Abstract
A number of particles in a multiplying medium under rather general conditions is asymptotically exponential with respect to time t with the parameter λ, i.e., with the index of power
In [1] the multiplication process of the particles in a random medium σ with asymptotically exponential (via the time t)
number
where
Note that formula (1) may be the basis for numerical investigations of concrete problems.
The law of increase, corresponding to formula (1), may be naturally called “super-exponential”. More total and practically convenient definition of super-exponentiality is related to the growth of the increase coefficient
In real problems the relation
The above results show that growth rate of mean number of the arbitrary nature particles (for example, microorganisms) may be super-exponential. This behavior corresponds to the rise of the increase coefficient α of the particle numbers
In particular, the novel coronavirus pandemic in the world behaved like this according to the WHO (World Health Organization) statistics [3] in the time interval from 9.03.2020 to 21.03.2020. The corresponding number of confirmed cases (via days) is approximated as in formula (1) with the error not exceeding 2 % as
for
Date (2020 year) | 9.03 | 12.03 | 15.03 | 18.03 | 21.03 |
WHO data | 109577 | 125260 | 153517 | 191127 | 266073 |
Approximation (2) | 109577 | 125757 | 152239 | 194400 | 261845 |
Approximation (2) was obtained as follows. Let
and therefore
Solving these equations by the least-squares method (i.e., by the linear regression method) gives us coefficients in formula (2). Comparison of the results obtained by this formula with statistical data is given in Figure 1 in logarithmic scale. Comparatively with formula (2) the rise of WHO statistics weakens over time, in particular, due to the medical activities. When
It is remarkable that accordingly to expression (1) formula (2) may be obtained by averaging the random number
So, if the estimates of the quantities
Funding source: Russian Foundation for Basic Research
Award Identifier / Grant number: 18-01-00356
Award Identifier / Grant number: 18-01-00599
Funding statement: This work was supported in part by the Russian Foundation for Basic Research, project numbers 18-01-00356 and 18-01-00599.
References
[1] G. Z. Lotova and G. A. Mikhailov, The study of time dependence of particle flux with multiplication in a random medium, Russian J. Numer. Anal. Math. Modelling 35 (2020), 11–20. 10.1515/rnam-2020-0002Search in Google Scholar
[2] A. P. Prudnikov, Y. A. Brychkov and O. I. Marichev, Integrals and Series (in Russian), Nauka, Moscow, 1981. Search in Google Scholar
[3] Website of the World Health Organization. ␣https://www.who.int/emergencies/diseases/novel-coronavirus-2019/situation-reports/. Search in Google Scholar
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