key: cord-1004278-hyjnlbxl
authors: Sharov, Konstantin S.
title: Creating and applying SIR modified compartmental model for calculation of COVID-19 lockdown efficiency
date: 2020-09-24
journal: Chaos Solitons Fractals
DOI: 10.1016/j.chaos.2020.110295
sha: f4dbe8f783508bcd401f236823be11befdc79782
doc_id: 1004278
cord_uid: hyjnlbxl

We propose a Susceptible–Infected–Recovered (SIR) modified model for Coronavirus disease – 2019 (COVID-19) spread to estimate the efficacy of lockdown measures introduced during the pandemic. As input data, we used COVID-19 epidemiological information collected in fifteen European countries either in private surveys or using official statistics. Thirteen countries implemented lockdown measures, two countries (Sweden, Iceland) not. As output parameters, we studied herd immunity level and time of formation. Comparison of these parameters was used as an indicator of effectiveness / ineffectiveness of lockdown measures. In the absence of a medical vaccine, herd immunity may be regarded as a factor of population adaptation to severe acute respiratory syndrome-related coronavirus-2, the viral pathogen causing COVID-19 disease (SARS-CoV-2), and hence COVID-19 spreading stop. We demonstrated that there is no significant difference between lockdown and no-lockdown modes of COVID-19 containment, in terms of both herd immunity level and the time of achieving its maximum. The rationale for personal and business lockdowns may be found in the avoidance of healthcare system overburdening. However, lockdowns do not prevent any virus with droplet transmission (including SARS-CoV-2) from spreading. Therefore, in case of a future viral pathogen emergence, lockdown measures efficiency should not be overestimated, as it was done almost universally in the world during COVID-19 pandemic.

 SIR compartmental model of COVID-19 spread may help to evaluate efficiency of lockdown measures  Epidemiological data collected in fifteen European countries, are taken as input parameters  Herd immunity level and the time of its formation are considered as factors indicating efficiency of lockdowns  Lockdown and no-lockdown mode of containment lead to roughly similar results. Lockdowns do not stop COVID-19 spread  Rationale for lockdowns is avoidance of healthcare system overburdening  Reliance on lockdowns as the only administrative measures that may contain a pandemic, is a misstep that should be avoided in the future

Efficiency and necessity of lockdown measures implemented on a scale of the whole world, cause much controversy [1] [2] [3] . In the current paper, we will analyse whether total lockdowns are helpful in stopping spread of Coronavirus disease -2019 (COVID- 19) and future similar global diseases, by means of investigating herd immunity formation to Severe acute respiratory syndrome-related coronavirus-2, the viral causative agent of COVID-19 disease (SARS-CoV-2). In the current absence of a vaccine, herd immunity remains the only way to stabilise human population reaction to the novel viral pathogen.

It was repeatedly emphasised that creating of so-called herd (population, block, natural)

immunity is important for slowing down the rate of COVID-19 spread in population and, actually, for stopping the pandemic [4] [5] [6] [7] [8] [9] [10] . However, as of 25 June 2020, not many official investigations of SARS-CoV-2 dissemination in the whole ecosystems and populations (e.g.

random mass testing, representative sample screening, closed ecosystem studies, etc.) have been performed and reported, that would allow us to estimate the level of herd immunity formation. Henrik Jarlov collected the most comprehensive list of all programmes of COVID-19 population mass screening [11] .

The evaluation of herd immunity formation may be highly important in clinical treatment of COVID-19 patients for several reasons. First, it may help to avoid excessive loads related to COVID-19 suspicious cases, on healthcare systems by differentiating COVID/non-COVID cases. This knowledge is relevant even in June 2020, as a second COVID-19 wave is not impossible. A threat of healthcare system overheat has been already reported for a number of countries [12] [13] [14] [15] . Second, it may give a more realistic picture of COVID-19 spread in the population and, therefore, provide more dependable statistical data on the number of the Currently, a lot of mathematical models of SARS-CoV-2 spread have been elaborated, both theoretical [16, 17] and computational, many of them already described in Special Issues of Chaos, Solitons and Fractals journal "Modeling and forecasting of epidemic spreading: the 6 case of Covid-19 and beyond" [18, 19] . By dint of our model, we hope to contribute to our mutual understanding of the lockdown measures efficacy.

We propose a Susceptible-Infected-Recovered (SIR) modified compartmental model. Using epidemiological data as input parameters for the model, we calculate times and levels of herd immunity formation for different modes of containment (lockdown and no-lockdown). Finally, we will evaluate the efficiency of lockdown measures.

