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Kinetic models for epidemic dynamics in the

presence of opinion polarization

Mattia Zanella

Department of Mathematics “F. Casorati”

University of Pavia, Italy

mattia.zanella@unipv.it

Abstract

Understanding the impact of collective social phenomena in epidemic

dynamics is a crucial task to eﬀectively contain the disease spread. In

this work we build a mathematical description for the assessing the in-

terplay between opinion polarization and the evolution of a disease. The

proposed kinetic approach describes the evolution of aggregate quantities

characterizing the agents belonging to epidemiologically relevant states,

and will show that the spread of the disease is closely related to consensus

dynamics distribution in which opinion polarization may emerge. In the

present modelling framework, microscopic consensus formation dynam-

ics can be linked to macroscopic epidemic trends to trigger the collective

adherence to protective measures. We conduct numerical investigations

which conﬁrm the ability of the model to describe diﬀerent phenomena

related to the spread of an epidemic.

Keywords: kinetic equations, mathematical epidemiology, opinion dy-

namics

Mathematics Subject Classiﬁcation: 92D30, 35Q20, 35Q84, 35Q92

Contents

1 Introduction 2

2 A kinetic model approach for consensus formation and epidemic

dynamics 3

2.1 Kinetic models for opinion formation ................ 5

2.2 Derivation of a Fokker-Planck model ................ 7

3 Macroscopic opinion-based SEIR dynamics 9

3.1 Derivation of moment based systems ................ 9

3.2 The macroscopic model with saturated incidence rate . . . . . . 12

4 Numerical examples 13

4.1 Test 1: large time behaviour of kinetic opinion formation models 13

4.2 Test 2: consistency of the macroscopic limit ............ 15

4.3 Test 2a: equilibrium closure ..................... 16

1

4.4 Test2b: the bounded conﬁdence case ................ 16

4.5 Test 3: the impact of opinion polarization on the infection dynamics 22

1 Introduction

During the outbreak of SARS-CoV-2 pandemic, we observed how, as cases es-

calated, collective compliance to the so-called non-pharmaceutical interventions

(NPIs) was crucial to ensure public health in the absence of eﬀective treatments,

see e.g. [3,6,8,33,51,53]. Nevertheless, the eﬀectiveness of lockdown measures

heavily depended on the beliefs/opinions of individuals regarding protective be-

havior, which are thus linked to personal situational awareness [27,47]. Recent

experimental results have shown that social norm changes are often triggered

by opinion alignment phenomena [50]. In particular, the perceived adherence

of individuals’ social network has a strong impact on the eﬀective support of

the protective behaviour. The individual responses to threat is a core question

to set-up eﬀective measures prescribing norm changes in daily social contacts

[21] and cases escalation is a factor that may be perceived in diﬀerent ways.

For these reasons, it appears natural to couple classical epidemiological models

with opinion dynamics in order to understand the mutual inﬂuence of these

phenomena.

In recent years the study of emerging properties of large systems of agents

have obtained a growing interest in heterogeneous communities in social and

life sciences, see e.g. [9,5,12,13,16,18,17,20,30,35,41]. In particular,

thanks to their cooperative nature, the dynamics leading to opinion formation

phenomena have been often described through the methods of statistical mech-

anics [7,14,36,46,52]. Amongst other approaches, kinetic theory provided a

sound theoretical framework to investigate the emerging patterns of such sys-

tems [28,29,48]. In this modelling setting, the microscopic, individual-based,

opinion variations take place through binary interaction schemes involving the

presence of social forces, whose eﬀects are observable at the macroscopic scale

[44]. The equilibrium distribution describes the formation of a relative con-

sensus about certain opinions [43,48,49]. In this direction, it is of paramount

importance to obtain reduced complexity models whose equilibrium distribu-

tion is explicitly available under minimal assumptions [31,48]. The deviation

from global consensus appears in the form of opinion polarization, i.e. the di-

vergence away from central positions towards extremes [38]. This latter feature

of the agents’ opinion distribution is frequently observed in problems of choice

formation [4].

The derivation of classical compartmental epidemiological dynamics from

particle systems have been recently explored as a follow-up question on the

eﬀectiveness of available modelling approaches. Indeed, epidemics, as well as

many other collective phenomena, can be easily thought as a result of repeated

interactions between a large number of individuals that eventually modify their

epidemiological state. The transition rates between epidemiologically relevant

states are furthermore inﬂuenced by several phenomena linked to the disease

itself, and to the social behaviour of individuals. Without attempting to revise

the whole literature, we mention [1,8,19,22,23,24,39,39] and the references

therein for an introduction to the subject. Amongst them, contact dynamics

are particularly relevant for contact-based disease transmissions.

2

In this work we introduce a novel kinetic model that takes into account

opinion formation dynamics of the individuals’ protective behaviour coupled

with epidemic spreading. These dynamics will result structurally linked due to

the mutual inﬂuence of opinion formation processes and the transmission of the

infection. In this direction, we mention the recent results in [34,54]. Thanks to

the kinetic approach, we can derive from microscopic agent-based dynamics the

observable macroscopic trends of the infection. The new derived model encodes

all the information of the opinion-based interactions, and describes coherent

transition rates penalizing agents clustering on a weak protective behaviour.

We will observe how opinion polarization can trigger an increasing spread of

infection in society.

In more details, the paper is organized as follows: in Section 2we introduce

a kinetic epidemic model where agents are characterized by their epidemiolo-

gical state and their opinion. Hence, a reduced complexity operator is derived

to compute the large time opinion distribution of the system of agents and we

discuss minimal assumptions to observe opinion polarization. In Section 3we

derive a macroscopic system of equations by considering an equilibrium closure

method. The derived macroscopic model expresses the evolution at the epi-

demic scale of the conserved quantities in the operator for opinion exchanges.

Finally, in Section 4we present several numerical tests showing the coherence of

the presented closure strategy with the initial kinetic model in suitable scales.

Furthermore, in the latter section we explore the possibility of considering more

complex interaction functions in the opinion exchange process together with the

inﬂuence of opinion polarization on the spreading of the disease.

