A Study on Model-based Optimization of Vaccination Strategies against

Epidemic Virus Spread

Zonglin Liu, Muhammed Omayrat and Olaf Stursberg

Control and System Theory, Dept. of Electrical Engineering and Computer Science, University of Kassel, Germany

Keywords:

Epidemic Modeling, Markov Process, Infection Control, Optimal Control, Biomedical Systems.

Abstract:

This paper aims at applying optimal control to investigate different vaccination strategies against the epidemic

spread of viral diseases. Background of the study is the situation in the ﬁrst half of 2021, when many countries

started their vaccination procedures against the COVID-19 disease, but suffered from shortages of vaccines,

such that the efﬁcient distribution of the available amount of vaccine turned out to be crucial to mitigate the

pandemic. The paper ﬁrst suggests an extended version of a known model of virus spread in order include

the vaccination process. Based on this model, the formulation and solution of optimization problems is used

to determine how available vaccine should be distributed over different age-groups of the population to min-

imize virus spread. Effectiveness of the obtained strategies compared to standard ones is demonstrated in

simulations.

1 INTRODUCTION

In 2020, the outbreak and rapid spread of COVID-

19 affected the life of almost everyone on the planet.

Facing high infection and mortality rates, as well as

the absence of efﬁcient treatment to patients, strict in-

tervention policies (such as lock-down of cities and

restricting social life) were deployed by the govern-

ments to control the spread of the virus. Effectiveness

of these policies, however, were barely satisfactory in

many countries for several months with respect to the

daily number of active cases and the number of fatal-

ities. The successful development of vaccines in the

beginning of 2021, led to the hope that life may re-

turn to normal rather quickly, but limited production

capacities of the vaccines led to the situation that only

a small shares of the population could be vaccinated

during the ﬁrst half of 2021. Thus, the question of

how to control the vaccination process subject to the

given capacity constraints turned out to be crucial to

mitigate the pandemic – and this, of course, is a cen-

tral question to any future pandemic virus disease.

Before the outbreak of COVID-19, different vac-

cination strategies had been proposed in literature,

including the uniform strategy, see (Pastor-Satorras

et al., 2003), the targeted strategy, see (Liu et al.,

2003), the random strategy, see (Zanette and Kuper-

man, 2002), and the acquaintance strategy, see (Co-

hen et al., 2003). (Preciado et al., 2013) considered

optimal vaccination strategies by taking the vaccina-

tion cost into account, using a Susceptible-Infected-

Susceptible (SIS) epidemic model (see (Kermack and

McKendrick, 1932), (Ganesh et al., 2005)) for de-

scribing the virus spread. (Peng et al., 2010) followed

a similar pattern to develop an optimal vaccination

strategy, but considered additional constraints in op-

timization of the SIS model. (Wan et al., 2007) fo-

cused on the case that the vaccination resources are

limited, and they exploited the topologicalstructure of

the network to eliminate the virus. The resulting vac-

cination strategy was then tested on a model for the

SARS transmission in 2003. The work by (Gourdin

et al., 2011) investigated how to efﬁciently distribute

available medicine to minimize the overall infection.

Unlike vaccines, which reduce the individual infec-

tion rate, the medicine in the latter work is assumed

to increase the individual curing rate.

Most of these strategies are based on the SIS

model, in which only two states of the individuals are

modelled, namely, to be susceptible for the virus or

being infected. Experiences from COVID-19 have

shown, however, that aspects like the questions of

whether an infected person has symptoms or not, or of

whether she/he can be tested in time (and is isolated

before infecting others) are decisive for the evolution

of the virus spread. Accordingly, the determination

of an optimal (or at least good) vaccination strategy

must also take these aspects into account. In addition,

630

Liu, Z., Omayrat, M. and Stursberg, O.

A Study on Model-based Optimization of Vaccination Strategies against Epidemic Virus Spread.

DOI: 10.5220/0010604006300637

In Proceedings of the 18th International Conference on Informatics in Control, Automation and Robotics (ICINCO 2021), pages 630-637

ISBN: 978-989-758-522-7

most of the work mentioned above also assumed that

the population can be modeled homogeneously in the

sense that the same rates of infections, courses of de-

veloping symptoms or being cured apply to anyone.

Observations for the ﬁrst year of COVID-19 clearly

showed differences, however, with respect to senior

members of the population have signiﬁcantly higher

mortality rates and lower curing rates than younger

generations.

