key: cord-1020330-225en1jb
authors: Trigger, S. A.; Ignatov, A. M.
title: Strain-stream model of epidemic spread in application to COVID-19
date: 2022-03-31
journal: nan
DOI: 10.1101/2022.03.26.22272973
sha: 48aa368addef6b4181656f8e35059723decc004f
doc_id: 1020330
cord_uid: 225en1jb

The recently developed model of the epidemic spread of two virus stains in a closed population is generalized for situation typical for the couple of strains delta and omicron, when there is high probability for omicron infection enough soon after recovering from delta infection. This model can be considered as some kind of weave of SIR and SIS models for the case of competition of two strains of the same virus having different contagiousness in a population.

the emergence of a second strain of high contagiousness (for example, omicron) against the background of an already developed epidemic with the dominance of the delta strain. To describe such a situation, it suffices to take into account the initial conditions, bringing them into line with the actual level of delta disease in a certain population, to the time the strain appeared in South Africa, which was subsequently named omicron by WHO. At the same time, if we are interested in the rivalry of two strains (for specificity, below we designate delta -1 and omicron -2) in any country, region, city or locality, we naturally must use the available statistical data on the incidence, caused by the 1 mutant when cases of the disease caused by the 2 strain appear. For different countries, the corresponding data are quite fully reflected in [19] . City data are presented on the websites of the respective countries (for example, the Robert Koch Institute in Germany, Stopcoronavirus in Russia, the Johns Hopkins Institute in the USA, etc.).

In this paper, the basic equations [18] are generalized to the case of "strain nonorthogonality". These generalization we named "strain-stream" equations. These equations are the development of a general mathematical approach to the "principle of competitive exclusion" (see, e.g., Murray [20] ) and the papers [21] [22] [23] , where the first applications of this principle was formulated for virus transmission.

This generalization reflects the observable property to be infected with a high probability by the strain 2 of the COVID-19 disease for those who have already been ill and recovered from infection caused by the strain 1. This means that immunity to strain 2 is not developed (or is only partially developed) after disease caused by strain 1. Obviously, for strains that cause COVID-19 (as well as for influenza viruses), there is only limited period of immunity, however much longer than the average disease duration. In fact, the property of "strain non-orthogonality" means that infection caused by strain 2 (omicron) can appear with some probability even immediately after recovering from the disease caused by strain 1 (e.g., delta). According to our knowledge, the disease COVID-19 caused by two strains which simultaneously coexist in one sick person was not observed (in contrast with the rare cases of COVID-19 and flu). At the same time, according to the existing statistical data after infection by strain 2 infection 1 was not observed. The additional reason for this is a fast . CC-BY-NC-ND 4.0 International license It is made available under a perpetuity.

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The copyright holder for this this version posted March 31, 2022. ; https://doi.org/10.1101/2022.03.26.22272973 doi: medRxiv preprint disappearance of the less contagious strain, as we demonstrate below.

As in [18] we denote S the number of never infected people in a closed population N , I 1 and I 2 are the number of strain carriers of type 1 and 2. Then, equations of the model of "strain non-orthogonality", which takes into account that after disease caused by strain 1 one can be immediately infected by strain 2 (but not vice versa) are

The values T 1 and T 2 are the average durations of the diseases caused by strains 1 an 2.

Parameters p 1 and p 2 are the characteristics of the contagiousness for two strains, which are determined as the product of the quantity of dangerous contacts n c of the infected people per day and the average susceptibility k of the healthy person on dangerous distance [10, 11] .

The new term in Eq.

(3) describes the infection process by strain 2 of the people recovered after the disease caused by strain 1.

The coefficient 0 ≤ γ 2 < 1, hereinafter referred to as the Viral Link Attenuation Factor (VLAF), describes a certain decrease in the probability of getting 2 after being infected with 1 (partial increase in immunity) compared to the probability of getting 2 without having been ill before 1 (i.e., directly from the group u). This is due to the production of antibodies after the disease caused by the 1 strain, which perform some protective function against the 2 strain (or after vaccination). The structure of the last term in (3) is obvious if we take into account that the proportion of strains 1, 2 recovered from diseases is equal to R 1,2 = −y 1,2 /T 1,2 respectively.

