Kinetic models for epidemic dynamics with social heterogeneity
SSocial contacts and the spread of infectious diseases
G. Dimarco ∗ , B. Perthame † , G. Toscani ‡ , and M. Zanella § Mathematics and Computer Science Department,University of Ferrara, Italy. Sorbonne Universit´e, CNRS, Universit´e de Paris, Inria,Laboratoire Jacques-Louis Lions, 75005 Paris, France. Mathematics Department, University of Pavia, Italy.
Abstract
Motivated by the COVID-19 pandemic, we introduce a mathematical description of the impactof sociality in the spread of infectious diseases by integrating an epidemiological dynamic with akinetic modeling of population-based contacts. The kinetic description leads to study the evolutionover time of Boltzmann type equations describing the number densities of social contacts of suscept-ible, infected and recovered individuals, whose proportions are driven by a classical compartmentalmodel in epidemiology. Explicit calculations show that the spread of the disease is closely relatedto the mean number of contacts, thus justifying the lockdown strategies assumed by governmentsto prevent them. Furthermore, the kinetic model allows to clarify how a selective control can be as-sumed to achieve a minimal lockdown strategy by only reducing individuals undergoing a very largenumber of daily contacts. This, in turns, could permit to maintain at best the economic activitieswhich would seriously suffer from a total closure policy. Numerical simulations confirm the abilityof the model to describe different phenomena characteristic of the rapid spread of an epidemic.A last part is dedicated to fit numerical solutions of the proposed model with experimental datacoming from different European countries.
Contents ∗ [email protected] † [email protected] ‡ [email protected] § [email protected] a r X i v : . [ phy s i c s . s o c - ph ] S e p Introduction
The recent spreading of the COVID-19 epidemic led in many countries to heavy lockdown measuresassumed by the governments with the aim to control and limit its effects on population. It is clearthat these severe social distancing measures have the main consequence to produce dramatic dam-ages in the real economy and society, even if their precise quantification is an extremely difficultproblem which also requires the knowledge/estimation of a large number of variables and relation-ships among them [5]. In this direction, a recent study of the economic consequences of this newepidemic and the related lockdown measures on western economies has been done by resorting ona coupling of a kinetic model for the wealth with a SIR model for the infectious disease [15]. Thismodel, while relying on few evident universal features, can be easily implemented to furnish an-swers on possible scenarios as consequence of epidemic and permitted to show that indeed lockdownmeasures may trigger an economic crisis leading to a drastic reduction of the middle class.A very interesting aspect of mathematical modeling of infectious diseases is that, unlike otherclassical phenomenological approaches, it allows a direct comparison with real data. This con-sequently permits evaluation of control and prevention strategies by comparing their cost witheffectiveness and support to public health decisions [17, 43]. Most of the models present in literat-ure make assumptions on transmission parameters [6,13] which are considered the only responsibleof the spread of the infection. However, special attention was recently paid by the scientific com-munity to the role and the estimate of the distribution of contacts between individuals as also arelevant cause of the potential pathogen transmission (cf. [3, 19, 37] and the references therein).In this direction, the results reported in [3] can be of great help when designing partial lock-down strategies. In fact, and optimal limitation of the pathogen transmission of the epidemic,which limitates the number of daily contacts among people taking into account contact numberheterogeneity, permits probably to reduce the severe impact on economy and society that equaldistributed closure policies generate. In particular, the detailed analysis performed in [3] put intoevidence that the number of social contacts in the population is in general well-fitted by a Gammadistribution, even if this distribution is not uniform with respect to age, sex and wealth. Gammadistributions belong to the wide class of generalized Gamma distributions [34,45], which have beenrecently connected to the statistical study of social phenomena [32, 42], and fruitfully describedas steady states of kinetic models in [14, 46], aiming to describe the formation of these profiles inconsequence of repeated elementary interactions.Starting from the above consideration and inspired by the recent development of kinetic modelsfor describing human behaviors [14, 46], in this paper we develop a mathematical framework toconnect the distribution of social contacts with the spreading of a disease in a multi agent system.This result is obtained by integrating an epidemiological dynamics given by a classical compart-mental model [6, 13, 29] with an approach based on kinetic equations determining the formation ofsocial contacts. In this paper, from the epidemiological point of view we concentrate on the classicalso-called SIR dynamic. However, we stress that the ideas here described are clearly not reducedto this model which represent an example, instead the methodology here discussed can be easilyextended to incorporate more complex epidemiological dynamics, like for instance the classical en-demic models discussed in [6, 13, 29]. Resorting to SIR model means for instance that we abstractaway from disease related mortality. Clearly, this in some situation can be a strong hypothesis, asit excludes demographic interaction. However, it is currently believed that the mortality rate ofCOVID-19 is likely low enough to justify this assumption. Moreover, we are particularly interestedin how the epidemic spreads in a country and not on studying the consequences of this diffusion.Thus, in this context, mortality has a low impact on the dynamics of the number of infected peopleand can be disregarded in first approximation. Other aspects, which certainly have a strongerimpact on how a virus spreads, are related to the presence of asymptomatic individuals [24] as wellas to a time delay between contacts and outbreak of the disease [10] which helps in the diffusion ofthe illness. These possible modeling improvements are the subject of future investigations and theywill not be treated in this work. Besides this simplifying assumptions, we will show that the basicfeatures considered and detailed in the rest of the article are sufficient in many cases to construct model which well fits with the experimental data.An easy way to understand epidemiology models is that, given a population composed of agents,they prescribe movements of individuals between different states based on some matching functionsor laws of motion. According to the classical SIR models [29], agents in the system are splitinto three classes, the susceptible, who can contract the disease, the infected, who have alreadycontracted the disease and can transmit it, and the recovered who are healed, immune or isolated.Inspired by the model considered in [14] for describing a social attitude and making use of theSIR dynamics, we present here a model composed by a system of three kinetic equations, each onedescribing the time evolution of the distribution of the number of contacts for the subpopulationbelonging to a given epidemiological class. These three equations are further coupled by takinginto account the movements of agents from one class to the other as a consequence of the infection.The interactions which describe the social contacts of individuals are based on few rules and canbe easily adapted to take into account the behavior of agents in the different classes in presenceof the infectious disease. Our joint framework is consequently based on two models which canbe considered classical in the respective fields. From the epidemic side, the SIR and its relatedmodels are widely used in most applications (cf. [29] and the references therein). From the sideof multi-agent systems in statistical mechanics, the linear kinetic model introduced in [14, 46] hasbeen shown to be flexible and able to describe, with opportune modifications, different problemsin which human behavior plays an essential role, like the formation