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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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