Communication strategies to contrast anti-vax action: a differential game approach

Abstract

Vaccination is one of the greatest discoveries of modern medicine, capable of defeating many diseases. However, misleading information on the effectiveness of vaccines has caused a decline in vaccination coverage in some countries, leading to the reappearance of related diseases. Therefore, a proper and well-planned pro-vax communication campaign may be effective in convincing people to get vaccinated. We formulate and solve a differential game with an infinite horizon played à la Nash. The players involved in the game are the national healthcare system and a pharmaceutical firm that produces and sells a certain type of vaccine. The former aims to minimize the healthcare costs that unvaccinated people would entail. In turn, the pharmaceutical firm wants to minimize the missed profits from unsold vaccines. The two players run suitable vaccination advertising campaigns to diminish the à-régime number of unvaccinated. The Hamilton-Jacobi-Bellman approach is used to determine a Markovian-Nash equilibrium, studying how communication strategies can be effective in reducing the strength of anti-vax word of mouth.

Full text

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Central European Journal of Operations Research
https://doi.org/10.1007/s10100-024-00917-0
1 3
ORIGINAL PAPER
Communication strategies tocontrast anti‑vax action:
adifferential game approach
AlessandraBuratto1 · RudyCesaretto1· MaddalenaMuttoni1
Accepted: 20 April 2024
© The Author(s) 2024
Abstract
Vaccination is one of the greatest discoveries of modern medicine, capable of
defeating many diseases. However, misleading information on the effectiveness of
vaccines has caused a decline in vaccination coverage in some countries, leading to
the reappearance of related diseases. Therefore, a proper and well-planned pro-vax
communication campaign may be effective in convincing people to get vaccinated.
We formulate and solve a differential game with an infinite horizon played à la Nash.
The players involved in the game are the national healthcare system and a pharma-
ceutical firm that produces and sells a certain type of vaccine. The former aims to
minimize the healthcare costs that unvaccinated people would entail. In turn, the
pharmaceutical firm wants to minimize the missed profits from unsold vaccines. The
two players run suitable vaccination advertising campaigns to diminish the à-régime
number of unvaccinated. The Hamilton-Jacobi-Bellman approach is used to deter-
mine a Markovian-Nash equilibrium, studying how communication strategies can be
effective in reducing the strength of anti-vax word of mouth.
Keywords Differential games· Stationary Markovian nash equilibrium· Vaccine
communication policy· Advertising
Mathematics Subject Classification 49N90· 90B60· 49N10
Rudy Cesaretto and Maddalena Muttoni have contributed equally to this work.
* Alessandra Buratto
alessandra.buratto@unipd.it
Rudy Cesaretto
rudy.cesaretto@gmail.com
Maddalena Muttoni
maddalena.muttoni@studenti.unipd.it
1 Department ofMathematics “Tullio Levi-Civita”, Università degli Studi di Padova, Via Trieste,
63, 35121Padova, Italy
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A.Buratto et al.
1 3
1 Introduction
Vaccination is one of the greatest discoveries of modern medicine. Thanks to vac-
cines, diseases such as poliomyelitis, tetanus, smallpox, diphtheria, and rubella
have been eradicated in many countries. Furthermore, vaccination has recently been
shown to reduce the incidence of some other diseases, for example, human papillo-
mavirus (HPV) infection (Takla etal. 2018), meningitis (Buonomo etal. 2022), and
some forms of cancer. The public health benefits brought by such discoveries are so
important that several countries have even devised mandatory childhood vaccination
policies1,2 .Committees on Vaccination have been constituted in many countries to
plan communication campaigns in favor of vaccination, such as the (STIKO) in Ger-
many by Robert Koch-Institut3 and the European Centre for Disease Prevention and
Control4 have done.
However, people often do not perceive the importance of vaccines, either because
some of the diseases that were eradicated are no longer visible or because their
effects may show up only after long periods. In case a given vaccination is not man-
datory, without memory of the damage the related disease can cause, the perceived
risks of vaccination among some people have begun to outweigh their perceived
benefits (Omer etal. 2009; Buonomo etal. 2013). Some people focus only on the
risk of side effects, which for them appears to be extremely high compared to the
risk associated with contracting the disease, (Salmon etal. 2006). Furthermore, vac-
cine efficacy has recently been debated by skeptics who try to spread the idea that
vaccines are ineffective and even dangerous (see, e.g., Shim etal. (2013), Carrillo
and Lopalco (2012), and Hotez (2017)).
In turn, media, such as magazines, television, the Internet, and social networks,
often present news related to vaccines without submitting them to strict verification
by scientific and health authorities. Sometimes, they spread alarming news without
any foundation, in the worst cases even claiming an association between vaccines
and serious diseases. Just to mention, even though Andrew Wakefield’s claim about
a causal relation between vaccines and autism was refuted by the scientific commu-
nity; see Taylor etal. (2014), such a conjecture has caused a decline in vaccination
coverage, especially in certain countries.
Due to these fake news, some diseases were taken too lightly and gave rise to
the vaccine hesitancy effect; see, e.g., Bozzola etal. (2018) and White etal. (2023).
Shim etal., Shim etal. (2013), assert that if the great benefits to society of measles
vaccination are to be maintained, the public must be educated about these benefits
in order to increase public confidence. When organizing vaccine administration,
national health systems must consider many aspects, and in the last decade the issue
of correct communication about the vaccination campaign has become crucial. The
authors observe that the effectiveness of vaccination programs can be jeopardized
1 https:// www. immun ize. org/ laws, retrieved on 2023/06/19.
2 https:// ijpon line. biome dcent ral. com, retrieved on 2023/06/19.
3 https:// www. rki. de/ EN/ Conte nt/ infec tions, retrieved on 2023/06/19.
4 https:// www. ecdc. europa. eu/ en, retrieved on 2023/06/19.
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1 3
Communication strategies tocontrast anti‑vax action: a…
by public misperceptions of vaccine risk. They illustrate “the importance of public
education on vaccine safety and infection risk to achieve vaccination levels that are
sufficient to maintain herd immunity.”
In this paper, we study the effect of a vaccination advertising campaign in sup-
porting countries to increase the coverage of measles, rubella, and other vaccines.
The idea arises from a recent report by the World Health Organization (WHO),
which underscores the impact of the COVID-19 pandemic on surveillance and
immunization efforts. The report highlights that the suspension of immunization
services has decreased the estimates of immunization coverage for many infectious