To capture such characteristics, this paper ﬁrst re-

views in Sec. 2 recent work on COVID-19 modeling,

and then extends the model proposed in (Giordano

et al., 2020) by including the vaccination process in

Sec. 2. Note that the selected model was proposed to

account for the course of the COVID-19 pandemic in

Italy and it includes already more states than the SIS-

model, but does not account for vaccination. Based

on the extended model, optimal control problems are

formulated and solved in order to determine optimal

vaccination strategies in Sec. 3. The solution takes

into account different age groups of the population

and heterogeneity with respect to infection and curing

rates, as well as different contact situations and limi-

tations of the amount of available vaccines. In Sec. 4,

the computed strategies are simulated exemplarily for

a mid-size city and compared to alternative strategies,

before Sec. 5 concludes the paper.

2 THE EPIDEMIC MODEL

Most literature on epidemic modeling uses the SIS

model to describe the spread of a virus. In this

rather simple model, only two states of any individ-

ual, susceptible and infected, are considered. After

the ﬁrst outbreak of the COVID-19 pandemic, dif-

ferent groups have invested effort into adapting the

SIS model to the characteristics of this disease: The

work in (Della et al., 2020) considered the networked

structure of different regions in Italy and extended the

SIS model with four more states, i.e., quarantined,

hospitalized, recovered, and deceased. In the work

of (Nda¨ırou et al., 2021), even more COVID-19 re-

lated states are included and the inﬂuence of super-

spreading individuals are studied. In (Chaturvedi

and Chakravarty, 2021), the authors applied the SIR

model (with R for recovered) to provide a predic-

tive analysis of when the pandemic can be ended

with the help of vaccination. Among these efforts,

the so-called SIDARTHE model, which was proposed

in (Giordano et al., 2020) and is shown by the part

marked in black in Fig. 1, has been widely adopted

in different work, see (K¨ohler et al., 2020; L´opez and

Rod´o, 2020).

Ineffective

vaccination

Effective

vaccination

Contagion

Diagnosis

Symptoms

Critical

Death

Healing

, β

, γ

, δ

(t)

Figure 1: The states and transitions in black represent the

original SIDARTHE model, while the parts in red are newly

added to model the vaccination process, as well as are the

indices i to model different groups of persons.

The SIDARTHE model, distinguishes the follow-

ing health status: being susceptible (S

), infected (I

diagnosed (D

), ailing (A

), recognized (R

), threat-

ened (T

), healed (H

), and extinct (E

). The corre-

sponding states represent the percentage of persons

of a given population with the corresponding health

status, and the changes of these percentages is mod-

eled stochastically by a continuous-time Markov pro-

cess. The ﬁrst extension of this model compared to

earlier papers is to refer to a set of n groups of per-

sons, where the index i ∈ N = {1, ·· · , n} refers to one

these groups. This extension will allow later to refer

to a certain age-group of the population. The sum of

the percentages over the discrete states of the Markov

process for a single group i is always one.

Note that, although the SIDARTHE model can

well represent the development of the pandemic, it

was proposed in March 2020, when vaccines were

still not yet in sight. This can be noticed from the

black part in Fig. 1, since the health status cannot

transition directly from S

to H

without being in-

fected. Accordingly, the SIDARTHE model is ex-

tended in this paper also to contain the vaccination

process (see the red part in Fig. 1). This includes the

addition of one state U

referring to the percentage of

people in group i, for whom the vaccine is ineffective.

In addition, three new transitions are also assigned,

namely, from S

to H

, from S

to U

, and from U

. In order to model the evolution of the probability

distribution over the discrete states for group i ∈ N,

the following nonlinear dynamics is selected:

(t) = −S

(t)

∑

j∈N

(α

(t) + β

(t) + γ

(t)

+ δ

(t)) − (χ

(t) + ψ

(t))S

(t) (1)

A Study on Model-based Optimization of Vaccination Strategies against Epidemic Virus Spread

631

(t) = −U

(t)

∑

j∈N

(

(t) +

(t)

(t)) + χ

(t)S

(t) (2)

(t) = S

(t)

∑

j∈N

(α

(t) + β

(t) + γ

(t)

+ δ

(t)) − (ε

+ ζ

+ λ

(t) +U

(t)

∑

j∈N

(

(t)

(t) +

(t)) (3)