2 A kinetic model approach for consensus form-

ation and epidemic dynamics

In this section we introduce a kinetic compartmental model for the spreading

of an infectious disease that is coupled with the evolution of the opinions’ of

individuals. We consider a system of agents that can be subdivided in the

following epidemiologically relevant states: susceptible (S) agents are the ones

that can contract the disease, infectious agents (I) are responsible for the spread

of the disease, exposed (E) have been infected but are still not contagious and,

ﬁnally, removed (R) agents cannot spread the disease. Each agent is endowed

of a continuous opinion variable w∈Iwhich varies continuously in I= [−1,1],

where −1 and 1 denote two opposite beliefs on the protective behaviour. In

particular, w=−1 means that the agents do not believe in the necessity of

protections (like wearing masks or reducing daily contacts) whereas w= 1 is

linked to maximal agreement on protective behaviour. We also assume that

agents characterized by high protective behaviour are less likely to contract the

infection.

With the aim to incorporate the impact of opinion evolution in the dynamics

of infection we denote by fJ(w, t) the distribution of opinions at time t≥0 of

agents in the compartment J∈ C ={S, E, I , R}. In particular, fJ=fJ(w, t) :

[−1,1]×R+→R+is such that fJ(w, t)dw represents the fraction of agents with

opinion in [w, w +dw] at time t≥0 in the Jth compartment. Furthermore, we

3

impose

X

J∈C

fJ(w, t) = f(w, t),Z1

−1

f(w, t)dw = 1,

while the mass fractions of the population in each compartment and their mo-

ment of order r > 0 are given by

ρJ(t) = Z1

−1

fJ(w, t)dw, ρJ(w, t)mr,J =Z1

−1

xrfJ(w, t)dw. (1)

In the following, to simplify notations, we will indicate with mJ(t), J∈ C, the

mean opinion in the compartment Jcorresponding to r= 1.

We assume that the introduced compartments of the model can have diﬀerent

impact in the opinion dynamics. The kinetic model for the coupled evolution

of opinions and infection is given by the following system of kinetic equations

∂tfS(w, t) = −K(fS, fI)(w, t) + 1

τQS(fS, fS)(w, t),

∂tfE(w, t) = K(fS, fI)(w, t)−σEfE(w, t) + 1

τQE(fE, fE)(w, t),

∂tfI(w, t) = σEfE(w, t)−γfI(w, t) + 1

τQI(fI, fI)(w, t),

∂tfR(w, t) = γfI(w, t) + 1

τQR(fR, fR)(w, t),

(2)

where τ > 0 and QJ(·,·) characterizes the evolution of opinions of agents that

belong to the compartment J∈ C. In the next section we will specify the form

of these operators describing binary opinion interactions among agents. The

parameter σE>0 is such that 1/σEmeasures the mean latent period for the

disease, whereas γ > 0 is such that 1/γ > 0 is the mean infectious period [26].

In (2) the transmission of the infection is governed by the local incidence rate

K(fS, fI)(w, t) = fS(w, t)Z1

−1

κ(w, w∗)fI(w∗, t)dw∗,(3)

where κ(w, w∗) is a nonnegative decreasing function measuring the impact of

the protective behaviour among diﬀerent compartments. A leading example for

the function κ(w, w∗) can be obtained by assuming

κ(w, w∗) = β

4α(1 −w)α(1 −w∗)α,(4)

where β > 0 is the baseline transmission rate characterizing the epidemics and

α > 0 is a coeﬃcient linked to the eﬃcacy of the protective measures. In Figure

1we represent the introduced function κ(·,·) for several values of α > 0. We

may observe how for α1 we We highlight that in the simple case α= 1 we

get

K(fS, fI)(w, t) = β

4(1 −w)fS(w, t)(1 −mI(t))I(t)≥0, I(t)≥0

with K(fS, fI)≡0 in the case mI≡1 or in the case where all susceptible agents

are concentrated in the maximal protective behaviour w= 1.

4

Figure 1: We sketch the function κ(w, w∗) in (4) for α=1

2(left) and α= 1

(right). In both cases, we ﬁxed the coeﬃcient β=1

2.

2.1 Kinetic models for opinion formation

The dynamics of opinion formation have often been described by resorting to

methods of statistical physics, see e.g. [14,32]. In particular, kinetic the-

ory provide a sound theoretical background to model fundamental interactions

among agents and to provide a convenient dynamical structure for related follow-

up questions on control problems and network formation [2,48]. In the afore-

mentioned kinetic models, the opinion variation of large systems of agents de-

pends on binary interactions whose are driven by social forces determining the

formation of consensus about certain opinions. The emerging distribution of

opinions can be evaluated at the macroscopic level [41,43]. Recent advance-

ments have been devoted to include external inﬂuences in opinion formation

models to capture realistic complex phenomena. Without intending to review

the very huge literature on the topic, we mention [7,25,28,29] and the refer-

ences therein.

The elementary interactions between agents weight two opposite behaviour,

the ﬁrst is the compromise propensity, i.e. the tendency to reduce the opinion

distance after interaction, and the second is the self-thinking, corresponding

to unpredictable opinion deviations. In details, an interaction between two

individuals in the compartments J∈ C with opinion pair (w, w∗) leads to an

opinion pair (w0, w0

∗) deﬁned by the relations

w0=w+λJP(w, w∗)(w∗−w) + D(w)ηJ

w0

∗=w∗+λJP(w∗, w)(w−w∗) + D(w∗)˜ηJ,(5)

where λJ∈(0,1) and P(w, w∗)∈[0,1] is an interaction function. In (5) we

further introduce the local diﬀusion function D(w), and ηJ,˜ηJare independent

and identically distributed centered random variables with ﬁnite variance hηJi=

hηJi=σ2

J, where we indicate with h·i the expected value with respect to the

distribution of the random variables.