diseases. In fact, during the last three years, the COVID-19 pandemic stopped sur-
veillance and immunization efforts, putting many children at risk for preventable
diseases. “Approximately 25 million infants missed at least one dose of the measles
vaccine through routine immunization in 2021.”5 Globally low immunization rates
increased the chances of outbreaks and endanger unvaccinated children. The World
Health Organization just decided to work towards regional measles elimination by
strengthening immunization programs (e.g., Measles and rubella strategic frame-
work: 2021-2030) and implementing effective surveillance systems.
Perceiving the same need for a focused public educational system, this paper aims
to formalize in a mathematical context the problem of planning a pro-vax commu-
nication campaign to convince people to get vaccinated. In what follows, we assume
that the vaccination we are dealing with is not mandatory.
Our research questions are as follows.
• How does negative word of mouth affect the evolution toward herd immunity?
• How can the pro-vax communication campaigns of the national health-care sys-
tem and of pharmaceutical firms speed up such an achievement?
• How can a proper pro-vaccination campaign sustain the vaccination efficiency of
the national health-care system?
The remainder of the paper is organized as follows. In Sect. 2 we briefly review
the literature and position our contribution. In Sect. 3 we introduce our model for
a controlled evolution of the unvaccinated population, together with the cost func-
tionals associated with such an attempt. We formalize the problem in a differential
game framework. In Sect.4 we determine the optimal communication strategies that
constitute the associated steady-state feedback Nash equilibrium. Section5 presents
some sensitivity analysis of the solution together with interpretations of the results.
Section6 concludes.
5 https://www.who.int/news-room/, retrieved on 2023-06-19.
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A.Buratto et al.
1 3
2 Brief literature background
A consistent stream of literature tackles the issue of controlling infectious dis-
eases based on epidemic models, in particular, on behavioral Susceptible-
Infected-Recovered (SIR) models. In the book Manfredi and d’Onofrio (2013)
a vast literature on vaccination and other influences of human behavior on the
spread of infectious diseases is presented. A detailed report reviewing models that
account for behavioral feedback and / or the spatial / social structure of the popu-
lation can be found in Wang etal. (2016). More recent publications among the
same stream of literature are e.g. d’Onofrio and Manfredi (2020) and Buonomo
etal. (2022). An interesting approach to the vaccination problem in the context
of game theory is tackled in Shim etal. (2013), where a game-theoretic model of
disease transmission and vaccination is formalized as a population game. Here,
the game-theoretic epidemiological analysis performed can yield insights into the
interplay between anti-vaccine behavior, vaccine coverage, and disease dynamics.
More recently in Matusik and Nowakowski (2022) a game-theoretical approach
has been used to model the control of COVID-19 transmission, always in the con-
text of SIR dynamics.
The so-called word-of-mouth effect plays a crucial role in the spread of anti-
vax beliefs. Bauch in Bauch (2005) studies the strategic interaction between
individuals when they decide whether or not to vaccinate, using an imitation
dynamic game. The role of word of mouth in voluntary vaccination planning is
presented in Bhattacharyya etal. (2015), where the synergetic feedback between
word of mouth and the epidemic dynamics controlled by voluntary vaccination
is analyzed. The authors present an epidemiological model with a social learn-
ing component that incorporates the reciprocal influence of population groups, as
well as the feedback that can occur from the incidence of diseases. They model
this social interaction through a game-theoretical framework using the concept of
payoff, adapted from applications of Game Theory to Economics.
Just because anti-vax behavior is often associated with word of mouth, rather
than with scientific data and information, it seems interesting to tackle the prob-
lem not necessarily from an epidemiological point of view but from a sociologi-
cal point of view. Therefore, the communication approach derived from the the-
ory of dynamic advertising models, described in Huang et al. (2012), suggests
a correct communication policy to defeat the spread of the anti-vax movement.
In a communication framework, El Ouardighi etal. in ElOuardighi etal. (2016)
consider two different types of word of mouth: negative (or adverse) and positive
(or favorable) according to the different reactions of satisfied and dissatisfied cus-
tomers. The authors stress how negative word of mouth is more influential than
positive, especially for brands with which potential customers are not familiar,
and this could be the case for the vaccine issue.
Within the literature stream based on dynamic advertising models, Grosset
and Viscolani (2021) formulate and solve an optimal control problem to deter-
mine a provaccination communication campaign that contrasts the effect of
antivaccination word of mouth, with the objective of minimizing the care cost
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1 3
Communication strategies tocontrast anti‑vax action: a…
induced by unvaccinated people and the cost resulting from the communication
campaign. The authors propose an upper bound to the final number of unvacci-
nated people to guarantee herd immunity. In such a model the programming inter-
val is fixed, while in Grosset and Viscolani (2020) the authors study a variable-
final-time optimal control problem to stress the importance of reaching the given
herd immunity threshold, rather than reducing costs in a fixed-length program-
ming period.
In Buratto etal. (2020), a model for the aforementioned national health problem
is proposed using a linear-quadratic differential game in a finite time horizon. The
model aspires to understand how an optimal interaction between the communication
campaigns of the healthcare system and of a pharmaceutical firm that produces a
given vaccine can help increase vaccination coverage. The healthcare system wants
to minimize the number of unvaccinated people at a minimum cost. The pharmaceu-
tical firm aims to maximize its profits while reducing the number of unvaccinated
people.
This paper belongs to the last group of models presented above, named Commu-
nication models, where the focus is on the communicative approach and considers
the “educational plan to vaccination” as a possible communication strategy that can
be planned by both the national healthcare system and pharmaceutical firms. Refer-
ences cited above Grosset and Viscolani (2020, 2021), and Buratto et al. (2020))
ssume that the unvaccinated population, affected by negative word of mouth,
diverges to infinity if it is not supported by a pro-vaccine campaign. Here, we adopt
a more realistic dynamics where the à-régime number of unvaccinated people is
increased by the adverse action of anti-vax negative word of mouth and converges to
a finite value, which can be deduced from the annual report published by the World
Health Organization (WHO).6
