The Math Of Epidemics

Guest Post by Willis Eschenbach

Watts Up With That is about more than just the climate. It’s also about interesting things in the world. So if you’re looking for just climate, this isn’t the post for you. However, it’s an interesting peregrination through the world of the mathematics of illness.

We’ve been hearing a lot about “exponential growth” in relation to the cases, spread, and mortality of the Wuhan Virus. And to be sure, it is indeed a global health crisis, one we need to take very seriously.

The curious part is, “exponential growth” doesn’t actually describe the progress of a given disease. Exponential growth never stops—it just gets larger and larger, going up and up without end. But that’s not what happens with a disease. 

For example, here’s what has been happening with the total number of Wuhan Virus cases in South Korea.

Figure 1. South Korean total cases of Wuhan Virus since the beginning of the epidemic. SOURCE

We see the same shape of curve regarding the total number of deaths in China.

Figure 2. Total deaths in China from the Wuhan Virus since the illness took hold. SOURCE

Now, this particular “s-shaped” curve is called a “Gompertz Curve”. It is a curious curve, in that it is not symmetrical. It goes up faster than it levels off. Here’s an example.

Figure 3. A typical Gompertz Curve, which describes the evolution of the number of total cases or the number of total deaths in an epidemic.

So I thought I’d see how well those two graphs in Figures 1 & 2 could be fit by a Gompertz Curve. First, here’s the graph of the South Korean total cases, along with the best-fit Gompertz Curve:

Figure 4. Total Wuhan Virus cases, South Korea, along with the best-fit Gompertz curve for the data.

You can see why the Gompertz Curve is used to describe epidemics—it’s a very good fit to real-world epidemiological data. And because any given Gompertz Curve ends up at some maximum value that it doesn’t exceed, it also allows us to estimate the part of the curve that hasn’t happened yet. So far, there have been some 7,362 cases in South Korea. The Gompertz Curve estimates that the final total will be on the order of some 8,100 cases or so. 

Now, that’s not a hard number, of course. All kinds of things can happen to bend the curve either up or down. But it’s better than just making a blind guess.

Next, here is the same kind of look, data plus best-fit Gompertz Curve, but at the deaths in China.

Figure 5. Total Wuhan Virus deaths, China, along with the best-fit Gompertz curve for the data.

Again, with the same caveats as before, we can take an educated guess at what the total number of deaths is likely to be. By the end, the Gompertz Curve estimates about 3,500 deaths. 

Finally, let’s take a look at the deaths in South Korea. It’s still early, deaths are still happening, so this will be more uncertain.

Figure 6. Total Wuhan Virus deaths, South Korea, along with the best-fit Gompertz curve for the data.

Although the uncertainty in this one is greater, it looks at present like the final total of deaths in South Korea will be on the order of one hundred, give or take.

Conclusions

On my planet at least, this is very good news. Deaths in China look like they will be on the order of 3,500 lives lost. Cases in South Korea are near to peaking. And although it’s early to do this kind of analysis on the number of deaths in South Korea, to date there have only been 60 deaths, and the best fit Gompertz Curve peaks out at a hundred deaths.

Please be clear, though, that I’m not minimizing the danger. A virus of this nature can do immense harm if we don’t stay ahead of it. What I’m saying is that China and South Korea show that we indeed can stay ahead of it.

So let me suggest that we take all precautions, wash hands, social distancing, canceling large gatherings, testing as and when required, self-quarantine, it’s very important to slow the virus down … and that we also dial way back on the hysteria and the politicization of the issue. I assure you, assigning blame to one political party or the other and buying six cases of toilet paper won’t help end the epidemic. Although I must confess, thanks to the web I finally understand the panic buying of toilet paper. I read that it’s because when one person sneezes, a hundred people soil their underpants … so don’t be one of them. 

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Here on my hillside, I don’t need social distancing. I’m a hermit anyhow, have been for a while now. I hate going to town, and I love my forest home. The plum trees in our entry were fooled into blossoming by the warm dry February.

But then the plum trees got surprised by the March rains. The ground underneath them looked like a local snowfall, there was not a blossom left on the limbs … we’ll see if we get any plums in the fall.

I spent the day using the weedwhacker to give a haircut to the flowers and greenery growing up through the bricks on the patio I built … perhaps I should have put down geotextile fabric, but I do like the way the plants never stop shouldering their way up through the tiny crevices. I take it as a sign of hope, that life endures no matter the obstacles.

My best to all, wash your hands, stay well in these parlous times.

w.

As Usual: I ask that when you comment you quote the exact words that you are responding to. This avoids endless misunderstandings and disagreements.

Data: All of the data is from the Worldometer site.

For Math Geeks Like Me: The Gompertz Curve is defined as

y(t) = α * exp(-β * exp(-k * t))

where

t = time
alpha = upper asymptote
beta  = growth displacement
k = growth rate

On Political Correctness: Yes, I know that the wokerati have taken up the cry that calling it the “Chinese Coronavirus” or the “Wuhan Virus” is RACIST!!!, their go-to accusation to try to discount anyone they don’t like. Consider it my small blow for freedom of speech and not bowing to political pressure. Me, I’m the least PC guy imaginable, and I was brought up in a household where racism was simply not tolerated in any form, so trying to bust me for being RACIST!! just doesn’t work. I know who I am.

And if you think that the people who are claiming that this is RACIST!!! actually believe it and it’s not just sleazy political maneuvering, consider the following two tweets, six weeks apart:

Followed by …

However, if you insist on taking that route, before you bust me for calling it the Wuhan Virus because it originated in Wuhan, please let me know your racism-free politically 100% correct alternative names for each of the following illnesses, all of which are named for their place of origination:

• Rocky Mountain Spotted Fever
• Lyme Disease
• Marburg Fever
• West Nile Virus
• Valley Fever
• Middle Eastern Respiratory Syndrome (MERS)
• Ebola Virus
• German Measles
• Lassa Fever
• Ross River Fever
• La Crosse Encephalitis
• Legionnaire’s Disease
• St. Louis Encephalitis
• Heartland Virus
• Bornholm Disease
• Junin Virus
• Nipah Virus
• Rift Valley Fever
• Zika Virus
• Norovirus

I mean, we don’t want to be RACIST!! regarding the people of St. Louis or the inhabitants of the Rocky Mountains, do we? …