Modelling the Spreading of the SARS-CoV-2 in Presence of the Lockdown and Quarantine Measures by a "Kinetic-Type Reactions" Approach
MModelling the Spreading of the SARS-CoV-2 inPresence of the Lockdown and QuarantineMeasures by a ”Kinetic-Type Reactions”Approach
SONNINO Giorgio, PEETERS Philippe, and NARDONE PasqualeUniversit´e Libre de Bruxelles (ULB), Facult´e de SciencesBvd du Triomphe, Campus Plaine CP 231, 1050 Brussels, Belgium
Emails: [email protected], [email protected], [email protected]
Abstract
We propose a realistic model for the evolution of the COVID-19 pan-demic subject to the lockdown and quarantine measures, which takes intoaccount the time-delay for recovery or death processes. The dynamicequations for the entire process are derived by adopting a kinetic-typereactions approach. More specifically, the lockdown and the quarantinemeasures are modelled by some kind of inhibitor reactions where suscep-tible and infected individuals can be trapped into inactive states. Thedynamics for the recovered people is obtained by accounting people whoare only traced back to hospitalised infected people. To get the evolu-tion equation we take inspiration from the Michaelis- Menten’s enzyme-substrate reaction model (the so-called
MM reaction ) where the enzyme isassociated to the available hospital beds , the substrate to the infected peo-ple , and the product to the recovered people , respectively. In other words,everything happens as if the hospitals beds act as a catalyzer in the hos-pital recovery process. Of course, in our case the reverse
MM reactions has no sense in our case and, consequently, the kinetic constant is equalto zero. Finally, the O.D.E.s for people tested positive to COVID-19 issimply modelled by the following kinetic scheme S + I ⇒ I 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 re-covered people previously hospitalised, subject to the lockdown and thequarantine measure, and the total number of deaths. The model foreseesalso the second wave of Infection by Coronavirus. The tests carried outon real data for Belgium, France and Germany confirmed the correctnessof our model. Key words : Mathematical model; COVID-19; Dynamics of population; a r X i v : . [ phy s i c s . m e d - ph ] J a n neumonia. Coronavirus disease 2019 (COVID-19) is caused by a new Coronavirus (SARS-CoV-2) that has spread rapidly around the world. Most infected people haveno symptoms or suffer from mild, flu-like symptoms, but some become seriouslyill and can die. In recent weeks coronavirus has had too many opportunitiesto spread again. After successfully tamping down the first surge of infectionand death, Europe is now in the middle of a second coronavirus wave as itmoves into winter [1], [2], [3], [4], [5], [6]. Even though several vaccines forCOVID-19 are actually been produced other ways of slowing its spread have tocontinue to be explored. One way of controlling the disease are the lockdownand the quarantine measures. The lockdown measures are emergency measuresor conditions imposed by governmental authorities, as during the outbreak ofan epidemic disease, that intervene in situations where the risk of transmittingthe virus is greatest. Under these measures, people are required to stay in theirhomes and to limit travel movements and opportunities for individuals to comeinto contact with each other such as dining out or attending large gatherings.The lockdown measures are more effective when combined with other measuressuch as the quarantine. Quarantine means separating healthy people from otherhealthy people, who may have the virus after being in close contact with aninfected person, or because they have returned from an area with high infectionrates. Similar recommendations include isolation (like quarantine, but for peoplewho tested positive for COVID-19) and physical distancing (people withoutsymptoms keep a distance from each other). Several governments have thendecided that stricter lockdown and quarantine measures are needed to bringdown the number of infections. In this work we shall propose interventionswhich are as targeted as possible. Unfortunately, the greater the number ofinfections, the more sweeping the measures have to be. Tightening the measureswill impact on our society and the economy but this step is needed for gettingthe coronavirus under control.The aim of this work is to model the dynamics of the infectious, recovered,and deceased people when population is subject to lockdown and quarantinemeasures imposed by governments. We shall see that the combined effect ofthe restrictions measures with the action of the Hospitals and Health Institutesis able to contain and even dampen the spread of the SARS-CoV-2 epidemic.The dynamics of the entire process will be obtained by taking into accountthe theoretical results recently appeared in literature [7] and [8] and by adopt-ing a kinetic-type reactions approach. In this framework, the dynamics of theHealth Institutes is obtained by taking inspiration from the Michaelis- Menten’senzyme-substrate reaction model (the so-called
