key: cord-0761908-qo8aksfj
authors: Trigger, Sergey A
title: Initial stage of the COVID-19 infection process in human population
date: 2020-04-17
journal: nan
DOI: 10.1101/2020.04.13.20063701
sha: b4208007d70786df38938938fb8ee8a9fc1afaf0
doc_id: 761908
cord_uid: qo8aksfj

The simplest approximation for the first stages of the infection spread is considered. The specific feature of the COVID-19 characterized by its long latent period is taken into account. Exponential increase of numbers of infected people is determined by the half period of the maximal latent time for the COVID-19. The averaging over latent period leads to additional increase of the infected numbers.

There are various scenario and models for the description of the epidemic situation. The most models are concentrated on the spontaneous development of the infection spread to describe all stages of the process. An example is, e.g., the susceptible-infected-susceptible (SIS) -one of the basic themes in the mathematical epidemiology [1, 2] . These models describe the balance between the susceptible and infected individuals in population under the various conditions of infection transfer [3] . Kinetic approach can be applied on the basis of works [4] , [5] . However, the situation with the COVID-19 has the specific features which should be taken into account.

There are essential efforts to confront the epidemic, to localize its spread. In general we can characterize its development as a non-spontaneous epidemic process. It is especially important also to slow down the propagation velocity to enable medicine to provide treatment for patients. Under these conditions the investigation of the initial stage of infection is needed. The long latent period of the COVID-19 is the feature which is considered in this paper.

If the number of dangerous (at distance less s ≃ 2 meters) contacts per day equals N c for one infected person and k is the coefficient of infection transfer (the ratio of the number of people accepting virus to the number of dangerous contacts N c ) we find that the average number of infected persons in population after l 0 days equals the half of the maximal latent time 2l 0 days for the COVID-19.

where l 0 is the average number of days between obtaining the virus and appearance of the illness after which persons have to be isolated.

As an example, let us consider N c = 50, k = 0.1 and l 0 = 7 (based on the half of the maximal latent period equals of 14 days). As easily seen, in this example N i (50; 0.1; 7) = 93750. If N c = 25, k = 0, 1 and l 0 = 7 we find N i (25; 0.1; 7) = 854. The value of k parameter is unknown. It can be roughly estimated but practically cannot be changed. The average value l = l 0 is well known, but of course, there is some distribution on l, which is not included in the simplest model under consideration. In Fig.1 the dependence of the infected cases . CC-BY-NC-ND 4.0 International license It is made available under a is the author/funder, who has granted medRxiv a license to display the preprint in perpetuity.

The copyright holder for this preprint . https://doi.org/10.1101/2020.04.13.20063701 doi: medRxiv preprint To generalize the above consideration for the case of distribution on l, which always exists, we use the Gauss normalized distribution around l 0 (the restriction for the maximal latent period is removed due to exponential decrease of the distribution) 

Using the stationary-phase method we arrive at the approximate result for the the integral

. CC-BY-NC-ND 4.0 International license It is made available under a is the author/funder, who has granted medRxiv a license to display the preprint in perpetuity.

The copyright holder for this preprint . https://doi.org/10.1101/2020.04.13.20063701 doi: medRxiv preprint . CC-BY-NC-ND 4.0 International license It is made available under a is the author/funder, who has granted medRxiv a license to display the preprint in perpetuity.

(which was not peer-reviewed) The copyright holder for this preprint . https://doi.org/10.1101/2020.04.13.20063701 doi: medRxiv preprint

The typical for the virus COVID-19 long maximal latent period 2l leads to the fast exponential increase of the average infected people with the exponent l. The subsequent restriction of the infection spread is related with the fast and true quarantine actions, which transfer first the exponential increase to saturation, as it is in China at the moment and to further decrease.

Stochastic and deterministic models for SIS epidemics among a population partitioned into households

Stochastic Dynamics of Nanoparticle and Virus Uptake

Fokker-Planck equation for Boltzmann-type and active particles: Transfer probability approach

SIS epidemics with household structure: the self-consistent field method

Author is thankful to M.V. Fedorov and A.M. Ignatov for the useful discussions.[1] H. W. Hethcote, Qualitative analysis of communicable disease models, Math. Biosci. 28, 335 (1976)