Wednesday, October 29, 2014

Logarithmic Growth 101

Riffing off a scenario at WRSA, the concept of logarithmic growth came up.
If you're RainMan with math, go watch Wapner, and skip the following.
If not, let me, as a non-mathematician, take a whack at it, as it relates to Ebola, and why you should know or care.

(And for those who'll bitch at me in Comments with degrees from MIT and CalTech, I feel your pain. After suffering through four, yes four math-incompetent nursing professors trying to teach Medication dosage calculations(!!), I noted aloud that having nurses teach math was like asking garbagemen to teach physics. Especially when the same university had perfectly adequate math professors already on staff. But I digress. Suffice it to say, you're stuck with a nurse teaching math here. If it makes you feel better, I never bounce a check.)

Quick and dirty: the earthquake Richter scale is base-10 logarithmic.
I.e., A 5.0 is 10 times bigger than a 4.0, a 6.0 is 100 times bigger than a 4.0, etc.
If you tried to straight line plot a logarithmic event, either the right side would require a piece of paper 20 feet tall, or the left side increases will be nearly flat unless examined under a microscope.
Like Mercator for maps, it's a good (not perfect) way to make things fit on a flat square/rectangular piece of paper small enough to hold in your hand.

The actual Wikipedia page for this Ebola outbreak has log scales for reference: instead of the rocket launch reverse parabola on the straight-line scales, the trend line seen is almost exactly 45 degrees up and right, indicating an almost perfect logarithmic R-naught of 2.
(Any departures owe more to shoddy math and stat collection than to any flaws in the disease's ability to spread. Nota bene the graph at top is an old one for this very outbreak, as we now have nearly triple that number of cases, and triple that again of likely official ones, i.e. pushing 40,000 cases.
To see a more current graph, with projections to the bitter end, Frozen Patriot has done one here:
I'd post it pictorially , but Blogger won't read the image. Go look for yourself.
h/t to Frozen Patriot for the work!)

100 new cases a week of something isn't that big a deal; we have 100 homicides in this country every few days, with no danger of running out of people. But if the homicide rate went logarithmic every month, in a year's time we'd be looking at 400,000 homicides a month. By mid-October of Year Two the homicide rate is larger than the population of the US, i.e. everyone's dead.

Ebola, playing with the entire planet, gets 33 doublings to get from 1 case to the 7.7B people on earth, less 10-25% or so survivors. Which would take the world's population back to where it was around 1700, more or less. It would take US population back to the 31M we had in 1860, except now those 31M would be distributed  thinly in all 50 states. Move your state's population decimal one place to the left, and that's where it'll be if Ebola keeps on marching at logarithmic growth worldwide. If we keep it out, we stop that from ever getting a toehold to start. That's why travel bans and quarantines matter!

People wonder why they should believe the graph. Fair enough. As far as suspicion, the graphists above has done the same thing the editors at Wikipedia did: plotted the numbers, and literally connected the dots. The graph is only as good as the data.

If the doublings speed up, the line goes farther to the right, faster. If we slow the speed of spread, it doesn't. Currently, Ebola is doubling every 2-4 weeks, depending on the country we're talking about. The average overall is every three weeks. If we even slowed that rate down to every four weeks, we'd add a year to the planetary "Game Over" date. It was at every four weeks in July/August, for reference.

It doesn't flatten until we get the R-naught to less than 2.
It doesn't turn downward at all until we have a treatment(s) or vaccine that starts cutting the R-number to less than 1.
For smallpox and polio, that took 5900 years or so of human history.
We don't have to replicate that, but it gives you an idea of what has to be accomplished.
We've been trying to do it with HIV/AIDS for decades, with zero success, because it mutates rapidly and becomes resistant to our efforts.
Hopefully Ebola doesn't mutate as fast as we come up with something to kill it, and a vaccine will work on it for more than five minutes. If not, we're boned.

What that means for our chances at getting ahead of this disease:
If aggressive fluid therapy in the initial day or two pulls the fangs out this disease, that would be HUGE. We're still learning. In the US, we only have an insignificant number of cases, with a lot of other variables in play, so we still don't know.
And we wouldn't know that before now, because they've never done that in Africa (no capabilities or resources) and still don't, and most people died before anyone could do much of anything, including the medical staffs.

Math teachers may leave red pencil marks in Comments now.


  1. No complaints from me.

    The Y axis is a log scale. Since log(exp(x)) = x, the graph of cases and deaths is exponential.

