Destroy to Save

I wrote last year about the fight in Worcester county, Massachusetts, against the invasive and destructive Asian Longhorn Beetle (ALB).

Starting in late 2008, a program was initiated that involved cutting down trees to slow the spread of this pest and that involved imposing a quarantine that made it illegal to move firewood from inside the county to outside.

I speculated that both cutting down trees and imposing a quarantine at the same time perhaps created an unintended consequence: an increased supply of firewood tends to drive down its price, making it profitable to (illegally) move wood across the quarantine boundary. (In fact, two Rhode Island companies were caught doing this and fined about $1800 each.)

Maybe a much stiffer fine could have helped to curtail this activity, but since I wrote that post I’ve been wondering if there might be other ways that reduce wood moving and help fight the infestation. The effort in Worcester is still underway, with a plan to spray trees with ALB-killing chemicals this spring. But while the rest of this post is inspired by the events in Worcester, I don’t intend for it to necessary reflect what actually happened nor be concerned with specific numbers of trees cut down or so forth. The beetle has been found in a variety of places in the US, Canada and Europe and attacks many common species of trees, so I’m not so concerned about making certain that the numbers I calculate are reasonable for Worcester.

My intention here is simply to make a qualitative, step-wise simulation to see how it might be possible to (1) minimize tree loss, (2) minimize the moving of infested wood, and (3) minimize the financial benefit for moving wood.

My spreadsheet-based model has many parts, but I hope that each part individually is simple to understand. There are only a few main factors: the biology of ALB infesting trees, the dynamics of fighting the infestation, and the economics driving people to move wood outside the quarantine zone.

The Biology of Infestation

Imagine an area initially has 200,000 trees. Time progresses in steps, each of which is 1 week long. At time 10 (weeks), a cluster of 15 trees is infested with Asian Longhorned Beetles.

Each infested tree infests other trees at a rate that decreases as the density of uninfested trees decreases. This is rate is 0.1 * d, where d is the ratio of uninfested trees to the initial number of all trees (at time=0). So, when nearly all of the trees are present and healthy, each tree will infest 0.1 other trees per week, on average. As the number of uninfested trees drops to 0, so does the rate at which healthy trees get infested.

Infested trees live an average of 26 weeks (6 months), so 1/26th of all infested trees die each week.

This formulation is not intended to be controversial in any way. Epidemics, fads and the diffusion of innovations have been studied using so-called SEIR models (Susceptible-Exposed-Infected-Recovered) and this formulation is conceptually similar, though simplified to be just Susceptible-Infested-Dead (SID).

Even this portion of the model by itself (where the infestation is allowed to run rampant) provides some interesting results. Left to its own devices, the ALB infestation would destroy 90% of all trees over a period of about 6 years after the outbreak.

Interestingly, if the average tree lived 52 weeks (instead of 26), more than 99% of all trees would be killed. And, if the average tree lived only 11 weeks, then only about 17% of trees would die. (Again, these numbers are not intended to predict what will actually happen.)

This result might sound counter-intuitive, but it’s really not: few diseases or parasites kill their hosts quickly, because if they did, they wouldn’t be able to spread to a new host. A genetically engineered ‘supervirus’ that killed within seconds wouldn’t wipe out humanity – it would only kill the first person it infected.

Yet rarely have I heard people suggest making harmful insects more destructive or weakening trees ability to fight pests as a way to slow the spread of invasive species. Instead, efforts tend to focus on cutting down infested (and potentially infested) trees. So, let’s add that to the model.

Fighting the Infestations

I’ve considered two ways to fight the ALB, which I call ‘harvesting’ and ‘culling’. Say that inspectors are able to check 1000 trees per week. Any infested tree is immediately cut down, chipped and incinerated. In addition, the 8 nearest trees are also automatically removed. However, since a quarantine is in place and these trees might not be infested, they are simply split and sold as firewood (for use within Worcester county only).

I’ve assumed that trees which are near an infested tree are more likely to also be infested themselves. Of the 8 trees cut down with each infested tree, the portion of them that happen to also be infested, I’ve hypothesized, is something like 1 – (1-i)^N, where i is the fraction of trees that are currently infested and N is a constant I’ve chosen to be equal to 4. (This might sound complicated, but for any value of i, it simply generates a number that is somewhat higher than i, but still between 0 and 1.)

In addition to harvesting infested and potentially infested trees, we can have a 1-time random culling of some fraction of the number of all trees. So, if we decide to cull 20% of trees then at, say, week 100, 20% of all trees are chopped down and sold as firewood. The intention here is to make it harder for the pest to spread, since the rate of spreading depends on the density of uninfested trees.

Obviously, culling 100% of the trees would completely eliminate the infestation in Worcester county, but it would also sort of defeat the purpose of trying to save the trees – and, it turns out, it might actually help the spread of the pest to areas outside of Worcester. The reason, I think, is basic economics.

The Economics of Smuggling Wood

Destroying all potentially infested wood might sound like it makes the most sense, but it would actually be very difficult and wasteful. It’s OK it burn potentially infested firewood, as long as the wood does not get moved any great distance (which might spread the beetle).

The problem, though, is that if we impose a quarantine on moving wood and then chop down large numbers of trees, the local price of firewood should drop significantly. And this creates and incentive for people to move wood out of Worcester, to places where it more expensive.

