growth Archives

Steady State Growth in SIR & SEIR Models

Beware of the interpretation of R0, and models that plug an R0 estimated in one context into a delay structure from another.

This started out as a techy post about infection models for SD practitioners interested in epidemiology. However, it has turned into something more important for coronavirus policy.

It began with a puzzle: I re-implemented my conceptual coronavirus model for multiple regions, tuning it for Italy and Switzerland. The goal was to use it to explore border closure policies. But calibration revealed a problem: using mainstream parameters for the incubation time, recovery time, and R0 yielded lukewarm growth in infections. Retuning to fit the data yields R0=5, which is outside the range of most estimates. It also makes control extremely difficult, because you have to reduce transmission by 1-1/R0 = 80% to stop the spread.

To understand why, I decided to solve the model analytically for the steady-state growth rate in the early infection period, when there are plenty of susceptible people, so the infection rate is unconstrained by availability of victims. That analysis is reproduced in the subsequent sections. It’s of general interest as a way of thinking about growth in SD models, not only for epidemics, but also in marketing (the Bass Diffusion model is essentially an epidemic model) and in growing economies and supply chains.

First, though, I’ll skip to the punch line.

The puzzle exists because R0 is not a complete description of the structure of an epidemic. It tells you some important things about how it will unfold, like how much you have to reduce transmission to stop it, but critically, not how fast it will go. That’s because the growth rate is entangled with the incubation and recovery times, or more generally the distribution of the generation time (the time between primary and secondary infections).

This means that an R0 value estimated with one set of assumptions about generation times (e.g., using the R package R0) can’t be plugged into an SEIR model with different delay structure assumptions, without changing the trajectory of the epidemic. Specifically, the growth rate is likely to be different. The growth rate is, unfortunately, pretty important, because it influences the time at which critical thresholds like ventilator capacity will be breached.

The mathematics of this are laid out clearly by Wallinga & Lipsitch. They approach the problem from generating functions, which give up simple closed-form solutions a little more readily than my steady-state growth calculations below. For example, for the SEIR model,

R0 = (1 + r/b₁)(1 + r/b₂)           (Eqn. 3.2)

Where r is the growth rate, b1 is the inverse of the incubation time, and b2 is the inverse of the recovery time. If you plug in r = 0.3/day, b1 = 1/(5 days), b2 = 1/(10 days), R0 = 10, which is not plausible for COVID-19. Similarly, if you plug in the frequently-seen R0=2.4 with the time constants above, you get growth at 8%/day, not the observed 30%/day.

There are actually more ways to get into trouble by using R0 as a shorthand for rich assumptions in models. Stochastic dynamics and network topology matter, for example. In The Failure of R0, Li Blakely & Smith write,

However, in almost every aspect that matters, R 0 is flawed. Diseases can persist with R 0 < 1, while diseases with R 0 > 1 can die out. We show that the same model of malaria gives many different values of R 0, depending on the method used, with the sole common property that they have a threshold at 1. We also survey estimated values of R 0 for a variety of diseases, and examine some of the alternatives that have been proposed. If R 0 is to be used, it must be accompanied by caveats about the method of calculation, underlying model assumptions and evidence that it is actually a threshold. Otherwise, the concept is meaningless.

Is this merely a theoretical problem? I don’t think so. Here’s how things stand in some online SEIR-type simulators:

Model	R0 (dmnl)	Incubation (days)	Infectious (days)	Growth Rate (%/day)
My original	3.3	5	7	17
Homer US	3.5	5.4	11	18
covidsim.eu	4	4 & 1	10	17
Epidemic Calculator	2.2	5.2	2.9	30*
Imperial College	2.4	5.1	~3**	20***

*Observed in simulator; doesn’t match steady state calculation, so some feature is unknown.

**Est. from 6.5 day mean generation time, distributed around incubation time.

***Not reported; doubling time appears to be about 6 days.

I think this is certainly a Tower of Babel situation. It seems likely that the low-order age structure in the SEIR model is problematic for accurate representation of the dynamics. But it also seems like piecemeal parameter selection understates the true uncertainty in these values. We need to know the joint distribution of R0 and the generation time distribution in order to properly represent what is going on.

