key: cord-0955282-305erfb0
authors: SONNINO, G.; PEETERS, P.; NARDONE, P.
title: Modelling the Spreading of the SARS-CoV-2 in Presence of the Lockdown and Quarantine Measures by a "Kinetic-Type Reactions" Approach
date: 2021-01-15
journal: nan
DOI: 10.1101/2021.01.14.21249797
sha: 0693dfeddfe847be7975c6bbc1e1b1dbcc51521f
doc_id: 955282
cord_uid: 305erfb0

We propose a realistic model for the evolution of the COVID-19 pandemic subject to the lockdown and quarantine measures, which takes into account the time-delay for recovery or death processes. The dynamic equations for the entire process are derived by adopting a "kinetic-type reactions" approach. More specifically, the lockdown and the quarantine measures are modelled by some kind of inhibitor reactions where susceptible and infected individuals can be "trapped" into inactive states. The dynamics for the recovered people is obtained by accounting people who are only traced back to hospitalised infected people. To get the evolution equation we take inspiration from the Michaelis- Menten enzyme-substrate reaction model (the so-called "MM reaction") where the "enzyme" is associated to the "available hospital beds", the "substrate" to the "infected people", and the "product" to the "recovered people", respectively. In other words, everything happens as if the hospitals beds act as a "catalyser" in the hospital recovery process. Of course, in our case the reverse "MM reactions" has no sense in our case and, consequently, the kinetic constant is equal to zero. Finally, the O.D.E.s for people tested positive to COVID-19 is simply modelled by the following kinetic scheme S+I=>2I with I=>R or I=>D, with "S", "I", "R", and "D" denoting the compartments Susceptible, Infected, Recovered, and Deceased people, respectively. The resulting "kinetic-type equations" provide the O.D.E.s, for elementary "reaction steps", describing the number of the infected people, the total number of the recovered people previously hospitalised, subject to the lockdown and the quarantine measure, and the total number of deaths. The model foresees also the second wave of Infection by Coronavirus. The tests carried out on real data for Belgium, France and Germany confirmed the correctness of our model.

the hospital recovery process [12] . In addition, the time-delay for recovery or death processes are duly taken into account. More specifically, in our model, we have the following 10 compartments: S = Number of susceptible people. This number concerns individuals not yet infected with the disease at time t, but they are susceptible to the disease of the population; I = Number of people who have been infected and are capable of spreading the disease to those in the susceptible category; I h = Number of hospitalised infected people; I Q = Number of people in quarantine. This number concerns individuals who may have the virus after being in close contact with an infected person; R = Total number of recovered people, meaning specifically individuals having survived the disease and now immune. Those in this category are not able to be infected again or to transmit the infection to others; r h = Total recovered people previously hospitalised; D = Total number of people dead people for COVID-19; d h = Total number of people previously hospitalised dead for COVID-19; L = Number of inhibitor sites mimicking lockdown measures: Q = Number of inhibitor sites mimicking quarantine measures. In addition, N , defined in Eq. (19), denotes the number of total cases. The manuscript is organised as follows. In Section 2 we derive the deterministic Ordinary Differential Equations (ODSs) governing the dynamics of the infectious, recovered, and deceased people. The lockdown and quarantine measures are modelled in Subsection 2.2. The dynamics of the hospitalised individuals (i.e., the infectious, recovered, and deceased people) can be found in Subsection 2.4. As mentioned above, the corresponding ODEs are obtained by considering the MM reaction model. The equations governing the dynamics of the full process and the related basic reproduction number are reported in Section 3 and Section 4, respectively. It is worth mentioning that our model foresees also the second wave of Infection by Coronavirus. As shown in Section 5, in absence of the restrictive measures and by neglecting the role of the Hospitals and the delay in the reactions steps, our model reduces to the classical Susceptible-Infectious-Recovered-Deceased-Model (SIRD-model) [13] . Finally, Section 6 shows the good agreement between the theoretical predictions with real data for Belgium, France and Germany. The last Section 7 presents the conclusions and perspectives of this manuscript.

