Optimal Resource Allocation for Fast Epidemic Monitoring in

Networked Populations

Paolo Di Giamberardino

1 a

, Daniela Iacoviello

1 b

and Federico Papa

2

1

Dept. of Computer, Control and Management Engineering A.Ruberti, Sapienza University of Rome, Rome, Italy

2

IASI, CNR, Rome, Italy

Keywords:

Epidemic Modeling, Optimal Resource Allocation, Monitoring.

Abstract:

The COVID-19 pandemic highlighted the fragility of the world in addressing a global health threat. The avail-

able resources of the pre-pandemic national health systems were inadequate to cope with the huge number of

infected subjects needing health care and with the rapidity of the infection spread characterizing the COVID-19

outbreak. Indeed, an adequate allocation of the resources could produce in principle a strong reduction of the

infection spread and of the hospital burden, preventing the collapse of the health system. In this work, taking

inspiration from the COVID-19 and the difﬁculties in facing the emergency, an optimal problem of resource

allocation is formulated on the basis of an ODE multi-group model composed by a network of SEIR-like sub-

models. The multi-group structure allows to differentiate the epidemic response of different populations or of

various subgroups in the same population. In fact, an epidemic does not affect all populations in the same way,

and even within the same population there can be epidemiological differences, like the susceptibility to the

virus, the level of infectivity of the infectious subjects and the recovery from the disease. The subgroups are

selected within the total population based on some peculiar characteristics, like for instance age, work, social

condition, geographical position, etc., and they are connected by a network of contacts that allows the virus

circulation within and among the groups. The proposed optimal control problem aims at deﬁning a suitable

monitoring campaign that is able to optimally allocate the number of swab tests between the subgroups of the

population in order to reduce the number of infected patients (especially the most fragile ones) so reducing the

epidemic impact on the health system. The proposed monitoring strategy can be applied both during the most

critical phases of the emergency and in endemic conditions, when an active surveillance could be crucial for

preventing the contagion rise.

1 INTRODUCTION

The COVID-19 pandemic has been modifying the

habits all over the world, inducing all the populations

to apply containment measures, such as social dis-

tancing, using masks, participating to swab test cam-

paigns, and getting the vaccination when it became

available. A huge amount of papers have been pub-

lished on this topic, focusing on the effects and the

effectiveness of such actions, see (Gatto et al., 2020),

(Di Giamberardino and Iacoviello, 2021), (Pung et al.,

2021), (Casares and Khan, 2020), (Espinoza et al.,

2020), (Liu et al., 2020), (Borri et al., 2021) only

as possible examples. In literature, different aspects

have been investigated, changing the focus of the

analysis as the pandemic conditions vary, ranging

a

https://orcid.org/0000-0002-9113-8608

b

https://orcid.org/0000-0003-3506-1455

from data analysis in the very ﬁrst period of the emer-

gency, (Tang et al., 2020), (Wu et al., 2020), to mod-

eling and control during 2020 − 2021, (Ivorra et al.,

2020), (Wang et al., 2020), (Silva et al., 2021), (Di Gi-

amberardino et al., 2021b) up to ex-post analysis of

what happened, in order to understand the dynam-

ics of the infections and to avoid possible delays in

decision making in case of new sanitary emergen-

cies, (Assefa et al., 2022), (Marziano et al., 2021),

(Di Giamberardino et al., 2021a). The main COVID-

19 speciﬁcities are the high infection rate, the signiﬁ-

cant percentage of asymptomatic subjects that can in-

fect, the different course of the illness in the various

categories, and not well identiﬁed reasons of sensi-

tivity with respect to the disease. The strong rela-

tionships among populations, and inside each popu-

lation, increase the probability of new infections and

therefore of the epidemic spread. The spread among

616

Di Giamberardino, P., Iacoviello, D. and Papa, F.

Optimal Resource Allocation for Fast Epidemic Monitoring in Networked Populations.

