Biosecurity and spread of an infectious animal disease.

Diminishing infectious animal disease prevalence amounts to a global public good. This has in part motivated the long tradition of public involvement in infectious animal disease control. But appropriately designed public intervention requires a clear understanding of failure in private incentives.

Two important features of communicable disease are spatial spread and the possibility of costly private actions to reduce spread. In addition, since routine biosecurity actions by growers are typically costly to verify by veterinary authorities, voluntary compliance is often essential. To this end, growers must recognize the consequences of their actions and must have adequate incentives to take socially desirable actions. The intent of this article is to provide a better understanding of private incentives to protect against the spread of infectious animal disease. We do so by developing a model that emphasizes spatial relations in infection, prevention technology, and externalities across agent payoffs.

The economics literature on public goods aspects of animal diseases is limited. Indeed, a surprisingly small body of work exists on the economics of infectious human diseases (Gersovitz and Hammer 2004). Animal diseases have been the subject of formal models (Chi et al. 2002), but the issue generally addressed is that of internal costs and not how farms inter-relate. A strongly related theme is that of controlling invasive species. Economic perspective on this issue is expanding, but has been confined largely to quarantine (Mumford 2002) and other public behaviors given an assumed exogenous stochastic dynamic process for infection (Olson and Roy 2002).

The direction in Hennessy, Roosen, and Jensen (2005) is closest to that in the present work. While theirs is not a spatial model, the biosecuring decisions involve whether to trade in young stock and the extent of production. Private benefits from trade are shown to lead to socially excessive losses from an endemic communicable disease. Furthermore, communicable disease is shown to alter the format and scale of production.

This article emphasizes protecting a farm's borders in a spatial model of private behavior to prevent the spread of infection. For farms arranged on a circle, we show how biosecuring actions are local substitutes and explain what this means for behavioral patterns under simultaneous-moves Nash equilibrium. Two insights obtained are that losses from disease-spread externalities are smaller when production is concentrated, and subsidies to small producers may exacerbate overall disease losses. We also consider a line topology for farms in order to show how the model can be adapted, and how locational asymmetries can affect incentives to protect farm boundaries. We find that more centrally located farms, which are more vulnerable, will take more care than other farms, all else equal. But they may not adopt enough protective measures for the social good. More isolated farms take less care, all else equal, but over protect.

Circle Topology

A region has N [greater than or equal to] 3 farms labeled n [member of] {1, 2, ..., N} = [[OMEGA].sub.N], where each seeks to protect potential production to the value of [V.sub.n] > 0. The N [greater than or equal to] 3 assumption on the extent of the outbreak is convenient, but could be relaxed at only the cost of substantially more tedious algebra. An infected farm loses all of [V.sub.n]. The farms are located on a circle (see figure 1). The circle topology was chosen because farms are located symmetrically on it. It enables a consideration of many of the article's main points but not the role of location asymmetry. Location asymmetry issues are examined in a separate section using a line spatial structure.

[FIGURE 1 OMITTED]

Infection is rare and can enter the region at some farm with probability [theta], where each farm is equally likely to be the first infected. By "rare" we mean it is almost certainly true that at most one farm inside the region becomes infected from outside the region at any time. The first farm to be infected within a region is labeled as the "originating farm." It will also be assumed that public authorities intervene to suppress a disease outbreak after the disease spreads to no more than the most proximate two farms (clockwise), if indeed it spreads at all. We allow infection to occur only in one (arbitrarily, the clockwise) direction to simplify algebra. Condition N [greater than or equal to] 3 was imposed to avoid double-counting on the circle, where the disease spreads clockwise back to the originating farm. The case where infection occurs in both clockwise and anticlockwise directions is available upon request.

Farm-level care taking is modeled through actions taken at the farm border. If infection has reached its direct anticlockwise neighbor, the nth farm will become infected with probability [a.sub.n]. The grower can change this probability at a cost. The nth farm is said to take comparatively less care when the value of [a.sub.n] is comparatively high. Before representing the cost of reducing probability [a.sub.n], consider the expected revenue loss.