As observed in [44] we have that the mean opinion is conserved for symmetric

interaction functions, P(w, w∗) = P(w∗, w) for all w, w∗∈[−1,1]. Indeed, from

(5) we get

hw0+w0

∗i=w+w∗+λJ(P(w, w∗)−P(w∗, w))(w∗−w),

5

which reduces to hw0+w0

∗i=w+w∗under the aforementioned assumptions.

Furthermore, if we consider the mean energy we get

(w0)2+ (w0

∗)2=w2+w2

∗+λ2

JP2(w, w∗) + P2(w∗, w)(w∗−w)2

+ 2λJ[P(w, w∗)w−P(w∗, w)w∗](w∗−w)

+σ2

J(D2(w) + D2(w∗)),

meaning that the energy is not conserved on average in a single binary inter-

action. In the absence of the stochastic component, σ2

J≡0, we get that for

symmetric interactions the mean energy is dissipated

(w0)2+ (w0

∗)2=w2+w2

∗−2λJP(w, w∗)(w∗−w)2+o(λJ)≤w2+w2

∗+o(λJ)

The physical admissibility of interaction rules (5) is provided if |w0|,|w0

∗| ≤ 1

for |w|,|w∗| ≤ 1. We observe that

|w0| ≤ |(1 −λJP(w, w∗))w+λJP(w, w∗)w∗+D(w)ηJ|

≤(1 −λJP(w, w∗))|w|+λJP(w, w∗) + D(w)|ηJ|,

since |w∗| ≤ 1, from which we get that the suﬃcient condition for |w0| ≤ 1 is

provided by

D(w)|ηJ| ≤ (1 −λJP(w, w∗))(1 − |w|),

which is satisﬁed if a constant c > 0 exists and is such that

(|ηJ| ≤ c(1 −λJP(w, w∗))

c·D(w)≤1− |w|,(6)

for all w, w∗∈[−1,1]. Since 0 ≤P(·,·)≤1 by assumption, the ﬁrst condition

in (6) can be enforced by requiring that

|ηJ| ≤ c(1 −λJ).

Therefore it is suﬃcient to consider the support of the random variables de-

termined by |ηJ| ≤ c(1 −λJ). The second condition in (6) forces D(±1) = 0.

Other choices for the local diﬀusion function have been investigated in [44,48].

The collective trends of a system of agents undergoing binary interactions

(5) are determined by a Boltzmann-type model having the form

∂tfJ(w, t) = 1

τQJ(fJ, fJ),(7)

with τ > 0 and

QJ(fJ, fJ)(w, t) = Z1

−11

0JfJ(0w, t)fJ(0w∗, t)−fJ(w, t)fJ(w∗, t)dw∗,

where (0w, 0w∗) are pre-interaction opinions generating the post-interaction opin-

ions (w, w∗) and 0Jis the Jacobian of the transformation (0w, 0w∗)→(w, w∗).

6

2.2 Derivation of a Fokker-Planck model

The equilibrium distribuion of the kinetic model (7) is very diﬃcult to obtain

analytically. For this reason, several reduced complexity models have been pro-

posed. In this direction, a deeper insight on the equilibrium distribution of the

kinetic model can be obtained by introducing a rescaling of both the interaction

and diﬀusion parameters having roots in the so-called grazing collision limit of

the classical Boltzmann equation [15,43]. The resulting model has the form

of an aggregation-diﬀusion Fokker-Planck-type equation, encapsulating the in-

formation of microscopic dynamics. For the obtained surrogate model, the study

of asymptotic properties is typically easier than the original kinetic model.

We start by observing that we can conveniently express the operators QJ(·,·)

in weak form. Let ϕ(w) denote a test function, thus for J∈ C we have

Z1

−1

ϕ(w)QJ(fJ, fJ)(w, t)dw

=Z1

−1

(ϕ(w0)−ϕ(w))fJ(w, t)fJ(w, t)dw∗dw,

where w0is deﬁned in (5). The prototype of a symmetric interaction function

Pis given by the constant function P≡1. In this case, we may obtain analytic

insight on the large time distribution of the system by resorting to a reduced

complexity Fokker-Planck-type model [48]. We introduce the so-called quasi-

invariant regime

λJ→λJ, σ2

J→σ2

J,(8)

where > 0 is a scaling coeﬃcient. We have

ϕ(w0)−ϕ(w)

=ϕ0(w)hw0−wi+1

2ϕ00(w)(w0−w)2+1

6ϕ000( ¯w)(w0−w)3,

where min{w, w0}<¯w < max{w, w0}. Plugging the above expansions in the

Boltzmann-type model we have

d

dt Z1

−1

ϕ(w)fJ(w, t)dw =

λJρJZ1

−1Z1

−1

ϕ0(w)(mJ−w)fJ(w, t)dw

+σ2

2Z1

−1

ϕ00(w)D2(w)fJ(w, t)dw +R(fJ, fJ),

(9)

where R(fJ, fJ) is a reminder term

R(fJ, fJ)(w, t) = 1

2Z1

−1

ϕ00(x)2λ2

J(w∗−w)2fJ(w, t)dw

+1

6Z1

−1Z1

−1

ϕ000(w)(λJ(w∗−w) + D(w)ηJ)3fJ(w, t)fJ(w∗, t)dw dw∗

7

Hence, in the time scale ξ=t, introducing the distribution gJ(w, τ ) = fJ(w, ξ /),

we have that ∂ξgJ(w, ξ) = 1

∂tfJand (9) becomes

d

dξ Z1

−1

ϕ(w)gJ(w, ξ)dw =λJZ1

−1Z1

−1

ϕ0(w)(mJ−w)gJ(w, ξ)dw

+σ2

J

2Z1

−1

ϕ00(w)D2(w)gJ(w, ξ)dw +1

R(gJ, gJ)(w, ξ),

where now 1

R(gJ, gJ)→0 under the additional hypothesis |ηJ|3<+∞, see

[18,48]. Consequently, for →0+, from the above equation we have

d

dξ Z1

−1

ϕ(w)gJ(w, ξ)dw =λJZ1

−1Z1

−1

ϕ0(w)(mJ−w)gJ(w, ξ)dw

+σ2

J

2Z1

−1

ϕ00D2(w)gJ(w, ξ)dw.

