being used in these alternative settings. 70 Increased time horizon In this case, we increase the time horizon of the simulation by 50%, i.e., the simulation is run for a 150 year period, in order to see whether the time horizon has any effect on the results. As you can see from Table 2.5, the social benefits of this setting are very close to the baseline, and vaccination still outperforms the test-and-slaughter strategy. For the steady state values, the vaccination strategy seems to result in very similar steady state values as the baseline, while the test-and-slaughter strategy ends up with more susceptible bison and less resistant ones than in the baseline. This is possibly because test-and-slaughter, which prevents natural recovery to resistant bison, is utilized for a longer time in the current scenario relative to the baseline scenario. 7] Figures 2.6a-b Bison Population Dynamics: The Partial Singular Solution for v when T=0(Optimistic Parameters) BhuutPopuhflkut A / 5000 /1 i ,- l ‘m, -4___ ’ 1“ _ 1‘ / ‘1/‘ .4 ./ ‘ 1/ r— . ’ I / I I / v l'fi .’ 1 ' ’ I ’ /' / I / " \I I /‘ / 30004 ' ——1. l; 4‘! ...... 7 l l / \/ 2000 U .— rlt 1000 {f 0 -H-r:‘-+r-H-r~v1fl~n—:-I'rrm-rfl inf-hfiTT‘rrrrI'rfi'rfl'rrfi‘rrw-wrrrrfirfi-rrfi-mm W’T‘TT‘T’TW’T’IT'TTT'T‘T‘TT‘YT‘H’J' 15 913172125293337414549535761656973778185899397101 Year [ — — Susceptible Infected Resistanfl Figure 2.6a 17 \5‘ :3»? g . . '§ “get? a ; talc ' 5»...- *5 '5 his? i it.“ at l 7 13 19 25 31 37 43 49 55 61 67 73 79 85 91 97 Year Figure 2.6b 72 Figures 2.7a-b Bison Population Dynamics: The Partial Singular Solution for Twhen v=0 (Optimistic Parameters) Bison Population T‘I'T‘TYTI m I-I‘r tar-1+1: 1 r I’T'Trt‘ I-IT-r‘rr rrr-rrm 15 913172125293337414549535761656973778185899397101 Year — — Susceptible —~——— Infected Resistant Figure 2.7a h 8 8 a 3 .1 'f a c 3 .2 '6 u 161116 2126 3136 4146 5156 6166 7176 8186 9196101 Year Figure2.7b 73 Figures 2.8a-b Bison Population Dynamics: The Partial Singular Solution for v when T-’=0 (Pessimistic Parameters) 3000 w~—--——v--«~—--———— ---—-..“--- _-__-_____. - --.-J 2500 -1~ 2000 1500 . 1ooo , — Bison Population -- la 15 91317 212529 3337 414549 53 57 616569 73 77818589 93 97101 Year — — Susceptible Infected Resistant J' Figure 2.8a = *7 .9. i g i I: 1 I8 7 o z 1' 4 > h o v a: 43 49 55 61 67 73 79 85 91 97 Year Figure 2.8b 74 Figures 2.9 Bison Population Dynamics: The Partial Singular Solution for Twhen v=0 (Pessimistic Parameters) 2500 ——- w — —- —-a m-“ ~ ”...--- — —~-—— —~-—— — - -.-_-._ — * --~——~:—~-~~1 2000 1 J‘ g . l o 1500 l; ‘5‘ 3 2 I a ' l o 1000 1. fl. 1 r: I ‘1. g 500 u - -—' I (‘q - . 0 LI I I "1' I I l I I ' I I I I I T I l V ':I 7 ITT :I ITI I—II T: T :7 I:T:T-If lfl’IT-I—‘I‘Tfl T7 Ti": ‘Tfi‘i:—;—:T.:-IT rift—H 1 5 9 13 17 21 25 29 33 37 41 45 49 53 57 61 65 69 73 77 81 85 89 93 97 101 Year [— — Susceptible —-~-— Infected Resistanli Figure 2.9 75 .5505 En memo» cw «we. 65 35 2:9 owns; no 833650 .Exo .: 82? 88m @6on a om» 82. m am e? we: :85 Bees”. : c _ S. .5 =85 Bees 2: Km 38 8% .5 :85 29683 85m oz 9% oZ mo> €35 mo cocofixm RED 26::2 6:73 203:: 2m m3 .8. 3A: “cocom 82 32% ans 5:8 at. 529 Sun 528 at. .629 a .5.— ..etiem h .5.— =e_.=_em a ..8 net—....m & ..8 acts—cm Lain—gm Etna— ..a_=w=_m 15.8; Lain—.5 .525— .._a_=w=_m Etna— .EeEEEem EcmueBQM c-a§§~n§k EuBEEek ~336ch 6.3.»...qu «macBEEQR ~356ch ctfiEMmmem BB SENEQQQ» meta—5% >31..— .aoaoh—E .5.— mfi—oaom .oZ Eoem 1.5 856850 35.2.3 he naming—=60 Md “...—«h. 76 5.6% 2m memo» cm 32 05 .85 02.; owns: an oflEofiO its a 82.; 03% £3on .. $3 82 33m 82 .9: 5.5 999.2 9 ca _ 8N .8 82m Bees mm: 82 :2 82 .5 5.5 2968.5 88m 02 mc> oZ mo> >9.on ho cocofixm— am: 2252 . 623/ 282.: gm 2:. as: :2 526m ...2 3.8 81.9 528 at. 528 Sun .629 $1. .628 a ..8 .u ..8 acts—em a .8.