Steady State Growth – SIR

Continue reading “Steady State Growth in SIR & SEIR Models”

Species Restoration & Policy Resistance

I’ve seen a lot of attention lately to restoration of extinct species. It strikes me as a band-aid, not a solution.

Here’s the core of the system:

Critters don’t go extinct for lack of human intervention. They go extinct because the balance of birth and death rates is unfavorable, so that population declines, and (stochastically) winks out.

That happens naturally of course, but anthropogenic extinctions are happening much faster than usual. The drivers (red) are direct harvest and loss of the resource base on which species rely. The resource base is largely habitat, but also other species and ecosystem services that are themselves harvested, poisoned by pollutants, etc.

Reintroducing lost species may be helpful in itself (who wouldn’t want to see millions of passenger pigeons?), but unless the basic drivers of overharvest and resource loss are addressed, species are reintroduced into an environment in which the net gain of births and deaths favors re-extinction. What’s the point of that?

If the drivers of extinction – ultimately population and capital growth plus bad management – were under control, we wouldn’t need much restoration. If they’re out of control, genetic restoration seems likely to be overwhelmed, or perhaps even to contribute to problems through parachuting cats side effects.

This is not where I’d be looking for leverage.

Tim Jackson on the horns of the growth dilemma

I just ran across a nice talk by Tim Jackson, author of Prosperity Without Growth, on BigIdeas. It’s hard to summarize such a wide-ranging talk, but I’d call it a synthesis of the physical (planetary boundaries and exponential growth) and the behavioral (what is the economy for, how does it influence our choices, and how can we change it?). The horns of the dilemma are that growth can’t go on forever, yet we don’t know how to run an economy that doesn’t grow. (This of course begs the question, “growth of what?” – where the what is a mix of material and non-material things – a distinction that lies at the heart of many communication failures around the Limits to Growth debate.)

There’s an article covering the talk at ABC.au, but it’s really worth a listen at http://mpegmedia.abc.net.au/rn/podcast/2010/07/bia_20100704_1705.mp3

The Law of Attraction

No, not that silly one.

Controlling Growth by Controlling Attractiveness

In Woodstock, Vermont, everyone’s mad about a highway. In other places the issue is a sewer system or a school. The real issue, of course, is growth. According to Jay Forrester’s Attractiveness Principle (Forrester is a professor of systems analysis at MIT) there’s only one way to control growth — control attractiveness.

…

In a free society if any place is unusually attractive, folks will — no surprise — be attracted there. The most mobile people (the young, the rich, the educated) will get there first. The place will grow until its attractiveness has been reduced by crowded highways, or unemployment, or scarce housing, or pollution, or just plain visual blight. (The most mobile people have moved on by then). When the place is no more attractive than anywhere else, then and only then will it stop growing. What else can stop it?

…

The attractiveness of a place is a complex combination of climate, economy, amenities, scenery. No one can define attractiveness exactly, but people make up their minds about it every day by deciding to move from Hartford or Boston or Westchester County to Vermont (that’s the direction they’re moving at the moment). Millions of human judgements weigh Vermont’s clean air against Boston’s job market and Manhattan’s cost of living. The very different mixes of attractiveness and unattractiveness in those places may seem incommensurable, but people make their comparisons, and eventually attractiveness evens out everywhere.

…

The normal instinct of public officials, including those of Woodstock, is to fix problems and make their community perfect. The more perfect they make it, the more new people show up. What Woodstock needs to do, Forrester would say, is decide what kinds of imperfection it’s willing to live with.

A crowded, unsafe highway? If that’s unacceptable, then choose something else. Super-restrictive zoning, perhaps, or an absolute limit on new curb cuts, or higher property taxes (I know, they’re already too high, but not high enough to stop people from moving in). Bad schools. Bad air. No jobs. Developments so ugly you might as well live in New Jersey. Some sort of whopping surcharge on those developers. Either Woodstock chooses its form of unattractiveness, or the growth process itself chooses.

It takes awhile to absorb the implications of the Attractiveness Principle, because it turns conventional thinking upside down (Forrester is good at doing that). Its implications are not good news for the sort of people who live in Woodstock. The Principle says you can’t live in a privileged bubble of attractiveness, unless you are perpetually young, rich, educated, and on the move at the head of the attractiveness wave. It says that growth is your problem wherever it occurs. It says the only way to be sure of living in an attractive place is to be committed to the attractiveness of every place.