Now, with a slight abuse of notation, we restore t≥0 as time variable. In view

of the smoothness of ϕ, integrating back by parts the terms on the right hand

side, we ﬁnally get the Fokker-Planck-type model

∂tfJ(w, t) = ¯

QJ(fJ)(w, t)

=∂wλJ(w−mJ)fJ(w, t) + σ2

J

2∂w(D2(w)fJ(w, t))(10)

complemented by the following no-ﬂux boundary conditions

λJ(w−mJ)fJ(w, t) + σ2

J

2∂w(D2(w)fJ(w, t))w=±1= 0

D2(w)fJ(w, t)w=±1= 0.

We can observe that the steady state of the Fokker-Planck-type model (10)

is analytically computable under suitable hypotheses on the local diﬀusion func-

tion. If D(w) = √1−w2, then the large time behavior of the model is given by

a Beta distribution having the form

f∞

J(w) = (1 + w)

1+mJ

νJ

−1(1 −w)

1−mJ

νJ

−1

22

νJ

−1B1+mJ

νJ,1−mJ

νJ, νJ=σ2

J

λJ

,(11)

where B(·,·) indicates the Beta function. It is worth to highlight that the ﬁrst

two moments of the obtained Beta distribution are deﬁned as follows

Z1

−1

wf ∞

J(w)dw =mJ;Z1

−1

w2f∞

J(w)dw =νJ

2 + νJ

+2

2 + νJ

m2

J.(12)

We can observe that the obtained model is suitable to describe classical

consensus-type dynamics. This behaviour is observed if the compromise force

is stronger than the one characterizing self-thinking, i.e. σ2

J< λJ. On the

other hand, if self-thinking is stronger than the compromise propensity, i.e.

σ2

J> λJ, we observe opinion polarization of the society. In Figure 2we depict

8

-1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 1

0

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

-1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 1

0

0.5

1

1.5

2

2.5

Figure 2: We depict the equilibrium distribution (11) for several choices of

the parameter νJ>0 and for mJ= 0 (left) or mJ= 0.2 (right). Opinion

polarization is observed for νJ>1 whereas consensus formation corresponds to

νJ1.

the equilibrium distribution (11) for several choices of the parameter νJ>0. In

the right ﬁgure we assume that mJ= 0 whereas, in the left ﬁgure, we consider

the asymmetric case with mJ= 0.2. We may observe that opinion polarization

is obtained in the case νJ>1 as discussed.

Remark 2.1.In the more general case where interactions between agents is

weighted by a nonconstant function P(w, w∗)∈[0,1], we may obtain the non-

local Fokker-Planck-type model

∂tfJ(w, t) = ∂wB[fJ](w, t)fJ(w, t) + σ2

2∂wfJ(w, t)

where

B[fJ](w, t) = Z1

−1

P(w, w∗)(w−w∗)fJ(w∗, t)dw∗.

In this case, it is diﬃcult to get an analytical formulation of the steady state

distribution.

3 Macroscopic opinion-based SEIR dynamics

Once the equilibrium distribution of the operators QJ(fJ, fH)(w, t) is charac-

terised, we can study the behaviour of the original system (2). In this section

we compute the evolution of observable macroscopic equations of the introduced

kinetic model for epidemic dynamics with opinion-based incidence rate.

3.1 Derivation of moment based systems

Let us rewrite the original model (2) with the reduced complexity Fokker-

Planck-type operators deﬁned in Section 2.2. We obtain the following model

9

∂tfS(w, t) = −K(fS, fI) + 1

τ¯

QS(fS)(w, t),

∂tfE(w, t) = K(fS, fI)−σEfE(w, t) + 1

τ¯

QE(fE)(w, t),

∂tfI(w, t) = σEfE(w, t)−γfI(w, t) + 1

τ¯

QI(fI)(w, t),

∂tfR(w, t) = γfI(w, t) + 1

τ¯

QR(fR)(w, t)

(13)

where K(·,·) has been deﬁned in (3). The system of kinetic equations (13) is

further complemented by no-ﬂux boundary conditions at w=±1 and contains

the information on the spreading of the epidemic in terms of the distribution of

opinions of a population of agents.

Integrating the model (2) with respect to the wvariable and recalling that

in the presence of no-ﬂux boundary conditions the Fokker-Planck operators are

mass and momentum preserving, we obtain the evolution of mass fractions ρJ,

J∈ C,

d

dtρS(t) = −β

4(1 −mI−mS+mSmI)ρSρI,

d

dtρE(t) = β

4(1 −mI−mS+mSmI)ρSρI−σEρE,

d

dtρI(t) = σEρE−γρI,

d

dtρR(t) = γρI,

(14)

observe that (1 −mI−mS+mSmI)ρSρI= (1 −mI)(1 −mS)ρSρI≥0 since

ρImI, ρSmS∈[−1,1]. Unlike the classical SEIR model, the system for the

evolution of mass fractions in (14) is not closed since the evolution of ρJ,ρJ∈ C

depends on the evolution of the local mean opinions mJ,J∈ C. The closure of

system (14) may be formally obtained by resorting to a limit procedure. The

main idea is to observe that the typical time scale of the opinion dynamics is

faster than the one of the epidemic, and therefore τ1. Consequently, for small

values of τthe opinion distribution of the Jth compartment reaches its local

Beta-type equilibrium with a mass fraction ρJand local mean opinion mJas

veriﬁed in Section 2.2. In particular, we observe exponential convergence of the

derived Fokker-Planck equation (10) towards the local Maxwellian parametrised

by the conserved quantities, i.e. ρJand mJ, see [31].

Hence, to get the evolution of mean values we can multiply by wand integ-

rate (13) to get system

d

dt(ρS(t)mS(t)) = −β

4ρI(1 −mI)Z1

−1

w(1 −w)fS(w, t)dw,

d

dt(ρE(t)mE(t)) = β

4ρI(1 −mI)Z1

−1

w(1 −w)fS(w, t)dw −σEmEρE,

d

dt(ρI(t)mI(t)) = σEmEρE−γmIρI,

d

dt(ρR(t)mR(t)) = γmIρI,

10

which now depends on the second order moment, making this system not closed.

