Understanding exponential growth

You may have been hearing that the Covid-19 cases are growing exponentially. That means that the number of new cases is proportional to the number of existing cases. So for instance, if you get 100 new cases when you have 1,000 existing cases, then when there are 10,000 cases, there will be 1,000 new cases that day. The numbers quickly blow up and we are overwhelmed.

The point of social distancing is to reduce the transmission rate and break that exponential growth. We will need to continue the shutdown until we have slowed down the transmission to the point that we can track all new cases. But how can we know if it is working?

Let’s look at a graph showing the number of cases per day in New York and California.  Everything is growing so fast that it is hard to see anything else. We can see that New York grew faster than California and that is about it. One way to help is to use a logarithmic scale. Now we can see that New York grew very fast at the beginning of the outbreak and that early burst of cases meant that its total number would end up much higher. But it is still hard to see if we are actually changing the rate with our social distancing.

A trick to see whether we have stopped exponential growth is to plot the daily number of cases against the total number of cases, with both scales being logarithmic.  If we are in exponential growth, we are almost plotting a number against itself and the slope of the curve will always be one. We need to use a running average (average of 7 days of data) because daily reporting is uneven.

Here you can see that during March, the slope of the curve in both New York and California is 1, because even though they were growing at different rates, they were still growing exponentially. We can also that the rate in both states is bending over as social distancing is stopping the exponential growth. California is turning over at a lower total number of cases because it started social distancing at a lower total number of cases – 1089 in California on the day social distancing began statewide compared to 10356 in New York.

Here are the same curves for 12 states and Los Angeles County (at more than 10 million residents, Los Angeles County is bigger than half of the states). I grouped them by states where the drop from exponential is clear and the second graph are states where the data is just starting to show the effect.  The number for each state is the number of cases it had on the day they started social distancing.

All the data come from the New York Times database of cases: https://github.com/nytimes/covid-19-data

The Covid19 pandemic: A slow-moving disaster

I am not an epidemiologist. But you don’t need to know the details of how the virus works, to understand the public health statistics. I am an educated layperson with more than four decades experience in statistics, and from that perspective, I want to share my thoughts on what we are going through and how to listen to the public health professionals. As a disaster scientist, I also look at what we should expect going forward.

Social Distancing

First, let’s look at why we are “social distancing.” The facts are:

  • This Corona virus is new, so essentially everyone is susceptible to it.
  • The virus spreads easily. It is more contagious than the flu although less contagious than chicken pox or measles.
  • According to health officials in China, about 80% of the cases are “mild” (meaning the patient does not need to be hospitalized), so people are spreading the virus without realizing they have it, or before they are really sick.
  • Many people are dying from Covid-19, with over 9,000 reported deaths and over 220,000 confirmed cases worldwide, as of Thursday 3/19. The real death rate depends on how many cases are being missed, an unknown, but all of the estimates are at least 10 times as deadly as the flu.

This situation — a virus that easily is transmitted and a completely susceptible population — leads to exponential growth.  This means the number of cases doubles in a set period of time. At this point in the United States, that time seems to be about 3 days. This may not scare you, but it terrifies me.  By Easter Sunday, April 12, at this rate, confirmed cases will have doubled eight times.  That is 2x2x2x2x2x2x2x2 = 256 times more than the 10,000 confirmed cases the US has right now.  That means there will be 2.5 million cases in the United States by Easter Day.

The Chinese experience suggests that 20% of those 2.5 million people will need to be in the hospital. That means 500,000 people needing to be hospitalized by Easter. In 24 days.

A vaccine could stop this, but it will not come in time.  The only way we can slow down the transmission is by keeping carriers from interacting with other people.  And because we all could be carriers and not know it, our only hope is to keep everyone apart. This is social distancing.

With social distancing, we can reduce the doubling time. If it is 6 days instead of 3, we would only have 160,000 cases instead of 2.5 million in 24 days. If we get it to stretch out to 8 days, we have only 80,000 cases by Easter. As we reduce the interactions, people have time to recover and stop being contagious before they infect others.