In the general case, passing to symmetric equations, we can consider the situation when after the disease 2 it is possible to get sick 1 with a certain probability γ 1 , but this mathematical generalization is not considered in this article as unrealizable for omicron and delta strains.

Equations (1)-(3) describing the epidemic spread for the case of two "non-orthogonal" strains in the closed population N can be rewritten in the form using the variables I 1 (t)/N = y 1 (t), I 2 (t)/N = y 2 (t), S/N = u(t)

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Here we use the same notations as in [18] . The value u(t) ≡ 1 − z(t) is the fraction of the population that is not affected by virus at all, z(t) = N tot (t)/N corresponds to the fraction of full population N that are affected (ill and recovered N tot (t) ≡ N 1 (t)+N 2 (t)) by the strain 1 (N 1 ) or the strain 2 (N 2 ) to the time t. The values y 1 (t) = I 1 (t)/N and y 2 (t) = I 2 (t)/N are the current fractions of population actively infected (viruses carriers) by strains 1, 2 respectively in a moment t.

The two-strain propagation model developed in [18] is the limiting case of the considered more general model (1)-(3) for γ 2 = 0. We also use the x i values for the proportion of those affected (recovered and sick) by the strain i = 1, 2 to the moment of time t

III. NUMERICAL SOLUTION FOR VARIOUS IMMUNITY PARAMETER VLAF An analysis of the stability of the stationary solution, carried out in [18] , showed that the necessary condition for the development of an epidemic process at γ 2 = 0 is the condition

This condition remains valid for equations (4)-(6).

As was revealed in [18] for γ 2 = 0, using the example of specific initial conditions and parameters p i and T i , the coexistence of two viruses of different contagiousness leads over time to the replacement of the less contagious strain by a more contagious one, even if the share of the latter at the beginning of the process was significantly smaller than is the author/funder, who has granted medRxiv a license to display the preprint in (which was not certified by peer review) preprint As is easy to see that strain 1 is effectively suppressed by strain 2 since the value of maximum for the solid curve in Fig. 2 is approximately five times lower than in Fig. 1 for u(0) = 0.8. Comparison of these figures shows that duration of strain 1 circulation is also effectively suppressed ( 4 times shorter for the used parameters) due to the appearance of strain 2. Comparison of Figs. 1 and 2 shows that circulation of strain 2 is essentially shorter than in the case of it absence. It is easy to see that for arbitrary parameters the maximum for strain 1 in Fig. 2 is shifted to earlier time in comparison with Fig. 1 . This property, mentioned in [18] , is valid also for the strain-stream model under consideration.

In this paper, we are interested in the impact of a possible infection with virus 2 after recovery from an infection caused by virus 1. This situation corresponds to the epidemic process observed with the appearance of the omicron strain. An important difference from the specific examples considered in [18] is the appearance of strain 2 under conditions of a developed epidemic of strain 1, which is characterized by rather large initial values of u(0) and y 1 (0).

The results of the numerical solution of equations (5-7) for the initial conditions simulating the situation of the appearance of omicron in already developed epidemic of the delta strain are shown in Fig. 3 -Fig. 5 .

The proportions of y 1 (t) and y 2 (t) infected with strains 1 and 2 are shown in Fig. 3 for different parameters γ 2 , left and right, respectively. As in Fig. 1 and Fig. 2 , the initial is the author/funder, who has granted medRxiv a license to display the preprint in (which was not certified by peer review) preprint condition for the proportion of the population that did not encounter either of the two considered strains was chosen at the level of u(0) = 0.8, which significantly exceeds the official statistics for, e.g., Germany at the time the omicron strain appeared in the country.

By such an overestimation, we take into account a significant number of unreported cases of diseases with the delta strain at the time of the appearance of the omicron strain. The same qualitative picture is observable also in other countries. The initial proportion of those infected with strain 1 is chosen to be very high y 1 (0) = 0.01, which also corresponds to the presence of a significant number of hidden virus carriers that can actively infect others.