It is now possible to close this expression by using the energy of the Beta-type

local equilibrium distribution as in (12). We have

m2,J =ρJ

νJ+ 2m2

J

2 + νJ

,(15)

where νS=σ2/λSand mJis the local mean opinion in the Jth compartment

(1)

Hence, we have

d

dt(ρS(t)mS(t)) = −β

4(1 −mI)ρIρSmS−νS+ 2m2

S

2 + νS

which gives

ρS(t)d

dtmS(t) = −β

4(1 −mI)ρIρSmS−νS+ 2m2

S

2 + νS−mS

d

dtρS

where the time evolution of the fraction ρShas been derived in the ﬁrst equation

of (14). The evolution of the local mean mSis therefore given by

d

dtmS(t) = β

4(1 −mI)ρIνS+ 2m2

S

2 + νS−m2

S.

We may apply an analogous procedure for the remaining local mean values in

the compartments of exposed, infected and recovered. to obtain

d

dtmS(t) = β

4

νS

2 + νS

(1 −mI)ρI1−m2

S.

d

dtmE(t) = β

4

ρSρI

ρE

(1 −mI)mS−νS+ 2m2

S

2 + νS−mE(1 −mS)

d

dtmI(t) = σE

ρE

ρI

(mE−mI)

d

dtmR(t) = γρI

ρR

(mI−mR).

(16)

Remark 3.1.In the case of consensus of the susceptible agents, i.e. for νS→0+,

we can observe that d

dt mS(t) = 0 which leads mS(t) = mS(0) for all t≥0. The

spread of the infection therefore depends only on the protective behavior of the

agents on the compartment I∈ C. Furthermore, the trajectory of the second

equation is decreasing in time since

d

dtmE(t) = −β

4(1 −mI)ρI(1 −mS)ρS

mE

ρE

,

and β

4(1 −mI)ρI(1 −mS)ρS/ρE≥0.

Remark 3.2.If the local incidence rate K(fS, fI) in (3)is such that κ(w, w∗)≡

β > 0 than we easily observe that the evolution of mass fractions are decoupled

11

with the local mean opinions since in this case integrating (2) we get

d

dt Z1

−1

fS(w, t)dw =−βZ1

−1

fS(w, t)dw Z1

−1

fI(w, t)dw,

d

dt Z1

−1

fE(w, t)dw =βZ1

−1

fS(w, t)dw Z1

−1

fI(w, t)dw −σEZ1

−1

fE(w, t)dw,

d

dt Z1

−1

fI(w, t)dw =σEZ1

−1

fE(w, t)dw −γZ1

−1

fI(w, t)dw,

d

dt Z1

−1

fR(w, t)dw =γZ1

−1

fI(w, t)dw.

Therefore, the model (2) for constant κ(w, w∗)≡βreduces to the classical SEIR

compartmental model.

3.2 The macroscopic model with saturated incidence rate

It is not restrictive to suppose that infected agents possess enforced situational

awareness. For this reasons, we may consider the case in which mI(t) = ¯mI∈

(0,1). From the ﬁrst equation of (16) we get

d

dtmS(t) = β

4ρI(t)(1 −¯mI)νS

2 + νS1−m2

S(t)

with initial condition mS(0) = m0

S∈[−1,1]. In particular, if m0

S=±1 then

mS(t) = m0

Sfor all t≥0, otherwise if −1< m0

S<1 we get

mS(t) = exp{2Rt

0J(ρI(s))ds} − exp{C0}

exp{C0}+ exp{2Rt

0J(ρI(s))ds},(17)

with C0= log 1−m0

S

1+m0

S

and J(ρI(s)) = β

4

νS

2+νS(1 −¯mI)ρI(s)≥0. We may easily

observe that from (17) we have mS(t)∈(−1,1) for all t≥0.

Hence, plugging (17) into the system for the mass fractions (14) we get

d

dtρS(t) = −¯

βH (t, ρI)ρS(t)ρI(t),

d

dtρE(t) = ¯

βH (t, ρI)ρS(t)ρI(t)−σEρE,

d

dtρI(t) = σEρE−γρI,

d

dtρR(t) = γρI

(18)

where

¯

βH (t, ρI) = ¯

β 1−e2Rt

0J(ρI(s))ds −eC0

e2Rt

0J(ρI(s))ds +eC0!∈(0,1),

and ¯

β=β

4(1 −¯mI). In this case, model (18) is a generalization of classical

models with saturated incidence rate, see [11,37]. In this setting, we derive the

basic reproduction number by deﬁning

D(ρS, ρI) = ¯

βH (t, ρI)ρSρI,

12

and the function D(ρS, ρI) is such that

∂D(ρS, ρI)

∂ρS

>0,∂D(ρS, ρI)

∂ρI

>0

and D(ρS, ρI) is concave since ∂2

∂ρ2

I

D(ρS, ρI)≤0 for all ρS, ρI>0. Hence, the

basic reproduction number R0of the model is given by

R0=1

γlim

ρI→0,ρS→1

∂D(ρS, ρI)

∂ρI

=β(1 −¯mI)

4γ.

4 Numerical examples

In this section we present several numerical examples to show the consistency

of the proposed approach. Based on classical direct simulation Monte Carlo

(DSMC) methods we will show how, in the quasi-invariant limit deﬁned in (8),

the large time distribution of the Boltzmann-type model (7) is consistent with

the one obtained from the reduced complexity Fokker-Planck model (10). In

the following, we will ﬁrst concentrate on the case of interactions leading to a

Beta distribution of the form (11). As a follow-up question we will explore the

observable eﬀects of nonlinear interaction functions. We point the interested

reader to [42] for a detailed discussion on DSMC methods for the Boltzmann

equation.