A great visual demonstration of this has been published by the Washington Post (https://www.washingtonpost.com/graphics/2020/world/corona-simulator/) and shows how social distancing can really slow the progression.

Some people wonder why we don’t just let the disease run its course. The problem is: our health care system doesn’t have the capacity to handle this many sick people at the same time. Many more people will die from this disease if they cannot be hospitalized. In Italy, where the health care system has been overwhelmed, over 8% of the 36,000 confirmed cases have already died. In South Korea and Taiwan, where aggressive testing and isolation have kept the disease in check, the death rate is only 1%. The lives of many, many people depend on spreading out the infections so not too many are sick at the same time.

The next question is: how long will this continue? I wish I could give a definitive answer, but there are too many unknowns. It could go on until we have a vaccine. It is too late to confine the disease, so until most people are immune, either from having contracted the disease or from a vaccine, the danger is with us.

What Is Next?

So, what does this mean for all of us going forward? We humans have more control over our behavior than a virus does, so I cannot predict everything that will happen. We will be making choices that determine the outcome. But we can look at what has happened during natural disasters – another type of social crisis – to get some idea.

First, like in earthquakes, most of us will live through this, but our reactions are driven by our personal fear of dying. The development of human intelligence was a response to our need to survive in a world filled with bigger and stronger predators. We evolved to respond to danger by trying to make a pattern that would allow us to make safer choices. When we could make the connection between movement in the grass and a hidden predator, or between our gastrointestinal distress and the mushrooms we just ate, we were more likely to survive to have children.

The problem is that our need for patterns is so strong, that we create them even when they do not exist. The transmission of the viruses, like the timing of earthquakes, has a very large random component, so we cannot predict when or if we will be hit. This is so unbearably stressful, that we create patterns to give us the illusion of control. Traditionally, humans have attributed random disasters, including pandemics, to the gods. This gives us a pattern to believe in that cannot be proven wrong. And it gives us someone to blame.

As human thought has matured and the fallacies of the divine retribution model become more obvious, we have moved on to find something else to blame. We blame FEMA for failing to respond, we blame builders for making bad buildings, we blame the government for not having tests – and many of our complaints have an element of truth, that the disaster could have been less with appropriate preventative action.

One of the most common and most problematic human reactions is to blame the victim. “If I can believe that the victims brought this on themselves, then maybe I can avoid the same fate.” We respond to a cancer diagnosis with speculation on lifestyle choices. We blamed the residents of New Orleans for the losses in Katrina and created a false narrative of the breakdown of social order to absolve ourselves of the guilt in how we responded to their need. Now, some are blaming individual Asians, many of whom were the first victims, and looking for ways to say it was their fault. Finding someone to blame is human nature. Recognizing that we are invoking this instinctual, irrational response is the first step in moving to a more rational, compassionate outlook.

We also need to recognize how the need to blame the victim colors our response to poverty. We have a strong need to believe that poor people somehow caused their own problems because otherwise, we have to believe that it could happen to us. And Covid-19 is making poverty much more likely for many of us. One silver lining might be that this experience will help many of us have more compassion for the poor.

Which brings me to the final point. Just like in other natural disasters, what is most at risk now is not an individual life, but the health of our communities. The next few months will strain the social fabric, as we have to face our fears in physical isolation. Our choices will matter. What can we do?

First and foremost, we need to take social distancing seriously. The lives of many people depend on it. Without social distancing, we are looking at millions of American victims within one month.

Second, we need to listen to the experts and support the governmental actions. For many decades, government has been called the problem. But now we are seeing how much we need our government. Public health systems that have been underfunded or cut completely have the expertise to help us get through this. They are telling us which actions can reduce the risk. It is too late to do many of them, but it is not too late to start respecting their expertise. We need the government to enforce the social distancing to save our lives and the buffering to help our economic system not collapse. Government is the mechanism by which we work together to support the shared community.

