On the optimal control of kinetic epidemic models with uncertain social features J. Franceschi1 A. Medaglia1 and M. Zanella1

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On the optimal control of kinetic epidemic models with uncertain

social features

J. Franceschi∗1, A. Medaglia†1, and M. Zanella‡1

1Department of Mathematics “F. Casorati”, University of Pavia, Italy

Abstract

It is recognized that social heterogeneities in terms of the contact distribution have a strong

inﬂuence on the spread of infectious diseases. Nevertheless, few data are available on the group

composition of social contacts, and their statistical description does not possess universal patterns

and may vary spatially and temporally. It is therefore essential to design robust control strategies,

mimicking the eﬀects of non-pharmaceutical interventions, to limit eﬃciently the number of infected

cases. In this work, starting from a recently introduced kinetic model for epidemiological dynamics

that takes into account the impact of social contacts of individuals, we consider an uncertain contact

formation dynamics leading to slim-tailed as well as fat-tailed distributions of contacts. Hence, we

analyse the eﬀects of an optimally robust control strategy of the system of agents. Thanks to clas-

sical methods of kinetic theory, we couple uncertainty quantiﬁcation methods with the introduced

mathematical model to assess the eﬀects of social limitations. Finally, using the proposed model-

ling approach and starting from available data, we show the eﬀectiveness of the proposed selective

measures to dampen uncertainties together with the epidemic trends.

Keywords: kinetic models, mathematical epidemiology, optimal control, non-pharmaceutical inter-

ventions, multi-agent systems

Mathematics Subject Classiﬁcation: 92D30, 35Q84, 35Q92

Contents

1 Introduction 2

2 Kinetic epidemic models with uncertain contact distribution 3

2.1 Contact formation dynamics ................................... 4

2.2 Fokker-Planck scaling and steady states ............................ 6

2.3 Uniqueness of the solution .................................... 8

3 Selective control of the kinetic epidemic model 11

3.1 The controlled model ....................................... 12

3.2 Damping eﬀects on the model uncertainties .......................... 13

3.3 Controlled kinetic epidemic model ............................... 16

∗jonathan.franceschi01@universitadipavia.it

†andrea.medaglia02@universitadipavia.it

‡mattia.zanella@unipv.it

arXiv:2210.09201v3 [math.OC] 8 Jun 2023

4 Observable eﬀects of non-pharmaceutical interventions 16

4.1 Derivation of the macroscopic model .............................. 16

5 Numerical examples 18

5.1 Stochastic Galerkin methods .................................. 18

5.2 Test 1: Uncontrolled model ................................... 20

5.3 Test 2: Consistency of the macroscopic limit .......................... 22

5.4 Test 3: Controlled model and uncertainty damping ...................... 22

5.5 Test 4: A data-oriented approach ................................ 26

5.5.1 Test 4a: Calibration of the model ............................ 26

5.5.2 Test 4b: Assessment of diﬀerent restriction strategies ................. 26

1 Introduction

In recent years extensive research eﬀorts have been devoted to design eﬀective non-pharmaceutical in-

terventions (NPIs) to mitigate the impact of the COVID-19 pandemics [4,7,23,27,32,39]. In particular,

several works in mathematical epidemiology shed light on the importance of the inner heterogeneity in

the social structure of a population, see [5,17,19,50]. In this direction, among the main factors shaping

the evolution of the epidemic, the contact structure of a population has been deeply studied especially

in relation to the age distribution of a population. Special attention was recently paid by the scientiﬁc

community to the role and the estimate of the distribution of contacts between individuals as also a

relevant cause of the potential pathogen transmission [6,9,25]. Nevertheless, we have often limited in-

formation on the real social features of a population, whose characteristics are structurally uncertain

and may frequently change due to exogenous processes that are also inﬂuenced by psychological factors,

determining diﬀerent responses in terms of individuals’ protective behavior, see e.g. [20,28].

Starting from the above considerations, recent works proposed kinetic-type models to connect the

distribution of social contacts with the spreading of a disease in multi-agent systems [15,17,35,49]. The

result is obtained by integrating a compartmental modeling approach for epidemiological dynamics with

a thermalization process determining the formation of social contacts. We highlight how the advantages

of kinetic modeling approaches for epidemiological dynamics rely on a clear connection between the

scales of the transmission of the infection, linking agent-based dynamics with the macroscopic observable

ones. Within this research framework, we mention [13,31] where epidemiological relevant states are

characterized by agent-based viral load dynamics.

In this paper, we concentrate on a classical SEIR compartmentalization of the population whose con-

tact distribution is uncertain. In particular, we introduce an interaction scheme describing the evolution

in the number of social contacts of individuals. The microscopic model is based on a simple transition

operator whose parameters are assumed to be uncertain. At the kinetic level, the aforementioned model

is capable to identify a variety of equilibrium distributions, ranging from slim-tailed Gamma-type dis-

tributions to power-law-type distributions depending on the introduced uncertainties. In the introduced

setting, the analysis of the emerging distribution is essential to deﬁne the evolution of the main mo-

ments of the system of kinetic equations via a closure approach determining the evolution of macroscopic

quantities. In particular, we will consider stationary states that depend on uncertain quantities thus,

the derived system of equations embeds an incomplete knowledge on the real distribution of contacts.