The statistical data on COVID-19 random testing were used: Diamond Princess cruise ship [20] [21] [22] [23] , evacuation flights data in Japan [24] , Japan [25], Republic of Korea [26] , Taiwan [27] , Austria [28] , Germany [29] , Iceland [30], Lombardy (Italy) [31] ; Portugal [32] ; Russia [33] ; and United Kingdom [34] .

COVID-19 primary statistical data sets were collected since 2 March to 21 April 2020. In

May-August 2020, additional data have been collected that changed our understanding of SARS-CoV-2 spread substantially. The Acknowledgement section contain a number of names of people who assisted in collecting the data and expressed their explicit consent for their contribution to be noted.

The primary information and its sources are summarised in Table 1 for the initial period (March-April 2020). 

We assume that the dissemination of COVID-19 may be explained by a continuous-time

Markov process model [35, 36] . This process can be generally described by a Susceptible-Infected-Recovered (SIR) compartmental model [37, 38] . Some authors used Susceptible-Exposed-Infected-Recovered (SEIR) model for their calculations [39] [40] [41] . We chose to use a common SIR compartmental model altered by introducing several significant modifications to the corresponding Markov process structure.

Regarding lockdown measures imposed on citizens, we used two models: a) common; b)

Swedish. In the former, full lockdown measures were implemented during the surge of 

Let us consider a human who may be in any of the five places in a full lockdown mode of life.

Then the probabilities of his/her stay in these five places are p 0 (home), p 1 (pharmacy), p 2 (grocery store), p 3 (street saunter with an animal, most commonly a dog), p 4 (hospital / any other medical institution, in inpatient / outpatient modes of treatment). It is obvious that

First, let us write a system of Kolmogorov linear differential equations for the movement of a human in Model 2 (in a country with a full lockdown mode). For the considerations of simplicity, we assume that a hospitalisation may be made only from his/her home, and intensities are not functions of t: Typical values of λ ij and μ ij are taken from the similar models of human relocation, as described in the works [42] [43] [44] [45] .

To estimate the number of the immune people and, therefore, the level of herd immunity created thus far, we have to observe how Kolmogorov equation systems 2-6 and 7, and the corresponding master equations behave (how the solutions p i (t) change) on varying the epidemiologic and demographic parameters. The solutions, i.e. probability functions p i (t) are complex functions of t, and they have these parameters as variables:

where r 0 is basic reproduction number, c drop contagiousness coefficient of droplet transmission, c surf contagiousness coefficient of touching surface transmission, N the total number of humans in a community, N imm the number of people in the community who already have the immunity to SARS-CoV-2.

Using coefficients of transmission is a rather new concept. In fact, these coefficients represent probabilities of a non-immune person of becoming infected through a direct contact with a person already infected or a contaminated surface: = , and, therefore, 0 ≤ c drop. + c surf. ≤ 1. In a more broad sense, 0 ≤ c drop. + c surf. ≤ 2, but the value 2 for the sum is not achievable even for the most contagious diseases known to humanity thus far (e.g. varicella for droplet transmission or Ebola haemorrhagic fever for body liquids transmission).

Basic reproduction number r 0 is estimated differently by different research groups. The difference is tremendous, from 1.6 [36] as low as 0.5-0.6 [29] . Such enormous difference in evaluating basic reproduction number may result from the fact that various research groups studied different samplings incomparable with each other in terms of closeness and rate of human contacts. We will use r 0 = 1.6 and 5.6 as reference points.

The sum c drop + c surf was initially estimated in 0.3-0.4 range (nearly 3 or 4 of 10 persons directly contacting an infected individual of surfaces with a full virus titre, will be infected by SARS-CoV-2 virus) [48] [49] [50] , where c drop may be 0.1-0.2 [50] . Values of N for different communities are taken from various demographic sources publicly available over the Internet.

ℎ( ) = . ( ) is required to be assessed in our model. 

where probability changes (first derivatives) in the left parts of the equations are calculated by means of separate probabilities p i and current intensities λ ij. A separate probability p i stands for a description that in moment t a human being will be in state S i (one of the described states). A major assumption of the model that oversimplifies it is that a said human cannot be in two states simultaneously. Plus signs are for straight direction (from initial events in the disease to further events, i.e. classical progression of the infection) (black arrows in Figure 2) and minus signs for backward events that return a human to previous states (blue arrows in Figure 2 ). The exact structure of possible connections and, therefore, currents, were taken from the book of Karin VanMeter and Robert Hubert [35] .