Hence, in order to approximate the dynamics of the kinetic SEIR model (2),

we resort to classical strong stability preserving schemes combined to recently

developed semi-implicit structure preserving schemes for nonlinear Fokker-Planck

equations [45], see also [40] for further applications. These methods are capable

to reproduce large time statistical properties of the exact steady state with ar-

bitrary accuracy together with the preservation of the main physical properties

of the solution, like positivity and entropy dissipation.

4.1 Test 1: large time behaviour of kinetic opinion form-

ation models

In this section we test the consistency of the quasi-invariant limit to obtain a

reduced complexity Fokker-Planck model. In particular, we concentrate on a

kinetic model for opinion formation where the binary scheme is given by (5) in

the simpliﬁed case P≡1 and for D(w) = √1−w2. As discussed in Section 2.2,

for quasi-invariant interactions as in (8) and in the limit →0+, the emerging

distribution can be computed through the Fokker-Planck model (10) and is

given be the Beta distribution (11).

We rewrite the Boltzmann-type model (7) as follows

∂tfJ(w, t) = 1

τQ+(f, f )(w, t)−f(w, t),

where τ > 0 is a positive constant and

Q+(f, f )(w, t) = Z1

−1

1

0Jf(0w, t)f(0w∗, t)dw∗,

13

Figure 3: Test 1. Comparison between DSMC solution of the Boltzmann-type

problem (7) and the Beta equilibrium solution of the Fokker-Planck model (10)

for several values of νJ= 0.25 (left column) νJ= 2 (right column) and choices

of the initial distribution. In particular we considered the choices in (19) (top

row) and (20) (bottom row). The DSMC scheme has been implemented with

N= 106particles over the time frame [0,5] with ∆t== 10−1,10−3.

where (0w, 0w∗) are the pre-interaction opinions generating the post-interaction

opinions (w, w∗) according to the binary interaction rule (5) and 0Jis the Jac-

obian of the transformation (0w, 0w∗)→(w, w∗). To compute the large time nu-

merical solution of the introduced Boltzmann- type model we consider N= 106

particles and we assume that τ= 1. The quasi-invariant regime of parameters

in (8) is considered for = 10−1,10−3.

In Figure 3we depict the densities reconstructed from the DSMC approach

with N= 106particles at time T= 5 and assuming ∆t== 10−3,10−1. In

the top row we considered the initial distribution

fJ(w, 0) = (1

2w∈[−1,1]

0w /∈[−1,1] (19)

such that mJ(0) = R1

−1f(w, 0)dw = 0 which is conserved in time. In the bottom

row we consider the initial distribution

fJ(w, 0) = (5

8w∈[−0.6,1]

0w /∈[−0.6,1] (20)

such that mJ= 0.2. We further assume that λJ= 1 and σ2

J= 0.25 in the left

column whereas σ2

J= 2 in the right column. Hence, under the introduced choice

14

of parameters we have considered νJ= 0.25 (left column) and νJ= 2 (right

column). The emerging distribution is compared with the Beta distribution

deﬁned in (11). We may observe how, for decreasing values of →0+, we

correctly approximate the large time solution of the surrogate Fokker-Planck-

type problem.

4.2 Test 2: consistency of the macroscopic limit

In this test we compare the evolution of mass and local mean of the distributions

fJ,J∈ C, solution to (2), with the evolution of the obtained macroscopic system

(14)-(16).

We are interested in the evolution fJ(w, t), J∈ C,w∈[−1,1], t≥0 solution

to (2) and complemented by the initial condition fJ(w, 0) = f0

J. We consider

a time discretization of the interval [0, tmax] of size ∆t > 0. We denote by

fn

J(w) the approximation of fJ(w, tn). Hence, we introduce a splitting strategy

between the opinion consensus step f∗

J=O∆t(fn

J)

∂tf∗

J=1

τ¯

QJ(f∗

J, f ∗

J),

f∗

J(w, 0) = fn

J(w), J ∈ C

(21)

and the epidemiological step f∗∗

J=E∆t(f∗∗

J)

∂tf∗∗

S=−f∗∗

S(1 −w)ρ∗∗

I(1 −m∗∗

I)

∂tf∗∗

E=f∗∗

S(1 −w)ρ∗∗

I(1 −m∗∗

I)−σEf∗∗

E

∂tf∗∗

I=σEf∗∗

E−γf ∗∗

I

∂tf∗∗

R=γf ∗∗

I,

f∗∗

J(w, 0) = f∗

J(w, ∆t).

(22)

The operator ¯

QJ(·,·) in (21) has been deﬁned in (10) together with no-ﬂux

boundary conditions. Hence, the solution at time tn+1 is given by the combin-

ation of the two described steps. In particular a ﬁrst order splitting strategy

corresponds to

fn+1

J(w) = E∆t(O∆t(fn

J(w))),

whereas the second order Strang splitting method is obtained as

fn+1

J(w) = E∆t/2(O∆t(E∆t/2(fn

J(w)))),

for all J∈ C. The opinion consensus step (21) is solved by means of a second-

order semi-implicit structure-preserving (SP) method for Fokker-Planck equa-

tions, see [45]. The integration of the epidemiological step (22) is performed with

an RK4 method. In the following, we will adopt a Strang splitting approach.

We consider the following artiﬁcial parameters characterizing the epidemi-

ological dynamics β= 0.4, σE= 1/2, γ= 1/12. These values are strongly

dependent on the infectious disease under investigation. We highlight that,

without having the intention to use real data for the calibration of the presen-

ted model, these values are coherent with several recent works for the COVID-19

pandemic [1,10,24].

15

4.3 Test 2a: equilibrium closure

In this test we assume a constant interaction function P(·,·)≡1 such that

the Fokker-Planck model is characterized by a Beta equilibrium distribution

(11) as shown in Section 2.2. To deﬁne the initial condition we introduce the

distributions

g(w) = (1w∈[−1,0]

0 elsewhere,h(w) = (1w∈[0,1]

0 elsewhere,

and we consider

fS(w, 0) = ρS(0)g(w), fE(w, 0) = ρE(0)g(w),

fI(w, 0) = ρI(0)h(w), fR(w, 0) = ρR(0)h(w),(23)

with ρE(0) = ρI(0) = ρR(0) = 10−2and ρS= 1 −ρE(0) −ρI(0) −ρR(0).