Therefore, the deﬁnition of eﬀective NPIs, generally based on a generalized reduction of the number

of contacts, should take into account the uncertain contact structure of a population. In particular, we

aim at giving a deeper understanding of the mitigation eﬀects due to the reduction of social interactions

among individuals. To this end, we develop an approach suﬃciently robust in terms of the introduced un-

certainties. This is done through a combination of a kinetic epidemiological model and a control strategy

whose target is to point the population towards a given target number of contacts. The development

of control protocols for kinetic and mean-ﬁeld equations has been deeply investigated in recent years,

without pretending to review the huge literature we mention [1–3,24,40] and the references therein. In

detail, we concentrate on modeling the lockdown policies through a selective optimal control approach.

In particular, we show how the form of the implemented control may result in very diﬀerent mitigation

eﬀects, that deeply depend on the heterogeneity in the contact distribution of the population. In the last

part, starting from the calibrated model at our disposal, we focus on the numerical study of the proposed

approach and we exploit accurate methods for the uncertainty quantiﬁcation of kinetic equations.

The rest of the paper is organized as follows. In Section 2we introduce a system of kinetic equations

with SEIR compartmentalization combining the dynamics of social contacts with the spread of an infec-

tious disease in a multi-agent system. The main features of the solution of a surrogate Fokker-Planck

model are studied in Section 2.3. In Section 3a control strategy is introduced at the kinetic level and

in Section 4we observe the eﬀects of the control on the corresponding second-order macroscopic model.

Finally, in Section 5we investigate numerically the relationship between the kinetic epidemic model

with uncertainties and its macroscopic limit. A second part is dedicated to the interface between the

introduced modeling approach and available data.

2 Kinetic epidemic models with uncertain contact distribution

In this section, we introduce a compartmental model describing the spreading of an infectious disease

coupled with a kinetic-type description of the contact evolution of a system of individuals [17,18,35,49].

In addition, we will also take into account uncertainties collecting the missing information on the contact

distribution.

In more details, we consider a system of agents that can be subdivided into the following relevant

epidemiological states [8,14,30]: susceptible (S) agents are the ones that can contract the disease,

infectious agents (I) are responsible for the spread of the disease, exposed (E) agents have been in

contact with infectious ones but still may or may not become contagious; ﬁnally, removed (R) agents

cannot spread the disease.

To incorporate the impact of contact distribution in the infectious dynamics, we denote by fJ=

fJ(z, x, t) the distribution of the number of contacts x∈R+at time t≥0 of agents in compartment J,

where J∈ C :={S, E, I, R}. The random vector z∈Iz⊆Rdz, with dz∈N, collects all the uncertainties

of the system and we suppose to know its distribution p(z) such that

Prob(z∈Iz) = ZIz

p(z)dz.

We deﬁne the total contact distribution of a society as

J∈C

fJ(z, x, t) = f(z, x, t),ZR+

f(z, x, t)dx = 1,

while the mass fractions of the population in each compartment and their moment of order r > 0 are

given by

ρJ(z, t) = ZR+

fJ(z, x, t)dx, ρJ(z, t)mr,J (z, t) = ZR+

xrfJ(z, x, t)dx.

In the following, to simplify notations we will indicate with mJ(z, t), J∈ C, the mean values correspond-

ing to r= 1.

Hence, we assume that the introduced compartments in the model could act diﬀerently at the level

of the social process constituting the contact dynamics. The kinetic model deﬁning the time evolution

Parameter Deﬁnition

βcontact rate between susceptible and infected individuals

1/ζ average latency period

1/γ average duration of infection

Table 1: Parameters deﬁnition in the SEIR model (1).

of the functions fJ(z, x, t) follows by combining the epidemic process with the contact dynamics. This

gives the system











∂fS(z, x, t)

∂t =−K(fS, fI)(z, x, t) + 1

τQS(fS)(z, x, t),

∂fE(z, x, t)

∂t =K(fS, fI)(z, x, t)−ζfE(z, x, t) + 1

τQE(fE)(z, x, t),

∂fI(z, x, t)

∂t =ζfE(z, x, t)−γfI(z, x, t) + 1

τQI(fI)(z, x, t),

∂fR(z, x, t)

∂t =γfI(z, x, t) + 1

τQR(fR)(z, x, t),

(1)

where the operators QJ(fJ) characterizes the emergence of the distribution of social contacts in the

compartment J∈ C. The transmission of the infection is governed by the local incidence rate deﬁned as