of social contacts.The study of the effects of the intensity of social contacts in the epidemic diffusion, amongother consequences, allows to obtain in a rigorous way a variety of non-linear incidence rates of theinfectious disease as for instance recently considered in [33]. It is also interesting to remark that thepresence of non-linearity in the incidence rate function, and in particular, the concavity conditionwith respect to the number of infected has been considered in [33] as a consequence of psychological effects. Namely, the authors observed that in the presence of a very large number of infected,the probability for an infective to transmit the virus further may decrease because the population tend to naturally reduce the number of contacts . This fact will be embedded in the kinetic modelproducing a change in the mean value of the number of daily contacts. The importance of reducingat best the social contacts to countering the advance of a pandemic is a well-known and studiedphenomenon [17]. While in normal life activity, it is commonly assumed that a large part of agentsbehaves in a similar way, in presence of an extraordinary situation like the one due to a pandemic,it is highly reasonable to conjecture that the social behavior of individuals is strictly affected bytheir personal feeling in terms of safeness. In this work, we focus on the assumption that it isthe degree of diffusion of the disease that changes people’s behavior in terms of social contacts,in view both of the personal perception and/or of the external government intervention. Moregenerally, the model can be clearly extended to consider more realistic dependencies between anepidemic disease and the social contacts of individuals. However, this does not change the essentialconclusions of our analysis, namely that there is a close interplay between the spread of the diseaseand the distribution of contacts, that the kinetic description is able to quantify. The encouragingresults described in the rest of the article, finally suggest that a similar analysis can be carried out,at the price of an increasing difficulty in computations, in more complex epidemiological modelslike the SIDARTHE model [25, 26], recently used to simulate the COVID-19 epidemic in Italy tovalidate and improve the eventual partial lockdown strategies of the government and to suggestfuture measures.The rest of the paper is organized as follows. Section 2 introduces the system of three SIR-type kinetic equations combining the dynamics of social contacts with the spread of infectiousdisease in a system of interacting individuals. Then, in Section 3 we show that in a suitableasymptotic procedure the solution to the kinetic system tends towards the solution of a systemof three SIR-type Fokker-Planck type equations with local equilibria of Gamma-type [3]. Oncethe system of Fokker-Planck type equations has been derived, in Section 4 we close the SIR-typesystem of kinetic equations around the Gamma-type equilibria to obtain a SIR model in whichthe presence and consequently the evolution of the social contacts leads to a non-linear incidencerate of the infectious disease satisfying the compatibility conditions introduced in [33]. Last, in ection 5, we investigate at a numerical level the relationships between the solutions of the kineticsystem of Boltzmann type, its Fokker-Planck asymptotics and the SIR model. These simulationsconfirm the ability of the model to describe different phenomena characteristic of the trend of socialcontacts in situations compromised by the rapid spread of an epidemic, and the consequences ofvarious lockdown action in its evolution. A last part is sacred to a fitting of the model with theexperimental data by first estimating the parameters of the epidemic through data and successivelyby using these parameters in the kinetic model. Our first goal is to build a kinetic system which suitably describes the spreading of an infectiousdisease under a dependence of the contagiousness parameters on the individual number of socialcontacts of the agents. As in classical SIR models [29], the entire population is divided into threeclasses: susceptible, infected and recovered individuals.Aiming to understand social contacts effects on the dynamics, we will not consider in the sequelthe role of heterogeneity in the disease parameters (such as the personal susceptibility to a givendisease), which could be derived from the classical SIR model, suitably adjusted to account for newinformation [12, 38, 48]. Consequently, agents in the system are considered indistinguishable [40].This means that the state of a person in each class at any instant of time t ≥ x ≥
0, measured in some unit.While x is a natural positive number at the individual level, without loss of generality we willconsider x in the rest of the paper to be a nonnegative real number, x ∈ R + , at the population level.We denote then by f S ( x, t ), f I ( x, t ) and f R ( x, t ), the distributions at time t > f ( x, t ) = f S ( x, t ) + f I ( x, t ) + f R ( x, t ) . As outlined in the Introduction, we do not consider for simplicity of presentation disease relatedmortality as well as the presence of asymptomatic individuals. Therefore, we can fix the totaldistribution of social contacts to be a probability density for all times t ≥ (cid:90) R + f ( x, t ) dx = 1 . As a consequence, the quantities J ( t ) = (cid:90) R + f J ( x, t ) dx, J ∈ { S, I, R } (1)denote the fractions of susceptible, infected and recovered respectively. For a given constant α > m α ( t ) the moment of order α of the distribution of the number of contacts m α ( t ) = (cid:90) R + x α f ( x, t ) dx, and the moments of order α for the distributions of the number of contacts for each class m αJ ( t ) = (cid:90) R + x α f J ( x, t ) dx, J ∈ { S, I, R } . Unambiguously, we will indicate the mean values, corresponding to α = 1, by m ( t ) and, respectively, m J ( t ), J ∈ { S, I, R } .In what follows, we assume that the evolution of the densities is built according to the classicalSIR model [29], and that the various classes in the model could act differently in the social process onstituting the contact dynamics. The kinetic model then follows combining the epidemic processwith the contact dynamics, as modeled in [14, 46]. This gives the system ∂f S ( x, t ) ∂t = − K ( f S , f I )( x, t ) + 1 τ Q S ( f S )( x, t ) ∂f I ( x, t ) ∂t = K ( f S , f I )( x, t ) − γf I ( x, t ) + 1 τ Q I ( f I )( x, t ) ∂f R ( x, t ) ∂t = γf I ( x, t ) + 1 τ Q R ( f R )( x, t ) (2)where γ is the constant recovery rate while the transmission of the infection is governed by thefunction K ( f S , f I ), the local incidence rate, expressed by K ( f S , f I )( x, t ) = f S ( x, t ) (cid:90) R + κ ( x, y ) f I ( y, t ) dy. (3)In (3) the contact function κ ( x, y ) is a nonnegative function growing with respect to the number ofcontacts y of the population of infected, such that κ ( x,
0) = 0. Then, the spreading of the epidemicdepends heavily on the function κ ( · , · ) used to quantify the rate of possible contagion in terms of thenumber of social contacts between the classes of susceptible and infected. The evolution of the massfractions J ( t ), J ∈ { S, I, R } then obeys to the classical SIR model by choosing κ ( x, y ) ≡ β > κ ( x, y ) = β x α y α , where α, β are positive constants, that is by taking the incidence rate directly proportional to theproduct of the number of contacts of susceptible and infected people. When α = 1, for example,the incidence rate takes the simpler form K ( f S , f I )( x, t ) = β xf S ( x, t ) m I ( t ) . (4)In equations (2) the operators Q J , J ∈ { S, I, R } characterize the thermalization of the distributionof social contacts in the various classes in terms of repeated interactions [14, 46]. The Q J , J ∈{ S, I, R } are integral operators that modify the distribution of contacts f J ( x, t ), J ∈ { S, I, R } .Their action on observable quantities is given by (cid:90) R + ϕ ( x ) Q J ( f J )( x, t ) dx = (cid:68) (cid:90) R + B ( x ) (cid:0) ϕ ( x ∗ J ) − ϕ ( x ) (cid:1) f J ( x, t ) dx (cid:69) . where B ( x ) measures the interaction frequency, (cid:104)·(cid:105) denotes mathematical expectation with respectto a random quantity, and x ∗ J denote the post-interaction value of the number x of social contactsof the J -th population. Last, the constant τ in front of the interaction integral measures the timescale of the thermalization of the distribution of social contacts. Remark 1.