DOI: 10.5220/0011299300003271

In Proceedings of the 19th International Conference on Informatics in Control, Automation and Robotics (ICINCO 2022), pages 616-625

ISBN: 978-989-758-585-2; ISSN: 2184-2809

Copyright

c

2022 by SCITEPRESS – Science and Technology Publications, Lda. All rights reserved

populations is studied in (Di Giamberardino et al.,

2021) where the mobility aspects after the lockdown

of 2020 in Italy are analysed; in that case the entire

Italian population has been split into three geograph-

ical groups (referring to North, Center and South of

Italy) showing the inﬂuence of controlled re-opening,

after the lockdown, on the distribution of the infec-

tion. The connections among groups of populations

can be faced considering as “population” not only the

habitants of a region but also, for example, a cate-

gory of workers, or an age-category; therefore, it is

possible to study the interconnections among social

groups with different characteristics, analysing the ef-

fects, on the entire population, of control actions ap-

plied separately in one group, (Contreras et al., 2020),

(Di Giamberardino et al., 2021), (Ndairou et al.,

2020). At the very beginning of the current pan-

demic, the available resources were not sufﬁcient for

the severity of the emergency and some choices had to

be taken. In absence of vaccination and ad hoc med-

ications, two main containment measures are, in gen-

eral, achievable: the social distancing (including the

use of masks) and the fast identiﬁcation of infected

patients. This latter issue can be fulﬁlled by means

of an adequate monitoring campaign based on swab

tests, which were not easily available at the begin-

ning of the COVID-19 pandemic. Moreover, also in

the endemic condition, an active surveillance action

could consider a regular swab test campaign to con-

tain the number of infected patients, especially in the

most fragile part of the population. The main goal of

this paper is to propose a strategy for optimizing the

swab test campaign, considering also limitations from

logistic, economic and material points of view.

The paper is organized as follows. In Section 2,

the total population is divided into n groups and a

compartmental model describing the disease dynam-

ics in each group is proposed and analysed. An op-

timal allocation control is proposed, which aims at

yielding a strategy for deﬁning a swab test campaign

in case of limited resources, both from economic and

technical point of view. In Section 3, the introduction

of the optimized strategy for fast epidemic monitor-

ing is studied, comparing its effects with the non con-

trolled case. Conclusion and future developments end

the paper.

2 MATERIALS AND METHODS

A population is here modeled as partitioned into n

groups; this could represent, for example, a classiﬁca-

tion with respect to the age, or to the geographical po-

sition, or to the working categories, or any other char-

acteristics that imply different response to the virus.

In each group, 7 compartments representing differ-

ent patient conditions w.r.t. the disease evolution are

introduced. In particular, the severity of the disease

is explicitly taken into account by distinguishing be-

tween asymptomatic (A) and symptomatic isolated

infected (I), and hospitalized (H) individuals. This

choice depends on the COVID-19 speciﬁcity charac-

terized by an high percentage of asymptomatic pa-

tients that could infect other people and by the high

heterogeneity of the disease severity, going from the

absence of symptoms to the need of health assistance.

In the following, three subsections will be devel-

oped; in the ﬁrst one the mathematical model is pro-

posed; then, the reproduction number is determined

analysing the system by means of the next generation

matrix approach. Finally, the control problem aim-

ing at the optimal resource allocation is introduced.

The optimal strategy addresses the fast identiﬁcation

of asymptomatic patients to reduce, as soon as possi-

ble, new infections and hospitalizations.

2.1 Mathematical Model

As previously said, each group is partitioned into 7

compartments as shown by Fig. 1. So, referring to

the h–th group, h = 1, . . . , n, the model accounts for

the following compartments:

• S

h

: susceptible subjects, that is the healthy peo-

ple within the population which are not vaccinated

yet;

• E

h

: exposed individuals, i.e. the subjects in the

incubation period; they are infected but cannot in-

fect;

• I

h

: infected isolated patients; this compartment

collects infected patients who are diagnosed as

COVID-19 positive (because of the presence of

recognizable symptoms or of a positive swab test).