— .Szflem h ...... Esta—em sets—em .5335 .3985 ....Eufim Eta.— ..a_=u=_w Eta.— ..a_=u=_m Eta.— EuBEEeR fianceeum cmfimfimnuum EeBEEek NESESN u-u~fi§§© «22656.8.» Seesaw ethafiamek 3:6 chfiEtQQ» meta—com hazem 25.3.59 he autonom— 32 132% ...—a moEeoaac 9:5.an— ue naming—:60 YN «Sea. 77 .5505 0.8 83% co. amm— ofi .95 33> $805 5 839850 4min. .: .825, 85m $5on a 88 Em «o: =85 2928“ m NS «5 :85 38$:— oo: 82 “6 =85 032.08% 02 mu; 88m @505 m0 oocoixm Em: 22:5 m 2: £3 as; 232% 52:5 .oz 368 3! age ans 5:2 a 8.. 5:28 h .8 3.335 3.5— § 8:28 3.35 35$ 3 Suntan ASSN «55.8% 255 38.3.85» Sta—Sow 5:9— 2.0.8::— .:¢ mace-.3— .oz 38m was 85839 £5.33 .3 acfluaA—Ecu m.~ 02.; . 78 Conclusion Previous studies on optimal wildlife disease management have mainly focused on solutions like non-selective population control. Disease control approaches that have been applied extensively in livestock disease management, such as vaccination and test- and-slaughter, have only received limited attention in wildlife disease management. This article developed a bioeconomic model to explore economically optimal management. The time-variant control strategies derived under this approach are chosen by considering the economic and ecological tradeoffs. This paper expands the disease ecology literature by integrating disease dynamics with economic choices in such a way that risks of infection are a function of optimized livestock disease management choices, and then economic choices are, in turn, a function of disease states. Our results suggest that a solution where both vaccination and test-and-slaughter controls are utilized is not optimal. The reason is that the two management options work in different directions in terms of reducing disease risks: test-and-slaughter reduces the force of infection on susceptible animals, while vaccination reduces the number of animals at risk of infection. The two controls are substitutes in disease control, as the marginal disease control benefits of each control are diminishing in the use of the other control. Hence, there are not incentives to use both controls. By examining the population and disease dynamics for solutions involving vaccination only or test-and-slaughter only, we illustrate that the bison disease and population dynamics vary among these two approaches. The optimal vaccination-only solution outperforms the optimal test-and-slaughter-only solution, as the former achieves higher social net benefits. The reason is that vaccination not only prevents infections. but 79 also helps contribute to the resistant population, in turn increasing the overall healthy population. Test-and-slaughter, on the other hand, removes infected animals that would have recovered, thereby reducing the population of healthy animals and also existence values, even though it decreases the infected population. The sensitivity analysis also suggests that the numerical results are robust to changes in the ecological and economic parameters, as well as changes in the time horizon. On the other hand, in the baseline scenario and in most of the alternative (sensitivity analysis) scenarios, the economic gain from using vaccination over test-and- slaughter is quite small. Moreover, there might be transaction costs (not modeled) associated with implementing the controls,— especially vaccination since the solution involves significant oscillations in the use of the control. Therefore, the decision of choosing vaccination or test-and-slaughter would depend on whether the transaction costs of running the program actually offset differences in social net benefits. For brucellosis management in the Greater Yellowstone Area, the problem is much more complicated than what is stated in this paper, as it involves multiple disease