    Of course, when it starts running out of victims, ohhh, in say a few years, it's going to flat line.

    Would it be too much to hope that lots of those in government are early victims?

  2. I join your fond wishes, but unfortunately, at current rate of spread, we run out of population NLT Christmas next year, and perhaps as early as late summer.
    And long before it starts running out of local victims, the victims start running out of their current countries, depositing new patients in spreading ripples.

    When a cluster develops in Kinshasa, Capetown, Cairo, Delhi, Mumbai, Hong Kong, Manila, etc., it's game over.
    Imagine someone infecting Muslims in Mecca, and then them returning to all points of the compass unbeknownst.

    The target population is everyone, and the disease is no respecter of persons or nationalities.

    In any crisis, you're at the mercy of the stupidest person in the group.
    That person is always Gilligan.

  3. I actually did go to MIT, and did fine on my math classes ($$$ prevented my finishing).

    Your math and explanation of it is fine.

    However, haven't we noticed in the real world that such steady exponential growth seldom sustains to the bitter end? For example, in the current 3 afflicted countries, we'd expect some islands of survival, simply those people who's habits, perhaps modified by what they've learned since the outbreak started, don't sustain a R-nought > 1. "Think of it as evolution in action" and all that.

    It's at least theoretically possible. Moving on to the rest of the world, we'd expect different results based on things like degree of acceptance of the germ theory of disease, how PC the government is, and continues to be as the infections and deaths pile up, how endemic corruption is and its nature (e.g. in the US small cash payments aren't the coin of that realm, more like Potemkin institutions), etc.

    At this point, it looks to me like old fashioned, that I've heard are in living memory quarantine measures would make all the difference in the world, as long as we're "heartless" enough to implement them (i.e. this is in alignment with your "Kids, don't try this at home" you're correctly pounding on).

    Then there's path dependence: for us, our first Ebola tourist dies, that's good from the viewpoint of discouraging others from coming over. The nurses he infects get relatively mild cases and recover, bad from the viewpoint of our taking it more seriously. Etc.

    1. That last is one of my personal fears: people see us successfully treat a handful of cases, and then we decide that "Ebola ain't so bad."

      Completely ignoring that the first few may be a statistical anomaly from a dearth of cases, we blunder into ten or twenty all at once, and everything predictably turns to crap.

      Having functional children in charge isn't helping, nor is having theoretically educated and intelligent health care workers who take multiple cross-country flights while febrile, board cruise ships, ride around town on subways while symptomatic, go out for drive-thru food, or rail against standard historical quarantines repetitively despite still being inside the known scientific window for developing active symptoms.

      If the kids would STFU and let the grown-ups take a hand, this would have never gotten here, and if it eventually did, it wouldn't be nearly the problem it is now.

      But the first hurdle when trying to take the hand grenade away from a child is trying to explain to them that they don't know what they don't know. That's usually the point at which recourse to blunt instruments best makes an appearance.

    2. "Having functional children in charge"

      Regarding this, and I am guessing this was a typo and that you mean to say dysfunctional, that being said; This is a common misconception that we find all too often regarding the actions of the state.

      Indeed the axiom about never attributing to malice anything that can be ascribed to incompetence is often true; that does not make it always true.

      Obama, to be sure, is a blathering idiot. But do not let that fool you, he does not act in a vacuum, and we all know the names of many of his publicly know advisors and officials in the WH power structure.

      Weaponizing the IRS for political reasons was not a mistake or lucky guess, this was deliberate calculated action. As was F&F, and the list goes on.

      Refusing to close the borders is nothing different. They are not acting like children, if they were they would be making mistakes across the board and failing to accomplish their larger goals. They do not fail.

      Underestimating these people is not wise.

      The flights from Ebola impacted countries cannot stop because that threatens amnesty; quarantines cannot be allowed because that threatens their ability to refuse implementing real travel restrictions.

      This can change of course in the event of domestic infections resulting in greater public pressure, but they are acting in the short term, with their eyes on the elections and the near term following that.

      They are not childishly blundering around and making foolish decisions; we would be far better off if they were! No, this is deliberate and reasoned, and that should scare the pants off of anyone who cares to pay attention.

      No I do not believe they are deliberately seeking to infect us; I cannot see that they gain anything from that. Of course there are those that believe that this is their goal, using that as a basis to justify martial law and thus to evade the elections and do whatever else it is that may want to do.

      I should hope that isn't true. Of course, these people have surprised us in their ability to go above and beyond what we thought they would try and get away with many times already.