I suspect that most wood isn’t moved by companies like ones from Rhode Island that got caught and were fined – Most likely, most wood is moved by guys with pickup trucks who think they are helping friends in this rough economy by bringing them cheap firewood. Some people don’t know the danger. Some might think the benefit is worth the risk.

I don’t know how wood-moving activity depends on the price of wood, so I’ve simply assumed that there are initially 10,000 units of wood in Worcester county and the price per unit inside and outside Worcester is $250. The price of wood in Worcester goes up and down simply as the ratio of available wood changes. So, if 5,000 trees are cut down in the culling, the number of available units of wood will jump from 10,000 to 15,000 and thus the price will fall from $250 to ($250/1.5) or $167.

Moving wood is assumed to be directly linear to the difference between the local price (which varies) and the price outside Worcester (which is held constant at $250). Additionally, as the local price falls, consumption of firewood rises linearly with the price difference. In each case, a $1 price difference causes 5 additional units of wood to be moved and 5 additional units to be consumed locally per week.

So, all of this might sound complicated, but there are actually only a few key aspects. To recap:

  1. Infested trees infect uninfested trees.
  2. Infested trees get harvested and nearby trees turned into firewood.
  3. Some fraction of all trees get culled in a one-time event.
  4. Extra firewood causes the local price to drop, spurring both the movement of wood outside the quarantine limit and the increased usage of firewood within the limit.

For a fairly straightforward model, I think some of the results are surprising. I’ve already mentioned finding that making the ALB more lethal to trees actually reduces the fraction of trees that will naturally get killed. Below are some others.

Results of the Model

The purpose of these calculations was to determine if there might be ways to reduce the amount of wood moved outside the county, based on the observation that culling trees while imposing a quarantine might actually encourage the movement of wood to places where its price is higher.

The graph below shows how the fraction of trees that are infested with Asian Longhorned Beetles and the price of a unit of firewood might change over time if there is a cull of 24% of all trees at week 100.

Obviously, the cull causes a sudden drop in local wood prices as firewood floods the market, but it doesn’t affect the fraction of trees that are infested (because the trees to be culled were selected randomly, and therefore no more or less likely to be infested than the total population as a whole).

There is a second dip in the price as the infestation reaches its peak around 2 years (100 weeks) later.

One thing that is surprising to me is the length of time over which this story plays out. The initial infestation was discovered in week 10 and harvesting of infected trees began then too, but by the time of the culling, 2 years later, only a small fraction of trees had been affected. Even after the culling, it still took 2 more years for the infestation to peak and then another 2 years for it to die down, even though each tree infects one other tree, on average, once every 10 weeks.

This 24% cull fraction is interesting because it minimizes the total value accrued in moving wood out of the county. However, for the parameters and equations I’ve chosen, it does not minimize the percentage of trees that are ultimately lost to the pest, nor the total amount of infested wood moved out of the county.

So, if you were a county manager and the identification and removal of infested trees is going as quickly as possible – 1000 trees inspected per week. You’re planning a county-wide culling and thought perhaps that choosing a cull fraction such that it made moving wood as low-profit as possible would save the most trees and reduce the movement of wood, you’d actually be better off choosing a higher cull fraction.

It turns out that, for a cull that occurs in week 100, the percentage of trees lost is minimized when the cull fraction is about 50%. To cull more or fewer trees ultimately results in more trees being lost. If we wait a year (maybe due to budget problems) and have the culling in week 150 instead, there is no selection of culling fraction that results in less than about 85% being lost. Having the cull only 1 year into the infestation still can only save about 50% of the trees, at most.

Interestingly, neither the 24% cull that minimizes the profit to wood movers nor the 50% cull that minimizes the percentage of trees lost actually minimizes the amount of infested wood moved out of the county. For a cull that occurs in week 100, the number of units of infested wood moved outside the county is at its lowest when the cull percentage is about 54%.

To have a cull fraction less than or greater than that amount results in more infested wood being moved out of the county. One lesson from these calculations is that it is not possible to simultaneously reach reach every goal – To minimize the economic benefit of moving wood means not minimizing the fraction of trees ultimately lost. And to minimize the total amount of infested wood moved out of the county means not minimizing either of the other two goals.

A second important lesson shows the real advantage of having the earliest cull possible: It turns out that one of the strongest levers available is not in the choice of which fraction of trees to cull, but rather when the cull takes place.

For a cull that takes place in week 150 (the 3rd year of the problem), the infestation is so widespread that no selection of cull fraction results in less than 10,000 units of wood (one ‘unit’ is one standard-sized urban tree) moving outside the country. If the cull can happen one year earlier, it is possible to have less than 1,000 units move. And for a cull in the first year of the problem, less than 100 units will be moved. Each extra 50 weeks waited actually makes the problem more than 10 times worse than it would otherwise have been.

A final lesson, I suppose, is to not wait until an infestation breaks out and is detected to begin proper management of trees in an urban environment. Perhaps if inspections had been much larger in number or a different number of nearby trees were taken whenever an infected tree was discovered or, maybe, if the fines for moving wood had been high enough to make moving wood unprofitable at any price, the aggressive actions Worcester is now taking against the Asian Longhorned Beetle night not have been necessary at all.

The best way to find out, I think is to download my ALB spreadsheet and try to make changes for yourself. You can change parameter settings, and also any of the equations. (There might be cases when errors occur, like ‘division by zero’ problems if the number of trees drops to 0. So you’ll have to check any results you get to see that they actually make sense.)


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