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The section of The Time occupies a prominent part in the news of all the chains. This section, apart from informing us of the weather of the day, has helped us to understand concepts such as probability of rain, or the inability to know on a Monday if Sunday we can go to the beach.
In the case of the epidemic , another phenomenon that is very difficult to explain, the uncertainty of when it will come (or came) the peak is and how many people will be entering the hospital next week invited to use a variety of mathematical models called chains of Márkov.
In our aim of bringing to the society the mathematical instruments which help us to cope with the current pandemic, we echo the figure of a mathematician, exceptional, Andrei Andréyevich Márkov , the inventor of the strings that bear his name.
Studies and beginnings in the probability
Márkov was born in Ryazan, Russia, on June 14, 1856. His father, son of a deacon, a rural, studied in a seminary, getting a job in the Forestry Department in St. Petersburg. Andrei was the eldest of the two sons in a large family. His younger brother, Vladimir , who died prematurely of tuberculosis , had in a short space of life a great reputation as a good mathematician.
The firstborn is a child with a delicate health that led to crutches to ten years but, already in high school he excelled in Mathematics , calling the attention of their teachers.
Portrait of Andrei Markóv. – Wikicommons
it Was obvious that Márkov was going to study this matter and did so at the prestigious faculty of Physics and Mathematics of St. Petersburg University. There he received a strong influence of the brilliant mathematician Pafnuty Lvovich Chebyshev (Okátovo 1821 – St. Petersburg 1894).
Márkov graduated in 1878, and began his work of master’s degree on the theory of numbers (approximation rational), a thesis that was highly commended and is considered one of the best treaties on the issue at that time. This enabled him to continue his career as a professor of the university and get the doctorate in 1884. Two years later, he became deputy of the Academy of Sciences of St. Petersburg at the proposal of Chebyshev, although it continued to maintain its links with the university.
it Was in 1900 when Márkov began to take an interest in the theory of probability, a subject in which he obtained results very bright, including the discovery of chains which bear his name.
Chebyshev, Kolmogorov, and Márkov are the big names that used elements of the theory of the measure to convert the theory of probability in one of the areas most rigorous and respected of mathematics.
Márkov not worked considering the potential practical applications of the chains discovered. In fact, the only area where the employed was in the literature, counting vowels and consonants, perhaps because of his great love of the poetry . However, as we will show below, applications of the chain Márkov are of great practical utility.
The chains of Márkov
intuitively, a chain of Márkov in discrete time (for simplicity) is a stochastic process that evolves in discrete time or in stages, and has the property markoviana that says: “the future depends on what happens in the present, but not the last strict ”.
So, we will have some states E₁, E₂, E₃,… so that it is passed from one to another by a transition matrix on a stage. The cardinality (number of elements) of the set of states is denumerable, that is to say, it is a finite set, or with the same cardinality than the natural numbers.
The transition matrix in each stage has as elements the probabilities of passing from one state to another when the process evolves from a stage n to the next stage n+1. Therefore, it is composed of positive real numbers between 0 and 1, so that the sum of each row or column, depending on the willingness of states initial (at stage n) and final (on the stage n+1), is 1.
Application in medicine
The following is a very simple example. In an intensive care unit, the patients are classified ( triage ) according to their status: critical, serious and stable. Every day updated ratings of agreement with the historical evolution of the patients admitted to the unit until that time, so that the relative frequencies of state changes of a patient are:
In the preceding provision, the entries for rows are associated with the state of the patient on day n, and the columns refer to their state at the day n+1. Then, we may take as a transition matrix in one step:
The rows sum to 1. A graphical representation such as the following can help us to better understand the dynamics of change between states of a patient on two consecutive days:
With a chart like this, we can calculate probabilities in more than one stage. For example, the probability of passing the critical state C at the stable And the in two days. There are three possible paths, depending on the state of C, S and E of the patient after the first day:
C –> C –> E
C –> S –> E
C –> E –> E
we Just have to multiply the probabilities and add the form:
0,6 × 0,1 + 0,3 × 0,2 + 0,1 × 0,5 = 0,17
that Is to say, a patient admitted in a critical state C will evolve to the stable state, And in two days in 17% of occasions. Of course, these probabilities may change as new treatments appear, or could be defined for the different states of the chain according to the age of the patient, etc, All of these generalizations would enrich the chain of Márkov, but the general idea would remain the same.
political and social Commitment
Márkov was a person engaged politically in an era, the beginning of the TWENTIETH century, transitional, turbulent Russia. For example, when Maksim Gorky he withdrew his appointment as an academician of the Academy of Sciences for political reasons, protested vigorously.
In 1913, the mathematician refused to endorse the celebration of the third centenary of the tsar to celebrate, by his account, the second anniversary of the Law of Large Numbers .
When he triumphed in the Russian Revolution of 1917, Márkov requested that sent him to a small town in the interior, Zaraisk, to teach in the local school free of charge and thus contribute to the improvement of the poor rural society.
Grave of A. Markóv in St. Petersburg. – A. Savin/Wikimedia Commons
Suffering from serious health problems, this brilliant mathematician Russian died in St. Petersburg on July 20, 1922. He died at 66 years of age due to generalized infection produced by one of the various surgical operations of the knee to which it was subjected.
Manuel de León Rodríguez is a Research Professor of the CSIC, Royal Academy of Sciences, Institute of Mathematical Sciences (ICMAT-CSIC)
Antonio Gómez Corral is a Professor in the Department of Statistics and Operational Research, Complutense university of Madrid
Mario Castro Ponce is a Professor and Researcher at the Higher Technical Engineering School (ICAI), Universidad Pontificia Comillas
A version of this article was originally published on the blog Mathematics and its borders, of the Foundation for the Knowledge madrid+d. Also another version is published at The Conversation.