MM reaction [9], [10], and [11])where the enzyme is associated to the available hospital beds , the substrate tothe infected people , and the product to the recovered people , respectively. Inother words, everything happens as if the hospitals beds act as a catalyzer in2he hospital recovery process [12]. In addition, the time-delay for recovery ordeath processes are duly taken into account. More specifically, in our model, wehave the following 10 compartments: S = Number of susceptible people. This number concerns individuals not yetinfected with the disease at time t , but they are susceptible to the disease of thepopulation; I = Number of people who have been infected and are capable of spreading thedisease to those in the susceptible category; I h = Number of hospitalised infected people; I Q = Number of people in quarantine. This number concerns individuals whomay have the virus after being in close contact with an infected person; R = Total number of recovered people, meaning specifically individuals havingsurvived the disease and now immune. Those in this category are not able tobe 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 determin-istic Ordinary Differential Equations (ODSs) governing the dynamics of theinfectious, recovered, and deceased people. The lockdown and quarantine mea-sures are modelled in Subsection 2.2. The dynamics of the hospitalised indi-viduals (i.e., the infectious, recovered, and deceased people) can be found inSubsection 2.4. As mentioned above, the corresponding ODEs are obtainedby considering the MM reaction model . The equations governing the dynamicsof the full process and the related basic reproduction number are reported inSection 3 and Section 4, respectively. It is worth mentioning that our modelforesees also the second wave of Infection by Coronavirus. As shown in Sec-tion 5, in absence of the restrictive measures and by neglecting the role of theHospitals and the delay in the reactions steps, our model reduces to the clas-sical
Susceptible-Infectious-Recovered-Deceased-Model (SIRD-model) [13]. Fi-nally, Section 6 shows the good agreement between the theoretical predictionswith real data for Belgium, France and Germany. The last Section 7 presentsthe conclusions and perspectives of this manuscript.
As mentioned in the Introduction, the population is assigned to compartmentswith labels S , I , R D etc. The dynamics of these compartments is generallygoverned by deterministic ODEs, even though stochastic differential equationsshould be used to describe more realistic situations [7]. In this Section, we3hall derive the deterministic ordinary differential equations obeyed by com-partments. This task will be carried out by taking into account the theoreticalresults recently appeared in literature [8], [14] and without neglecting the delayin the reactions steps.
If a susceptible person encounters an infected person, the susceptible person willbe infected as well. So, the scheme simply reads S + I µ −→ I (1) The lockdown measures are mainly based on the isolation of the susceptiblepeople, (eventually with the removal of infected people by hospitalisation), butabove all on the removal of susceptible people.
Subsection 2.1. Modelling the Lockdown and Quarantine Measureswith Chemical Interpretation
It is assumed the lockdown and quarantine measures are modelled by somekind of inhibitor reaction where the susceptible people and the infected canbe trapped into inactive states S L and I Q , respectively. Indicating with L and Q the Inhibitor sites mimicking the lockdown and the quarantine measuresrespectively, we get S + L k L −−−−−−− (cid:42)(cid:41) −−−−−−− k LMax − k L S L (2) I k Q −−→ I Q k QR , t QR ====== ⇒ R In the scheme (2), symbol = ⇒ stands for a delayed reaction just like enzymedegradation processes for instance. Here, L max = S L + L hence, if L (cid:39) L Max ,an almost perfect lockdown measures would totally inhibit virus propagation byinhibiting all the susceptible people S and the infected people I . A not so perfectlockdown measures would leave a fraction of I free to spread the virus. Thenumber 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 ourmodel. For simplicity, we have chosen a parameter which is constant k LMax (cid:54) =0 inside the time-interval t ≤ t ≤ t and vanishes outside it. The inverseLockdown efficiency parameter is k − L = k LMax − k L , which is equal to k LMax outside the door and vanishes inside the the interval t ≤ t ≤ t . Finally, fromSchemes (1) and (2), we get the O.D.E.s for S , L , Q , and I Q :˙ S = − µSI − k L S ( L Max − S L ) + (1 − k L )( L Max − L ) (3)˙ S L = k L SL − k − L S L ˙ I Q = k Q I − χI Q ( t − t R ) (4)with the dot above the variables denoting the time derivative .4 L t k LMax t t t Figure 1:
Lockdown Efficiency Parameter.