As diagnosed patients, they are isolated and can-

not transmit the infection;

• H

h

: isolated hospitalized patients; it is assumed

that they are isolated in hospital, with serious

health conditions;

• A

h

: infected non isolated patients; they are sub-

jects unaware of their infection status, since they

show only light or no symptoms at all; therefore,

they can infect other people of the same or of dif-

ferent groups;

• R

h

: removed subjects, immunized because of the

healing from the infection; they are aware of their

previous infection status;

Optimal Resource Allocation for Fast Epidemic Monitoring in Networked Populations

617

• R

Ah

: removed subjects, immunized because of the

healing from the infection; they are unaware of

their previous infection status.

Note that the immunization assumed for compart-

ments R

h

and R

Ah

is actually a temporary condition

that could be reached after healing, and its presence

and duration substantially depend on the particular

virus strain that is dominant within the infected pop-

ulation. Since the majority of virus strains appeared

during the COVID-19 evolution have produced some

transitory protection from the reinfection, it is reason-

able to assume a sort of immunization in the problem

formulation, at least for a short period after the recov-

ery. Note also that the distinction among the healed

patients between those aware of their previous infec-

tion state (R

h

) and those unaware (R

Ah

), allows to ex-

clude the ﬁrst class R

h

from the swab test campaign,

at least until few months after healing. The class of

people indicated by R

Ah

is not measurable, of course,

as the class of A

h

, but they could be approximately

inferred from some extrapolation based on the test re-

sults related to speciﬁc populations or to particular pe-

riods.

Figure 1: Block diagram of the proposed model for the h–th

group.

It is assumed that only the subjects in the class A

h

of each group h can infect the subjects S

j

of any group

j. Therefore, it is mandatory to recognize as soon as

possible the subjects in each class A

h

, h = 1, . . . , n, by

means of a suitable swab test campaign, taking into

account the operative limitations.

The unique subjects really sure of their health con-

ditions are the ones that have received a positive di-

agnosis of COVID-19 after a swab test and/or after

having developed the symptoms, i.e. the subjects in

the classes I

h

, H

h

and R

h

, whereas the remaining ones

in the classes S

h

, E

h

, A

h

and R

Ah

need to be tested.

After a test, the subjects in S

h

and R

Ah

don’t change

their class, whereas the subjects in E

h

and A

h

move to

I

h

; this justiﬁes the weight coefﬁcient

ρ

h

E

h

S

h

+E

h

+A

h

+R

Ah

of the control action term inﬂuencing the subjects in

E

h

and, correspondingly, the weight

τ

h

A

h

S

h

+E

h

+A

h

+R

Ah

re-

garding the evolution of A

h

.

The dynamic equations describing the epidemic

spread of each group h can be written as

˙

S

h

=N

h

− S

h

n

∑

k=1

β

hk

A

k

− d

Sh

S

h

(1)

˙

E

h

=S

h

n

∑

k=1

β

hk

A

k

− d

Eh

E

h

− k

h

E

h

−

ρ

h

E

h

S

h

+ E

h

+ A

h

+ R

Ah

u

h

(2)

˙

I

h

=−d

Ih

I

h

+ p

h

k

h

E

h

− γ

Ih

(1 − o

h

)I

h

+ ν

h

A

h

+

ρ

h

E

h

+ τ

h

A

h

S

h

+ E

h

+ A

h

+ R

Ah

u

h

− o

h

¯o

h

I

h

(3)

˙

A

h

=−d

Ah

A

h

+ (1 − p

h

)k

h

E

h

− γ

Ah

A

h

− ν

h

A

h

−

τ

h

A

h

S

h

+ E

h

+ A

h

+ R

Ah

u

h

(4)

˙

H

h

=−d

Hh

H

h

− γ

Hh

H

h

+ o

h

¯o

h

I

h

(5)

˙

R

h

=−d

Rh

R

h

+ γ

Ih

(1 − o

h

)I

h

+ γ

Hh

H

h

(6)

˙

R

Ah

=−d

RAh

R

Ah

+ γ

Ah

A

h

(7)

The parameters appearing as coefﬁcients in (1)–

(7) have the following meanings:

• d

∗h

denotes the death rate in compartment ∗ of the

h −th group;

•

1

k

h

is the mean incubation time in E

h

;