The relaxation constant τ plays an important role in the time evolution of the model(2), since it relates the time scale of the epidemic evolution with that of the statistical formationof social contacts. Small values of the constant τ will correspond to a fast adaptation of people toa steady situation. Hence, since it is reasonable to assume that adaptation of people is faster thatthe dynamics of epidemic, in what follows we will assume τ (cid:28) . The process of formation of the social contacts distribution is obtained by taking into accountthe typical aspects of human being, in particular the search, in absence of epidemics, of oppor-tunities for socialization. In addition to that, social contacts are due to the common use of publictransportations to reach schools, offices and, in general, places of work and to basic needs of inter-actions due to work duties. As shown in [3], this leads individuals to stabilize on a characteristicnumber of daily contacts depending on the social habits of the different countries, (represented by suitable value ¯ x M of the variable x , which can be considered as the mean number of contactsrelative to the population under investigation).Then, following a consolidated path in the kinetic theory of social phenomena, one aims inobtaining the characterization of the distribution of social contacts in a multi-agent system (themacroscopic behavior), starting from some universal assumption about the personal behavior ofagents (the microscopic behavior). Indeed, as in many other human activities, the daily amount ofsocial contacts is the result of a repeated upgrading based on well-established rules. To this extent,it is enough to recall that the daily life of each person is based on a certain number of activities, andeach activity carries a certain number of contacts. Moreover, for any given activity, the number ofsocial contacts varies in consequence of the personal choice. The typical example is the use or notof public transportations to reach the place of work or the social attitudes which scales with theage. Clearly, independently of the personal choices or needs, the number of daily social contactscontains a certain amount of randomness, that has to be taken into account. Also, while it is veryeasy to reach a high number of social contacts attending crowded places for need or will, goingbelow a given threshold is very difficult, since various contacts are forced by working or schoolactivities, among others causes. This asymmetry between growth and decrease, as exhaustivelydiscussed in [14, 27, 28], can be suitably modeled by resorting to the value function description ofthe elementary variation of the x variable. Remark 2.
It is important to outline that, in the presence of an epidemic, the characteristic meannumber of daily contacts ¯ x M reasonably changes in time, even in absence of an external lockdownintervention, in reason of the perception of danger linked to social contacts that people manifest.Consequently, even if not always explicitly indicated, we will assume ¯ x M = ¯ x M ( t ) . Also, this timedependent value can be different depending on the class to which agents belong. A further non secondary aspect of the formation of the number of social contacts is that theirfrequency is not uniform with respect to the values of x . Indeed, it is reasonable to assume that thefrequency of interactions is inversely proportional to the number of contacts x . This relationshiptakes into account that it is highly probable to have at least some contacts, and the rare situationin which one reaches a very high values of contacts x .The choice of a variable interaction frequency has been fruitfully applied in a different contextin a different situation [22]. This was done to better describe the evolution in time of the wealthdistribution in a western society. There, the frequency of the economic transactions has beenproportionally related to the values of the wealth involved, to take into account the low interestof trading agents in transactions with small values of the traded wealth. As discussed in [22],the introduction of a variable kernel into the kinetic equation does not modify the shape of theequilibrium density, but it allows a better physical description of the phenomenon under study,including an exponential rate of relaxation to equilibrium for the underlying Fokker-Planck typeequation.Following [14, 27, 28, 46], we will now illustrate the mathematical formulation of the previouslydiscussed behavior. In full generality, we assume for the moment that individuals in differentclasses can have a different mean number of contacts. The microscopic updates of social contactsof individuals in the class J , J ∈ { S, I, R } will be taken of the form x ∗ J = x − Φ (cid:15) ( x/ ¯ x J ) x + η (cid:15) x. (5)In a single update (interaction), the number x of contacts can be modified for two reasons, expressedby two terms, both proportional to the value x . In the first one, the coefficient Φ (cid:15) ( · ), which canassume both positive and negative values, characterizes the typical and predictable variation of thesocial contacts of agents, namely the personal social behavior of agents. The second term takes intoaccount a certain amount of unpredictability in the process. The usual choice is to assume thatthe random variable η is of zero mean and bounded variance, expressed by (cid:104) η (cid:105) = 0, (cid:104) η (cid:105) = λ , with λ >
0. Small random variations of the interaction (5) will be expressed simply by multiplying η bya small positive constant √ (cid:15) , with (cid:15) (cid:28)