Moreover, the literature cited above considers only finite-time horizon optimal
control problems, while in this paper, being interested in a long-term plan for the
provaccination campaign, we consider the plan over an infinite horizon.
We study the interaction between the communication campaigns of the healthcare
system and of a pharmaceutical firm that produces a vaccine for a given disease,
formulating a differential game played à la Nash. We use the Hamilton-Jacobi-Bell-
man approach to determine a Markovian-Nash equilibrium. In recent years, with the
introduction of big data solutions, healthcare management can rely on accurate and
up-to-date reports. Mobile contact track & trace apps, recently urged by the Covid-
19 emergency, provide constant information on the number of vaccinated (and con-
sequently of unvaccinated) individuals.7 Therefore, feedback strategies based on the
number of unvaccinated are not only credible but also more reliable than Open-Loop
ones, as the former are time-consistent. It is commonly known, as reported also in
6 To be precise, WHO reports annually the report on global vaccination coverage for each infectious
disease in https://www.who.int/news-room. The number of unvaccinated people for a given disease can
be deduced by subtracting the WHO declaration from the size of the population interested in the related
vaccination.
7 see e.g. https:// theva ccine app. com/, retrieved on 2023/06/19.
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A.Buratto et al.
1 3
Della-Marca and d’Onofrio (2021), that in health economics, it is preferable, when-
ever possible, to control a system by means of a feedback-based strategy, which aims
to minimize the economic and human burden of some health-related phenomena.
On the basis of these considerations, we look for communication strategies that con-
stitute a Markovian Nash equilibrium.
3 The model
We assume that the population to which vaccination is devoted is stationary. The
dynamics we analyze describes the evolution of the unvaccinated population, as is
done in the related literature cited above (see Grosset and Viscolani (2020, 2021),
and Buratto etal. (2020)).
For this purpose, let x(t) be the number of unvaccinated individuals at time t, and
let
x0
be its value at the initial time.
The World Health Organization annually publishes data related to global trends
and the total number of reported cases of vaccine-preventable diseases (VPD).8 In
an idealistic vaccination system, the number of unvaccinated should decrease and
tend to zero (neglecting people who cannot be vaccinated due to more serious
immunodeficences). However, it can be observed from WHO’s data that there is
always a significantly high average number of unvaccinated. This residual number of
unvaccinated can be attributed to the presence of antivax action. On the other hand,
in each time unit, a fixed percentage of the unvaccinated population voluntary gets
the vaccine, convinced by peers and networks, or even by the spread of the disease.
We can formalize this type of scenario with the following dynamics of the unvac-
cinated people
The term
w>0
represents the constant number of people that at a given time may
be considered as new-unvaccinated, because they’ve just entered into the subset of
population ought to be vaccinated and did not get the vaccine. The reasons for such a
decision can be varied, including the negative information about vaccines and vacci-
nation that no-vax movement spreads throughout a word-of-mouth mechanism. For
simplicity, in what follows we will call the parameter w a word-of-mouth parameter.
The parameter
r>0
represents the instantaneous vaccination rate. It measures
the intensity of vaccination and in some way it describes the efficiency of the
national vaccination system: the faster the national healthcare system in the vac-
cination issue, the higher r. From the dynamics we can observe that the number
of unvaccinated increases with w, however, it decreases due to the effect of spon-
taneous vaccinations. Unlike in Grosset and Viscolani (2021), where the number
of unvaccinated tends to explode in the absence of a communication action, due
to negative word of mouth, here we assume, in a more realistic formalization,
(1)
x(t)=w−rx(t).
8 https:// www. who. int/ news- room/ fact- sheets/ detail/ immun izati on- cover age, retrieved on 2023/06/19.
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1 3
Communication strategies tocontrast anti‑vax action: a…
that the unvaccinated group converges to a finite positive value. Indeed, as t goes
to infinity in dynamics (1), the number of unvaccinated tends to the following
à-régime level
This value, which increases in w and decreases in r, is in any case finite, and it cor-
responds to the nonnegative residual of unvaccinated people that can be deduced by
the annual World Health Organization report.
However, as Wang etal. stress in Wang etal. (2016), there are scenarios in
which voluntary vaccination is not sufficient to provide herd immunity. If word
of mouth w is too high or the effectiveness of vaccination r is too low, then
xSS
may be too large; far from the level requested for herd immunity. This issue may
become too expensive for the national healthcare system to sustain; therefore, a
pro-vaccination communication campaign is needed.
In this paper, we formulate a dynamic model assuming to be exactly in the
situation in which the negative word of mouth is so high that a pro-vaccination
campaign is necessary. Since we are interested in the scenario where communica-
tion is necessary, in what follows we will assume that the parameter w is greater
than a given threshold, which guarantees nontrivial equilibrium strategies. The
value of such a threshold will be specified later in the next section.
We assume that both the national healthcare system (S) and the pharmaceu-
tical firm (F) are independently planning their own communication campaign
to promote vaccination. Let
𝜙S(t)∈US
and
𝜙F(t)∈UF
be the pro-vax advertis-
ing efforts of the national healthcare system and the pharmaceutical company,
respectively. We assume that feasible communication strategies are non-negative
and reasonably bounded. In what follows, we assume that the upper bounds for
such strategy functions are sufficiently high. This will permit us to concentrate on
the characteristics of non-trivial inner solutions. Let
𝛿S,𝛿F>0
be the effective-
ness of the two communication intensities, as in Buratto etal. (2020). The evolu-
tion of the unvaccinated is affected by these controls according to the following
dynamics
At first glance, we can observe that if the number of unvaccinated x is zero at a given
time, then a positive communication effort may drive the state function below zero,
and this would not be meaningful. However, it will be proved that under optimal
conditions, given our assumption of a significantly high value for the variable w, the
positivity of the state function is ensured.
After describing the dynamics, let us focus on the payoff functions of the two
players. The national healthcare system and the pharmaceutical firm both seek to
minimize their respective costs associated with the number of unvaccinated indi-
viduals, along with reducing their communication costs. For any
t≥0,
we assume
the following cost flows for the two players respectively
(2)