(t) = ε

(t) − (η

+ ρ

(t) (4)

(t) = ζ

(t) − (θ

+ µ

+ κ

(t) (5)

(t) = η

(t) + θ

(t) − (υ

+ ξ

(t) (6)

(t) = µ

(t) + υ

(t) − (σ

+ τ

(t) (7)

(t) = λ

(t) + ρ

(t) + κ

(t) + ξ

(t) + σ

(t)

+ ψ

(t)S

(t) (8)

(t) = τ

(t) (9)

For consistent initialization of the model in an initial

time t

, it is required for any i ∈ N that the sum of

all states is one. Note that the model also satisﬁes

(t) +

(t) = 0 at any time t for any choice of pa-

rameters. These are denoted by α

, β

, γ

, δ

, ε

, ζ

, η

, θ

, υ

, ξ

, τ

, ρ

, σ

, µ

, and model the transi-

tion rates in between the pairs of corresponding states.

The different contagion parameters α

, β

, γ

, δ

in the

transition from S

to I

are due to contacts between a

susceptible person and an infected, a diagnosed, an

ailing, or a recognized person according to the orig-

inal paper in (Giordano et al., 2020). These param-

eters together with other transitions rates in Fig. 1

are assumed to be known

, but differ among the dif-

ferent groups i ∈ N. In detail and compared to the

original SIDARTHE model, the new model here cov-

ers the following effects in addition: 1.) a success-

ful vaccination step (transition from S

to H

) with

rate ψ

(t) ∈ R

≥0

. The rates ψ

(t) will be used as in-

puts determining the vaccination strategy to be opti-

mized in the coming section; 2.) an ineffective vac-

cination step (from S

to U

) with rate χ

(t) ∈ R

≥0

where χ

(t) is assumed to be proportional to ψ

(t),

i.e., χ

(t) = q · ψ

(t), q ∈ R

; 3.) an infection step

(from U

to I

) with respective contagion parameters

, and

. Note that the labeling of this tran-

sition with four rates refers to the shares that a per-

son in state U

gets infected by contact with a person

in state I

, D

, A

or R

respectively (the same princi-

In practice, one can only estimate these parameters,

leading to model uncertainty. In order to consider e.g. pa-

rameter intervals, extensions to techniques of robust predic-

tive control could be employed, see (Campo and Morari,

1987), but this is outside of the scope of the present paper.

ple underlies the assignment of four parameters to the

transition from S

to I

). Note further that the conta-

gion parameters assigned toU

→ I

are different from

those assigned to S

→ I

to account for the possibil-

ity that a vaccinated person, which is unaware of the

fact that the vaccination was not successful, will likely

follow relaxed contact patterns, compared to persons

that still wait to be vaccinated.

In order refer to the above model in brief, let it be

denoted by:

˙x

(t) = f(x

(t), ψ

(t)), x

) = x

i,0

(10)

with state vector x

(t) := (S

(t),U

(t), I

(t),

(t), A

(t), R

(t), T

(t), H

(t), E

(t)), and the ini-

tial state x

i,0

. Furthermore, the model aims at

considering the spread of the virus across different

age groups of the population. In order to formalize

the contacts between different age groups, and thus

the possibility that infections occur across the groups,

an undirected graph G = {N, E} is set up. In here,

the set N of nodes models the different age groups,

with indices i ∈ N as before. The set E of undirected

edges models the contact between a pair of groups.

To an edge e

i, j

∈ E between the groups with indices

i and j, a positive weight W

∈ R

is assigned,

representing that infections in group j can affect the

infection in group i (and vice versa), see also (1) - (3)

in the model. More precisely, a weight W

models

the average share of time that a person from group

i spends with a person from group j. Accordingly,

a self-loop transition with weight W

is introduced

for any node, and the matrix W ∈ R

n×n

is chosen as

doubly stochastic matrix, i.e.,

∑

j∈N

= 1 for all

i ∈ N, and

∑

i∈N

= 1 for all j ∈ N.

Note that the model in the present form does not

explicitly account for the necessity that two (or more)

shots of vaccine may be necessary to get from state

to H

. This could be included by introducing inter-

mediate state in between S

→ H

and S

→ U

, but is

omitted here to not overload the model.