• β

i j

represents the infection rate constant between

the healthy subjects S

i

and the infectious ones A

j

,

with i = 1, .., n, j = 1, ..., n;

•

1

ν

h

is the mean period of natural transition from A

h

to I

h

;

•

1

¯o

h

represents the mean period required for an in-

fected patient to be hospitalized, i.e. to transit in

H

h

; it is assumed that it is immediately evident

that the patient is in need of hospitalization; in the

following this value will be set equal to 1;

•

1

γ

A

h

,

1

γ

I

h

and

1

γ

H

h

represent the mean periods re-

quired for the healing of patients in the classes A

h

,

I

h

and, respectively, H

h

;

• ρ

h

and τ

h

are normalization coefﬁcients related

with the reliability of the swab tests;

• p

h

∈ [0, 1] is the probability that a subject in E

h

becomes an infected patient in I

h

after developing

COVID-19 symptoms;

• o

h

∈ [0, 1] is the probability that a subject in I

h

re-

quires hospitalization because of health worsen-

ing;

ICINCO 2022 - 19th International Conference on Informatics in Control, Automation and Robotics

618

• N

h

represents the rate of new incomers.

The different n groups inﬂuence each other by

means of the infection rates depending on the pecu-

liarities of the population at hand. In the following,

the total model of the n groups is analysed, ﬁnding, in

particular, the expression of the reproduction number

R

0

.

2.2 Model Analysis

This model will be analysed considering the whole

population with the n groups connected; the state of

the h-th group is denoted by

X

h

=

S

h

E

h

I

h

A

h

H

h

R

h

R

Ah

T

(8)

Therefore, setting X as the 7n–dimensional column

vector

X =

X

1

X

2

.

.

.

X

n

and observing that the equations describing the trend

of each group depend on all the state variables of all

the groups (because of the infection mechanism sus-

tained by the asymptomatic subjects), the h–th dy-

namical system can be written as

˙

X

h

= Γ

h

(X) + Ω

h

(X

h

)u

h

(9)

with the obvious meaning of the notation, and the

global system can be described as

˙

X = Γ(X ) + Ω(X)U (10)

In (10)

Γ(X) =

Γ

1

(X)

.

.

.

Γ

n

(X)

(11)

is the 7n ×1 vector function representing the non con-

trolled part of the state equations, while Ω(X ) is the

7n × n matrix

Ω(X) =

Ω

1

(X

1

)··· 0

.

.

.

.

.

.

.

.

.

0 ···Ω

n

(X

n

)

(12)

multiplying the control vector

U =

u

1

u

2

... u

n

T

The disease free equilibrium point

P

e

=

P

e

1

P

e

2

.

.

.P

e

n

is easily obtained and it is given by

P

e

h

=

S

e

h

0 0 0 0 0 0

T

with S

e

h

=

N

h

d

S

h

.

A useful parameter that provides information on

the evolution of the pandemic, in the absence of con-

trol actions, is the reproduction number R

0

, that is

the mean number of secondary cases per unit time

that a unique infected subject produces in a totally

susceptible population. An estimation of R

0

can

be obtained by means of the next generation matrix

method, (Van Den Driessche and Watmough, 2002);

it requires the deﬁnition of the vector Z that collects

the variables of the global system describing the evo-

lution of the infected people (in all the phases) in the

whole population

Z =

Z

1

Z

2

.

.

.

Z

n

where

Z

h

=

E

h

I

h

A

h

H

h

T

The reduced dynamical system

˙

Z can be described by

distinguishing the terms that include new infections

(F ) from all other changes (V ) in the population

˙

Z = F − V

where

F =

F

1

F

2

.

.

.

F

n

V =

V

1

V

2

.

.

.

V

n

with

F

h

=

S

h

∑

n

k=1

β

hk

A

k

0

0

0

V

h

=

d

Eh

E

h

+ k

h

E

h

d

Ih

I

h

− p

h

k

h

E

h

+ γ

Ih

(1 − o

h

)I

h

− ν

h

A

h

d

Ah

A

h

− (1 − p

h

)k

h

E

h

+ ν

h

A

h

+ γ

Ah

A

h

d

Hh

H

h

+ γ

Hh

H

h

− o

h

I

h

Starting from F and V , new matrices, say F and V ,

are deﬁned as follows

F =

∂F

∂Z

P

e

=

F

1,1

···F

1,n

.