xSS =
w
r
.
(3)
{
x(t)=w−rx(t)−𝛿S𝜙S(t)−𝛿F𝜙F(t)
,
x(0)=x0.
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A.Buratto et al.
1 3
For what concerns the costs due to the number of unvaccinated individuals, from
the national healthcare system point of view, the number of unvaccinated individ-
uals affects national healthcare costs because a relevant percentage of them need
medications and sometimes even hospitalization. While in Grosset and Viscolani
(2020, 2021) these costs are assumed to be proportional to the number of unvac-
cinated people and therefore linear in the variable x, in our model we assume that
national healthcare costs are quadratic and convex, formalized by the term
𝛽
2
x2(t
)
(with
𝛽>0),
to underscore the significant expenses associated with hospital therapy.
Furthermore, this assumption highlights the crucial point that vaccine refusal not
only puts those who decline vaccination at risk but also increases the chances of dis-
ease transmission for people who interact with unvaccinated individuals. A similar
quadratic assumption for national health system costs due to the number of unvac-
cinated has also been considered in Buratto etal. (2020), where a linear-quadratic
differential game is formulated and solved, although in a finite time horizon.
From the pharmaceutical firm point of view, the number of unvaccinated affects
the revenue of the pharmaceutical firm, although in a different way. Assuming
𝜃>0
to be the unit profit of a given vaccine, the consequent missed revenue for each
unsold vaccine is here formalized as the linear cost
𝜃x(t).
Finally, both players sustain the communication costs associated to their pro-vax
advertising efforts, here assumed to have the following quadratic and convex formu-
lation
k
S
2
𝜙2
S
(t
)
and
k
F
2
𝜙2
F
(t
)
(with
kS
>
0
and
kF>0)
for the healthcare system and the
firm respectively.
All the models cited above neglect the cost of vaccination, as we also do; never-
theless, few papers in the related literature evaluate vaccination costs; among them,
we mention the real option approach used in Favato etal. (2013).
We consider the problem over an infinite horizon. In such a setting, taking into
account the dynamics (3), and discounting the costs
CS(t)
and
CF(t)
in (4), we for-
mulate the following differential game played à la Nash9
healthcare System (S) pharmaceutical Firm (F)
min
𝜙
S∈US∫
+∞
0
e−𝜌t
(
𝛽
2x2(t)+ 𝜅S
2𝜙2
S(t)
)dt
min
𝜙
F∈UF∫
+∞
0
e−𝜌t
(
𝜃x(t)+ 𝜅F
2𝜙2
F(t)
)dt
The table below collects the meaning of the parameters:
(4)
C
S(t) =
𝛽
2
x2(t)+
𝜅
S
2
𝜙2
S(t),CF(t)=𝜃x(t)+
𝜅
F
2
𝜙2
F(t
)
(5)
{
x(t)=w−rx(t)−𝛿S𝜙S(t)−𝛿F𝜙F(t)
,
x(0)=x0.
9 Observe that in this formulation in infinite horizon we are discounting the costs that appear in Buratto
etal. (2020), obviously neglecting the “residual value functions”.
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1 3
Communication strategies tocontrast anti‑vax action: a…
𝜌
discount rate (
𝜌>0
)
𝛽
health care and social cost (
𝛽>0
)
𝜃
missed profit due to each unsold vaccine (
𝜃>0
)
𝜅S,𝜅F
communication cost parameters (
𝜅S,𝜅F>0
)
wword-of-mouth coefficient (
w>0
)
rinstantaneous constant vaccination rate,
𝛿S,𝛿F
pro-vax communication effectiveness (
𝛿S,𝛿F>0
)
It is interesting to observe the asymmetry of the game due to the different types
of costs associated with the unvaccinated group. This asymmetry will emerge in the
different forms of the value functions of the two players and, consequently, in the
equilibrium strategies of (S) and (F), respectively. The problem above is formulated
over an infinite horizon and is autonomous (because there is no other explicit time
dependence in its formulation, apart from the discount factors). Therefore, we are
interested in looking for a stationary solution. Stationarity means that each player’s
strategy is determined as a function of the state variable only:
𝜙∗
j(x)
j∈{S,F}
as
stated in Dockner etal. (2000),p.210.
4 The solution (Markovian nash equilibrium)
We are interested in feedback advertising strategies based on the number of unvac-
cinated individuals, so we look for a Markovian Nash equilibrium that, being the
game autonomous, turns out to be subgame perfect (Dockner etal. 2000,p.105).
Proposition 1 The Markovian Nash Equilibrium Feedback strategies that represent
the pro-vax communication strategies are
where
Proof See Appendix1.
◻
(6)
𝜙
∗
S(x)= 1
𝛿S
(𝜂−(𝜌+r))
[
x+
(
w−𝛿2
F𝜃
𝜅F𝜂
)
∕𝜂
],
(7)
𝜙
∗
F(x)=
𝛿
F
𝜃
𝜅
F
𝜂
,
(8)
𝜂
=
√[
𝜌
2+r
]
2
+𝛽
𝛿2
S
𝜅
S
+𝜌
2
.
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A.Buratto et al.
1 3
It can be immediately observed that
𝜙∗
F(
⋅
)
is strictly positive (being
𝛿F,𝜃,kF>0,
and
𝜂>𝜌+r>𝜌>0
). Furthermore,
𝜙∗
F(
⋅
)
does not depend on x, therefore it con-
stitutes a degenerate feedback strategy. It represents the constant positive communi-
cation contribution of the pharmaceutical firm to contrast the antivax action.10 From
now on, we will denote it by
𝜙∗
F.
As an economic interpretation, since the firm is
interested in minimizing its missed profits due to unvaccinated, its optimal commu-
nication strategy is adopted at a positive constant rate, independently of the number
of unvaccinated individuals.
Remark 1 For what concerns the communication strategy of the national healthcare
system
𝜙∗
S(
⋅
)
, it is easy to prove that
𝜙∗
S(x)>0
for all feasible x, since
𝜂>𝜌+r>0.
Observing that if
w
>𝛿
2
F𝜃
𝜅F𝜂,
then
𝜙∗
S(x)>0
for all
x≥0,
we can conclude that if the
negative word of mouth is so high that it compromises herd immunity, as assumed
in the problem formulation, then the national healthcare system needs to support the
firm’s pro-vaccination campaign.
4.1 Steady state
These kinds of problems are known in the literature as “Discounted Autonomous
Infinite Horizon Models” (DAM), (Grass etal. 2008, p.159). The main computa-
tional effort to solve optimal control models of this type is calculating the stable
manifolds of the occurring saddles. Let us substitute the optimal strategies (6) and
(7) in the dynamics of the unvaccinated individuals, then we obtain
Proposition 2 The steady-state number of unvaccinated people, while adopting the
optimal pro-vax communications turns out to be
Proof Differentiating the dynamics in (9), we obtain the optimal state trajectory
It is easy to verify that
x∗(t)≥0
for any starting point
x0
≥
0
and, since
𝜂>𝜌,
the
state converges to the stable positive steady state
xSS
in (10).
◻
(9)

