## Mathematics for an epidemic - Propagation of SARS-CoV-2During these days of domestic confinement due to the Covid-19 pandemic, the media do not stop offering data and statistics. Those infected are counted by hundreds of thousands and will undoubtedly number in the millions. The number of deceased is predicted to be very high, although in some countries more than others. Some are more scrupulous than others, noting the number of sick and the number of dead. In some nations, appearance weighs more than science and they hide deaths to pretend they have the disease under control. Some politicians are in favour of falsifying the rates of affectation to show social organization, sanitary strength and technical efficiency abroad. Internally, panic is avoided and economic damages are mitigated, as people resign themselves to contagion to safeguard their jobs. Others prefer to be realistic and count cases as strictly as they can to find out what they are doing and thus make sound decisions in order to contain the virus. I think the last attitude is more accurate.
I have read some theories that try to predict the evolution of this disease and I have had time to reflect on the conditions of its spread. I follow
the official statistics from the beginning and it has struck me that the data of the cases accumulated up to a date are prioritized over the data of active cases per day. Daily data from many countries where the SARS-CoV-2 virus has spread is collected on a well-known statistical information website,
Not all countries, as I said before, have the same reliability, so I focused on those that diagnosed the disease early and took measures to stop it soon after detecting it. One of those countries that has shown the most effectiveness and transparency is South Korea. There it took little time to carry out massive tests to know for sure how the epidemic was spreading. Due to its effectiveness, they took the right measures that made South Korea the first country in the world to contain Covid-19, with a very small margin of affected and deceased.
For this reason, I wrote down the data for South Korea and adjusted it to a curve that, by intuition, seemed the most appropriate. I reflected on the variables on which the behavior depends and the consequent analysis was quite adjusted to reality. I kept recording the data daily ... and I didn't have to correct: the graph was still valid day after day.
So I started to do the math and I concluded that in all the regions of the world where data is reliably collected, the affected curve fits very well with the following formula:
The function s" parameter evaluates the transmissibility of the disease and "t" is the day in which the maximum number of affected "_{m}A" occurs.
_{max}
Here is the forecast for Spain:
Here is the forecast for Galicia:
Next I will explain the origin of this function.
We have assumed that the net amount / quantity of people who are likely to get sick,
Where "
We have also assumed that the daily variation of those affected is proportional to the number of susceptible people
In this way you have to:
At the beginning of the epidemic it happens that
Whose solution is a growing exponential function, typical of other growth models that occur in nature, such as uncontrolled and unlimited reproduction or nuclear chain reactions.
When the disease has already affected a large population and immunity has been generated, the
It turns out that A(t) from which it drops to zero.
_{m}
We see clearly that both considerations have to come together if we want to explain the behavior of the current epidemic. The first thing to keep in mind is that the product α = β the product is worth zero. In this case, the function must reach its maximum value (the famous affectation "peak"), since it will turn out that dA(t)/dt = 0.
On the other hand, we know that at time zero the parameter
At
Another consideration to keep in mind is that the disease must decrease after a certain time since many people will acquire immunity (and some will die), so they are people who will not be able to catch it again. So I thought that there should be a negative factor in the exponential function. I propose it in the following way:
I have chosen this form "
Consequently, as parameterization of the product
This quadratic form brings together all of the above considerations and is easy to deal with analytically. Earlier I said that:
so that:
Now we proceed to solve the starting differential equation, that is:
This equation is solved by changing members, as I did previously, the function A (t) and the term dt, and then integrating both parts.
And, as a consequence of the calculation, the initially proposed function is reached. It looks like the typical Gaussian bell ... but with the denominator of the exponent dependent on time, as if the standard deviation of a Gaussian curve increases with time.
Now the problem lies in getting the parameters right. We must ensure that the function sticks well to the daily data. Next, I review the initial differential equation to verify that the function A(t) fits correctly:
As obtained, it appears that the calculation is consistent.