For simplicity, in our modelthe lockdown efficiency parameter k L is a door-step function. This function isconstant, K LMax (cid:54) = 0 ,within the range t ≤ t ≤ t and zero outside it. At the first approximation, the O.D.E. for the total recovered people R (i.e. thetotal individuals having survived the disease) is trivially obtained by consideringthe following kinetic scheme : I χ, t R ==== ⇒ R (5) I Q k QR , t QR ====== ⇒ R That is, the rate of R t is approximatively proportional to the number of theinfected people I at time t i.e. .˙ R = χI ( t − t R ) + χR ( t − t R ) (6)where we have introduced the time-delay t R (the number of the recovered peopleat time time t is proportional to the infected people at time t − t R ). However, itis useful to clarify the following. In Eqs (5), R stands for the total number of therecovered people (i.e. the number of the recovered people previously hospitalised,plus the number of the asymptomatic people, plus the infected people who havebeen recovered without being previously hospitalised). The natural question is: how can we count R and compare this variable with the real data ? . The currentstatistics, produced by the Ministries of Health of various Countries, concernthe people released from the hospitals. Apart from Luxembourg (where theentire population has been subject to the COVID-19-test), no other Countries Notice that the first reaction in the scheme Eq. (5) is the dynamic equation for the totalrecovered people adopted in the SIRD-model [13]. R , is not useful sinceit is practically impossible to compare R with the experimental data. We thenproceed by adopting approximations and to establish the differential equationwhose solution can realistically be subject to experimental verification. Morespecifically:Firstly, we assume that R is given by three contributions: R = r h + r A + r I (7)with r h , r A , and r I denoting the total number of the recovered people previouslyhospitalised , the total number of asymptomatic people , and the total number ofpeople immune to SARS-CoV-2 , respectively.Secondly, we assume that the two contributions r A and r I are negligible i.e. weset r A ≈ r I ≈ . 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 : I + b h k −→ I h k r , t r ==== ⇒ r h + b h (8) I h k d , t d ==== ⇒ d h + b h with b h denoting the number of available hospital beds , I the number of infectedpeople , I h the number of infected people blocking an hospital bed , r h the numberof recovered people previously hospitalised , and d h the number of people deceasedin the hospital . Of course, I h + b h = C h = const. where C h = Total hospital (cid:48) s capacity (9)The dynamic equations for the processes are then:˙ I h = k I ( C h − I h ) − k r I h ( t − t r ) − k d I h ( t − t d ) (10)˙ r h = k r I h ( t − t r ) ˙ d h = k d I h ( t − t d ) where t r and t d are the average recovery time delay and the average death timedelay , respectively, and we have taken into account Eq. (9) i.e., b h = C h − I h .In general t r (cid:54) = t d (cid:54) = 0. Of course, the variation of r ( t ) over a period ∆ t is:∆ r ht = r ht − r h ( t − ∆ t ) (11) We consider that the SARS-CoV-12 has just appeared for the first time. So, we donot consider the asymptomatic people who are immune to the virus without any medicaltreatment. Our model is inspired by Michaelis-Menten’s enzyme-substrate reaction. Of course, thereverse
MM reaction has no sense in our case and, consequently, the kinetic constant is equalto zero. .5 O.D.E. for People Tested Positive to COVID-19 The number of the infected people may be modelled by the following kineticscheme S + I µ −→ I (12) I χ, t R ==== ⇒ RI α, t D ==== ⇒ DI + b k −→ I h I k Q −−→ I Q The scheme (12) stems from the following considerations a) If a susceptible person encounters an infected person, the susceptible personwill be infected ; b) The infected people can either survive and, therefore, be recovered after anaverage 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 then˙ I = µSI − k Q IQ − k I ( C h − I h ) − χI ( t − t R ) − αI ( t − t D ) (13) In this model, we assume that the rate of death is proportional to the in-fected people, according to the scheme (12). By also taking into account thescheme (2), we get I α, t D ==== ⇒ D (14)and the corresponding O.D.E. for deaths reads˙ D = αI ( t − t D ) (15) By collecting the above O.D.E.s, we get the full system of differential equationsgoverning the dynamics of the number of the infected people, the total number7f the recovered people previously hospitalised and the total number of deceasedpeopled, when the lockdown and the quarantine measures are adopted˙ S = − µSI − k L S ( L Max − S L ) + k − L S L with k − L = k Max − k L (16)˙ S L = − k L S ( L Max − S L ) + k − L S L ˙ I = µSI − k Q I − k I ( C h − I h ) − χI ( t − t R ) − αI ( t − t D ) ˙ I h = k I ( C h − I h ) − k r I h ( t − t r ) − k d I h ( t − t d ) ˙ I Q = k Q I t − χI Q ( t − t R ) ˙ r h = k r I h ( t − t r ) ˙ R = χI ( t − t R ) + χI Q ( t − t R ) ˙ d h = k d I h ( t − t d ) ˙ D = αI ( t − t D ) From Eqs (16) we get S + S L + I + I Q + I h + R + r h + D + d h = const. (17)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 S T ot. + I T ot. + R T ot. + D T ot. = const. (18)The number of total cases N is defined as N = I T ot. + r h + D T ot. (19)