Besides, the condition (1) is met for this system, as ∑ =

, where m is the number of Markov process knots, in our case 11, and we consider that the number of deaths is small.

Several obvious border conditions and definitions may be further explicated, such as p 0 (0) = N(0) / N < 1; the inequality assumes that at the initial moment of time there is a portion of persons that already have the immunity to SARS-CoV-2.

p 10 (T) + p 9 (T) = N immune (T) / N, where T >> t (in fact, we may take T → ∞ for the simplicity), is the final herd immunity level; while p 10 (t) + p 9 (t) may be not equal to N immune (t) / N for any moment of time. For any t the intermediary herd immunity level may be measured as

Likewise, p 8 (T) = N d (T) / N, is total population fatality rate (TPFR). For any t we use

instead. It is population fatality rate (PFR) that shows the percentage of the deceased in the total population.

for any t, is population infection rate (PIR);

and 8 ( ) 5 ( ) + 6 ( ) + 7 ( ) + 8 ( )

is the infection fatality rate (IFR). We composed the system (9-19) just for one person. Solving the system numerically for each person for each state in (2-6) and 7 with a subsequent averaging, and observing how the solutions would change on varying epidemiological parameters, is not a best algorithm. In a community there are N people, and this number is constant (the number of deaths is small in regard to the total population size). Therefore, we have a multinomial distribution. We may assume that the time of a contact with a carrier is negligibly small in comparison with other times (e.g. the time of self-isolated life or a stay in a hospital). Then, according to Sanov theorem about large deviations for a multinomial distribution [51] further explicated by Borovkov [52] , we receive the equation: Switching in (29) to the limit at t → ∞ (we assume that the virus spread may be as prolonged as we prefer and not restricted by time), we receive After that, we proceed to the limit at N → ∞ (the human population size where SARS-CoV-2 is spreading may be as large as we prefer). Using Stirling formula for factorial approximation

we obtain (with compensating ( 1 ) with an approximate equality and simplifying it) 

It is obvious that the right part may be re-written using logarithmic functions of probabilities ψ(p i ) =p 1 lnp 1p 2 lnp 2 and exponent, and then we have: We investigate our system of SARS-CoV-2 dissemination for stabilityprecisely what we do with studying the behaviour of its Lyapunov functions. Here the stability may be defined as Sanov [51] , Borovkov [52] and Gasnikov [54] define it, viz. a stable state of virus spread system (p 1 , p 2, … , p m ) in whose little neighbourhood stationary measure (probability of frequencies ν m receiving values m, m + 1, m + 2, …) is being concentrated, independently of t.

It is simpler to consider stationary states with t → ∞, but non-stationary states may be obtained as easily. Maximising ℙ( ( ) = 1 , +1 ( ) = 2 ) on condition that p 1 + p 2 = 1,

is the same as minimising ψ(p i ). Differentiating ψ(p i ) by t, we

It is not difficult to see that we received exactly our system of equations (9) (10) (11) (12) (13) (14) (15) (16) (17) (18) (19) with

Lyapunov functions ψ(p i ), Q.E.D. The same procedure may be carried out first for considering limit at N → ∞ and only after that at t → ∞. In such a case, the completely similar result may be received using the theorem formulated by Thomas G. Kurtz [53] . Now, having assured ourselves that system (36) is in fact system (9) (10) (11) (12) (13) (14) (15) (16) (17) (18) (19) , we can analyse the behaviour of our multinomial distribution (26) without solving system (9) (10) (11) (12) (13) (14) (15) (16) (17) (18) (19) for each state in 

Microsoft Visual Studio 2010, OriginLab OriginPro 8.1 and PTC Mathcad 6.0 were used.

The paper was prepared in its initial state in April 2020 and was re-considered several times in May, June and August with serious corrections of the model proposed. The final revision was completed by 24 August 2020.

In addition to the mean population infection rate (PIR) level, its change over time is of importance. In Figure 3 , the time dependence of instantaneous PIR is shown for United c. disease peak has been already passed unnoticed by the world community, by the moment of the pandemic announcement (early March 2020).

Taking into account these three scenarios, creating herd immunity may become an even more important factor for population adaptation to COVID-19. It was so in April and continues to be in August 2020.

To estimate times and values of maximums of herd immunity formation in the countries studied, PC numerical solving the Kolmogorov equation systems described in Materials and Methods section with modelling the Lyapunov functions behaviour for the corresponding master equations (finding global attractors) has been performed for the three scenarios described above, for full-lockdown and no-lockdown modes of SARS-CoV-2 containment.