From the Donella Meadows Archive

Another Look at Limits to Growth

I was just trying to decide whether I believed what I said recently, that the current economic crisis is difficult to attribute to environmental unsustainability. While I was pondering, I ran across this article by Graham Turner on the LtG wiki entry, which formally compares the original Limits runs to history over the last 30+ years. A sample:

Industrial output in Limits to Growth runs vs. history

The report basically finds what I’ve argued before: that history does not discredit Limits.

The Growth Bubble

I caught up with my email just after my last post, which questioned the role of the real economy in the current financial crisis. I found this in my inbox, by Thomas Friedman, currently the most-emailed article in the NYT:

Let’s today step out of the normal boundaries of analysis of our economic crisis and ask a radical question: What if the crisis of 2008 represents something much more fundamental than a deep recession? What if it’s telling us that the whole growth model we created over the last 50 years is simply unsustainable economically and ecologically and that 2008 was when we hit the wall ’” when Mother Nature and the market both said: ‘No more.’

Certainly there are some parallels between the housing bubble and environment/growth issues. You have your eternal growth enthusiasts with plausible-sounding theories, cheered on by people in industry who stand to profit.

There’s plenty of speculation about the problem ahead of time:

Google news timeline – housing bubble

People in authority doubt that there’s a problem, and envision a soft landing. In any case, nobody does anything about it.

Sound familiar so far?

However, I think it’s a bit of a leap to attribute our current mess to unsustainability in the real economy. For one thing, in hindsight, it’s clear that we weren’t overshooting natural carrying capacity in 1929, so it’s clearly possible to have a depression without an underlying resource problem. For another, we had ridiculously high commodity prices, but not many other direct impacts of environmental catastrophe (other than all the ones that have been slowly worsening for decades). My guess is that environmental overshoot has a lot longer time constant than housing or tech stock markets, both on the way up and the way down, so overshoot will evolve in more gradual and diverse ways at first. I think at best you can say that detecting the role of unsustainable resource management is like the tropical storm attribution problem. There are good theoretical reasons to think that higher sea surface temperatures contribute to tropical storm intensity, but there’s little hope of pinning Katrina on global warming specifically.

Personally, I think it’s possible that EIA is right, and peak oil is a little further down the road. With a little luck, asset prices might stabilize, and we could get another run of growth, at least from the perspective of those who benefit most from globalization. If so, will we learn from this bubble, and take corrective action before the next? I hope so.

I think the most important lesson could be the ending of the housing bubble, as we know it so far. It’s not a soft landing; positive feedbacks have taken over, as with a spark in a dry forest. That seems like a really good reason to step back and think, not just how to save big banks, but how to turn our current situation into a storm of creative destruction that mitigates the bigger one coming.

Will the Chinese Miracle End Soon?

Just after writing my last post on China, I found this remarkably candid SpiegelOnline interview with Pan Yue, China’s Deputy Minister of Environment. A few excerpts:

…
Pan: Of course I am pleased with the success of China’s economy. But at the same time I am worried. We are using too many raw materials to sustain this growth. To produce goods worth $10,000, for example, we need seven times more resources than Japan, nearly six times more than the United States and, perhaps most embarrassing, nearly three times more than India. Things can’t, nor should they be allowed to go on like that.
…
Pan: This miracle will end soon because the environment can no longer keep pace. Acid rain is falling on one third of the Chinese territory, half of the water in our seven largest rivers is completely useless, while one fourth of our citizens does not have access to clean drinking water. One third of the urban population is breathing polluted air, and less than 20 percent of the trash in cities is treated and processed in an environmentally sustainable manner. Finally, five of the ten most polluted cities worldwide are in China.

Pan: …Because air and water are polluted, we are losing between 8 and 15 percent of our gross domestic product. And that doesn’t include the costs for health. …

While I was building an electric power model for China in 2005, I saw estimates of 7% GDP losses from health impacts of air pollution, so this strikes me as plausible. One could argue that even 20% of GDP lost to pollution would not be a big deal, because it represents less than three years of growth. But that is to ignore a fundamental valuation problem: that increased material consumption is probably a poor substitute for lost environmental services and especially health problems. In a utopian world where China’s development path reflected individual preferences, are these the choices we would see?