As mentioned in the Introduction, the population is assigned to compartments with labels S, I, R D etc. The dynamics of these compartments is generally governed by deterministic ODEs, even though stochastic differential equations should be used to describe more realistic situations [7] . In this Section, we shall derive the deterministic ordinary differential equations obeyed by compartments. This task will be carried out by taking into account the theoretical results recently appeared in literature [8] , [14] and without neglecting the delay in the reactions steps.

If a susceptible person encounters an infected person, the susceptible person will be infected as well. So, the scheme simply reads

The lockdown measures are mainly based on the isolation of the susceptible people, (eventually with the removal of infected people by hospitalisation), but above all on the removal of susceptible people.

It is assumed the lockdown and quarantine measures are modelled by some kind of inhibitor reaction where the susceptible people and the infected can be trapped into inactive states S L and I Q , respectively. Indicating with L and Q the Inhibitor sites mimicking the lockdown and the quarantine measures respectively, we get

In the scheme (2), symbol =⇒ stands for a delayed reaction just like enzyme degradation processes for instance. Here, L max = S L + L hence, if L L M ax , an almost perfect lockdown measures would totally inhibit virus propagation by inhibiting all the susceptible people S and the infected people I. A not so perfect lockdown measures would leave a fraction of I free to spread the virus. The number of inhibitor sites maybe a fraction of the number of the infected people. Fig. 1 . shows the behaviour of the lockdown efficiency parameter adopted in our model. For simplicity, we have chosen a parameter which is constant k LM ax = 0 inside the time-interval t 1 ≤ t ≤ t 2 and vanishes outside it. The inverse Lockdown efficiency parameter is k −1 L = k LM ax − k L , which is equal to k LM ax outside the door and vanishes inside the the interval t 1 ≤ t ≤ t 2 . Finally, from Schemes (1) and (2), we get the O.D.E.s for S, L, Q, and I Q :

with the dot above the variables denoting the time derivative.

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The copyright holder for this preprint this version posted January 15, 2021. ; https://doi.org/10.1101/2021.01.14.21249797 doi: medRxiv preprint Figure 1 : Lockdown Efficiency Parameter. For simplicity, in our model the lockdown efficiency parameter k L is a door-step function. This function is constant, K LM ax = 0,within the range t 1 ≤ t ≤ t 2 and zero outside it.

At the first approximation, the O.D.E. for the total recovered people R (i.e. the total individuals having survived the disease) is trivially obtained by considering the following kinetic scheme:

That is, the rate of R t is approximatively proportional to the number of the infected people I at time t i.e. 1 .

where we have introduced the time-delay t R (the number of the recovered people at time time t is proportional to the infected people at time t − t R ). However, it is useful to clarify the following. In Eqs (5), R stands for the total number of the recovered people (i.e. the number of the recovered people previously hospitalised, plus the number of the asymptomatic people, plus the infected people who have been recovered without being previously hospitalised). The natural question is: how can we count R and compare this variable with the real data ?. The current statistics, produced by the Ministries of Health of various Countries, concern the people released from the hospitals. Apart from Luxembourg (where the entire population has been subject to the COVID-19-test), no other Countries 1 Notice that the first reaction in the scheme Eq. (5) is the dynamic equation for the total recovered people adopted in the SIRD-model [13] .

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The copyright holder for this preprint this version posted January 15, 2021. ; https://doi.org/10.1101/2021.01.14.21249797 doi: medRxiv preprint are in a condition to provide statistics regarding the total people recovered by COVID-19. Hence, it is our opinion that the equation for R, is not useful since it is practically impossible to compare R with the experimental data. We then proceed by adopting approximations and to establish the differential equation whose solution can realistically be subject to experimental verification. More specifically: Firstly, we assume that R is given by three contributions:

with r h , r A , and r I denoting the total number of the recovered people previously hospitalised, the total number of asymptomatic people, and the total number of people immune to SARS-CoV-2, respectively. Secondly, we assume that the two contributions r A and r I are negligible i.e. we set r A ≈ 0 and r I ≈ 0 2 .

Now, let us determine the dynamics for the recovered people in the hospitals. So, we account people who are only traced back to hospitalised infected people.

We propose the following model 3 : 

where C h = Total hospital s capacity (9) The dynamic equations for the processes are then:

where t r and t d are the average recovery time delay and the average death time delay, respectively, and we have taken into account Eq.