Note, that the purpose of this work is to identify the general patterns of the development of the epidemic in the presence of two strains, and not a calculation based on a detailed analysis of the changing situation from day to day and incomplete statistical data.

As follows from Fig. 3 , the impact of the appearance of strain 2 capable of infecting those who have been ill with strain 1 depends significantly on the value of VLAF γ 2 . The more 0 ≤ γ 2 ≤ 1, the faster the process of infection with strain 1 is suppressed, i.e. it is forced out faster than in the original model with γ 2 = 0 [18] (see also Fig. 1 ). At the same time, as γ 2 grows, the current proportion of strain 2 carriers grows, exceeding by a factor of 4.5 at the maximum proportion of strain 1 carriers under the chosen parameters. is the author/funder, who has granted medRxiv a license to display the preprint in (which was not certified by peer review) preprint The effect of a non-zero value γ 2 on the fraction u(t) of non-affected by strains at all is shown on Fig. 4 . Possibility to become infected with strain 2 soon after disease caused by strain 1 is high. There is a much faster and complete depletion of the share of nonaffected. This means that with a certain parameter γ 2 , herd immunity becomes practically unattainable and almost everyone must get sick due to strain 2. It is of interest to determine values γ 2 for which the stationary value of the proportion of the population not affected by any of the viruses is reached. It can be considered as a numerical characteristic of herd immunity. The calculation carried out up to 1000 days (not shown in Fig. 4) showed that with the selected parameters, the solid curve corresponding to the absence of strain is the author/funder, who has granted medRxiv a license to display the preprint in (which was not certified by peer review) preprint

The copyright holder for this this version posted March 31, 2022. ; https://doi.org/10.1101/2022.03.26.22272973 doi: medRxiv preprint restrictions, vaccination process etc. In fact, after recovering from strain 1, a person may not immediately become infected with strain 2, and taking into account this factor associated with time shifts is beyond the scope of this work. Also the cases of death, re-infection with the same strain long time after recovery, limited time for vaccination efficiency and other known factors can be included in a more elaborated models. Above we restricted our consideration by the case of the free running epidemic under two "non-orthogonal" strains of a same virus.

This assumption can be considered as realistic for very fast developing epidemic caused by, e.g., the omicron strain (or another highly contagious virus strain) appeared in a population affected earlier by a less contagious virus strain. However, the considered model clarifies the main specific features of competition of two "non-orthogonal" viruses in population.

The principal picture of the replacement of one strain by another has already been revealed in the recently considered mathematical model [18] , where the basic equations were proposed that describe the replacement of a less contagious virus by a more contagious one. Further development of the theory is connected with taking into account the incomplete "orthogonality" of the strains under consideration. This is manifested in the fact that with a significant mutation of the virus, leading to a different molecular structure, a different virulence, and a different clinical picture of the disease, both strains, spreading in the population, are mutually more dependent. Immunity to one of them (for example, due to a previous disease), generally speaking, does not means the presence of immunity in relation to another. So, for example, omicron can infect those who have recovered from the delta strain, but not vice versa.

Thus, the situation cannot be described in the framework of SIR and similar models, where all recovered patients have a long immunity, nor within the SIS model, where immunity disappears immediately after recovery. This important property is taken into account by transferring to the "strain-stream" equations by modification of Eq. (3) and respectively (6). The additional term includes the new VLAF parameter γ 2 ≤ 1, due to the development of partial immunity to strain 2 as a result of the disease caused by strain 1, or to the effective vaccination against strain 1, giving partial protection also against strain 2.

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is the author/funder, who has granted medRxiv a license to display the preprint in (which was not certified by peer review) preprint Erratum in [18] Line 6 below Eq. (1) instead p 1 = 0 it should be p 1 = 0.1. Line 3 above Fig. 1 instead T 1 = 0.1 it should be T 1 = 20. is the author/funder, who has granted medRxiv a license to display the preprint in (which was not certified by peer review) preprint is the author/funder, who has granted medRxiv a license to display the preprint in (which was not certified by peer review) preprint

The copyright holder for this this version posted March 31, 2022. ; https://doi.org/10.1101/2022.03.26.22272973 doi: medRxiv preprint

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