3 DETERMINATION OF

OPTIMIZED VACCINATION

STRATEGIES

Based on the model introduced before, this section

proposes to determine vaccination strategies for dif-

ferent age-groups of the population based on optimal

control principles. The underlying assumption is that

there is a shortage of available vaccine (as was true for

the COVID-19 pandemic in the ﬁrst half year after de-

veloping the vaccine). This, together with the obser-

vation that the infection has, on average, more severe

ICINCO 2021 - 18th International Conference on Informatics in Control, Automation and Robotics

632

effects the older the infected person is, motivates to

divide the population into n groups according to their

age. For any age-group with index i ∈ N = {1, ··· , n},

let its share of the total population be denoted by v

and

∑

i∈N

= 1. The vaccination of different age

groups (represented in the model from Sec. 2 by the

rates ψ

(t) and χ

(t)) as well as contacts between age

groups (modelled by the weightsW

) can be expected

to have signiﬁcant impact on the evolution of the epi-

demic.

3.1 Optimization Objectives and

Constraints

In order to set up the optimal control formulation, ﬁrst

let an initial time t

be given as well as a time inter-

val [t

+ H], where H is the number of days over

which the vaccine is distributed. Assume that the vac-

cination strategy can only be adjusted every T days of

the horizon, leading to totally

decision steps. Let

k ∈ { 0, 1, ··· ,

−1} index these steps. Once the strat-

egy is determined for k it is held constant for the com-

ing T days. Accordingly, the vaccination rate applied

to an age-group i ∈ N on the interval [k· T, (k+ 1) · T[

is referred to by ψ

i,k

. The maximum amount of vac-

cine available for step k over all groups i is denoted

by Ψ

max,k

, leading to the following constraint:

∑

i∈N

P · v

· (ψ

i,k

+ χ

i,k

) ≤ Ψ

max,k

, (11)

with a population size P , and P · v

representing the

size of age-group i. Corresponding to the relation be-

tween χ

(t) and ψ

(t) as mentioned in Sec. 2, it fol-

lows that: χ

i,k

:= q · ψ

i,k

Given the constraint (11), the task is

now to optimize the vaccination strategy

ψ := (ψ

, ψ

, .. . ,ψ

−1

) with ψ

= (ψ

1,k

, .. . ,ψ

n,k

)

for all age-groups in all decision steps, in order to

minimize the infected share of the population (i.e., all

persons assigned to the states I

, D

, A

, R

, T

in Fig.

1), as well as the death cases (referring to the state

). To consider, in addition, the vaccination costs,

an additional term can be introduced, leading to the

following cost function:

J(ψ) :=

∑

i∈N



P v

· (I

(t) + D

(t) + A

(t)

+ R

(t) + T

(t)) + c

· E

(t))dt

∑

k∈{0,···,

−1}

i,k





. (12)

The parameters c

, c

and c

(all positive) denote ap-

propriate weights of the terms of the cost function.

Note that J(ψ) may be extended to additional cost

terms to account, e.g., for costs of testing. Likewise,

additional constraints, as limitations in available staff

for vaccination could be considered.

3.2 The Optimization Problem

In order to determine an optimized vaccination strat-

egy, denoted by ψ

∗

, the aforementioned objectives

and constraints together with the model (10) are cast

into the following optimization problem:

Problem 1.

min

J(ψ) (13)

s.t. for all i ∈ N and given G :

) = x

i,0

, t ∈ [t

+ H],

dynamics (10),

for all k ∈ {0, . . . ,

− 1} : (14)

i,k

≥ 0, (15)

∑

i∈N

P · v

· (1+ q)ψ

i,k

≤ Ψ

max,k

. (16)

The optimized strategy ψ

∗

is then determined by

solving this nonlinear continuous-time optimization

problem. The constraint (14) is a local constraint of

group i, while (16) determines a coupling constraint.

The dynamics of the groups is also coupled through

the matrix W affecting (1) - (3).

To solve Problem 1, one can apply, e.g., tech-

niques of multiple shooting, see e.g. (Bock et al.,

2000), which casts the original problem into a ﬁnite

dimensional nonlinear programming problem by pa-

rameterizing the input and state space. One should

note, however, that one can only hope to determine a

sub-optimal strategy for the given non-convex prob-

lem (this is why it is referred to an optimized strategy,

rather than the optimal one).

4 SIMULATION

To illustrate the computation of optimized vaccination

strategies and to discuss a number of effects, consider

a midsize city with a population of P = 200, 000. Ac-

cording to a statistics of distribution of the population

over the age groups i ∈ N (in Germany for 2021), con-

sider the following values for the city:

Table 1: Distribution over age groups.