1, which produces the new (small) variance (cid:15)λ . Notice that his formulation makes an essential use of the choice x ∈ R + . This scaled variable is the variable η (cid:15) appearing in (5).The function Φ (cid:15) plays the role of the value function in the prospect theory of Kahneman andTwersky [30, 31], and contains the mathematical details of the expected human behavior in thephenomenon under consideration. In particular, the hypothesis on which it is built is that it isnormally easier to increase the value of x than to decrease it, in relationship with the mean value¯ x J , J ∈ { S, I, R } . In terms of the variable s = x/ ¯ x J we consider as in [14] the class of valuefunctions given by Φ (cid:15)δ ( s ) = µ e (cid:15) ( s δ − /δ − e (cid:15) ( s δ − /δ + 1 , s ≥ , (6)where 0 < δ ≤ < µ < (cid:15) > (cid:15) (cid:28) (cid:104) x ∗ J − x (cid:105) and making it common toboth effects, randomness and adaptation, permits to equilibriate their effects as can be seen inSection 3. In (7), the value µ denotes the maximal amount of variation of x that agents will beable to obtain in a single interaction. Note indeed that the value function Φ (cid:15)δ ( s ) is such that − µ ≤ Φ (cid:15)δ ( s ) ≤ µ. Clearly, the choice µ < x ∗ J remains positiveif x is positive. As proven in [14], the value function (6) satisfies − Φ (cid:15)δ (1 − s ) > Φ (cid:15)δ (1 + s ) , and dds Φ (cid:15)δ (1 + s ) < dds Φ (cid:15)δ (1 − s ) . These properties are in agreement with the expected behavior of agents, since deviations from thereference point ( s = 1 in our case), are bigger below it than above. Letting δ → (cid:15) ( s ) = µ s (cid:15) − s (cid:15) + 1 , s ≥ . (7)introduced in [27, 28] to describe phenomena characterized by the lognormal distribution [2, 35].Once the elementary interaction (5) is given, for any choice of the value function, the studyof the time-evolution of the distribution of the number x of social contacts follows by resorting tokinetic collision-like models [8,40], that quantify at any given time the variation of the density of thecontact variable x in terms of the interaction operators. For a given density f J ( x, t ), J ∈ { S, I, R } ,the action on the density of the interaction operators Q J ( f )( x, t ) in equations (2) is fruitfullywritten in weak form. The weak form corresponds to say that for all smooth functions ϕ ( x ) (theobservable quantities) the action of the operator Q J on ϕ is given by (cid:90) R + ϕ ( x ) Q J ( f )( x, t ) dx = (cid:68) (cid:90) R + B ( x ) (cid:0) ϕ ( x ∗ J ) − ϕ ( x ) (cid:1) f J ( x, t ) dx (cid:69) . (8)Here expectation (cid:104)·(cid:105) takes into account the presence of the random parameter η (cid:15) in the microscopicinteraction (5). The function B ( x ) measures the interaction frequency. The right-hand side ofequation (8) quantifies the variation in density, at a given time t >
0, of individuals in the class J ∈ { S, I, R } that modify their value from x to x ∗ J (loss term with negative sign) and agents thatchange their value from x ∗ J to x (gain term with positive sign). In [14], the interaction kernel B ( x ) has been assumed constant. This simplifying hypothesis, which is not always well justifiedfrom a modeling point of view, is the common assumption in the Boltzmann-type description ofsocio-economic phenomena [21, 40]. In particular, the role of a non constant collision kernel B ( x )has been analyzed in its critical aspects in [22]. ollowing the approach in [22, 46], we express the mathematical form of the kernel B ( x ) byassuming that the frequency of changes which leads to increase the number x of social contacts isinversely proportional to x . Hence, we consider collision kernels in the form B ( x ) = ax − b , for some constants a > b >
0. This kernel assigns a low probability to interactions in whichindividuals have already a large number of contacts and assigns a high probability to interactionsin which the value of the variable x is small.The constants a and b can be suitably chosen by resorting to the following argument [46]. Forsmall values of the x variable, the rate of variation of the value function (6) is given by ddx Φ (cid:15)δ (cid:18) x ¯ x J (cid:19) ≈ µ (cid:15) ¯ x − δJ x δ − . (9)Hence, for small values of x , the mean individual rate predicted by the value function is proportionalto x δ − . Then, the choice b = δ would correspond to a collective rate of variation of the systemindependent of the parameter δ which instead characterizes the individual rate of variation of thevalue function.A second important fact is that the individual rate of variation (9) depends linearly on thepositive constant (cid:15) , and it is such that the intensity of the variation decreases as (cid:15) decreases. Then,the choice a = 1 (cid:15) , is such that the collective rate of variation remains bounded even in presence of very small valuesof the constant (cid:15) .With these assumptions, the weak form of the interaction integrals in (8), is given by (cid:90) R + ϕ ( x ) Q J ( f )( x, t ) dx = 1 (cid:15) (cid:68) (cid:90) R + x − δ (cid:0) ϕ ( x ∗ J ) − ϕ ( x ) (cid:1) f ( x, t ) dx (cid:69) . (10)Note that, in consequence of the choice made on the interaction kernel B ( · ), the evolution of thedensity f ( x, t ) is tuned by the parameter (cid:15) , which characterizes both the intensity of interactionsand the interaction frequency. This choice implieslim (cid:15) → (cid:15) Φ (cid:15)δ (cid:18) x ¯ x J (cid:19) = µ δ (cid:34)(cid:18) x ¯ x J (cid:19) δ − (cid:35) , lim (cid:15) → (cid:15) (cid:104) η (cid:15) (cid:105) = λ. (11)Consequently, the actions of both the value function and the random part of the elementary inter-action in (5) survive in the limit (cid:15) → Remark 3.
The approach just described can be easily adapted to other compartmental models inepidemiology like SEIR, MSEIR [6, 13, 29] and/or SIDHARTE [25, 26]. For all these cited models,the fundamental aspects of the interaction between social contacts and the spread of the infectiousdisease we expect not to change in a substantial way.
