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About math and pandemics

Disclaimer 1

I am a mathematician, NOT A DOCTOR, and I am not a specialist epidemiologist, and I wrote my last scientific work on the topic of epidemiological modeling 20 years ago. For all questions of health, coronaviruses and the meaning of life, consult your doctor, do not be stupid people.
Disclaimer 2

Below will be a number of graphs. Before building them, I deliberately calibrated and simplified the model, detuning from the parameters of COVID-19. These graphs demonstrate the development of an epidemic of a conditional virus in a conditional population in a conditional time. Do not make predictions about the course of the current pandemic, relying on my pictures, do not be stupid people.
Well, now let's go! For obvious reasons, now interest in any pandemic has pretty much jumped up, and all kinds of mathematical and not very mathematical models roam in social networks in packs. The number of epidemiologists and specialists in systems of differential equations completely exceeded all conceivable limits. Nevertheless, in all this information riot, percolation, stochastic imitation models are strangely ignored. We will immediately correct this shortcoming. By the way, for the first time about such models (as well as many other things ) I read in a wonderful book by Gould and Tobochnik “Computer modeling in physics”.

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The percolation model is deceptively simple. To begin with, we are creating a general computer model of an individual participating in an epidemic. Something not too complicated: healthy, sick, recovered, died, and the conditions for the transition between conditions. Based on statistical data on the studied population, each specific instance is randomly endowed with certain characteristics, à la age, gender (if it is important), immunity strength, etc. Having done a handful of such instantiated instances, we place them at the tops of a certain graph imitating social connections. After this, it remains to set the conditions for the transmission of infection between individuals, infect the first few lucky ones and start the epidemic.

The big advantage of this approach is the ease of modification and freedom from numerous a priori assumptions, allowing you to quickly run the model for a variety of, even exotic scenarios. We will see examples below. The flip side of this method is that a rigorous mathematical description and analysis of the effects arising in such a system is often very complicated (to put it mildly). However, this does not hurt us to experiment.

Enough talk to the point! Take 16 million patsaks and place them evenly on the hypersphere. Connect the neighbors in the graph with some regular pattern. This is a fair simplification for the social graph, but, thank God, we are not the Ministry of Health. We will spread the infection every day in two ways. Firstly, at every step, the patsak may with some probability become infected from sick neighbors. Secondly, with some other probability, he can become infected at every step from an accidental patsaka who is not included in his environment (the effect of “public transport”). And finally, the disease itself. We take a period of asymptomatic carriage 10 steps long, after which the patsak shows symptoms and is not involved in the spread anymore. The next 10 steps, he is sick, having at each step some chance to accumulate. After that, he recovers (if he survives,of course) and acquires persistent immunity. The initial sowing will take 100 patsaks.

Under these conditions, we get the following picture:



Violet shows the percentage of uninfected patsaks, yellow - sick, green - recovered, black - you understand.

Let’s take a closer look at the patients:



Here, at every step, the percentage of sick, but still asymptomatic patsaks is shown in red, and in blue - showing symptoms.

Now back to the first schedule and take a closer look at the initial phase of the epidemic (the legend is the same):



Yes Yes. This is the very exhibitor that the media has already eaten all the way around. If on the fingers, then the origin of this very exponent is as follows: in conditions when the number of carriers is small, and public life supplies the carrier with new random uninfected patsaks, the number of new infections is directly proportional to the number of carriers. Mathematically, this is written as a differential equation



the solution of which is, you won’t believe, an exhibitor. Such a thing is found in many places in nature, in particular, one of the brightest examples, in every sense, is an uncontrolled chain reaction. Then, with an increase in the number of carriers, the hybriol for the infectious agent ends, but in the framework of the current pandemic, for example, this phase has not yet been passed. If the above equation is slightly complicated and twisted, to take into account the exhaustibility of resources for reproduction, then we get the classical Verhulst equation (aka logistic equation):



the cornerstone of population dynamics. If you have ever heard of r-strategies and K-strategies of reproduction, this is named after the coefficients from the above equation. The solutions of the logistic equation by eye are completely indistinguishable from the graphs from the first figure (which is not too surprising), so I will not give them separately. Unfortunately, the Verhulst equation for our problems is an oversimplification, so we say goodbye to him and move on.

Let's take action now, say, we will send patsaks for the weekend and close public transport and events until the end of the epidemic. In the framework of the model, this will mean that the infection now spreads only along the edges of the social graph, from relative to relative, from friend to friend. Well, yes, obviously, Chatlane did not immediately catch up, so we will take action when 1000 patsaks get sick.

On the same time scale:



And



notice, in the last experiment, the epidemic managed to go out without “restrictive measures”, and here it even grows to its peak.

Let's take a look at the whole epidemic:



As you can see, the time of the epidemic has stretched for many times.

The step-by-step schedule is especially important:



Measures taken are an order of magnitudereduced the number of sick at the same time. Why this is important will be seen below.

Perhaps the graphs are not so noticeable, but the main effect is that after taking measures, the exponential increase in the number of cases almost instantly changes to a power-law one. Roughly on the fingers, this can be explained as follows: each new patient becomes a source of infection itself and begins to infect everyone around (a kind of Huygens infectious principle). But this “around” is limited only by a few uninfected neighbors who, when infected themselves, transmit the infection further. Thus, a “wave front” forms around the outbreak, which spreads out at some constant speed in all directions (who said “eikonal” is a fine fellow), and the number of infected people is the volume of “space” marked by the wave front, which (pure geometry) proportional to some degree of distance traveled by the front.

Well, the last experiment for today. We will be generous to the healthcare system, but at the same time add realism. Let the saturation threshold be 10% of the population at one time (this is obviously much cooler than reality) and let the probability of gluing fins for a patsaka who did not get a bed increase by 10 times. Let, finally, the chatlan not take care of the holidays for the patsaks (there is no point in calculating such a scenario for the holidays, the Ministry of Health will have a triple margin of safety at the peak moment). Then we get:



The saturation point is reached at the 75th step, just above the letter i. So that you do not suddenly think that “the Ministry of Health is not needed”, here are some more schedules for the case when the medicine wasn’t enough to be oversaturated, but it wasn’t originally (welcome to the Middle Ages):



So it goes. Do not be ill!

To be continued.

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