Finally, we need to help each other emotionally even as we stay physically distant. No one likes uncertainty and just about everything is uncertain right now; our physical, emotional and economic security are all threatened. But in our modern era, we have the tools to stay connected. Every day, you can call someone you know who may be feeling isolated.

Right now, we need to remember that we are all in this together.Edit

Leave a Reply

BLOG AT WORDPRESS.COM.Privacy & Cookies: This site uses cookies. By continuing to use this website, you agree to their use.
To find out more, including how to control cookies, see here: Cookie Policy

Normal aftershocks

I have been deluged with questions since the Ridgecrest earthquake that are some version of “What does [quake A at location B] mean?”

  • “Why are there so many aftershocks near Coso?”
  • “What does it mean that there are earthquakes near the Garlock fault?”
  • “We just had another M4.5? Does this mean we need to be ready for a big one?”
  • “Are the Bay Area quakes related to Ridgecrest?”
  • Etc.

First, for the basics about how one earthquake triggers another, look down in this blog to my post from February 2018. It describes what seismologists know about earthquake triggering. We actually know quite a bit and can describe the rate of earthquakes reasonably accurately. The variables are time from the largest earthquake (the rate goes down approximately as 1/time), the magnitude of the triggering event and the triggered event (small earthquakes are always much more common than big ones and big earthquakes trigger more new events), and distance (most triggered events are very near the fault that produce the mainshock and the rate dies off as 1/distance^2). If you want the full details, go to the 2018 post.

In other words, we have equations to describe the rate and that we can use to compare one aftershock sequence to another.  But this is a rate.  The time of an individual earthquake is randomly distributed about that rate. So for instance we can say there will probably be 25 to 35 earthquakes of M3 or greater in the next week, but no idea whether the next one will be in 1 minute or tomorrow.

The USGS is creating and posting daily updates of these predictions here.

People are also asking about the risk because we have the big Garlock fault near these aftershocks. Does that make our risk worse?  If another M7 were to be triggered (the current weekly chance of that is about 1 in 300), it would likely be on the Garlock fault so that risk is already included in the probability numbers.

People also ask about the San Andreas. It is much farther away from the Ridgecrest earthquakes so the chance of an earthquake on it has not changed.

Compared to the average California aftershock sequence, the Ridgecrest sequence is about average in total number but is dying off more quickly than average. That means the chance of bigger ones is going down fast which is why the USGS probabilities are quite a bit lower now than they were last week.

All of the questions are some version of “Are these aftershocks normal?” and the answer is “Yes.”

But don’t make the mistake of thinking that “normal” means another big earthquake can’t happen. A large late aftershock is also pretty normal. Magnitude 6 earthquakes every couple of years somewhere in Southern California are normal. “Normal” does not mean it cannot trigger another earthquake. We are trying to make patterns in a random distribution. Just like constellations, we can make the patterns but they are not going to be any use in figuring out what is coming next.

You can’t live in Southern California worrying which day will have the big earthquake. You wear your seat belt every day, not knowing when the next accident will be. You need to make your house safe, get your water supply and be ready for the earthquake on any day.

The Music of Climate Change

I am an experimental seismologist. That means I spend a lot of time looking at data about the earth. I am not an atmospheric scientist so I cannot make detailed models for the future of the world’s climate, but I can look at data and recognize alarming trends. And the basic data for the temperature of the earth, averaged over the whole atmosphere is terrifying. The world is warming and the rate at which it is warming is getting faster.

Change in the average temperature of the Earth since 1880 (data from NOAA).
Temperature is given in °F change from 1880.