8 to 15 percent is quite a bit more than the 3% in China’s Green GDP accounts, which in any case have been put on hold due to lack of support. On other measures, China’s HDI (a measure of life expectancy, literacy, educational attainment, and GDP per capita) is going up, but its GPI peaked in 2002. The World Bank shows adjustments shaving 16 percentage points off China’s national savings, but the net remains positive. China’s ecological footprint is in deficit territory.

SPIEGEL: But the economic growth fanatics in Beijing will still likely carry on just as before.

Pan: They’re still playing the lead role — for now. For them, the gross domestic product is the only yardstick by which to gauge the government’s performance. But we are also making another mistake: We are convinced that a prospering economy automatically goes hand in hand with political stability. And I think that’s a major blunder. The faster the economy grows, the more quickly we will run the risk of a political crisis if the political reforms cannot keep pace. If the gap between the poor and the rich widens, then regions within China and the society as a whole will become unstable. If our democracy and our legal system lag behind the overall economic development, various groups in the population won’t be able to protect their own interests.

This is crucial. Even communities in the developed world that experience rapid growth go through substantial pain. In part, that’s because growth puts pressure on resources (like open space, freedom from noise and pollution) previously regarded as free, for which property rights or other control mechanisms must be established. If institutions don’t keep up, conflict ensues.

And there’s yet another mistake in this thinking…..

SPIEGEL: Which one?

Pan: It’s the assumption that the economic growth will give us the financial resources to cope with the crises surrounding the environment, raw materials, and population growth.

SPIEGEL: Why can’t that work?

Pan: There won’t be enough money, and we are simply running out of time. Developed countries with a per capita gross national product of $8,000 to $10,000 can afford that, but we cannot. Before we reach $4,000 per person, different crises in all shapes and forms will hit us. Economically we won’t be strong enough to overcome them.

Counting on future growth to solve the problems of past growth is a classic escalation trap – Herman Daly’s “Hair of the Dog that Bit You” from the Catechism of Growth Fallacies in Steady State Economics. Daly cites Wallich, “Growth is a substitute for equality of income. So long as there is growth there is hope, and that makes large income differentials tolerable.” When the growth engine sputters, the social repercussions will be serious.

On Limits to Growth

It’s a good idea to read things you criticize; checking your sources doesn’t hurt either. One of the most frequent targets of uninformed criticism, passed down from teacher to student with nary a reference to the actual text, must be The Limits to Growth. In writing my recent review of Green & Armstrong (2007), I ran across this tidbit:

Complex models (those involving nonlinearities and interactions) harm accuracy because their errors multiply. Ascher (1978), refers to the Club of Rome’s 1972 forecasts where, unaware of the research on forecasting, the developers proudly proclaimed, “in our model about 100,000 relationships are stored in the computer.” (page 999)

Setting aside the erroneous attributions about complexity, I found the statement that the MIT world models contained 100,000 relationships surprising, as both can be diagrammed on a single large page. I looked up electronic copies of World Dynamics and World3, which have 123 and 373 equations respectively. A third or more of those are inconsequential coefficients or switches for policy experiments. So how did Ascher, or Ascher’s source, get to 100,000? Perhaps by multiplying by the number of time steps over the 200 year simulation period – hardly a relevant measure of complexity.

Meadows et al. tried to steer the reader away from focusing on point forecasts. The introduction to the simulation results reads,

Each of these variables is plotted on a different vertical scale. We have deliberately omitted the vertical scales and we have made the horizontal time scale somewhat vague because we want to emphasize the general behavior modes of these computer outputs, not the numerical values, which are only approximately known. (page 123)

Many critics have blithely ignored such admonitions, and other comments to the effect of, “this is a choice, not a forecast” or “more study is needed.” Often, critics don’t even refer to the World3 runs, which are inconvenient in that none reaches overshoot in the 20th century, making it hard to establish that “LTG predicted the end of the world in year XXXX, and it didn’t happen.” Instead, critics choose the year XXXX from a table of resource lifetime indices in the chapter on nonrenewable resources (page 56), which were not forecasts at all. Continue reading “On Limits to Growth”