x
(t)=w−rx(t)−𝛿S𝜙∗
S(x)−𝛿F𝜙∗
F= −(𝜂−𝜌)x(t)+ (𝜌+r)
𝜂
(
w−𝛿2
F𝜃
𝜅F𝜂
).
(10)
x
SS =
(𝜌+r)
(
w−𝛿
2
F𝜃
𝜅F𝜂
)
𝜂(𝜂−𝜌)
.
(11)
x∗
(t)=x
SS
+(x
0
−x
SS
)e
−(𝜂−𝜌)t
=x
SS
(1−e
−(𝜂−𝜌)t
)+x
0
e
−(𝜂−𝜌)t.
10 Mathematically, such a constant structure could be predicted from the linear form of the value func-
tion of player F.
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1 3
Communication strategies tocontrast anti‑vax action: a…
As expected, the steady-state level of unvaccinated individuals under the actions
of pro-vax communication campaigns of the two involved players increases with
negative word of mouth w, and it is lower than the à-régime unvaccinated level
observed in (2) without pro-vax communications. In fact, for any set of values of the
problem parameters, it holds
As a result, in case the word of mouth is too high, then the joint effects of the pro-
vax communication campaigns (6) and (7) may contribute to lower the number of
unvaccinated and to bring it down to the stationary level (10). This will help to reach
herd immunity.
5 Sensitivity analysis andsimulations
In the previous section, we obtained the analytical form for the strategies that consti-
tute the stationary Markovian Nash equilibrium of the pro-vax communication game
and the associated steady-state level of unvaccinated people. The simple dependence
of the solution on some of the parameters that characterize the problem allows us to
perform a sensitivity analysis.
In Table 1, we report the existing monotonicity properties, when analytically
computable, of the equilibrium strategies at the steady state, of the steady state level
of the unvaccinated, and of the optimal costs for the two players. For the reader’s
convenience, since the optimal solution contains the constant
𝜂,
in the first column
of the table, the dependencies of the constant
𝜂
on all the analyzed parameters are
also included. The arrows indicate either increasing monotonicity (
↗
) or decreasing
monotonicity (
↘
), while the lines “−" mean constant behavior.
Impact of Word-of-Mouth Effectiveness w (Table1)
Parameter w takes into account the word-of-mouth effects on the evolution of the
unvaccinated people. Here we assume, as stated in the previous section, that the neg-
ative word of mouth is so high that it requires nonzero pro-vax communications,
more precisely
w
>𝛿
2
F
𝜃
𝜅F𝜂
.
The effect of the negative word of mouth w drives the national healthcare sys-
tem to increase its pro-vax communication strategy (
𝜙∗
S
), with increased asso-
ciated costs. Moreover, as expected, the steady state number of unvaccinated
(12)
x
SS ≤xSS =
w
r
.
Table 1 Sensitivity analysis
with respect to main parameters
𝜂
𝜙∗
S
𝜙∗
F
xSS
CostS
CostF
w–
↗
–
↗
↗
↗
𝜃
–
↘
↗
↘
↘
↗
𝛿F
–
↘
↗
↘
↘
↘
𝜅F
–
↗
↘
↗
↗
↗
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A.Buratto et al.
1 3
(
xSS
) increases in w. These results are in line with the ones obtained in Buratto
etal. (2020), where the final level of unvaccinated people x(T) is increasing in
the word-of-mouth parameter. Another expected result is that both players have
increased costs as anti-vax word of mouth increases. However, it becomes inter-
esting to note that the optimal pharmaceutical strategy
𝜙∗
F
is not dependent on w;
the firm conducts its communication campaign independently of word of mouth
because its cost is not an effective loss, but a missed income.
Impact of vaccine unit profit
𝜃
(Table1)
Parameter
𝜃
represents the unit profit of a vaccine and is considered as the
virtual cost of each unsold vaccine. From Table1 we can observe that the higher
the unit profit, the higher the strategy communication of the firm (
𝜙∗
F
) and conse-
quently its cost. With a higher pro-vax campaign, the steady state level of unvac-
cinated individuals (
xSS