We solve numerically (21)-(22) over the time frame [0, tmax] and we introduce

the grid wi∈[−1,1] with wi+1−wi, where ∆w > 0, i= 1, . . . , Nw. We introduce

also a time discretization such that tn=n∆t, ∆t > 0, and n= 0, . . . , T with

T∆t=tmax. For all the details on the considered numerical scheme we point

the interested reader to [45]. Hence, for several values of τ > 0, we compare the

evolution of the computed observable quantities deﬁned as

ρτ

J(t) = Z1

−1

fJ(w, t)dw, mτ

J(t) = 1

ρτ

J(t)Z1

−1

wfJ(w, t)dw (24)

with the ones in (14)-(16) whose dynamics has been determined through a suit-

able kinetic closure in the limit τ→0+. In (24) we have highlighted the

dependence on the scale parameter τ > 0 through a superscript. It is import-

ant to remark that the introduced closure strategy is essentially based on the

assumption that opinion dynamics are faster than the ones characterizing the

epidemic. Furthermore, we ﬁx as initial values of the coupled system (14)-(16)

the values ρJ(0) and mJ(0), for all J∈ C.

In Figure 4we present the evolution of the macroscopic system (14)-(16)

and of the observable quantities (24) for several τ= 10−5,1,100. The consensus

dynamics is characterized by λJ= 1, σ2

J= 10−3for all J∈ C, such that νS=

10−3. We can easily observe how, for small values of τ1, the macroscopic

model obtained through a Beta-type equilibrium closure is coherent with the

evolution of mass and mean of the kinetic model (2).

In Figure 5we show the evolution of the kinetic distributions fS(w, t) and

fI(w, t) for t∈[0,100]. The parameters characterizing the opinion and epidemic

dynamics are coherent with the ones chosen for Figure 4. We may easily observe

how for τ= 100 the distributions are far from the Beta equilibrium (11) whereas

for τ= 10−5the kinetic distributions fJare of Beta-type. Therefore, for small

τ1, the opinion exchanges are faster than the epidemic dynamics and we are

allowed to assume a Beta-type closure as in (15).

4.4 Test2b: the bounded conﬁdence case

In this test we consider an interaction function of the form

P(w, w∗) = χ(|w−w∗| ≤ ∆), w, w∗∈[−1,1],(25)

16

Figure 4: Test 2a. Evolution of the macroscopic quantities deﬁned in (14)-

(16) and the ones extrapolated from the kinetic model (2) for several values

τ= 10−5,1,102, see (24). Discretization of the domain [−1,1] obtained with

Nw= 201 gridpoints, discretization of the time frame [0,100] obtained with

∆t= 10−1. The initial distributions have been deﬁned in (23) whereas we ﬁxed

λJ= 1 and σ2

J= 10−3for all J∈ C.

17

Figure 5: Test 2a. Evolution of the kinetic distributions fSand fIover the

time interval [0,100] for τ= 100 (top row) and τ= 10−5(bottom row). The

epidemic dynamics have been characterized by β= 0.4,σE= 1/2, γ= 1/12.

The solution of the Fokker-Planck step (21) has been performed through a semi-

implicit SP scheme over the a grid of Nw= 201 nodes and ∆t= 10−1. Initial

distributions deﬁned in (23).

18

where χ(·) is the indicator function, and ∆ ∈[0,2] is a conﬁdence threshold

parameter above which the agents’ with opinions wand w∗do not interact.

In the case ∆ = 0 only agents sharing the same opinion interact, whereas for

∆ = 2 the interaction function is such that P(·,·)≡1 since |w−w∗| ≤ 2 for

all w, w∗∈[−1,1]. Bounded conﬁdence-type dynamics have been introduced

in [36] and have been studied to observe the loss of global consensus. Indeed,

for large times, the agents’ opinion form several clusters whose number and size

depends on the parameter ∆ >0 and the initial opinions. We highlight that,

since bounded conﬁdence interactions (25) are symmetric, the mean opinion is

preserved in time [44].

Proceeding as in Section 2.2, the Fokker-Planck description of a system of

agents in the compartment J∈ C characterized by bounded conﬁdence interac-

tions is given by the following nonlocal operator

¯

QJ(fJ, fJ)(w, t)

=∂wλJZ1

−1

χ(|w−w∗| ≤ ∆)(w−w∗)fJ(w∗, t)dw∗fJ(w, t)

+σ2

J

2∂w(D2(w)fJ(w, t))

(26)

cf. Remark 2.1. The equilibrium distribution of the corresponding nonlocal

model is not explicitly computable and the resulting macroscopic models for

the evolution of observable quantities may deviate from the ones deﬁned in

Section 3. Let us consider the densities

g(w) = (1

2w∈[−1,1]

0 elsewhere,h(w) = (1w∈[0,1]

0 elsewhere

and we consider the initial distributions

fS(w, 0) = ρS(0)g(w), fE(w, 0) = ρE(0)g(w),

fI(w, 0) = ρI(0)h(w), fR(w, 0) = ρR(0)h(w)(27)

with ρE(0) = 0.01, ρI(0) = 0.01, ρR(0) = 0.01 and ρS(0) = 1 −ρE(0) −ρI(0) −

ρS(0).

In Figure 6we show the evolution of the kinetic distributions fS(w, t) and

fI(w, t), t∈[0,100] determined by bounded conﬁdence interactions described

by the nonlocal Fokker-Planck-type operator (26), with ∆ = 1

2,λJ= 1, and

σ2

J= 10−3for all J∈ C. We may observe how the opinion dynamics lead

to two separate clusters centered in −0.5 and in 0.5. Furthermore, coherently

with the modelling assumptions characterizing the incidence rate K(fS, fI)(w, t)

in (3)-(4), the cluster with negative opinions looses mass since it is linked to

agents with weak protective behaviour. The infection is therefore propagated

to these agents and the kinetic distribution fI(w, 0) gains mass for w < 0. We

highlight how the approximated equilibrium density is not coherent with a Beta

distribution. Therefore the evolution of the macroscopic quantities cannot be

obtained through a classical closure method and we need to solve the full kinetic

model.