On maximising herd immunity first derivative

we receive the inflection points for herd immunity curves. The schematics for Sweden is shown in Figure 4 . For the interval 0-120 days from Day Zero, confirmed positive cases dynamics (official WHO data) may be approximated by an extreme function

and for Sweden the coefficients are: Different peak functions may be used for confirmed cases dynamics approximation [53] [54] [55] [56] .

The extreme function provided above was chosen of the list of the most widely used functions by criterion of achieving the best approximation results (the lowest χ dim. 2 and highest R adj.

where the parameters were calculated as follows:

m is the numbers of points; n -1 is the number of degrees of freedom; x i are experimental epidemiological data; C i are approximated values; and σ 2 is variance.

The adjusted coefficient of determination

where R is Pearson correlation coefficient between experimental data and the approximation and k is the number of explanatory terms (descriptors). Of course, another approximation could be used instead of the proposed one.

Using private data on Sweden PIR, taking into account its no-lockdown mode of containment, r 0 = 5.6 and maximising herd immunity first time derivative by Lyapunov function analysis in the positive quadrant, give us a double exponential function ℎ( ) (green curve in Figure 4) :

The coefficients: broad and much more representative cohort of European population commenced to be tested [66] [67] [68] [69] [70] . Using the May and June data as input parameters for the model enabled us to augment its quality and predicting force. June iteration based on June European epidemiological situation resulted in the herd immunity level prediction for August approximately four times less than the April prediction.

Different predicting force of the model has been observed for different types of ambience:

closed communities, semi-open premises and open space ( Figure 6 ). In Figure 6 , the June modelling iteration is provided that was performed at the time when the paper was returned for revision. It was based on the data collected by us during surveillance procedures in UK, Russia, the Netherlands and Sweden. As modelling error estimations show (χ dim. 2 and R adj. 

where C x i is the largest by module of {C mod i ; C real i }; C mod i is a modelled value in point i; and C real i is an observed value in this point. In Figure 7 , one can see that the model changed essentially since the beginning of the pandemic.

In March 2020, PIR = 19.19% was taken as the maximum of herd immunity that may be achieved without any lockdown and restrictive measures. In August 2020, it is already becoming clear that that value is the maximum only for closed communities (Figure 6 

The analysis of both epidemiological data and simulation results indicates that the initially anticipated herd immunity level for SARS-CoV-2 of nearly two thirds of a population or even higher, is hardly ever achievable. The real herd immunity for the current virus is three-six times as less. Therefore, COVID-19 contagiousness is not so high as it was initially thought in January-March 2020. Almost universal and worldwide implementation of lockdown measures and complete switching off the economies, as it has been done in Germany, France, Russia or UK, may be reckless. Despite some governmental and administrative assumptions that only strict quarantine might lead to diminishing SARS-CoV-2 spread, our study does not confirm it.

Neither modelling, nor PIR statistical data on the European countries collected and studied by us so far, may corroborate that full-lockdown modes are any better than the Swedish nolockdown mode in terms of the virus dissemination ( Figure 8 , Table 2 ). Figure Was any sense in lockdowns at all? We think yes. They helped to avoid healthcare system collapse in many countries. However, their near and distant harm to health of different groups of population, economies, business and world supply chains is still to be assessed in the future.

We suppose that a real hazard of COVID-19 lockdowns is associated with the common governmental belief that it is the lockdowns that saved humanity from excessive mortality connected with COVID-19. Our research proves that it is a very dangerous misbelief with farreaching consequences. It is SARS-CoV-2 relatively low contagiousness and case fatality rate that led to avoidance of millions of deaths, not lockdowns. We agree with the Editors that the health consequences of the pandemic are devastating [77] . The inflection point of herd immunity is the maximum of its first derivative. Table 2 . Estimation of herd immunity formation in Europe (95% confidence interval) during the "first wave" of SARS-CoV-2 according to our SIR modified modelling with the privately collected input data on different European countries, for different possible basic reproduction numbers r0. Negative time values stand for the time before the beginning of the infection process in a given community.

* Scenarios: a. world disease peak ("first wave" of SARS-CoV-2) has not yet been passed on 11 March 2020 (announcement of the pandemic by World Health Organisation (WHO)); b. there are two peaks a and c; a current situation is their overlapping; c. disease peak has been already passed unnoticed before the pandemic was announced by WHO.

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