Age 0-19 20-39 40-59 60-79 80+

18.5% 24.6% 28.8% 21.6% 6.5%

A Study on Model-based Optimization of Vaccination Strategies against Epidemic Virus Spread

633

Table 2: Contacts between different age-groups during the

epidemic.

Age 0-19 20-39 40-59 60-79 80+

0-19 31% 34% 25% 8% 2%

20-39 34% 45% 15% 5% 1%

40-59 25% 15% 30% 20% 10%

60-79 8% 5% 20% 50% 17%

80+ 2% 1% 10% 17% 70%

The estimated average time (in percentage) per-

sons of one group spend with those of another is listed

in Table 2, representing the entries of matrix W. Note

that Table 2 also reﬂects certain intervention policies

deployed in certain phases in Germany, such as that

visiting nursing homes was forbidden (such that the

most senior group had to spent most of their time with

its own), while schools were open during that phase.

The parameters contained in the dynamics (1) - (9)

are chosen similar to (Giordano et al., 2020), except

for the following:

• The contagion parameters

for the transition

from U

→ I

are larger than for S

→ I

. This aims

at reﬂecting the effect that people are less cautious

to the virus after being vaccinated, although the

vaccination may be ineffective.

• Effectiveness of the vaccine is assumed to be 90%

for all age-groups

, i.e., χ

:= 0.11ψ

, i ∈ N.

The initial states for t

are chosen according to the

pandemic situation of the city at the end of 2020. The

considered horizon is H = 90 days, and the vaccina-

tion strategy can be changed every T = 30 days (thus,

in total 3 decision steps). The weights in J satisfy

> c

≫ c

, i.e., the main goal is to reduce the over-

all number of casualties, while the vaccine costs can

be neglected.

4.1 Uniform Vaccination Strategy

Before the optimized strategy is tested, a uniform

vaccination strategy is ﬁrst simulated: When assum-

ing that the availability of the vaccine is constant in

all decision steps, i.e., Ψ

max,k

:= Ψ

max

, ∀k ∈ {0, 1, 2},

the ψ

i,k

in the uniform vaccination strategy takes a

value of: ψ

i,k

max

P ·(1+q)

for all i ∈ N. The outcome

by adopting this strategy is illustrated in Fig. 2 and

3. Obviously, most of the casualties occur in the

senior groups 60 - 79 and 80+, although the number

of infections (over the states infected (I

), diagnosed

), ailing (A

), recognized (R

) and threatened (T

))

This may be extended to considering different effec-

tiveness rates of the vaccine for different age-groups, or to

different rates for different types of vaccines.

Figure 2: Number of infections when applying the uniform

strategy over H = 90 days.

Figure 3: Casualties for each age group with the uniform

strategy at the end of the 90 days.

in these groups are not the highest. This is mainly

due to the high mortality rate of the two groups.

4.2 Optimized Vaccination Strategy

In the second test, the Problem 1 is solved to deter-

mine the optimal vaccination strategy. The resulting

∗

i,k

for each group is shown in Fig. 4. By adopting

this strategy, the evolution of the infections and ca-

sualties are illustrated in Fig. 5 and 6. Compared to

the uniform strategy, more infected cases occur in the

groups 20 - 39 and 40 - 59, while less in the groups 60

- 79 and 80+. The casualties for the two senior groups,

however, is signiﬁcantly reduced by adopting the op-

timized strategy, and this also leads to a decrease of

the total number of casualties from 329 to 174, see

Fig. 7 and 8.

The optimized strategy is further compared to the

popular strategy of ﬁrst vaccinating the older genera-

tions, see Fig. 9 and 10. In this strategy, all available

vaccine is ﬁrst provided to the group 80+ in the steps

ICINCO 2021 - 18th International Conference on Informatics in Control, Automation and Robotics

634

Figure 4: The optimized vaccination strategy from Problem

1: in k = 0, the complete available vaccine is distributed to

the groups 60 - 79 and 80+; in k = 1, most of the vaccine

is distributed to the group 40 - 59, 60 - 79 and 0 - 19; in

the last step k = 2, the vaccine is distributed to all groups

besides the group 40 - 59.

Figure 5: Infections with the optimized strategy.

