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04262020, 07:33 AM  #1 
Orbinaut

Numerical Epidemics
While we've currently being bombarded with arguments based on R0 and fear of exponential growth in the news media, I eventually got curious why we do not see exponential growth in any country. Could it really be that they're all equally effective in containment procedures such as to reduce R0 to exactly 1 and get a roughly constant daily number of new infections (and an everdecreasing growth percentage). Or is there more at play?
As a result of a discussion I had, I did what I usually do when a problem bothers me  I start doing some theory to research it. So here's a piece of GPLlicensed software where you can simulate the spread of an epidemic for a population on a square grid  and do things like limit social contacts to the local environment rather than assume they're all across the grid. See here for the download and a (growing) tutorial. It's not overly sophisticated (yet), but you can already do a nice range of instructive scenarios. Here's a few pictures: Exponential growth followed by logistic turnover  what you're used to seeing: Dramatic slowdown by restricting mobility (a person is only allowed 80 social contacts rather than 40.000) Spatial propagation of infection hotspots on the grid: (The answer to the initial question is  exponential growth on large scale can't happen because populations can't mix fast enough and the local social contacts saturate too quickly to sustain it  thus even with no or insufficient containment measures, the model predicts an everdecreasing daily grows percentage and a roughly constant daily number of new infections after an initial rapid growth phase  but you don't need to believe me, you can simply inspect the code and run it yourself ). 
04262020, 10:55 AM  #2 
Not funny anymore

That is why the R0 is now getting more important, the measurement how many people a contagious person can inject.
Also, that the number of new infections stays constant or drops is not automatically a good achievement, since this could also happen by other causes  for example lack of testing capacity. A cellular automata could maybe also model this well, by including testing rules. 
04262020, 01:09 PM  #3 
Orbinaut

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Last edited by Thorsten; 04262020 at 01:17 PM. 
04282020, 05:08 PM  #4 
Orbinaut

My major insight today  we all know that opening up completely after a lockdown gets us necessarily back to exponential growth and a second wave of infections, right?
Actually not. it's a possible outcome, but there's many scenarios in which the propagation of the infection is fundamentally changed after the lockdown and never picks up the prior speed no matter that the restrictions are no longer in place. (Currently I'm implementing temporary measures and restrictions to the software  that's not published yet, but I'll make it available soon ) 
04282020, 05:54 PM  #5 
Not funny anymore

Well  technically it is still exponential growth, but the size of the population got reduced, that still can be infected. Thus, the net reproduction rate should be much lower.
Measles for example have a base reproduction rate of 1518, but vaccinating 94% of the population reduces the net reproduction rate to less than one. Would of course get interesting, how the plot would look like, if the immunity against SARSCoV2 gets lost after 6 months, as feared. 
04292020, 05:44 AM  #6 
Orbinaut

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Which is why it doesn't make sense to fit a base reproduction number to a power law  wrong functional form. Quote:
So that's not it  the reason has to do with the spatial (de)correlation of immune people with respect to the active spreading front  you can't see that without a grid. Quote:
Anyway, as you can also convince yourself, in a steep growth scenario most people have become immune only recently. 
04292020, 07:30 AM  #7 
Not funny anymore

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Still you describe an exponential function. Even if you assume that there is just a low number of contacts that an contagious person does not share with the person who infected it, it is still an exponential function and no polynomial. The base is (relatively) constant (of course it should drop when more people are getting infected), while the exponent is a function of time. 
04292020, 08:42 AM  #8 
Enthusiast !

You guys are discovering the difference between academical discussion and reality
Scientific projections are always a simplification and a worst case scenario. And cumulative curves are misleading. Just imagine a cumulative curve for more than 100 years of flu, with reinfections. Probably to total of cases would be higher than the total population. It would be better to only consider active cases. 
05032020, 06:38 AM  #9 
Orbinaut

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It's a very simple an testable criterion  and it says 'no exponential'. You can also try to get a good fit with an exponential  you'll fail miserably. Quote:
The starting point for me was that there's no exponential seen in nature, so I was very curious what needs to be added to a model to see that nonexponential behavior. Quote:

05032020, 07:33 AM  #10 
OBSP developer

If the virus infected 25% of the population and the the cured could not get reinfected, the average distance between a sick person and a susceptible one is increased significantly, but currently only a few percent of the population got infected with the new virus and we're not even sure you can't get reinfected. If we behave as if the epidemic is over, exponential growth will return.
Well, except in Sweeden. They just let the epidemic infect everyone 
05032020, 08:45 AM  #11 
Not funny anymore

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So, you would say that the atmospheric density is a linear function of altitude, because the approximation works fine sometimes? 
05032020, 05:26 PM  #12 
Orbinaut

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05062020, 10:38 AM  #13 
Orbinaut

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Here's the derivative of the same data (the daily number of new infections) from the same wiki page: That's a constant with statistical fluctuations. When you integrate it, you get an approximately linear increase. Whereas the derivative of an exponential function would be... an exponential. Anyway  I at least know how exponential growth and its derivative looks like, if people insist in calling something that's manifestly not exponential an exponential, then it's a bad use of my time to convince them otherwise. Quote:
And  people have strong antibody response after 3 months (that's how long cases have been tracked)  and according to the people tracking these cases, there's no reason to think the response will decay quickly (i.e. within the next months). So I've yet to see any evidence of reinfection or faulty immunity  people getting well again is because they get immune  both are other words for 'antibody kills virus''. Coming back to the model (which is what this thread is about)  that says that once you've slowed the first wave with a hard intervention, any further spread will be permanently slowed  and can be stopped dead with a comparatively lowlevel intervention. It also says a relatively low number of prior immunity level (or from a first wave) is sufficient to make a big effect. So I'm going to venture the guess that if we haven't seen much exponential growth till now, it's unlikely to 'return' in any way  especially now that people are more aware of the disease and a significant fraction is more careful. For anyone who wants to give it a try  here's version 0.2 of the code for download. 
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