) decreases. At the same time, the national healthcare sys-
tem can count on the firm’s action, so, as strategic substitutes do, it can reduce its
communication campaign (
𝜙∗
S
) and, therefore, its total cost.
Impact of communication effectiveness and marginal costs (
𝛿F,
𝜅F
) of the firm
(Table1)
It is straightforward to prove that
𝜙∗
F
increases in
𝛿F
and decreases in
𝜅F
: The
more effective (or less costly) the firm pro-vax campaign, the more intensive it
will be. This result is typical for dynamic advertising models Huang etal. (2012):
each optimal communication strategy increases in its corresponding communica-
tion efficacy and decreases in its corresponding marginal cost. Moreover, since
both strategies act jointly to decrease the number of unvaccinated people, they are
strategic substitutes, so that each optimal communication strategy decreases with
the communication efficacy of the other player and increases with the communi-
cation marginal cost of the other player. In particular,
𝜙∗
S
is decreasing in
𝛿F
: The
more effective the firm’s pro-vax campaign, the less the national healthcare sys-
tem needs to implement its own pro-vax communication. Finally, the steady state
(
xSS
) computed in (10) decreases in
𝛿F,
while it increases in the marginal cost
𝜅F.
Table 2 Sensitivity analysis
with respect to
𝛿S
and
𝜅S
𝜂
𝜙∗
S
𝜙∗
F
𝛿S
↗
no
↘
𝜅S
↘
↘
↗
Table 3 Sensitivity analysis
with respect to
𝛽
and r
𝜂
𝜙∗
S
𝜙∗
F
𝛽
↗
↗
↘
r
↗
no
↘
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1 3
Communication strategies tocontrast anti‑vax action: a…
Impact of communication strategies to communication effectiveness and mar-
ginal costs (
𝛿S,
𝜅S
) of national health-care system (Table2)
Let us observe in Table(2) that
𝜂
is increasing in
𝛿S
and decreasing in
𝜅S,
there-
fore
𝜙∗
F
, which is decreasing in
𝜂
(see (7)), turns out to be decreasing in
𝛿S
and
increasing in
𝜅S.
Furthermore,
𝜙∗
S
turns out to be decreasing in
𝜅S,
while no kind
of monotonicity nor regularity can be verified with respect to
𝛿S,
(a “no” appears
in the corresponding cell of Table2).
Sensitivity of communication strategies to social cost
𝛽
and vaccination rate r
(Table3)
The parameter
𝛽
represents the social cost for the national healthcare sys-
tem. Analytical results in Table3 highlight how a severe social cost motivates
the healthcare system to increase its pro-vaccination campaign, thus allowing the
pharmaceutical firm to reduce its own.
The parameter r represents the vaccination rate, in other words the efficiency
of the national vaccination system. Analytical results highlight how an increase
in the efficiency of vaccination in the national healthcare system allows the phar-
maceutical firm to lighten its own communication campaign. On the other hand,
no kind of monotonicity can be analytically proved in the national health-care
system communication effort with respect to r.
Fig. 1 Steady state
xSS
w.r.t.
𝛿S
(
w=0.1, 𝜃=0.1, 𝜅S=4, 𝛿F=0.6, 𝜅F=0.8, 𝜌=0.05, 𝛽=0.3, r=0.1
)
Fig. 2 Steady state
xSS
w.r.t.
𝜅S
(
w=0.1, 𝜃=0.1, 𝛿S=0.6, 𝛿F=0.6, 𝜅F=0.8, 𝜌=0.05, 𝛽=0.3, r=0.1
)
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A.Buratto et al.
1 3
Numerical analysis of the steady state
The dependence of the steady state
xSS
with respect to the parameters
𝛿S,
kS,
𝛽,
and r cannot be proved analytically, therefore, we performed some numerical sim-
ulations fixing, for each analysis,
n−1
parameters and letting one parameter vary
within its feasible values (such as to guarantee non-zero advertising strategies).11 In
the following Figs.1, 2, 3, 4, it is evident, as analytically proven in (12), that in the
long run, the steady-state number of unvaccinated people while adopting pro-vacci-
nation communication (
xSS
- solid lines) is smaller compared to the values à-régime
without such communication policies (