I am also a musician and I sometimes hear the data. This data is like a graceful minuet accelerating into a frantic jig. Of course, the increase is to be expected because our burning of fossil fuels for energy has been increasing the amount of CO2 in the Earth’s atmosphere. The heat in the atmosphere equals the energy that comes in minus the energy that goes out. Energy in comes from the sun with minor variations from rotational wobble, sunspots, etc.  Energy out is the infrared heat radiation that is given off by any body minus the amount that our atmosphere can trap. “Greenhouse gasses” are the chemicals that absorb energy in the infrared spectrum and thus keep some of the heat here on earth and if you increase the concentration of greenhouse gasses, you keep more heat. Details are complex with positive and negative feedback mechanisms, transport of heat through the ocean currents, effects of clouds and ice, all being studied by other types of earth scientists.  But I can look at this graph and see that the amount of carbon dioxide is up substantially and that it correlates well with the temperatures. I find these graphs terrifying and I struggle to understand why so many people accept this without fear.

CO2 concentration in the atmosphere. Data from NOAA

I turn to music to process and express my emotions. I realized early on in life that although I love music, I would never be a great musician – one who can make a living at it – but music continues to be the part of my life where I can experience my emotions in more depth. I saw someone turn the temperature data into pitch and play it on a cello. You could hear what I saw in the temperature data. It is not a piece of music – it is a string of notes without chords or harmony. But I started to think about using this to convey my emotions about climate change.

I play an old string instrument called the viol (or in Italian, the viola da gamba). A “consort of viols” was the 17th century predecessor of the string quartet, a small group ensemble with closely interlocking harmonies (an example here). One style of music for this ensemble is the In Nomine, where one instrument plays a drawn out melody and the others play intricate polyphonic lines around that slow steady line, called the cantus firmus, or fixed song. I decided to try to use the climate data as the cantus firmus and create music for the earth.

Turning that vision to reality has taken a few years, but I finally completed In Nomine Terra Calens: In the name of a warming Earth. This music video uses the visual and auditory arts to experience the warming Earth.  It is played by four viols with one of the bass viols playing the temperature data. Thanks to Josh Lee and Ostraka for recording the piece. A friend and colleague from the Art Center College of Design, Ming Tai, led a team to create an animation of temperature data to go with the music.

I realize that just knowing the Earth is warming is not enough. As scientists, we understand a lot about how earthquakes, floods and other disasters will affect people, but we have found that it is much easier for people to understand the problem and take action when we clearly connect the consequences to the problem.  I spent a decade creating science-based scenarios of natural disasters to provide people with the information to make better decisions.

With climate change, if we do nothing, if we continue to add CO2 to the atmosphere through burning fossil fuels, we will change life on earth. Even though there have been warmer times on Earth, those were times with different ecologies – that had time to adapt to the changes in climate.  The speed with which we are changing the climate is unprecedented. The world is already warmer by almost 2°F.  Everywhere on Earth, ecosystems are experiencing a different climate than that in which they evolved. Within the lifetime of children now born, with no action, the climate could be 5-10 °F warmer. What does this mean?

  • More disasters. Heat is energy and with more energy in the atmosphere, storms will be more severe. The strongest hurricane, the biggest daily rainfall, the longest duration of Cat 5 winds, have all happened in the last few years. Storms will be bigger and more devastating.
  • More wildfires. Every ecosystem in the world will be stressed by a different climate, mostly warmer. Wildfires are spreading through the boreal forests in Scandinavia and Alaska. The California wildfires of the last few years are just the beginning.
  • Food shortages. Farmers are already having to adapt their techniques to the changing climate. Some places will become unfarmable.
  • Water availability crises. Some places are getting wetter and some drier. Getting water to people will require different delivery systems than we currently have.
  • Climate refugees. Some countries will be underwater as the polar ice melts. Others will lose the ability to grow enough food for their populations. Millions, maybe hundreds of millions of people will need to move to stay alive.
  • Social disruption? We humans have many choices to make about how we respond to these disruptions. How will we respond to 50 million refugees?

I think many people, especially in America, think that dealing with climate change means giving up modern life. We talk about individuals driving an electric car, forgoing an airplane trip or reducing plastic use, as though that is the solution. We don’t want to give up modern life so we don’t think about climate change or we try to believe it isn’t really true. 