The limit procedure induced by (11) corresponds to the situation in which elementary interactions(5) which produce an extremely small modification of the number of social contacts (quasi-invariantinteractions) are prevalent in the dynamics. At the same time, the frequency of these interactionsis suitably increased to still permitting to see their effect.In kinetic theory, this is a well-known procedure with the name of grazing limit . We point theinterested reader to [11, 20, 40] for further details. Expanding the difference ϕ ( x ∗ J ) − ϕ ( x ) in Taylorseries, and then passing to the limit into the right-hand side of (10), one obtains, for J ∈ { S, I, R } lim (cid:15) → (cid:15) (cid:68) (cid:90) R + x − δ (cid:0) ϕ ( x ∗ J ) − ϕ ( x ) (cid:1) f J ( x, t ) dx (cid:69) = (cid:90) R + Q δJ ( f J )( x, t ) ϕ ( x ) dx = R + (cid:40) − ϕ (cid:48) ( x ) µ x − δ δ (cid:34)(cid:18) x ¯ x J (cid:19) δ − (cid:35) + λ ϕ (cid:48)(cid:48) ( x ) x − δ (cid:41) f J ( x, t ) dx If we impose at x = 0 the boundary conditions ∂∂x ( x − δ f J ( x, t )) (cid:12)(cid:12)(cid:12) x =0 = 0 , f J (0 , t ) = 0 , J ∈ { S, I, R } , (12)and a suitable rapid decay of the densities f J ( x, t ) at x = + ∞ [21], the limit operators Q δJ , J ∈ { S, I, R } , coincide with the Fokker–Planck type operators Q δJ ( f J )( x, t ) = µ δ ∂∂x (cid:40) x − δ (cid:34)(cid:18) x ¯ x J (cid:19) δ − (cid:35) f J ( x, t ) (cid:41) + λ ∂ ∂x ( x − δ f J ( x, t )) , (13)characterized by a variable diffusion coefficient and a time-dependent drift term. Indeed, in viewof Remark 2 we have ¯ x J = ¯ x J ( t ). Thus, system (2), in the limit (cid:15) → ∂f S ( x, t ) ∂t = − K ( f S , f I )( x, t ) + 1 τ Q δS ( f S )( x, t ) ,∂f I ( x, t ) ∂t = K ( f S , f I )( x, t ) − γf I ( x, t ) + 1 τ Q δI ( f I )( x, t ) ,∂f R ( x, t ) ∂t = γf I ( x, t ) + 1 τ Q δR ( f R )( x, t ) . (14)The Fokker-Planck system (14) is complemented with the boundary conditions (12) at x = 0.Clearly, the steady distributions satisfy the ordinary differential equations corresponding toequations (14) with time derivatives set equal to zero. It is interesting to remark that the explicitform of the equilibria is easily obtained when both β and γ are set equal to zero, and the contactfunction κ ( x, y ) = 0, so that the incidence rate K ( f S , f I ) vanishes. In this simple case, by assumingthat the mean values ¯ x J , J ∈ { S, I, R } are constant, and by setting ν = µ/λ , the equilibria aregiven by the functions f ∞ J ( x ) = f ∞ J (¯ x J )¯ x − δJ x ν/δ + δ − exp (cid:40) − νδ (cid:32)(cid:18) x ¯ x J (cid:19) δ − (cid:33)(cid:41) , (15) J ∈ { S, I, R } . By fixing the mass of the steady state (15) equal to one, in agreement with [3], theconsequent probability densities are generalized Gamma f ∞ ( x ; θ, χ, δ ) defined by [34, 45] f ∞ ( x ; θ, χ, δ ) = δθ χ
1Γ ( χ/δ ) x χ − exp (cid:110) − ( x/θ ) δ (cid:111) . (16)characterized in terms of the shape χ >
0, the scale parameter θ >
0, and the exponent δ > χ = νδ + δ − , θ = ¯ x J (cid:18) δ ν (cid:19) /δ . (17)It has to be remarked that the shape χ is positive, only if the constant ν = µ/λ satisfies the bound ν > δ (1 − δ ) . (18)Note that condition (18) holds, independently of δ , when µ ≥ λ , namely when the variance of therandom variation in (5) is small with respect to the maximal variation of the value function. Notemoreover that for all values δ > x J , J ∈ { S, I, R } , the variance λ of the random effects and the values δ and µ characterizing the value function φ (cid:15)δ defined in (6).The standard Gamma and Weibull distributions are included in (15), and are obtained bychoosing δ = 1 and, respectively δ = χ . In both cases, the shape χ = ν , and no conditions arerequired for its positivity. A SIR-type evolution of spreading
The Fokker-Planck system (14) contains all the information on the spreading of the epidemicin terms of the distribution of social contacts. Indeed, the knowledge of the densities f J ( x, t ), J ∈ { S, I, R } , allows to evaluate by integrations all moments of interest. However, in reason of thepresence of the incidence rate K ( f S , f I ), as given by (3), the time evolution of the moments of thedistribution functions is not explicitly computable, since the evolution of a moment of a certainorder depends on the knowledge of higher order moments, thus producing a hierarchy of equations,like in classical kinetic theory of rarefied gases [8]. We will come back to this question later on.Before discussing the closure, we establish some choices in the model in order to obtain resultswhich are in agreement with the ones reported in [3]. We fix the value δ = 1, and the incidencerate as in (4). In particular, the choice δ = 1 gives, for J ∈ { S, I, R } Q J ( f )( x ) = µ ∂∂x (cid:20) (cid:18) x ¯ x J − (cid:19) f ( x ) (cid:21) + λ ∂ ∂x ( xf ( x )) . In this case (17) imply χ = ν and θ = ¯ x J /ν , and the steady states of unit mass, for J ∈ { S, I, R } ,are the Gamma densities f ∞ J ( x ; θ, ν ) = (cid:18) ν ¯ x J (cid:19) ν
1Γ ( ν ) x ν − exp (cid:26) − ν ¯ x J x (cid:27) . (19)With this particular choice, the mean value and the variance of the densities (19), J ∈ { S, I, R } ,are given by (cid:90) R + x f ∞ J ( x ; θ, ν ) dx = ¯ x J ; (cid:90) R + x f ∞ J ( x ; θ, ν ) dx = ν + 1 ν ¯ x J . We now go back to the problem of modeling the time evolution of the densities f J ( x, t ). We assumethat due to the presence of the epidemic, the population tends to reduce the typical average numberof contacts which exhibits in standard situations. This can happen due to two main reasons: onvoluntary basis for preventing contagion or by authorities decision through lockdown measures. Thisaverage reduction effect can be introduced by assuming that the mean number of daily contacts¯ x J ( t ), J ∈ { S, I } depends on the proportion of infected¯ x J ( t ) = x J H ( I ( t )) , (20)where the function H ( r ) is decreasing with respect to r , 0 ≤ r ≤