x
SS
- dotdashed lines).
All the simulations conducted demonstrate a consistent pattern, specifically a
quasi-concave shape characterized by an initial convex-concave increase followed
by a subsequent decrease. We can interpret these results by noting the presence of a
threshold effect: Only sufficiently high values of the parameter with respect to which
the analysis is carried out make it possible for the national health-care system to
reduce the number of unvaccinated individuals to the extent necessary to achieve
herd immunity.
It should be noted in Fig. 4 that the dotdashed line, representing the à-régime
unvaccinated level

x
SS
without the pro-vax communications, is not constant (as the
Fig. 3 Steady state
xSS
w.r.t.
𝛽
(
w=0.1, 𝜃=0.1, 𝛿S=0.6, 𝜅S=4, 𝛿F=0.6, 𝜅F=0.8, 𝜌=0.05, r=0.1
)
Fig. 4 Steady state
xSS
w.r.t. r (
w=0.1, 𝜃=0.1, 𝛿S=0.6, 𝜅S=4, 𝛿F=0.6, 𝜅F=0.8, 𝜌=0.05, 𝛽=0.3)
11 Most of parameter values come from Buratto etal. (2020), as a reference.
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1 3
Communication strategies tocontrast anti‑vax action: a…
corresponding dotdashed lines in the previous figures) because
xSS,
defined in (2),
decreases in r. Furthermore, the gap between the two lines is large in correspond-
ence to small values of r. This permits to conclude that if the national healthcare
system is characterized by a low vaccination rate, then investing in effective and tar-
geted communication strategies becomes crucial in order to significantly slow down
the number of unvaccinated individuals and mitigate the potential consequences.
6 Conclusion
Vaccination can reduce the incidence of many diseases, but unfortunately, a negative
word of mouth based on fake news recently caused a decline in vaccination cover-
age in many countries. This phenomenon has increased the residual level of unvac-
cinated people reported each year by the World Health Organisation. In this paper,
we tackle the problem of planning a pro-vaccination communication campaign to
convince hesitant individuals to get vaccinated, so as to reduce the residual level of
unvaccinated people.
We formulate and solve an asymmetric differential game model over an infinite
horizon. Two players are involved in the provaccination communication campaign:
the national healthcare system and a pharmaceutical firm. The Hamilton-Jacobi-
Bellman approach is used to determine the optimal communication strategies that
constitute a Markovian-Nash equilibrium for the game. The two players act as stra-
tegic substitutes; the smaller the healthcare system campaign, the higher the firm’s
one. Sensitivity analyzes are performed with respect to the parameters of the prob-
lem to study their impact on the equilibrium strategies and steady-state solutions.
Let us answer the research questions declared in the Introduction: Our model
confirms that the national healthcare system needs to increase its investment in vac-
cine communication to contrast the effect of negative antivax word of mouth and
to aim at herd immunity. On the other hand, the firm’s pro-vaccine communication
campaign is not affected by negative word of mouth.
The proposed model also includes the vaccination rate as an index that char-
acterizes the efficiency of the national healthcare system. As far as we are aware,
such a parameter is not taken into account in the related literature. Our results show
that immunization can be obtained either by increasing vaccination efficiency, i.e.,
vaccinating at a high rate, or implementing a well-planned pro-vaccination com-
munication campaign. These results emphasize the importance of the “Immuniza-
tion Agenda 2030 Measles & Rubella Partnership”
(M&RP)
12 led by the American
Red Cross, United Nations Foundation, Centers for Disease Control and Prevention
(CDC), Gavi, the Vaccines Alliance, the Bill and Melinda French Gates Foundation,
UNICEF and WHO, to achieve the IA2030 measles and rubella specific goals.
These results also confirm the importance of national healthcare management in
maintaining a sufficiently high vaccination rate or, at least, in designing an efficient
provaccination campaign.
12 https:// measl esrub ellai nitia tive. org/ learn/ about- us, retrieved on 2023/07/06.
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A.Buratto et al.
1 3
Different extensions of this work can be envisioned. The first considers the
game played à la Stackelberg, where the leader may be the national healthcare
system and the follower the pharmaceutical company. A specific analysis of the
cases where the optimal communication strategy can turn out to be zero is an idea
that deserves investigation in order to take into account the various types of costs
that the national healthcare system must bear.
The important role that parameter r plays in the results may suggest consider-
ing the vaccination rate as a decision variable to be optimally set by the national
healthcare system together with the communication campaign for vaccination.
It can be interesting to analyze the stochastic evolution of the number of unvac-
cinated people. This goal could be obtained by introducing a stochastic effect in
communication campaigns and changing the ordinary differential equation (3)
into a stochastic one (see Wang etal. 2016).
Appendix A:Proof ofProposition 1
Let
VS(x)
and
VF(x)
be the stationary value functions associated with the health-
care system and the pharmaceutical firm, respectively. Let us assume that these
functions are differentiable, and let us denote by
V�
S(
x
)
and
V�
F(
x
)
their derivatives
w.r.t. x. They must solve the following Hamilton Jacobi Bellman equations asso-
ciated to the problems of the two players
Maximising the r.h.s. of (A1) and (A2) with respect to
𝜙S
and
𝜙F
respectively, we
obtain
Being interested in a non-trivial Nash equilibrium, we look for the best response
strategies in the region
{
x∈ℝ∶V
�
S
(x),V
�
F
(x)≤0}
,
in such a case
After substituting (A4) in (A1) and (A2), the HJB equations for the two players can
be rewritten as the following system
(A1)
𝜌
VS(x)=max
𝜙S≥0