In general t r = t d = 0. Of course, the variation of r(t) over a period ∆t is: 2 We consider that the SARS-CoV-12 has just appeared for the first time. So, we do not consider the asymptomatic people who are immune to the virus without any medical treatment. 3 Our model is inspired by Michaelis-Menten's enzyme-substrate reaction. Of course, the reverse MM reaction has no sense in our case and, consequently, the kinetic constant is equal to zero.

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The copyright holder for this preprint this version posted January 15, 2021. ; https://doi.org/10.1101/2021.01.14.21249797 doi: medRxiv preprint

The number of the infected people may be modelled by the following kinetic scheme

The scheme (12) stems from the following considerations a) If a susceptible person encounters an infected person, the susceptible person will be infected ;

b) The infected people can either survive and, therefore, be recovered after an average time-delay t R , or die after an average time-delay t D ;

c) The schemes (2) and (8), respectively, have been taken into account.

The differential equation for the infected people is reads theṅ

In this model, we assume that the rate of death is proportional to the infected people, according to the scheme (12) . By also taking into account the scheme (2), we get I

and the corresponding O.D.E. for deaths readṡ is the author/funder, who has granted medRxiv a license to display the preprint in perpetuity.

The copyright holder for this preprint this version posted January 15, 2021. ; https://doi.org/10.1101/2021.01.14.21249797 doi: medRxiv preprint of the recovered people previously hospitalised and the total number of deceased peopled, when the lockdown and the quarantine measures are adopteḋ

From Eqs (16) we get

or, by taking into account that S + S L = S T ot. , R + r h = R T ot. , D + d h = D T ot. , and I + I Q + I h = I T ot. we get

The number of total cases N is defined as

We note that, in absence of the lockdown and the quarantine measures, the dynamics of the infectious class depends on the following ratio:

with N T ot. denoting the Total Population. R 0 is the basic reproduction number. This parameter provides the expected number of new infections from a single infection in a population by assuming that all subjects are susceptible [2] , [3] . The epidemic only starts if R 0 is greater than 1, otherwise the spread of the disease stops right from the start.

The Susceptible-Infectious-Recovered-Deceased-Model (SIRD-model) is one of the simplest compartmental models, and many models may be derived from 8 . 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 this version posted January 15, 2021. ; this basic form. According to the SIRD model, the dynamic equations governing the above compartments read [13] S = −µSI (21)

It is easily checked that Eqs (16) reduce to Eqs (21) by adopting some assumptions. In particular: 1) The system is not subject to the lockdown and quarantine measures;

2) The average times-delay may be neglected;

3) Hospitals do not enter in the dynamics. Under these assumptions, Eqs (16) reduce to the SIRD equations: . It should be noted that this measures does not generally provide the true new cases rate but reflect the overall trend since most of the infected will not be tested [18] . It should also be specified that real data provided by ECDC refer to the new cases per day, which we denote by ∆I new (t). By definition, ∆I new (t) corresponds to the new infected people generated from step I + S µ − → 2I solely during 1 day, and not to the compartment I. Hence, the ECDC data have to be confronted vs the theoretical predictions provided by the solutions for S(t) and S L (t) of our model, according to the relation ∆I new (t) = −∆S(t) − ∆S L (t). The values of the parameters used to perform these comparisons are shown in Table 1 . Initial µ en k 1 values have been estimated (fitted) from the measurements using the short period at the start of the pandemic using simple solution valid during that period. I(60) (from March 1, 2020). Hospital capacity is evaluated from the different Countries published capacity. However, we are aware that the interpretation may vary from one Country to another. During the first lockdown, Countries have taken various 9 . 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 this version posted January 15, 2021. ; 10 . 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 this version posted January 15, 2021. ; actions to limit Coronavirus spreading (social distancing, wearing masks, reducing high density hotspots etc.). In order to include these measures in a simple way, we assumed that the net effect is to reduce the actual infection kinetic rate µ by some constant factor. This is given in the table as µ after L 1 . Note that the transition occurs instantaneously in our model hence the sharp drop in the total infected at that time. Other parameters are tuned to account for the actual variability of ∆I new (but not its absolute value) and official number of deaths (D(t) + d(t)). The delay for recovery or death processes has been estimated from the measurements of hospitalisation recovery in a Country. For instance, Fig. 2 shows the estimation of the recovery time-delay for Belgium: it corresponds to the time-interval between the peak of the new admission and the peak of the recovered people from hospitals. Such a procedure has been adopted for estimating the recovery and death time-delays also for France and Germany. • Belgian Case. Figs (3) refer to the Belgian case. In particular, Fig (3) shows the solutions of our model for the infectious (I), total recovered (R) and total deceased (D) people. Fig. (4) illustrates the theoretical solutions for hospitalised infectious (I h ), the total recovered (r h ) and total deceased (d h ) people previously hospitalised. Figs (5) and (6) shows the comparison between the theoretical predictions for ∆I new (t) and deaths and real data for Belgium (according to the database Sciensano). Notice in Fig. 5 the prediction of the second wave of infection by SARS-CoV-2 • French Case. Figs (7) and (8) shows the comparison between the theoretical predictions for 11 . 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 this version posted January 15, 2021. ; https://doi.org/10.1101/2021.01.14.21249797 doi: medRxiv preprint ∆I new (t) and deaths and real data for Belgium (according to the database Santé Publique France). Notice in Fig. 7 the prediction of the second wave of infection by SARS-CoV-2 • German Case. Figs (9) and (10) shows the comparison between the theoretical predictions for ∆I new (t) and deaths and real data for Belgium (according to the database (Robert Koch Institut). Country data from Worldbank.org). Notice in Fig. 9 the prediction of the second wave of infection by SARS-CoV-2

