The Optimal Lockdown Strategy Against Virus Propagation and Economic Loss
The present paper is my personal contribution to the volume Math in the Time of Corona whose title voluntarily reminds the novel El amor en los tiempos del cólera (Gabriel García Márquez, El amor en los tiempos del cólera, 1985, Love in the time of choler
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1 Introduction The present paper is my personal contribution to the volume Math in the Time of Corona whose title voluntarily reminds the novel El amor en los tiempos del cólera [12, Love in the time of cholera] by Gabriel García Márquez (Colombia, 1927– Mexico, 2014), worldwide known because of his Nobel Prize and of his originality. His novel [11] starts as follows. El día que lo iban a matar, Santiago Nasar se levantó a las 5:30 de la mañana para esperar el buque en que llegaba el obispo. Another of his novels [9] ends as follows. La mujer se desesperó. “Y mientras tanto qué comemos?”, preguntó, y agarró al coronel por el cuello de franela. Lo sacudió con energía. “Dime, qué comemos”. El coronel necesitó setenta y cinco años - los setenta y cinco años de su vida, minuto a minuto - para llegar a ese instante. Se sintió puro, explícito, invencible, en el momento de responder: “mierda”. In honor of Márquez, this paper could have started and ended with the very same sentences: El día que lo iban a matar... mierda. But, at the very last minute, I decided to organize it as I did below, by recalling, from time to time, the novels of Márquez.. This paper gives some hope and some suggestions on how to behave in case of future pandemic. It also shows that mathematics can explain everything, including virus propagation and lockdown strategies. This fact is well-known since Leonardo da Vinci (Italy, 1452–France, 1519) who said Nessuna certezza delle scienze è dove non si pò applicare una delle scienze matematiche, ovver che non sono unite con esse matematiche. Incidentally, we observe that Leonardo failed to receive the Nobel Prize only because Alfred Bernhard Nobel (Sweden, 1833–Italy, 1896) was born almost four centuries later. Have a nice time while reading this paper!
F. Gazzola (*) Dipartimento di Matematica, Politecnico di Milano, Milano, Italy e-mail: fi[email protected] © The Author(s), under exclusive license to Springer Nature Switzerland AG 2020 A. Wonders (ed.) Math in the Time of Corona, Mathematics Online First Collections, https://doi.org/10.1007/16618_2020_20
F. Gazzola
2 The Logistic Equation In 1798, in his study of the “future improvements of Society”, the English economist Thomas Robert Malthus [14] suggested a model for the dynamics of populations. His monograph was published under the pseudonym of J. Johnson, only much later discovered to be Malthus himself. The Malthus model assumes the existence of infinite quantities of both space and food, and that the variation of a population of individuals (e.g. viruses) merely depends on the natality rate n > 0 and on the mortality rate m > 0, taken as fixed constants. The relevant parameter is their difference ρ ¼ n m. If the population is initially (at time t ¼ 0) made by y0 > 0 individuals and if we denote by y(t) the population at time t, the model states that, in average, in any interval of time Δt there is a quantity of individuals born which is proportional to the population and to the interval of time, that is, equal to ny(t) Δt. Similarly, it
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