w−rx −𝛿F𝜙F(x)−𝛿S𝜙S

V�
S(x)−

𝛽
2
x2+𝜅S
2𝜙2
S

(A2)
𝜌
VF(x)=max
𝜙F
≥0

w−px −𝛿F𝜙F−𝛿S𝜙S(x)

V�
F(x)−

𝜃x+
𝜅
F
2
𝜙2
F

(A3)
𝜙
S(x)=max
{
0, −𝛿S
𝜅
S
V�
S(x)
}
,𝜙F(x)=max
{
0, −𝛿F
𝜅
F
V�
F(x)
}.
(A4)
𝜙
i(x)=−
𝛿
i
𝜅i
V�
i(x),i∈{S,F}
.
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1 3
Communication strategies tocontrast anti‑vax action: a…
Notice the asymmetry in the resulting equations with respect to the state variable x:
This entails the corresponding value functions featuring the same asymmetry; there-
fore, we assume the following forms for
VS
and
VF
:
where the coefficients A,B,C,D, and E are assumed constant, as we focus on sta-
tionary strategies.
After substituting the value functions (A6) into the HJB equations (A5) and com-
paring the coefficients of the resulting polynomials in the variable x, the real coeffi-
cients A,B,C,D, and E satisfy the following system of algebraic equations (known
as algebraic Riccati equations)
From the first equation we get the two solutions
where
A+>0
and
A
−
<0.
We discard the positive root because once substituted
into the second equation in (A7) to obtain the coefficient D, it would be incompat-
ible with assumption
V�
F(
x
)
≤
0.
From now on, let us denote
A=A−<0.
Observe
that if we define
it is easy to prove that
𝜂>𝜌+r>𝜌>0,
and that the system (A7) admits the fol-
lowing unique solution
(A5)







𝜌VS(x)=w−rx +𝛿2
F
𝜅F
V�
F(x)V�
S(x)− 𝛽
2
x2+
𝛿2
S
2𝜅S
(V�
S(x))2
,
𝜌VF(x)=

w−rx +
𝛿2
S
𝜅S
V�
S(x)

V�
F(x)−𝜃x+𝛿2
F
2𝜅F
(V�
F(x))2.
(A6)
V
S(x)=
1
2
Ax2+Bx +C,VF(x)=Dx+E
,
(A7)











𝛿2
S
2𝜅S
A2−

𝜌
2+r

A−𝛽
2=0
𝜌+r−𝛿2
S
𝜅S
AD+𝜃=0
𝜌+r−𝛿2
S
𝜅S
AB−w+𝛿2
F
𝜅F
DA=
0
𝜌E−w+𝛿2
S
𝜅S
B+𝛿2
F
2𝜅F
DD=0
𝜌C−

w+

𝛿2
F
𝜅F
D+𝛿2
S
2𝜅S
B

B=0
A±=
𝜅S
𝛿2
S





𝜌
2+r±

𝜌
2+r2
+𝛽
𝛿2
S
𝜅S





,
(A8)
𝜂
=

𝜌+r−
𝛿2
S
𝜅S
A

=

𝜌
2+r

2
+𝛽
𝛿2
S
𝜅S
+𝜌
2
,
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A.Buratto et al.
1 3
Substituting these values into (A6) we obtain the continuously differentiable value
functions, whose derivatives are
Observe that indeed the constant A is negative; moreover, from the second equation,
we can infer that if the word-of-mouth level w is high,
(
w>𝛿
2
F𝜃
𝜅
F
𝜂
)
, then the positivity
of the term inside the round brackets also guarantees the negativity of the constant
B. This implies that if we restrict the domain to nonnegative values of x, which are
those that have a physical meaning, then
V�
S(x)<0
for all
x∈ℝ+
.
The Markovian Nash equilibrium strategies representing the optimal pro-vax
communication efforts can be obtained in feedback form by substituting coefficients
(A9) into (A10), and in turn into (A4), obtaining
Once substitutes the values of parameters A,B,C, and D, we obtain (6) and (7).
Funding Open access funding provided by Università degli Studi di Padova within the CRUI-CARE
Agreement.
Declarations
Conflict of interest We confirm that the manuscript is the authors’ original work and the manuscript has
not received prior publication and is not under consideration for publication elsewhere. All authors have
contributed to this paper, reviewed and approved the current form of the manuscript to be submitted. We
confirm that all authors of the manuscript have no conflict of interest to declare.
Open Access This article is licensed under a Creative Commons Attribution 4.0 International License,
which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long
as you give appropriate credit to the original author(s) and the source, provide a link to the Creative
Commons licence, and indicate if changes were made. The images or other third party material in this
article are included in the article’s Creative Commons licence, unless indicated otherwise in a credit line
to the material. If material is not included in the article’s Creative Commons licence and your intended
use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permis-
sion directly from the copyright holder. To view a copy of this licence, visit http://creativecommons.org/
licenses/by/4.0/.
(A9)









A=
𝜅
S
𝛿2
S
{𝜌+r−𝜂}<0,
B=w+𝛿2
F
𝜅F
DA∕𝜂=w−𝛿2
F𝜃
𝜅F𝜂A∕𝜂
,
C=w+𝛿2
F
𝜅F
D+𝛿2
S
2𝜅S
BB∕𝜌,
D=−𝜃∕𝜂<0,
E=

w+𝛿2
F
2
𝜅F
D−𝛿2
S
𝜅S
B

D∕𝜌.
(A10)
V
�
S(x)=Ax +B=A
[
x+1
𝜂
(
w−𝛿2
F𝜃
𝜅F𝜂
)]
,V�
F(x)=D<
0.
𝜙
∗
S(x)=−
𝛿
S
𝜅
S
(Ax+B)𝜙∗
F(x)=−
𝛿
F
𝜅
F
D
.
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1 3
Communication strategies tocontrast anti‑vax action: a…
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