We showed that our model is able to produce predictions not only on the first but also on the second or even the third waves of SARS-CoV2 infections. The theoretical predictions are in line with the official number of cases with minimal parameter fitting. We discussed the strengths and limitations of the proposed model regarding the long-term predictions and, above all, the duration of how long the lockdown and the quarantine measures should be taken in force in order to limit as much as possible the intensities of subsequent SARS-CoV-2 infection waves. This task has been carried out by taking into account the theoretical results recently appeared in literature [7] and without neglecting the delay in the reactions steps. Our model has been applied in two different situations: the spreading of the Coronavirus in a small Country (Belgium) and in big Countries (France and Germany). It is worth noting the degree of the flexibility of our model. For example, let us suppose that we need to set up a model able to distinguish old population (over 12 . 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 this version posted January 15, 2021. ; https://doi.org/10.1101/2021.01.14.21249797 doi: medRxiv preprint 65 year old) from the young one (with age not exceeding 35 years), by assuming that the older population is twice as likely to get infected by Coronavirus with respect to the younger one. In this case, it is just sufficient to replace the scheme I + S µ − → 2I with the scheme

with S Y and S o denoting the susceptible young people and the susceptible old people, respectively. Another example could be the following. Let us suppose that we need to distinguish two class of infected individuals: 1) infected people (denoted by I 1 ) able to transmit the Coronavirus to susceptible according to the (standard) scheme I 1 + S → 2I; 2) Infected people (denoted by I 2 ) having the capacity to transmit the virus, say, 7 times higher with respect to the category 1). In this case, the corresponding scheme reads:

It is then easy to write the ordinary differential equations associated to schemes (23) and (24). Let us now consider another aspect of the model. In the Subsection (2.2), we have introduced scheme (2) that models the lockdown measures. As mentioned,

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The copyright holder for this preprint this version posted January 15, 2021. ; https://doi.org/10.1101/2021.01.14.21249797 doi: medRxiv preprint such measures are imposed by national governments to all susceptible population. However, we can also take into consideration the hypothesis that these measures are not rigorously respected by the population and this for various reasons: neglect of the problem, depression due to prolonged isolation, lack of confidence in the measures adopted by the Government, desire to attend parties with friends and relatives, etc. Scheme (2) still adapts to describe these kind of situations with the trick of replacing Fig. 1 with a curve that models the emotional behaviour of susceptible people. The O.D.E.s reaḋ

Finally, we mention that in ref. [14] we have incorporated real data into a stochastic model. The goal is to obtain a comparative analysis against the deterministic one, in order to use the new theoretical results to predict the number of new cases of infected people and to propose possible changes to the measures of isolation. 

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