Get best of the week

About Mathematics, Pandemics, and Quarantines, continued

Start here.

Disclaimer 1

I am a mathematician, not a doctor. On all issues of health, coronaviruses and the meaning of life, consult with doctors, do not be stupid people.

Disclaimer 2

For ethical reasons, the results of the model calibrated by the COVID-19 parameters will not be published. You may not agree with my decision, but you will have to live with it.
As we saw in the last part, the self-isolation mode is quite effective, in particular, it reduces the exponential increase in the number of cases to a power-law and thereby reduces the burden on healthcare to an acceptable level (“flatten the curve”, huh). Nevertheless, the self-isolation regime lasts a very long time, causes enormous damage to the economy, and a reasonable question arises: is it possible to get by with more stringent, but short-term measures?

For clarity of the experiments, I slightly modified the infectious agent, making it slightly less contagious, for more illustrative effects of interest to us. In addition, I lowered the saturation threshold of the Ministry of Health to 5% of the population (this is still very, very much). And yes, in order not to hurt sensitive souls too much, the chance of the death of an individual who “did not get a bed” now grows three times, and not ten, as before. Appreciate my humanism! The remaining parameters are the same (the most important: the incubation period, when the patch is contagious, lasts 10 days and the treatment period lasts the same).

For these parameters, the calculation of the course of the epidemic in the absence of any restrictive measures looks like this:


The legend is the same: purple shows the percentage of uninfected patsaks, yellow - sick, green - recovered, black - died. Additionally, the brown curve shows the number of deaths in the ideal case of unlimited health resources (we will strive for it).

A step-by-step layout looks like this:

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

Both of these pictures will be further used by us as a reference, to assess the effectiveness of the measures taken.

We now consider the measures that will be applied. The condition when not only random contacts are minimized (“public transport” is turned off), but also measures are taken to reduce the intensity of infection promotion by social connections (which we have not previously considered), will be conditionally called a “quarantine”, quoted, please note. The term “quarantine without quotes” is not only medical, but also legal, and rather harsh, therefore “not needed”. The parameters of infectiousness are selected in such a way that under the conditions of “quarantine” the infection slowly and sadly dies in a few hundred steps, gaining several percent of the population in the process.

So, for starters, let's try to introduce “naive quarantine”. Lock everyone and everything by 15 steps (it’s logical, the incubation period and half more just in case). Moreover, in responsible Chatlan ears at the top of the head, measures were introduced already at the 20th step of the epidemic (0.012% of the population is infected). What could go wrong? All.



and the

dashed line shows the results of the “action”

This seemingly paradoxical result is quite natural. “Quarantine” measures significantly reduce the number of pathogens circulating in the population, but alas, not enough. After the restrictions are lifted, the exponential growth begins again and the epidemic, practically unchanged, simply turns out to be a little delayed (let me remind you that our “minimum task” is to keep the blue chart under the blue line, preventing overloading the healthcare system).

It is interesting that if the only means at our disposal is such a 15-step “naive quarantine”, then it is more effective to apply it at later stages. Compare, for example, the option when we introduce it in step 70:





The situation has improved, but not radically. The swing was a ruble. Maybe we were greedy for the term of “quarantine” in vain? Let's immediately increase three times, for sure. Give "quarantine" for 45 days!



What will you do, huh ?!

Another try? Let's try the tactics of “dotted quarantine” by entering it three times, at step 70, 110 and 150 (the same 15 steps of “quarantine”):





And here we already see some clearance. The death rate for the case of the Ministry of Health of infinite power has not yet been achieved, but it is clear that by continuing to cut our boa in parts, the goals can be achieved, while really reducing the duration of the epidemic (the economic feasibility of such a push-pushing is another question).

Is it possible to qualitatively improve this result? I hasten to please: yes, and significantly. But for this we will have to introduce an additional factor.

Until now, in our constructions the governing structures of chatlan have behaved quite passively. To the best of their ability, they organized treatment, persuaded the patsaks to stay at home and, perhaps, the etsilop beat a little especially negligent at night. Yes, this happens in reality, but usually measures to combat the epidemic include finding and isolating the contacts of the patient. This activity is extremely laborious, but the effect of it is difficult to overestimate. Indeed, let us now quarantine all the neighbors of the sick according to the social graph. After that, he will not be able to infect anyone if he is already infected, and he will not be able to become infected if he was healthy. Under such conditions, we will receive the following course of the epidemic (isolation measures begin at step 20):





ABOUT! As they say, well, were you silent before? Glory to the hard work of the Etsilops, “quarantine is not needed”! The peak burden on healthcare is within acceptable limits; it needs to work in this mode for a relatively short time, and the mortality curve has fallen even below our brown target. Happiness to all, and no one will leave offended!

Unfortunately, this picture is too good to be true. The resources of special services are also exhaustible, no less than medical ones. In order not to produce additional lines on the graphs, let the saturation of special services occur with all the same 5% symptomatic (this is the same hyperoptimism as with doctors, you can estimate for yourself how much population will have to be controlled at this stage):





Hmm, everything is not so rosy. It is important to note the following. The course of the epidemic is much more sensitive to the overload of special services than to the overload of health care.
A short overload of health care will lead to some growth going to the upper tundra, which is sad. But even a slight overload of isolation and control services (pay attention to the previous pair of graphs, the blue dashed curve slightly exceeds the critical threshold, not even a half-bit) leads to a significantly more severe epidemic, increasing the total number of deaths by several times (I’m silent about the other unkind consequences )

The most important conclusion from the above: it is critically important to prevent overloading services that track and isolate contact patsaks. Let's try to enter “naive quarantine” again and see what comes of it (start at 70 steps, 30 steps long, so as not to be limited to half measures):



This is guaranteed, this is isolated! The epidemic is dead !!!

The following happened: thanks to “quarantine” measures, the sewn-on ecylopes got a respite and managed to catch the front of the epidemic, which was running ahead, and when they caught it, they gave everyone who had contact with him with nails and thus blocked the further spread of the infection. Glory to Comrade Major! Besides jokes.

481151952
The author of the picture is Vasya Lozhkin (if you suddenly did not know)

Looking ahead, I note that it was this scenario that was realized in China and several other countries that are less by ear. Judging by the actions of the Russian leadership, such an option is being worked out.

The question immediately arises: maybe we didn’t wave so well, let's do a “quarantine” of 15 steps?




Yes, but no. Do not be greedy!

I will no longer torment those who are tired of additional schedules, I will only note that the sooner you start taking measures, the shorter the “quarantine” will be. Within the framework of the described model, at the 50th step it is enough to lock up at 25 steps, at the 30th step it is enough to 20. And, say, at the 120th step even get rid of it, control over the spread of the infection will already be lost.

Summarizing: the “quarantine” measures introduced in time in combination with the active work of special services allow not only to “smooth the curve”, but also to completely stop the development of the epidemic. A striking historical example is the elimination of the outbreak of smallpox in the USSR in 1960, made possible by the unique joint operation of the KGB, the Ministry of Internal Affairs and the USSR Ministry of Health.

So it goes. Do not be ill!

More articles:


All Articles