We note that, in absence of the lockdown and the quarantine measures, thedynamics of the infectious class depends on the following ratio: R = µχ + α SN T ot. (20)with N T ot. denoting the
Total Population . R is the basic reproduction number .This parameter provides the expected number of new infections from a singleinfection in a population by assuming that all subjects are susceptible [2], [3].The epidemic only starts if R is greater than 1, otherwise the spread of thedisease stops right from the start. The
Susceptible-Infectious-Recovered-Deceased-Model (SIRD-model) is one ofthe simplest compartmental models, and many models may be derived from8his basic form. According to the SIRD model, the dynamic equations govern-ing the above compartments read [13]˙ S = − µSI (21)˙ I = µSI − χI − αI ˙ R = χI ˙ D = αI It is easily checked that Eqs (16) reduce to Eqs (21) by adopting some assump-tions. In particular: The system is not subject to the lockdown and quarantine measures; The average times-delay may be neglected; Hospitals do not enter in the dynamics.Under these assumptions, Eqs (16) reduce to the SIRD equations:˙ S (cid:39) − µSI (22)˙ I (cid:39) µSI − χI − αI ˙ R = χI ˙ D = αI Let us now apply our model to the case of a small Country, Belgium, and to othertwo big Countries, France and Germany. Real data are provided by the variousNational Health agencies (Belgium -
Sciensano [15]; France -
Sant´e PubliqueFrance [16]; Germany -
Robert Koch Institut. Country data from Worldbank.org [17]) and compiled, among others, by European Centre for Disease Preventionand Control (ECDC). It should be noted that this measures does not generallyprovide the true new cases rate but reflect the overall trend since most of theinfected will not be tested [18]. It should also be specified that real data pro-vided 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 fromstep I + S µ −→ I solely during 1 day, and not to the compartment I . Hence,the ECDC data have to be confronted vs the theoretical predictions providedby 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 performthese comparisons are shown in Table 1. Initial µ en k values have been esti-mated (fitted) from the measurements using the short period at the start of thepandemic using simple solution valid during that period. I (60) (from March1, 2020). Hospital capacity is evaluated from the different Countries publishedcapacity. However, we are aware that the interpretation may vary from oneCountry to another. During the first lockdown, Countries have taken various9able 1: List of the ParametersParameters Belgium France GermanyDensity [ km − ] 377 119 240Surface [ km ] 30530 547557 348560 µ [ d − km ] 0.00072 0.002 0.00093 µ after L χ [ d − ] 0.062 0.062 0.0608 α [ d − ] 0.05 χ χ χk L [ d − ] 0.07 0.06 0.06 k Q [ d − ] 0.02 0.01 0.01 L m [ km − ] 377.0 119 240 k [ d − km ] 0.01 0.01 0.01 k d + k r [ d − ] 0.2 0.2 0.21 k d k r t r [ d ] 7 7 7 t d [ d ] 7 7 7 t R [ d ] 8 8 8 t D [ d ] 8 8 8 C [ km − ] 0.0655 0.0091 0.023 I (60) [ km − ] 0.0023 0.0018 0.0014Start L [ d ] 77 71 76End L [ d ] 124 131 125Start L [ d ] 306 303 30610ctions to limit Coronavirus spreading (social distancing, wearing masks, reduc-ing high density hotspots etc.). In order to include these measures in a simpleway, we assumed that the net effect is to reduce the actual infection kineticrate µ by some constant factor. This is given in the table as µ after L . Notethat the transition occurs instantaneously in our model hence the sharp dropin the total infected at that time. Other parameters are tuned to account forthe actual variability of ∆ I new (but not its absolute value) and official numberof deaths ( D ( t ) + d ( t )). The delay for recovery or death processes has beenestimated from the measurements of hospitalisation recovery in a Country. Forinstance, 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 andthe peak of the recovered people from hospitals. Such a procedure has beenadopted for estimating the recovery and death time-delays also for France andGermany. New admissionsRecovered from hospitalsMaximum8 days P a t i en t den s i t y Figure 2:
Estimation of the time-delay. The time-delays have been estimated byconsidering the time-interval between the peak of the new admission and the peakof the recovered people from hospitals. This figure corresponds to the Belgiancase. • Belgian Case .Figs (3) refer to the Belgian case. In particular, Fig (3) shows the solutions ofour model for the infectious ( I ), total recovered ( R ) and total deceased ( D ) peo-ple. 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 Sci-ensano ). Notice in Fig. 5 the prediction of the second wave of infection bySARS-CoV-2 • French Case .Figs (7) and (8) shows the comparison between the theoretical predictions for11