Figure 6: Casualties with the optimized strategy.

k = 0 and k = 1, and then to the group 60 - 79 for

k = 2. One can notice, however, that only the number

of casualties in the group 80+ is slightly reduced com-

pared to the uniform strategy, while much more occur

in the group 60 - 79. This is due to the larger size of

the latter group than the 80+ group, and to the more

frequent contacts of this group with younger persons.

Uniform strategy

Optimized strategy

Figure 7: Infections over all age-groups by adopting the

uniform vaccination strategy and the optimized one.

Uniform strategy (329)

Optimized strategy (174)

Figure 8: Casualties over all age-groups by adopting the

uniform vaccination strategy and the optimized one.

Figure 9: Infections when applying the older-ﬁrst strategy.

The immunity of the group 60 - 79 is, however, only

slightly better than for the 80+ group, but much worse

than for the junior groups. Accordingly, the group 60

- 79 should not be ignored in the ﬁrst step of the vac-

cination process based on the simulation results.

4.3 Increasing Availability of Vaccine

In the third test, it is assumed that more vaccine

can be provided over time, i.e., Ψ

max,k

:= 2

· Ψ

max

A Study on Model-based Optimization of Vaccination Strategies against Epidemic Virus Spread

635

Figure 10: Casualties when applying the older-ﬁrst strategy.

k ∈ {0, 1, 2}. The new outcome from solving Prob-

lem 1 is illustrated in Fig. 11 and 12. Compared with

the last test, in which the available amount vaccine is

constant over time, the total number of casualties has

been barely reduced, but a decrease of the total num-

ber of infections can be observed after 30 days. In an-

other test with Ψ

max,k

:= 2

·Ψ

max

, the infections rates

for the transitions S

→ I

and U

→ I

are reduced by

20% compared to the test before for all k ∈ {0, 1, 2}.

This may be achieved by additional intervention poli-

cies such as lock down. Hereby, the number of infec-

tions as well as the number of casualties can be sig-

niﬁcantly reduced, as shown in Fig. 11 and 12 – this

result shows that one should not only rely on vaccina-

tion to mitigate the epidemic fast, i.e., the vaccination

strategy and intervention policies should be optimized

at the same time.

4.4 Effectiveness of the Vaccine

In the last test, a lower effectiveness of the vaccine

(compared to before) is investigated. By assuming

that the effectiveness is χ

:= 0.5ψ

instead of χ

Increased availability

Constant availability

Increased + Lockdown

Figure 11: Infections over all age-groups when: 1.) the

availability of vaccine is constant over time; 2.) the amount

of vaccine increases over the steps; 3.) additional interven-

tion policies are deployed and the amount of vaccine in-

creases over k.

Increased availability (171)

Constant availability (174)

Increased + Lockdown (131)

Figure 12: Casualties by applying the strategies in Fig. 11.

0.11ψ

, i ∈ N, the outcome from solving Problem 1 is

shown in Fig. 13 and 14. As could be expected, this

change will lead to less healed cases at the end of the

horizon, as well as an increase of casualties.

5 CONCLUSION

In this paper, the known SIDARTHE model has been

extended in different respects in order to be tailored to

the study of vaccination procedure against epidemic

virus spread. The motivation of these extensions is

the typical shortage of vaccine, if a new virus spreads

and the vaccine ﬁrst has to be developed. To effec-

tively combat the epidemic with a limited amount of

vaccine, this paper has shown that the solution of opti-

mal control problems serves to determine strategies to

optimally distribute the vaccine among different age-

groups. The contacts between the age-groups, hetero-

geneous infection and mortality rates over the groups,

as well as limited effectiveness of the vaccine can all

be taken into account in the optimization. The model

proposed here can also be adapted straightforwardly

to study some additional problems occurring during

vaccination procedures, such as the effect that a vac-

:= 0.5ψ

:= 0.11ψ

Figure 13: Number of healed persons of all age-groups for

different effectiveness of the vaccine.

ICINCO 2021 - 18th International Conference on Informatics in Control, Automation and Robotics

636

:= 0.5ψ

(215)

:= 0.11ψ

(174)

Figure 14: A less effective vaccine leads to an increase of

the total number of casualties.

cine is less effective or bears danger for certain age

groups, or that a vaccinated person would not get ill

but may still be contagious in case of infection.