hosts (bison, elk and cattle) as well as different stakeholders with different interests in wildlife and livestock management. X ie and Horan (2009) investigate private responses and ecological impacts of policies proposed to confront the problem of brucellosis being spread from elk to cattle in Wyoming. However, their paper does not take into account the economic benefit from wildlife, nor include bison as part of their analysis. More research in this area is needed. There is further need for analyses that combine those disease ecology models with economic decision models. Without it, models such as the one presented here can only provide general insights — not detailed guidance on how to 80 manage disease problems. However, as the full multi-host problem is extremely complex, there is value to analyzing simpler variations of the problem first, as we have done here. 8] APPENDIX 82 APPENDIX Here we illustrate the derivation of the double singular solution and two partial singular solutions that arise in the numerical example. To simplify notation, we assume the change of susceptible, infected, resistant population are S = f(S,I, R) — va/N ;1' = g(S, 1, R) — TI/N ; R = h(S,I,R) + va/N respectively. (A )Double Singular Solution A fully unconstrained double-singular solution is optimal when conditions (Al) (i.e., @— = -23- — lSK-LS- + 2.in = O ) and (A2) 211 = -c—T— - 11 —I— = Osimultaneously 6v N N N 67‘ N N vanish, so that AV = 2,9 and A] = A? . (A2) solves for 11,, 11 = 377-. By taking time derivative and plug into the arbitrage conditions 21 = p21 — .6511!“ we get condition c S,I,R c T c c c T 2g( 2 )‘ 2 =P(-—;)+9-3Sf1 “lg! ‘ARhR “—2'“ I 1 I 1 (A3) 1 As the term for T can be canceled out, equation (A3) could be simplified as 628(5, 1. R) _ (A4) 12 C C p(—%) +6-1st “7ng 4»th Since both (A4) and (A l) are linear in is ,AR, the two equations together could solve for 15(5, 1, R),/1R (S, I, R) . First Take a time derivative of ASJR , we get: (A5) . 6,1 . ‘. . 15(S,I,R,v,T) =—S—S+-a—A§-I+2’—i§-R =‘Ir’l(S,I,R)+v‘1’2(S,I,R)+7‘P3(S,I,R) 6X5 6X1 6X1; 83 (A6) 2R(S,1,R,v,T)=-a—’15—S+§flii+§iR—R =‘1’4(S,1,R)+v‘I’5(S,I,R)+T‘P6(S,I,R) 6X5 5X} 6X}; Note that both is and i R are linear in both v and T. Then plug (A5) and (A6) into the arbitrage conditions (A7) 11.8 = p13 —%%= ‘I’7(S,I,R)+v‘1’8(S,I,R)+T‘I’9(S,I,R) (A8) 2R = p/iR $1]; = ‘1’10(S,I,R)+v‘1’11(S,I,R)+7‘P12(S,I,R) We then have the following two conditions, (A9) ‘l’1(S,I,R) —‘¥7(S,1,R) = v{‘{’8(S, I, R) —‘1’2 (S, I,R)} + T{‘1’9 (S, I,R) - ‘1’3(S,1,R)} (A10) ‘i’lo(XS,X1,XR)-‘I’4(XS,X1,XR) = v{‘l’1 1(5, 1, R) - ‘115 (5,1, R)} + T(‘l’lz (S, 1, R) — ‘116 (S, 1, R)} Since both (A9) and (A10) are linear in v and T, together they should solve for v and T. However, our analysis found that (Al 1) Ll’3(S,I,R)-‘1’2(S,I,R) = ‘1’11(S,1,R)—‘P5(S,I,R) (A12) ‘l’9(S, I, R) —‘1’3(S,I, R) = ‘1’12(S,I,R)-‘P6(S,I,R) (A13) ‘Pl(S,1,R)-‘Y7(S,I,R)¢‘Plo(XS,X1,XR)-‘I’4(XS,X1aXR) Equations (A1 l)-(Al3) suggest that equation (A9) and (A10) contradict each other. Essentially they suggest that the adjoint conditions of (A7) and (A8) cannot be satisfied at the same time. Therefore, the double singular solution doesn’t exist. 84 (B) Partial singular solution when only v is constrained As we have shown that the double-singular feedback for v and Tdoesn’t exist, let’s take a look at the partial singular solutions. Partial singular solution occurs when one of the controls values are outside the boundaries for v and T, i.e., [0,1]. The singular solution in this case involves constraining v=0 or v=I, and also setting condition (7) equates zero to derive the optimality condition. Therefore, the partial singular could be considered as a second best solution rather than first best, as v and T