50 100 150
Days P op . D en s i t y InfectedRecoveredDead
Figure 3:
Theoretical solutions for in-fectious ( I ), cumulative number of re-covered people ( R ) and deaths ( D ) forBelgium. Days -0.00500.0050.010.0150.020.0250.030.0350.040.045 P op . D en s i t y Infected hospitalizedRecovered from hospitalDead in hospital
Figure 4:
Theoretical solutions forhospitalised infectious ( I h ), total re-covered ( r h ) and total deceased ( d h )people, previously hospitalised, forBelgium. ∆ I new ( t ) and deaths and real data for Belgium (according to the database Sant´ePublique France ). Notice in Fig. 7 the prediction of the second wave of infectionby SARS-CoV-2 • German Case .Figs (9) and (10) shows the comparison between the theoretical predictionsfor ∆ 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. 9the 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 firstbut also on the second or even the third waves of SARS-CoV2 infections. Thetheoretical predictions are in line with the official number of cases with minimalparameter fitting. We discussed the strengths and limitations of the proposedmodel regarding the long-term predictions and, above all, the duration of howlong the lockdown and the quarantine measures should be taken in force in orderto limit as much as possible the intensities of subsequent SARS-CoV-2 infectionwaves. This task has been carried out by taking into account the theoreticalresults recently appeared in literature [7] and without neglecting the delay inthe reactions steps. Our model has been applied in two different situations: thespreading 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 ussuppose that we need to set up a model able to distinguish old population (over12
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Figure 5:
Comparison between the the-oretical prediction for ∆ I New with realdata provided by the data base Scien-sano, for Belgium.
Days since 2020-3-1 T o t a l dea t h s den s i t y ModelECDC data
Figure 6:
Comparison between thetheoretical solution of our model forDeaths with real data provided by thedatabase Sciensano, for Belgium.
65 year old) from the young one (with age not exceeding 35 years), by assumingthat the older population is twice as likely to get infected by Coronavirus withrespect to the younger one. In this case, it is just sufficient to replace the scheme I + S µ −→ I with the scheme I + S Y µ y −−→ I (23) I + 2 S O µ o −→ IS = S Y + S O with S Y and S o denoting the susceptible young people and the susceptible oldpeople , respectively. Another example could be the following. Let us supposethat we need to distinguish two class of infected individuals: infected people (denoted by I ) able to transmit the Coronavirus to suscep-tible according to the (standard) scheme I + S → I ; Infected people (denoted by I ) having the capacity to transmit the virus, say,7 times higher with respect to the category . In this case, the correspondingscheme reads: I + S µ −→ I (24) I + 7 S µ −→ II = I + I 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), wehave introduced scheme (2) that models the lockdown measures. As mentioned,13
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Figure 7:
Comparison between the the-oretical prediction for ∆ I New with realdata provided by the data base Sant´ePublique France, for France.
Days since 2020-3-1 T o t a l dea t h s den s i t y ModelECDC data
Figure 8:
Comparison between thetheoretical solution of our model forDeaths with real data provided by thedatabase Sant´e Publique France, forFrance. such measures are imposed by national governments to all susceptible popula-tion. However, we can also take into consideration the hypothesis that thesemeasures are not rigorously respected by the population and this for variousreasons: neglect of the problem, depression due to prolonged isolation, lack ofconfidence in the measures adopted by the Government, desire to attend partieswith friends and relatives, etc. Scheme (2) still adapts to describe these kindof situations with the trick of replacing Fig. 1 with a curve that models the emotional behaviour of susceptible people. The O.D.E.s read˙ S = − µSI − k E S ( E Max − S E ) + (1 − k E )( E Max − E ) (25)˙ S E = k E SE − k − E S E where E stands for Emotional .Finally, we mention that in ref. [14] we have incorporated real data into astochastic model. The goal is to obtain a comparative analysis against thedeterministic one, in order to use the new theoretical results to predict thenumber of new cases of infected people and to propose possible changes to themeasures of isolation.
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Figure 9:
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