1, starting from H (0) = 1. Inparticular, in our model, we do not consider a time-dependent mean number of social contacts forthe class of recovered. This is due to the fact that the behavior of epidemic does not depend on it.For J ∈ { S, I, R } , and ¯ x J ( t ) as in (20), we now define¯ Q J ( f J ( x, t ) = µ ∂∂x (cid:20) (cid:18) x ¯ x J ( t ) − (cid:19) f ( x ) (cid:21) + λ ∂ ∂x ( xf ( x ))With these notations, system (14) with δ = 1 reads ∂f S ( x, t ) ∂t = − βx f S ( x, t ) m I ( t ) + 1 τ ¯ Q S ( f S )( x, t ) ,∂f I ( x, t ) ∂t = βx f S ( x, t ) m I ( t ) − γf I ( x, t ) + 1 τ ¯ Q I ( f I )( x, t ) ,∂f R ( x, t ) ∂t = γf I ( x, t ) + 1 τ ¯ Q R ( f R )( x, t ) . (21)Integrating both sides of equations in (21) with respect to x , and recalling that the Fokker-Plancktype operators are mass-preserving, we obtain the system for the evolution of the proportions J ( t ) efined in (1), J ∈ { S, I, R } ∂S ( t ) ∂t = − β m S ( t ) m I ( t ) ,∂I ( t ) ∂t = β m S ( t ) m I ( t ) − γI ( t ) ,∂R ( t ) ∂t = γI ( t ) . (22)As anticipated, unlike the classical SIR model, system (22) is not closed, since the evolution of themass fractions J ( t ), J ∈ { S, I, R } depends on the mean values m J ( t ).Similarly to the derivation of macroscopic equations in kinetic theory, the closure of system(22) can be obtained by resorting, at least formally, on a limit procedure. In fact, as outlined inthe Introduction, the typical time scale involved in the social contact dynamic is τ (cid:28) τ (cid:28) f S ( x, t )towards the Gamma equilibrium density with a mass fraction S ( t ) and momentum m S ( t ) = ¯ x S S ( t ) H ( I ( t )) , (23)as it can be easily verified from the differential expression of the interaction operator ¯ Q S .Indeed, if τ (cid:28) ∂f S ( x, t ) ∂t = 1 τ ¯ Q S ( f S )( x, t ) (24)towards the equilibrium f ∞ S ( x ; θ, ν ) (see [47] for details), that the solution f S ( x, t ) remains suffi-ciently close to the Gamma density with mass S ( t ) and momentum given by (23) for all times. Thisequilibrium distribution f ∞ S ( x ; θ, ν ) can be plugged into the first equation of (21) and then one canintegrate it with respect to the variable x . This procedure gives formally the first equation of (22).Analogous analysis and formal limit procedure can be done with the second equation in system(21), which leads to relaxation towards a Gamma density with mass fraction I ( t ) and momentumgiven by m I ( t ) = ¯ x I I ( t ) H ( I ( t )) . (25)Consequently, we obtain the closure ∂S ( t ) ∂t = − ¯ β S ( t ) I ( t ) H ( I ( t )) ,∂I ( t ) ∂t = ¯ β S ( t ) I ( t ) H ( I ( t )) − γI ( t ) ,∂R ( t ) ∂t = γI ( t ) . (26)In system (26) we defined ¯ β = β ¯ x I ¯ x S . This identifies the classical transmission parameter of theSIR model, where however now the difference is that this quantities are not postulated but insteadderived starting from microscopic considerations. In the following, we refer to this model as thesocial contact-SIR model (S-SIR). Remark 4.
The outlined closure strategy can be obtained also by resorting to the splitting method,a very popular numerical approach for the Boltzmann equation [23, 39]. If at each time step ( t, t +∆ t ) we consider sequentially the population-based interaction and relaxation operators in the firstequation in (14), during this short time interval we recover the evolution of the density from thejoint action of relaxation ∂f S ( x, t ) ∂t = 1 τ ¯ Q S ( f S )( x, t ) (27) nd SIR interaction ∂f S ( x, t ) ∂t = − βx f S ( x, t ) m I ( t ) (28) We recall once again that the typical time to consider is τ (cid:28) , which identifies a faster adaptationof individuals to social contacts with respect to the evolution time of the epidemic disease. Sincethe operator ¯ Q S is mass preserving, in the considered time interval the relaxation (27) with a value τ (cid:28) pushes the solution of equation (24) towards the Gamma equilibrium density with the samemass fraction S ( t ) of the initial datum, and momentum m S ( t + ∆ t ) = ¯ x S S ( t ) H ( I ( t )) , (29) as it can be easily verified from the differential expression of the interaction operator ¯ Q S .Indeed, if τ (cid:28) is sufficiently small, one can easily argue from the exponential convergence ofthe solution to (24) towards the equilibrium [47], that the solution f S ( x, t + ∆ t ) is sufficiently closeto the Gamma density with mass S ( t ) and momentum given by (29), and this density can be usedinto the SIR step (28) to close the splitted system (24)-(28). Analogous procedure can be done withthe second equation in system (21), which leads to relaxation towards a Gamma density with massfraction I ( t ) and momentum given by m I ( t + ∆ t ) = ¯ x I I ( t ) H ( I ( t )) . (30) Consequently, substituting into system (22) we obtain the closed system (26) . It remains to quantify the action of the social contacts on the evolution of the epidemic. Forthis reason, let us introduce for a given constant N (cid:29) H ( r ) = 1 √ N r , ≤ r ≤ , (31)which describes a possible way in which, in presence of the spread of the disease, the susceptibleand the infected population tend to reduce the mean number of daily social contacts ¯ x J , J ∈ { S, I } .This choice produces a SIR model with global incidence rate D ( S, I )( t ) = (cid:90) R + K ( f S , f I )( x, t ) dx = ¯ βS ( t ) I ( t )1 + N I ( t ) , (32)that fulfils all the properties required by the non-linear incidence rates considered in [33]. Indeed, D ( S,
0) = 0, and the function D ( S, I ) satisfies ∂D ( S, I ) ∂I > , ∂D ( S, I ) ∂S > S, I >
0. Moreover D ( S, I ) is concave with respect to the variable I, i.e. ∂ D ( S, I ) ∂I ≤ , (34)for all S, I >
0. In addition to the form (31), we can also consider the following function H ( t, r ) = 1 (cid:115) N r ( t ) (cid:90) t r ( s ) ds , ≤ r ≤ . (35)This function satisfies the same properties than (31) in terms of the incidence rate requests detailedin [33] and takes into account memory effects on the population’s behavior. In fact, people mayadapt their life style in terms of possible daily contacts to answer to the actual pandemic situation as Test 1 . Initial data and convergence of the Boltzmann dynamics towards the equilibriumstate of the corresponding Fokker-Planck equation. expressed by (31), however further effects based on the global dynamics of the disease can influencethe choices of the population.To conclude, let us introduce the basic reproduction number R of this model, i.e. the av-erage number of secondary cases produced by a single infected agent introduced into an entirelysusceptible population. This is given by R = 1 γ lim I → ,S → ∂D ( S, I ) ∂I = ¯ βγ , (36)where γ is the recovery rate of infected. According to the analysis of [33], an autonomous compart-mental epidemiological model with the non-linear incidence rate (32) under the constant populationsize assumption is stable. Such system has either a unique and stable endemic equilibrium state orno endemic equilibrium state at all. Since the incidence rate D ( S, I ) satisfies the conditions (33)-(34), if R >
1, then the endemic equilibrium state Q ∗ = ( S ∗ , I ∗ ) of system (26) is asymptoticallystable. If R ≤
1, then there is no endemic equilibrium state, and the infection-free equilibriumstate is asymptotically stable.