K(fS, fI)(z, x, t) = fS(z, x, t)ZR+

κ(x, x∗)fI(z, x∗, t)dx∗(2)

where κ(x, x∗) is a nonnegative contact function measuring the impact of contact rates among diﬀerent

compartments. A leading example for κ(x, x∗) is obtained by choosing

κ(x, x∗) = βxαxα

∗,

with β > 0 and α > 0. In the following, we will stick to the case α= 1 for simplicity so that

K(fS, fI)(z, x, t) = βxfS(z, x, t)mI(z, t)ρI(z, t).(3)

This choice formalizes an incidence rate that is proportional on the product of the number of contacts of

susceptible and infected people. The other epidemiological parameters characterizing the spread of the

disease are ζ > 0, the transition rate of exposed individuals to the infected class and γ > 0, the recovery

rate. The introduced parameters have been summarized in Table 1. Finally, the relaxation parameter

0< τ ≪1 represents the frequency at which the agents modify their contact distribution in response to

the epidemic dynamics. As we will see, we are assuming that the social dynamics is much faster than

the epidemic dynamics [50].

2.1 Contact formation dynamics

The total number of contacts can be viewed as a result of the superimposition of repeated updates and

possible deviations due to aleatoric uncertainty, see [26,37]. In particular, similarly to [17,18] we consider

the following microscopic scheme

x′

J=x−Φδ

ε(z, x/mJ)x+ηεx, (4)

where x′

J−xis the elementary variation of the number of contacts and Φδ

εdeﬁnes the transition function

Φδ

ε(z, s) = µeε(sδ−1)/δ −1

eε(sδ−1)/δ + 1, s =x/mJ,(5)

with ε > 0. In (5) we introduced a constant µ > 0 linked to the maximum variability of the function

and the centered random variable ηεsuch that η2

ε=εσ2, being ⟨·⟩ the expectation with respect to the

introduced random variable. The constant ε > 0 tunes the strength of interactions. We remark that the

microscopic model (4) depends on a parametric uncertainty and δ=δ(z), such that δ(z)∈[−1,1], for

any z∈Rdz. The transition function (5) is deﬁned such that it is simpler to reach a high number of

daily contacts while it is very unlikely to go under a certain threshold. This type of asymmetry is typical

of human and biological phenomena as shown e.g. in [17,18,29,33,41,43]. In the regime ε≪1 we have

Φδ

ε(z, x/mJ)≈εµ

2δ(z) x

mJδ(z)

−1:=εΦδ(z, x/mJ).(6)

Note also that the function Φδ

εis such that

−µ≤Φδ

ε(z, x/mJ)≤µ

for all δ(z)∈[−1,1] and ε > 0. Clearly, the choice µ < 1 implies that, in absence of randomness, the

value x′

Jremains positive if xis positive. It is interesting to observe that Φδ

εis asymmetric around that

value x/mJwith respect to diﬀerent distributions of δ. In particular, Φδ

εis increasing and convex for

any x/mJ≤1 if δ > 0 whereas, if δ < 0, the transition function becomes concave in an interval [0,¯x],

¯x/mJ<1, and then convex.

Once the microscopic process (4) is given, the time evolution of the distribution of the number of

social contacts ffollows by resorting to kinetic collision-like approaches, see [11,37], that quantify the

variation of the density of the contact variable in terms of an interaction operator, for any time t≥0.

The time evolution of fis given by the following kinetic equation written in weak form

dt ZR+

φ(x)fJ(z, x, t)dx =1

εZR+

φ(x)Q(fJ)(z, x, t)dx

where ZR+

φ(x)Q(fJ)(z, x, t)dx =ZR+

B(z, x)⟨φ(x′

J)−φ(x)⟩fJ(z, x, t)dx, (7)

where we indicated with φ:R+→R,φ(x)∈ C∞(R+) an observable quantity. In the following, we

will consider an uncertain interaction kernel expressing a multiagent system in which the frequency of

changes in the number of social contacts depends on xthrough the following law

B(z, x) = x−α(δ(z)),(8)

being in particular

α(δ(z)) = 1 + δ(z)

2≥0,for any δ(z)∈[−1,1].

We observe that the kernel (8) mimics the fact that a priori information on the frequency of interaction

of a system of agents is missing, see [34].

Remark 2.1.If we consider φ(x) = 1 in (7) we easily get the conservation of the mass. Furthermore, if

φ(x) = xwe have

dtmJ(z, t) = −1

εZR+

x1−α(δ)Φδ

ε(z, x/mJ)fJ(z, x, t)dx.

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OntheoptimalcontrolofkineticepidemicmodelswithuncertainsocialfeaturesJ.Franceschi∗1,A.Medaglia†1,andM.Zanella‡11DepartmentofMathematics“F.Casorati”,UniversityofPavia,ItalyAbstractItisrecognizedthatsocialheterogeneitiesintermsofthecontactdistributionhaveastronginfluenceonthespreadofinfectiousdiseases...

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On the optimal control of kinetic epidemic models with uncertain social features J. Franceschi1 A. Medaglia1 and M. Zanella1

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