      Fighting for amnesty is plenty bad enough, and this is what I think makes the most sense. Let's hope it isn't any worse.

    3. No, they're functioning exactly as children would be expected to; nothing dysfunctional there at all.

      As for "weaponizing" Ebola, the evidence for that is non-existent.
      Non. Existent.

      And you can't have a bunch of Gilligans running this clown parade seven days a week, and then have them suddenly morph into Professor Moriartys to hatch their Fiendish Secret Plot.

      They have evil desires, certainly. Amnesty and Open Borders being one such design. Sometimes they even work to plan -- albeit briefly. But overall, they are asstards hatching asstarded ideas which end up on the ashheap of asstardery, exactly as everyone with two working brain cells tries to tell them. And then they go try a brand new scheme, because they're asstards. With ADHD.

      I require no malign secret plan to discern that this outbreak is exactly what one should expect from the morons pulling the levers.
      My main impression is pure shock that it took them six years to get to this; I expected they'd be here much sooner.

      If you locked them in a padded room that was wall-to-wall sandbox for ten minutes with three crowbars, when you got back one of the crowbars would be broken, and one would be missing. It's just how they roll.

    4. They are not so much evil as incompetent, ideology driven stupids.
      Like Dorn and Ayers were incompetent terrorist wannabes these are wannabe evil socialist.
      The Stephen King novel Tommyknockers is apt, an alien race of idiot handymen invades earth, hilarity ensues.
      There is a video of the Taliban attempting to destroy a US truck on the road, the evil terrorist hold off pushing the mines detonator till some local tribesmen walk out of the impact area, and while urging them on do not notice the US truck has already driven by and is safe, so they decide to set off the mine and shoot an RPG and take down the Police chiefs truck, the gunner doesn't wait for the mine to blow and shoots the RPG and misses. He tells the leader he killed the truck, the leader says then why are you here and not there? He berates the mine emplacement man for building a mine that fails. The mine em placer blames the detonator, Then the leader berates the mine detonator for not detonating the mine, the detonator says I pushed the button and it did not go off, how is this my fault and then to show this is a fact, pushes the button again and the mine, useless at this point blows up harmlessly.
      This is what the Obama administration is in a nutshell, incompetent, evil dumb asses attempting to be what they just cannot manage to be, the three stooges of revolutionary socialism. They would like to kill off half of us, but being stupid cannot mange to achieve this in spite of themselves.
      Never ascribe to malice what can be nothing more than incompetence and luck.

    5. RE: "No, they're functioning exactly as children would be expected to; nothing dysfunctional there at all.

      As for "weaponizing" Ebola, the evidence for that is non-existent.
      Non. Existent.

      And you can't have a bunch of Gilligans running this clown parade seven days a week, and then have them suddenly morph into Professor Moriartys to hatch their Fiendish Secret Plot."

      Hang on a tic! I did not say that they have weaponized Ebola, I used that term related to the IRS and am only making the point that they have very effectively perverted and directed the agents of various state agencies to be used as tools - weapons - against their enemies, which of course the evidence for is well known and clear.

      I am saying they are not acting childishly; what does a child do? He acts without thinking, does foolish and dangerous things, and makes many mistakes. The goals of the statist are simple; to gain and preserve their power above all.

      Obamacare was their biggest win, and they have been working towards this end for literally decades. And they got it. This is the single largest big government program that has even been enacted against a people in all history, not to mention the largest tax that has even been levied. We have yet to see the impact of this program on our daily lives and pocketbooks and it is going to be truly devastating. This achievement was brutal and diabolical requiring congressional rule bending and trickery, bribery and no doubt behind the scenes threats and likely extortion. They have done this knowing the impact it would have on future elections of Democrats sacrificing their seats and careers. All of this was done knowing that in the end this would be an epic win for the statist.

      These things are not the act of children, these are the acts of power mad ruthless tyrants. Moriarty's, if you will, and remember this is not Obama, he is just a proxy, For every bumbling fool we see in the political spotlight there are ten calculating strategists that we do not see. They want us to assess them as children, as stupid and aimless - this is not by chance, this is also one of their deceptions.

      You see failures and mistakes, I understand this, these things do happen. It is the end goals that I am speaking of, and they do succeed. To the statist these failures do not matter. Lerner getting called out by conservatives and being forced out of her position is meaningless, the IRS continues in its work. Benghazi may look like a failure on many levels, but again, the ends justify any loss of life.