But the answer to climate change is not a Prius and reusable grocery bag.  With 7 billion people on Earth, we could give up every aspect of modern life and not solve the problem. We still need to keep warm in winter and move our food to where people are. The only solution is to move forward, to a world with a carbon neutral energy system, a society where the production of energy does not increase the carbon in the atmosphere. Until we get to carbon neutral energy, any life, let alone modern life, increases the CO2 in the atmosphere.

Dealing with climate change means technological innovation to create a better world. It can be done. Solar energy is already much cheaper than it was a decade ago. When we decided to go to the moon, we solved innumerable technical challenges and we were proud of doing it. We did that together and we need to do it again.

As an individual, you might be one of the technical innovators. But every one of us can say we want our government to fund and support the innovation. Why would we want someone else to own the technologies that are going to be needed by the future world? Dealing with climate change means investing in the future.  Look again at the consequences of our current trajectory. The true threat to modern life is not dealing with climate change. 

I end In Nomine Terra Calens with a stripping away of harmonies to finally land on one, lone, very high note. I end without direction to represent the uncertain future. We stand at a decision point where the future of the world really rests on our decisions.


Volcanoes Cool the Earth… at Least Temporarily

Could the eruption of Mount Agung in Indonesia cool the earth?

Very big volcanoes can disrupt the climate around the earth. In 1783, the Laki Craters eruption in Iceland produced more than 6 times as much sulfur dioxide as Mount Pinatubo. It also got more into the stratosphere, because the bottom of the stratosphere is lower near the poles – only eight miles up. In the next year, a great freeze settled over Europe, leading to a famine that contributed to the French Revolution. The continents did not heat up as much the next summer, so the monsoons didn’t develop. This contributed to droughts that killed one-sixth of the population of Egypt, 11 million people in India and over 1 million in Japan. Could the same thing happen because of the current eruption of Mount Agung in Indonesia?

The last time Mount Agung erupted in 1963, the average global temperature dropped for a few years.  This graph shows the average global temperature for the last 140 years, marking the times of five large volcanic eruptions. You can see how the world’s temperature drops for a few years after those eruptions.  Some, but not all, big volcanic eruptions have caused a drop in the global temperature. What makes the difference?

The answer is in the gases that come out of the lava. Magma often has a large amount of trapped gases that get released when the it reaches the surface. The most common are water, carbon dioxide, and sulfur dioxide. Water and carbon dioxide are already so common in the atmosphere that the volcanoes do not change the global concentrations. Mount Pinatubo in 1991 released 50 million metric tons of carbon dioxide. Human activity in the United States release 100 times that – 5 billion metric tons every year.  Besides, as we all know, carbon dioxide traps the infrared 

radiation of heat rising form the earth and tends to make the earth hotter.

The cooling effect of volcanoes comes from pushing larger particles into the stratosphere where they block sunlight coming in. Although volcanic ash blocks sunlight, it is heavy (it is rock, even though in very small pieces) and it falls back to the earth in a few days or weeks. The biggest culprit is the sulfur dioxide. Sulfur dioxide oxidizes into sulfuric acid and condenses into sulfate aerosols. In the lower atmosphere, sulfates are washed out of the atmosphere relatively quickly by rain. But above the main climate systems, in the much drier stratosphere, particles could be transported around the world, staying aloft for years. These sulfate particles are just the right size to scatter incoming sunlight, sending some of it back into space and, consequently, cooling the ground below. Volcanic eruptions that send a lot of sulfur into the stratosphere can have a substantial impact on the global temperature. Mount Pinatubo, erupting in 1991, cooled the world by 1.5°F, with an impact that could still be felt three years later.

Mount Agung in 1963 had a similar impact to Mount Pinatubo, and it could again. The next eruption would need to also have a high concentration of sulfur dioxide, and it would need to be powerful enough to force that sulfur dioxide up into the stratosphere, 12 miles above the earth’s surface. And as we see in the graph, it doesn’t stop the global warming we are creating with greenhouse gases; it just gives us a short break in the otherwise upward trend.

You can read more about the Laki eruption and other catastrophic natural disasters in my upcoming book, The Big Ones, available for pre-order here.