We go one step further in interpreting the function:
On the other hand, it would be nice if we could find out when the "peak" of the curve will happen and how much it opens at the beginning of everything. This would be ideal for taking containment measures and quarantine terms.
During the first days this will be difficult to guess since there is a lot of variability in the data obtained. As time passes (and as long as data is collected exhaustively), predictions will improve. To guess the maximum point tm, I operate as follows, taking increments instead of differentials, and always as an approximation.
Thus, I have obtained the time in which the maximum of the function s. Both parameters will be adapted according to daily data. It would be nice to have computer software that could do the interpolation properly. I will also say that the official data suffers from systematicity: sometimes they are counted less, other times they change their criteria and, sometimes, "weekend" jumps appear that prevent an effective computer correlation. "Craft" work may be even more suitable than "automatic" work.
I've left the meaning of the "
The variable "
If we look at the dimensionality of the formula, "
Since this value is a divisor of the exponent, we understand that it is a very delicate variable to adjust. The number of people affected is very sensitive to this parameter and, therefore, it changes markedly with small variations of "
Let us remember that, when knowing the form of
With all of the above I am going to consider the special case in which
We also know that the factor
According to this, the disease peak occurs when dA(t)/dt is canceled. But this happened if β = 1. What happens if 0 < β < 1 In this case I proceed to simplify expressions and reorder the terms:
And we get new values for the latency time and the maximum time:
With what we observe two direct alterations in the formula of
It also increases the value of latency time, making the effects of the epidemic more lasting. Indirectly we know that the value of
Here is a revealing example: if
This reflection leads us to think that there are not many cases of asymptomatic patients that are excluded from the computation. I refer to cases that have not done the PCR tests (or any other test), to find out if they have contracted Covid-19 but if they have it or have had it. So I don't think the infection is as widespread as some say. Rather, I believe that this disease is spreading more in environments where it is in contact with affected cases, that is, in medical centers, nursing homes, prisons and other places of risk due to confinement with high traffic of people.
It would be necessary to calculate
Here I leave the complementary formulas that complete the study:
There will be a time when the security measures taken in the face of the epidemic (spread according to this model) are relaxed. It is not possible to maintain strict isolation from the population for a long time. So, with a certain probability, there will be a sufficient number of cases that escape from the sanitary control and the people who are not yet immunized will be exposed again. Some of them will contract the disease again and expand it exponentially in a covert manner (more so in this case of SARS-CoV-2, which has a period when the infected person has no symptoms but is contagious).
One of the starting hypotheses was that the majority of the population at the beginning of the epidemic was not immunized. For this reason, and as long as the hypothesis is valid, there will be a recurrence of the epidemic and more or less powerful peaks will occur again.
I consider that this recurrence has a certain monotony if the strict measures are resumed when the new promotions are detected. If we do not resume restrictive measures, we would have dramatic and extensive evolution over time, since this disease has a long period of convalescence. Active cases that develop moderate and severe symptoms take a long time to leave the affected list (at this point we believe that the average is around 30 days).
Below is the graph of the possible evolution of the active cases "
Considering that the first peak occurred after about 45 days, and that the security measures could be moderated 20 days after the maximum, it would be feasible for the next peak to occur 45 days after significantly lifting the containment measures. . This leads to an approximate frequency of 65 days between peaks. As some of the anti-propagation measures will continue to be in force, we will observe that the first curve does not relax as much as expected and that the second enters the count as a stabilization at higher levels than initially expected with a single peak.
If the measurements are even more relaxed we will see how more or less smooth peaks develop over the frequency time that is increasingly difficult to determine. Ultimately, we will have a sustained spread of the epidemic, either until the population is immunized or until a vaccine or other effective system is found to exterminate the pathogen.
Note that the active case function A and _{max}s magnitudes for each new case.
NOTE:
That it presents a steeper rise at the beginning of the epidemic that does not fit empirical data as well. |