Farm 1 may be the first infected, where the probability of first infection is [theta]. Or it may contract the disease from its neighbor. Anticlockwise is farm N. If farm N is infected first, then farm 1 becomes infected through farm N with probability [theta][a.sub.1]. The originating farm's probability does not enter the calculation because we assume a farm has no incentive to try preventing the disease from exiting the farm. Farm N - 2 may also be the source of infection to farm 1, where the probability this occurs is [theta][a.sub.1][a.sub.N]. Since infection is rare, the overall probability that farm 1 is infected is approximately [theta] + [theta][a.sub.1] + [theta][a.sub.1][a.sub.N].

In order to develop a general expression for each farm's infection risk, define [n + i] = n + i - zN where z is an integer chosen such that [n + i] [member of] [[OMEGA].sub.N]. That is, clock algebra (also called modular algebra) is used. For the nth farm, the infection probability is approximately

[[omega].sub.n] = [theta] + [theta][a.sub.n] + [theta][a.sub.n][a.sub.[n-1]]. (1)

The overall expected loss in revenue to the region is [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

Finally, a prevention technology exists. Farms differ in their capacity to protect themselves, and the protection cost at entry probability level [a.sub.n] is [C.sup.n]([a.sub.n]), a decreasing function. Private profit to a farm is [L.sub.n] = [V.sub.n] - [V.sub.n] [[omega].sub.n] - [C.sup.n]([a.sub.n]), while the overall expected profit to the region is [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. The region is assumed to produce a small share of overall market output, so that consumer surplus may be ignored. Thus, L represents social surplus. Since actions by farm [n - 1] enter [L.sub.n] through [[omega].sub.n], externalities exist and one should not expect market competition to support the maximization of L.

In order to better understand protection incentives, the nth farm's cost of protection is posited as -[[alpha].sub.n]Ln([a.sub.n]) where [[alpha].sub.n] > 0. This ensures that the cost of not protecting at all is -[[alpha].sub.n]Ln(1) = 0, while the cost of complete protection--where [a.sub.n] = 0--is infinite. This, we believe, reflects reality to the extent that not protecting at all requires no expenditure, complete protection is prohibitively expensive, and the protection cost increases with the extent of protective action.

Private Incentives and Responses

The nth farm's profit is

[L.sub.n] = [V.sub.n] - [V.sub.n][[omega].sub.n] + [[alpha].sub.n] Ln([a.sub.n]), n [member of] [[OMEGA].sub.N]. (2)

Substituting (1) into (2) and differentiating, we obtain [[partial derivative].sup.2][L.sub.n]/[partial derivative][a.sub.k] [partial derivative][a.sub.s] [less than or equal to] 0 [for all] k, s [member of] [[OMEGA].sub.N], k [not equal to] s. Thus, an increase in care by the nth farm reduces the marginal net benefit of own care by farms other than the nth. This leads to the following:

PROPOSITION 1. If the probability of infection is as given in equation (1), then farm biosecurity actions to prevent the spread of infection are strategic substitutes.

This observation shows that the game being played is not of the type involving global strategic complementarities (Vives 2005). (1) Given substituting strategic interactions, the possibility exists for a public intervention to do harm by indirectly discouraging important actions while directly encouraging less important actions.

With Nash conjectures on payoffs (2), protective actions are chosen as solutions to

[partial derivative][L.sub.n]/[partial derivative][a.sub.n] = - [theta] [V.sub.n] (1 + [a.sub.[n-1]]) + [[alpha].sub.n]/[a.sub.n] = 0. (3)

Solutions are denoted by [a.sup.*.sub.n]. The system is illustrated for N = 3, so that profits are

Farm 1: [V.sub.1] - (1 + [a.sub.1] + [a.sub.1][a.sub.3])[theta][V.sub.1] + [[alpha].sub.1] Ln([a.sub.1])

Farm 2: [V.sub.2] - (1 + [a.sub.2] + [a.sub.1][a.sub.2])[theta] [V.sub.2] + [[alpha].sub.2]Ln([a.sub.2])

Farm 3: [V.sub.3] - (1 + [a.sub.3] + [a.sub.2][a.sub.3])[theta] [V.sub.3] + [[alpha].sub.3]Ln([a.sub.3]). (4)

From (3), ith farm Nash private conjectures are given by [a.sup.*.sub.i] + [a.sup*.sub.i][a.sup.*.sub.[i-1]] = [[lambda].sub.i] where [[lambda].sub.i] = [[alpha].sub.i]/[[theta][V.sub.i]]. The unique pure strategy interior solution is (2)