      Remember that the statist truly believes 'the needs of the many outweigh the needs of the few, or the one'. A few careers ruined, lives lost matter not at all as long as their goals, the election of "progressives", the passage or certain laws, gets done.

      And you should well know how they are passing laws that benefit the statist, and ignoring laws left and right, all while being called to the carpet on none of their actions. They are wildly successful at what they do.

      These people play the long game, and they win. These are not children, they are political chess masters who are winning big time. We would be wise to not underestimate them. That is my point.

    6. I don't think it's possible to underestimate them. However low you place the bar, they'll fail to meet it. That is not the behavior of long-term planning masters.
      Their chief assets are the short attention span of the electorate, and the pathological ability to lie non-stop without hindrance from conscience. No more, no less.
      They are not Sith Lords with vast reserves of patience and cunning wiles, they're 4-year-olds with better suits, doing what such children do, and taking advantage of a dearth of corporal punishment.
      Lord Of The Flies in real life.

    7. "They are not Sith Lords with vast reserves of patience and cunning wiles, they're 4-year-olds with better suits, doing what such children do, and taking advantage of a dearth of corporal punishment."

      With all due respect, and I do mean that, we have to look at history and learn from it. The progressive, or more accurately, the statist, has been relentlessly pushing for socialised medicine literally for more than a century.

      If this does not look like patience and long term planning, then I do not know what would.

      "1912: Former President Theodore Roosevelt champions national health insurance as he unsuccessfully tries to ride his progressive Bull Moose Party back to the White House.

      1929: Baylor Hospital in Texas originates group health insurance. Dallas teachers pay 50 cents a month to cover up to 21 days of hospital care per year.

      1935: President Franklin D. Roosevelt favors creating national health insurance amid the Great Depression but decides to push for Social Security first.

      1942: Roosevelt establishes wage and price controls during World War II. Businesses can't attract workers with higher pay so they compete through added benefits, including health insurance, which grows into a workplace perk.

      1945: President Harry Truman calls on Congress to create a national insurance program for those who pay voluntary fees. The American Medical Association denounces the idea as "socialized medicine" and it goes nowhere."

      And it goes on...

      Politics in many ways is a war without violence. The statist push for power is not a recent development, and they have been winning battles, some small and some larger throughout all of our history. Most of these things are thankfully stopped by the constitution, but this firewall has been breached little by little such that by now the constitution is effectively just a piece of paper that people talk about.

      They are well organized and have large amounts of money behind them, most of it ours. They have the media firmly in their pocket, they have public sector unions, the educational system, political organizations, corporate backing - what is misleadingly termed crony capitalism but is rightly called plain old corruption, not to mention the vast resources of the federal and state bureaucracies. This is a vast and powerful force!

      And they are winning, relentlessly.

    8. No, it's a century of failure, and a momentary opportunistic success.
      And even that "victory" has been pyrhhic in the extreme.
      They've done more to undermine that tenet in a few years of "winning" than we could have gained by beating it back for another two, or twenty, years.

      There's two teams (theoretically) on the field, and the bigger problem is whether the other one will leverage that into a more permanent rollback of the lunacy, or show themselves incapable of such.

  4. We've been trying to do it with HIV/AIDS for decades, with zero success, because it mutates rapidly and becomes resistant to our efforts.
    Hopefully Ebola doesn't mutate as fast as we come up with something to kill it, and a vaccine will work on it for more than five minutes. If not, we're boned.

    There is evidence that Ebola sees more mutation (changes in base-pairs) than Influenza. What it can't do, since it does not have segmented RNA, is reassortment.

    There is reason to be skeptical of the mutation argument as well, because any change to its functional proteins is likely to lead to non-operation or reduction in virility given how lethal it currently seems to be.

    1. As a practitioner, I'm simply hoping we can develop a vaccine.
      As to the nuts and bolts, I'm only mildly interested, but I want the average person to know how unlikely and fraught with unknowns pulling a vaccine out of our collective butts is.
      My main concerns are pragmatic: does it work? Safely? And how fast can you make us a few billion doses?

  5. Congratulations on writing what is probably the most persuasive piece on why we need flight bans that I have ever seen.

    I am a numbers guy (finance) and your math certainly looks sound to me.

    This is what has bothered me from the beginning. I have no medical background but I do understand numbers. When I see exponential growth patterns, coupled with a 70% mortality rate, it is simply terrifying.

    What President Obola is doing is absolutely depraved indifference.

  6. The plain fact is, If Ebola was weaponized, we would know it because it would be even more infectious and lethal.
    We are just lucky as of yet.