Finally, for the situation that the transition rates or

model parameters are only knownwithin certain inter-

vals, the optimal control problem could be modiﬁed

into one of optimization under uncertainty, e.g. adopt-

ing techniques of robust optimal control. To account

for the change of the parameters over time (e.g. due

to a virus mutation), the solution of the optimal con-

trol problem could be recursively solved over time,

leading to a strategy of model predictive control.

REFERENCES

Bock, H., Diehl, M., Leineweber, D., and Schl¨oder, J.

(2000). A direct multiple shooting method for real-

time optimization of nonlinear DAE processes. In

Nonlinear model predictive control, pages 245–267.

Springer.

Campo, P. and Morari, M. (1987). Robust model predictive

control. In American control conf., pages 1021–1026.

IEEE.

Chaturvedi, D. and Chakravarty, U. (2021). Predictive anal-

ysis of COVID-19 eradication with vaccination in In-

dia, Brazil, and USA. Infection, Genetics and Evolu-

tion, 92:104834.

Cohen, R., Havlin, S., and Ben-Avraham, D. (2003).

Efﬁcient immunization strategies for computer net-

works and populations. Physical review letters,

91(24):247901.

Della, F., Salzano, D., Di Meglio, A., De Lellis, F., Corag-

gio, M., Calabrese, C., Guarino, A., Cardona-Rivera,

R., De Lellis, P., Liuzza, D., et al. (2020). A network

model of Italy shows that intermittent regional strate-

gies can alleviate the COVID-19 epidemic. Nature

communications, 11(1):1–9.

Ganesh, A., Massouli´e, L., and Towsley, D. (2005). The ef-

fect of network topology on the spread of epidemics.

In 24th Joint IEEE Conf. of the Computer and Com-

munications Societies., volume 2, pages 1455–1466.

Giordano, G., Blanchini, F., Bruno, R., Colaneri, P., Di Fil-

ippo, A., Di Matteo, A., and Colaneri, M. (2020).

Modelling the COVID-19 epidemic and implementa-

tion of population-wide interventions in Italy. Nature

Medicine, pages 1–6.

Gourdin, E., Omic, J., and Van Mieghem, P. (2011). Op-

timization of network protection against virus spread.

In 8th Int. IEEE Workshop on the Design of Reliable

Communication Networks, pages 86–93.

Kermack, W. and McKendrick, A. (1932). Contributions to

the mathematical theory of epidemics. ii.—the prob-

lem of endemicity. Proc. of the Royal Society of Lon-

don. Series A, 138(834):55–83.

K¨ohler, J., Schwenkel, L., Koch, A., Berberich, J., Pauli,

P., and Allg¨ower, F. (2020). Robust and optimal pre-

dictive control of the COVID-19 outbreak. Annual

Reviews in Control.

Liu, Z., Lai, Y., and Ye, N. (2003). Propagation and immu-

nization of infection on general networks with both

homogeneous and heterogeneous components. Physi-

cal Review E, 67(3):031911.

L´opez, L. and Rod´o, X. (2020). The end of social con-

ﬁnement and COVID-19 re-emergence risk. Nature

Human Behaviour, 4(7):746–755.

Nda¨ırou, F., Area, I., Nieto, J., Silva, C., and Torres, D.

(2021). Fractional model of COVID-19 applied to

Galicia, Spain and Portugal. Chaos, Solitons & Frac-

tals, page 110652.

Pastor-Satorras, R., Vespignani, A., et al. (2003). Epidemics

and immunization in scale-free networks. Handbook

of Graphs and Networks, Wiley-VCH, Berlin.

Peng, C., Jin, X., and Shi, M. (2010). Epidemic thresh-

old and immunization on generalized networks. Phys-

ica A: Statistical Mechanics and its Applications,

389(3):549–560.

Preciado, V., Zargham, M., Enyioha, C., Jadbabaie, A., and

Pappas, G. (2013). Optimal vaccine allocation to con-

trol epidemic outbreaks in arbitrary networks. In 52nd

IEEE Conf. on Decision and Control, pages 7486–

7491.

Wan, Y., Roy, S., and Saberi, A. (2007). Network design

problems for controlling virus spread. In 46th IEEE

Conf. on Decision and Control, pages 3925–3932.

Zanette, D. and Kuperman, M. (2002). Effects of immu-

nization in small-world epidemics. Physica A: Statis-

tical Mechanics and its Applications, 309(3-4):445–

452.

A Study on Model-based Optimization of Vaccination Strategies against Epidemic Virus Spread

637