In addition to analytic expressions, numerical experiments allow us to visualize and quantify theeffects of social contacts on the SIR dynamics used to describe the time evolution of the epidemic.More precisely, starting from a given equilibrium distribution detailing in a probability setting thedaily number of contacts of the population, we show how the coupling between social behaviorsand number of infected people may modify the epidemic by slowing down the number of encountersleading to infection. In a second part, we discuss how some external forcing, mimicking politicalchoices and acting on restrictions on the mobility, may additionally improve the reduction of theepidemic trend avoiding concentration in time of people affected by the disease and then decreasingthe probability of hospitalization peaks. In a third part, we focus on experimental data for thespecific case of COVID-19 in different European countries and we extrapolate from them the mainfeatures characterizing the contact function H ( · ). The starting point is represented by a population composed of 99% of susceptibles and 0 .
01% ofinfected. The distribution of the number of contact is described by (15) with ν = 4, δ = 8 and Test 1 . Top: distribution of the social contacts at equilibrium over time for the two choicesof the function H . Middle: SIR dynamics corresponding to the different choices of the different meannumber of daily contact (left constant case, right as a function of the epidemic). Bottom left: finaldistribution of the number of daily contact for the susceptible and recovered agents. Bottom right:time evolution of the contact function H . 14igure 3: Test 1. SIR dynamic corresponding to different choices for the mean number of daily contacts(left constant case, right as a function of the epidemic) with parameters k ( x ) = 0 . /x J x and γ = 1. x J = 15 while the epidemic parameters are k ( x ) = . x J x and γ = 0 .
8. The kinetic model (14) issolved by a splitting strategy together with a Monte Carlo approach where the number of samplesused to described the population is fixed to M = 10 . The time step is fixed to ∆ t = 10 − andthe scaling parameter is τ = 10 − . These choices are enough to observe the convergence of theBoltzmann dynamics to the Fokker-Planck one as shown in Figure 1 where the analytical equilibriumdistribution is plotted together with the results of the Boltzmann dynamics. We considered alsouniform initial distribution f ( x ) = 1 c χ ( x ∈ [0 , c ]) , c = 30 , (37)being χ ( · ) the indicator function. In the introduced setting, we then compare two distinct cases: inthe first one we suppose that the social contacts do not affect the solution, meaning H ( I ( t )) = 1,while the second includes the effects of the function H ( I ( t )) given in (31) with N = 10. The resultsare depicted in Figure 2. The top right images show the time evolution of the distribution of thenumber of contacts for the two distinct cases, while the middle images report the correspondingevolution of the epidemic. For this the second case, the function H ( I ( t )) as well as the distributionof contacts for respectively the susceptible and the recovered are shown at the bottom of the samefigure. We clearly observe a reduction of the peak of infected in the case in which the dynamicsdepends on the number of contacts with H ( I ( t )) given by (31).We now repeat the same simulation by only changing the considered epidemic parameters. Inparticular, we consider a lower infection and recovery rate given by k ( x ) = 0 . /x J x and γ = 1,respectively. In Figure 3, we show the evolution of two epidemic profiles in time for the case inwhich social contacts do not affect the solution and for the case in which H ( I ( t )) is a function ofthe number of infected as for the previous situation. Results show the same qualitative behavior:peak reduction and spread of the number of infected over time is observed when sociality is takeninto account. The total number of infected is also reduced in the second case. Next, we compare the effects on the spread of the disease when the population adapts its habitswith a time delay with respect to the onset of the epidemic. This kind of dynamics corresponds to amodeling of a possible lockdown strategy whose effects are to reduce the mobility of the populationand, correspondingly, to reduce the number of daily contacts in the population.The setting is similar to the one introduced in Section 5.1 and we consider a switch between H = 1 to H ( I ( t )) = 1 / (cid:112) N I ( t ) when the number of infected increases. The social parameters re ν = 4, δ = 8 and x J = 15, as before, while the epidemic parameters are k ( x ) = 0 . /x J x and γ = 0 .
6, the final time is fixed to T = 15. The initial distribution of contacts is also assumed to beof the form (37).We consider three different settings, in the first one H = 1 up to t < T /
4, in the second one upto t < T / < t < H = 1. The results are shown in Figure 4 for both the distributionof daily contacts over time and the SIR evolution. We can identify three scenarios. The first onthe top gives a slightly change to the standard epidemic dynamics. Indeed, we can observe around t = 7 . t = 4, while for the third case we first observe inversion and then theresurgence of the number of infected when the lockdown measures are relaxed. In this part, we consider data about the dynamics of COVID-19 in three European countries:France, Italy and Spain. For these three countries, the evolution of the disease, in terms of re-ported cases, evolved in rather different ways. The estimation of epidemiological parameters ofcompartmental models is an inverse problem of generally difficult solution for which different ap-proaches can be considered. We mention in this direction a very recent comparison study [36]. Itis also worth to mention that often the data are partial and heterogeneous with respect to theirassimilation, see for instance discussions in [1, 7, 9, 44]. This makes the fitting problem challengingand the results naturally affected by uncertainty.The data concerning the actual number of infected, recovered and deaths of COVID-19 arepublicly available from the John Hopkins University GitHub repository [16]. For the specific case ofItaly, we considered instead the GitHub repository of the Italian Civil Protection Department [41].In the following, we present the adopted approach which is based on a strategy with two optimisationhorizons (pre-lockdown and lockdown time spans) depending on the different strategies enacted bythe governments of the considered European countries.In details, we considered first the time interval t ∈ [ t , t (cid:96) ], being t (cid:96) the day in which lockdownstarted in each country (Spain, Italy and France) and t the day in which the reported cases hit200 units. The lower bound t has been imposed to reduce the effects of fluctuations caused bythe way in which data are measured which have a stronger impact when the number of infected islow. Once the time span has been fixed, we then considered a least square problem based on theminimization of a cost functional J which takes into account the relative L norm of the differencebetween the reported number of infected and the reported total cases ˆ I ( t ), ˆ I ( t ) + ˆ R ( t ) and theevolution of I ( t ) and I ( t ) + R ( t ) prescribed by system (26) with H ≡