are bounded. When condition (7) vanish, x1} = :31. First, take time derivatives of this condition and plug into the arbitrage conditions 11 = p11 -33; , we would get (Bl) ‘78 C C 1 = p(-—IT—)+6-lsf1 +-1Lg1 -/1RhR Since 115 is linear in(.Bl), we could then solve for its as is (S, I , R, ,1 R) . Then we can take the time derivative of AS , is (S, I , R, T, J. R ) , and plug'into the . . . - 6H ' S arbitrage conditions/15 = p15 -—6—S— = q (S, I, R,/1R) , We then have the following condition, (82) iS(s,1,R,T,RR)—q5(s,1,R,/1R) =0 Since 11,, is linear in (32), we could then solve for 11 R (T ,S, I, R). Taking the time derivative of A R , we get 2: R (T , T',S, I , R) . Then plug into the arbitrage conditions ’lR = p/iR —%%— = qR(S, I, R,T) 85 We then have the following condition, (33) 1R(T,T',s, 1, R) —qR(S,I, R, T) = 0 Notice that (B3) is no longer a linear function in T, so we cannot solve for the partial singular solution for T explicitly. Instead we need to use differential equation together with the differential equations for the disease dynamics, to solve an ordinary differential system. (C )Partial singular solution when only T is constrained The singular solution in this case involves constraining T =0 or T =1 , and also setting condition (6) but not condition (7) equal to zero to derive the optimality condition: 6H c. S S Cl —=-————A K—+}. K—=0. ( )av N S R N N c (Cl) solves for A“, is = AR _EV cv Let AZ =1R _’?'S :3 Taking time derivative of 2.2 , we get (C2)iz = RR ~15 =f(v,S,I,R) . . .. ' - 6H 3 and plug mto the arbitrage conditions is = pl ~35 = q (S, I,R,v, 1.1.1.13) and - 6H 1R =le --a7e-=qR(S,I,R,21,/1R) “v” is cancelled out, then this leads to an implicit function. (c3) WZ(RI,AR,S,1,R)=0 86 Since A, is linear in ‘1’2 (15,1135, 1, R), from (C3), we could then solve for A, as it, (1R,S,I, R) .We can take the time derivative of ,1] , 21 (S, I, R,v,lR) and plug into . . . - 6H 1 , the arbitrage conditions 2.1 = p21 - E = q (S, I, R, AR), We would get the following condition, (C4) i](S,I,R,v,,1R)—qI(S,1,R,AR)=0 Since AR is linear in (C4), we could then solve for AR (v, XS , X1 ,XR ) from (B4). 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(http://www.nps.gov/yell/naturescience/bison.htm) :The unit cost is 7.27 dollars per bison head. {Total cost of test and slaughter approximately 800 bison is $181,000, lll . . . We assume the exnstence value take the form of aln(S+R). According to the Yellowstone national park environmental assessment, the non market benefits associated with aggressively reducing seroprevalence to 0% through park wide capture, test and slaughter or vaccination would be 3.57 million. We assume thataln(Sl +R1)- aln(S()+R0)=3.57 million. Where S 1 and R , are the population of susceptible and resistant population after vaccination and test-slaughter, S a and R0 are the population before action was taken. According to the reports, a mean population of 3700 with 0 seroprevalence rate is predicted under the plan, while the infection rates is 35% with 3100 population in 2006. a1n(3700) -a1n(3 100*65) =3570000. This gives us a calibrated value for alpha is 587447748 w The US. Fish and Wildlife Service and National Park Service (2007) conclude that costs of losing brucellosis free status in Wyoming is about 1.2-1.7 million per year statewide. The total population of elk in Wyoming is 12,904, the total number of bison is 5000. For simplicity, we assume the cost is proportional to the number of animals; infected bison and elk have same damage to the livestock industry. The proportion of infected elk is 20%, and the infected bison is 35%. Therefore a unit disutility created by the infected animal is 1,700,000/(12904*0.2+0.35*4000)= 427. 94 "'lllljltllflljlfl”ZilllzllllllllfllllEs