19

Figure 6: Test 2b. We consider a the bounded conﬁdence interaction function

(25) with ∆ = 1

2. Top row: evolution of mass fractions (left) and mean values

(right) for the agents in compartments Cwith τ= 1 and extrapolated from

the kinetic model (2) with a Fokker-Planck operator ¯

Q(·,·)(w, t) of the form

(26). Bottom row: evolution of the kinetic distributions for the compartments

S, I ∈ C. The solution of the Fokker-Planck step (21) has been performed

through a semi-implicit SP scheme over the a grid of Nw= 201 gridpoints and

∆t= 10−1. Initial distributions deﬁned in (27).

20

Figure 7: Test 2b. We consider a the bounded conﬁdence interaction function

(25) with ∆ = 1

4. Top row: evolution of mass fractions (left) and mean values

(right) for the agents in compartments Cwith τ= 1 and extrapolated from

the kinetic model (2) with a Fokker-Planck operator ¯

Q(·,·)(w, t) of the form

(26). Bottom row: evolution of the kinetic distributions for the compartments

S, I ∈ C. The solution of the Fokker-Planck step (21) has been performed

through a semi-implicit SP scheme over the a grid of Nw= 201 gridpoints and

∆t= 10−1. Initial distributions deﬁned in (27).

21

Figure 8: Test 3. Impact of the coeﬃcient νSin the large time behaviour of

the system (14)-(16) assuming diﬀerent initial conditions on the mean opinions

of the compartments, mJ(0) = −0.5 (left) and mJ(0) = 0.5 (right) for all

J∈ C. The epidemiological parameters are the same of the previous tests

and ﬁxed as follows β= 0.4, σE= 1/2, γ= 1/12. Furthermore we ﬁxed

ρE(0) = ρI(0) = ρR(0) = 0.01 and ρS(0) = 1−ρE(0)−ρI(0)−ρR(0). The system

of ODEs is solved through RK4 over a time interval [0,300] with ∆t= 10−2.

4.5 Test 3: the impact of opinion polarization on the in-

fection dynamics

In this test we exploit the derived macroscopic system of mass fractions and

mean opinions (14)-(16) to investigate the relation between opinion polariza-

tion and large number of recovered individuals. We recall that, assuming P≡1,

opinion polarization is observed if νS>1, see Section 2.2. Hence, we consider

two main cases, supposing that the mean agents’ opinions in all the compart-

ments are exactly alike: the case mJ(0) = −0.5, meaning that the agents in

each compartment have a bias towards weak protective behaviour, and the case

mJ(0) = 0.5, meaning that all the agents are biased towards protective beha-

viour.

In Figure 8we present the large time mass fractions of recovered individuals

ρR(T) obtained as solution to (14)-(16) over the time interval [0, T ], T= 300,

∆t= 10−2, where we ﬁxed the value νS∈[0,10]. In the left ﬁgure we consider

the case mJ(0) = −0.5, whereas in the right ﬁgure we consider the case mJ(0) =

0.5. We can observe how the eﬀect of opinion polarization strongly depends

on the macroscopic initial opinion of the population on protective behaviour.

In details, if the mean opinion is biased towards the adoption of protective

behaviour, i.e. mJ(0) = 0.5, large values of νStrigger an increasing number of

recovered individuals, meaning that the infection have a stronger eﬀect on the

society in the presence of polarized opinions.

On the other hand, if the initial opinion of the population is biased towards

the rejection of protective behaviour, i.e. mJ(0) = −0.5, opinion polarization

is a factor that can dampen the asymptotic number of recovered individuals.

Indeed, opinion polarization in this case pushes a fraction of the population

towards the two extreme positions and a fraction of agents will stick towards a

maximal protective behaviour.

22

Conclusion

In this work we considered the eﬀects of opinion polarization on epidemic dy-

namics. We exploit the formalism of kinetic theory for multiagent system where

a compartmentalization of the total number of agents is coupled with their

opinion evolution. Kinetic models for opinion formation have been developed

in details and are capable to determine minimal conditions for which we can

observe polarization of opinions, i.e. the divergence of opinions with respect

to a neutral center. Agents’ opinions on the adoption of protective behaviour

during epidemics is a central aspects for the collective compliance with non-

pharmaceutical interventions. Thanks to classical methods of kinetic theory we

derived a system of equations that describe the evolution in time of observable

quantities that are conserved during the opinion formation process. In par-

ticular, considering suﬃciently simple interaction functions and local diﬀusion

functions, we get a second order system of equations for the evolution of mass

fractions and mean opinions. This macroscopic system takes into account the

social heterogeneities of agents in terms of their opinions and is derived from

microscopic dynamics in a SEIR compartmentalization. Thanks to recently de-

veloped structure preserving numerical methods, we showed the consistency of

the approach by comparing the system of kinetic equations with the set of mac-

roscopic equations. Furthermore, we analysed more complex interaction func-

tions based on conﬁdence thresholds. The eﬀects of opinion polarization on the

asymptotic number of recovered is measured and strongly depends on the initial

mean opinion of the population. Indeed, if a positive bias towards protective

behaviour is observed opinion polarization is capable to worsen the infection,

whereas, if the population tends to reject protective mechanisms, opinion po-

larization may dampen the total number of infectious agents. Future works

will regard more complex opinion formation processes based on leader-follower

dynamics and dynamics opinion networks. In future works we will tackle the

calibration of the introduced modelling approach.

Data availability statement

The datasets generated during the current study is available from the corres-

ponding author on reasonable request.

Acknowledgements

MZ is member of GNFM (Gruppo Nazionale per la Fisica Matematica) of IN-

dAM, Italy., Italy. MZ acknowledges the support of MUR-PRIN2020 Project

No.2020JLWP23 (Integrated Mathematical Approaches to Socio–Epidemiological

Dynamics).

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