1. In practice, we solved thefollowing constrained optimisation problemmin β,γ J ( ˆ I, ˆ R, I, R ) (38)where the cost functional J is a convex combination of the mentioned norms and assumes thefollowing form J ( ˆ I, ˆ R, I, R ) = p (cid:107) ˆ I ( t ) − I ( t ) (cid:107) (cid:107) ˆ I ( t ) (cid:107) + (1 − p ) (cid:107) ˆ I ( t ) + ˆ R ( t ) − I ( t ) − R ( t ) (cid:107) (cid:107) ˆ I ( t ) + ˆ R ( t ) (cid:107) We choose p = 0 . ≤ β ≤ . , . ≤ γ ≤ . . In Table 1 we report the results of the performed parameter estimation together with the resultingattach rate R defined in (36).Once that the contagion parameters have been estimated in the pre-lockdown time span, wesuccessively proceeded with the estimation of the shape of the function H from the data.To estimate Test 2 . Comparisons of different lockdown behaviors. Top: late lockdown. Middle: earlylockdown. Bottom: early lockdown and successive relaxation.17igure 5:
Test 3 . Fitting of the parameters of model (26) where β, γ > H ≡
1. The parameters characterizing the function H ( · ) in (26) havebeen computed during and after lockdown at regular interval of time, up to July 15. The lockdownmeasures change in each county (dashed line).France Mar 5-Mar 17
Italy
Feb 24-Mar 9
Spain
Mar 5-Mar 14 β . . . γ . . . R . . . Test 3 . Model fitting parameters in estimating attack values for the COVID-19 outbreakbefore lockdown in various European countries. this latter quantity, we solved the following optimization problemmin H J (39)in terms of H where J is the same functional of the previous step and where in the evolution ofthe macroscopic model the values β, γ have been fixed as a result of the first optimization in thepre-lockdown period. The parameters chosen for (39) are p = 0 . ≤ H ≤ . The second optimisation problem has been solved up to last available data for each country withdaily time stepping h = 1 and over a time window of three days. This has been done with thescope of regularizing possible errors due to late reported infected and smoothing the shape of H .Both optimisation problems (38)-(39) have been tackled using the Matlab functions fmincon incombination with a RK4 integration method of the system of ODEs.In Figure 5, we present the result of such fitting procedure between the model (26) and theexperimental data. The evolution of the estimated H ( t ), t ∈ [ t (cid:96) , T ], is instead presented in the leftcolumn of Figure 6. From this figure, it can be observed in the case of Italy how, even if the dailynumber of infected decreases after May 1st, the estimated H remains quite stable after this day.This behavior cannot be reproduced by using a function H ( I ( t )) as the one given in (31). Instead,a function H ( t, I ( t )) of the form (35), which takes into account both the instantaneous number ofconfirmed infections N I ( t ) and the total number of infected in the population N (cid:80) s Mar 18 - Jul 15 Italy Mar 10 - Jul 15 Spain Mar 15 - Jul 15 a · − . · − . · − b . . . R . . . H France Mar 18 - Jul 15 Italy Mar 10 - Jul 15 Spain Mar 15 - Jul 15 a . 41 0 . . b . . . R . . . H i , i = 1 , H ( t ) solution of the optimisation problem (39). The corresponding R coefficient is also reported. H = H ( t, N I ( t ) (cid:80) s 25) in the case of Italy. The case of Spain will bediscussed later. Test 3 . Estimated shape of the function H in several European countries (left plots) andits dependency on the variables N I ( t ) and N I ( t ) (cid:80) s Figure 7: Test 4 . Left: relative number of infected over time for the S-SIR model when memoryeffects are taken into account in the contact function. Right: corresponding contact function overtime. Top: the French case. Bottom: the Italy case. In Figure 7 we show the profiles of the infected over time together with the shape of the function H again over time. The results show that with the choices done for the contact function, it is possibleto reproduce at least qualitatively the shape of the trend of infected during the pandemic observedin Italy and in France.It is worth to remark that the considered social parameters have been estimated only the inthe case of France, see [3], and we assumed that the initial contact distribution is the same for theItalian case. We now consider the case of Spain. For this country, according to Figure 5, the trendof infected undergoes a deceleration during the lockdown period. This can be also clearly observedin Figure 6 where the extrapolated shape of the contact function H is shown. Let also observe thatwhile the global behavior of this function is captured by the fitting procedure, we however lose theminimum which takes place around end of April. This minimum is responsible of the decelerationin the number of infected and can be brought back to a strong external intervention in the lifestyleof Spain country with the scope of reducing the hospitalizations. This effect can be reproduced byour model by imposing the same behavior in the function H . To that aim, the Figure 8 reportsfinally the profile of the infected over time together with the shape of the function H again overtime for this last case. The results show that also in this case, the S-SIR model is capable toqualitatively reproduce the data. -3 I S-SIRI Exp Figure 8: Test 4 . Left: relative number of infected over time for the S-SIR model when memoryeffects are taken into account in the contact function. Right: corresponding contact function overtime. Spain case. Conclusions The development of strategies for mitigating the spreading of a pandemic is an important publichealth priority. The recent case of COVID-19 pandemic has seen as main strategy restrictive meas-ures on the social contacts of the population, obtained by household quarantine, school or workplaceclosure, restrictions on travels, and, ultimately, a total lockdown. Mathematical models representpowerful tools for a better understanding of this complex landscape of intervention strategies andfor a precise quantification of the relationships among potential costs and benefits of different op-tions [17]. In this direction, we introduced a system of kinetic equations coupling the distributionof social contacts with the spreading of a pandemic driven by the rules of the SIR model, aimingto explicitly quantify the mitigation of the pandemic in terms of the reduction of the number ofsocial contacts of individuals. The kinetic modeling of the statistical distribution of social contactshas been developed according to the recent results in [3], which present an exhaustive descriptionof contacts in the France population, divided by categories. The numerical experiments then showthat the kinetic system is able to capture most of the phenomena related to the effects of partiallockdown strategies, and, eventually to maintain pandemic under control. Acknowledgement This work has been written within the activities of GNFM group of INdAM (National Instituteof High Mathematics), and partially supported by MIUR project “Optimal mass transportation,geometrical and functional inequalities with applications”. The research was partially supportedby the Italian Ministry of Education, University and Research (MIUR): Dipartimenti di EccellenzaProgram (2018–2022) - Dept. of Mathematics “F. Casorati”, University of Pavia. 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