.

.

.

.

.

.

.

.

F

n,1

···F

n,n

(13)

Optimal Resource Allocation for Fast Epidemic Monitoring in Networked Populations

619

where

F

h, j

=

∂F

h

∂Z

j

P

e

=

0 0 S

e

h

β

h j

0

0 0 0 0

0 0 0 0

0 0 0 0

(14)

and

V =

∂V

∂Z

P

e

(15)

The matrix V , of dimension 4n×4n, is block diagonal

with the non null blocks equal to

V

j j

=

d

Eh

+k

h

0 0 0

−p

h

k

h

d

Ih

+γ

Ih

(1−o

h

) −ν

h

0

−(1−p

h

)k

h

0 d

Ah

+γ

A

h

+ν

h

0

0 −o

h

0 d

Hh

+γ

Hh

The reproduction number R

0

is deﬁned as the dom-

inant eigenvalue of the next generation matrix FV

−1

.

Due to the diagonal block structure of the next gen-

eration matrix, the reproduction number R

0

is the

maximum of the set {R

0h

}, h = 1, ..., n, where R

h

is the reproduction number of group h and it is given

by

R

0h

=

S

e

h

β

hh

(1 − p

h

)k

h

(d

Eh

+ k

h

)(d

Ah

+ γ

Ah

+ ν

h

)

(16)

This implies that, if we consider the entire popula-

tion constituted by the n interacting groups, the total

reproduction number R

0

is less than 1, thus leading

to the reduction of the pandemic, if and only if the

reproduction number of each group R

0h

, h = 1, ..., n,

is smaller than 1. Note that each R

0h

, h = 1, ..., n,

increases with β

hh

and with (1 − p

h

); this is due to

the fact that the unique population that could infect is

constituted by the asymptomatic subjects in A

h

, that,

in fact, increase as (1 − p

h

) does.

2.3 Optimal Control

In a pandemic scenario characterized by a severe lim-

itation on the swab test availability, like for instance

in the ﬁrst phase of the epidemic outbreak or during

a prolonged surveillance strategy, it is mandatory to

optimally allocate the available resources to decrease

the total number of infected patients (infected, asymp-

tomatic, hospitalized). The following cost index is

proposed

J(U)=

Z

t

f

t

i

L(X,U)dt

=

1

2

Z

t

f

t

i

n

∑

k=1

[α

A

k

A

2

k

+ α

I

k

I

2

k

+ α

H

k

H

2

k

+ µ

k

u

2

k

]dt

with the constraints

u

k

∈ [u

m

k

, u

M

k

], u

m

k

< u

M

k

, k = 1, . . . , n

The weights α

A

k

, α

I

k

, α

H

k

are chosen between 0 and 1

depending on the relative importance of the various

terms. The use of the Pontryagin principle requires

the introduction of the 7n-dimension costate vector,

Λ = (λ

1

... λ

7n

)

T

, satisfying the costate vectorial

equation

˙

Λ = −

∂H

∂X

T

(17)

being H the Hamiltonian of the process

H (X,U) = L(X,U) + Λ

T

[Γ(X) + Ω(X)U] (18)

According to the Pontryagin inequality, the opti-

mal control action u

o

h

of group h satisﬁes the condi-

tion:

1

2

µ

h

u

o2

h

+

4

∑

j=2

λ

7(h−1)+ j

Ω

7(h−1)+ j,h

u

o

h

≤

1

2

µ

h

w

2

h

+

4

∑

j=2

λ

7(h−1)+ j

Ω

7(h−1)+ j,h

w

h

(19)

for any admissible control w

h

∈ [u

m

h

, u

M

h

]. The quantity

Ω

l,m

denotes the the row l and the column m of the

matrix Ω deﬁned in (12).

Therefore, the n optimal control actions are equal

to

u

o

h

(t) = Φ

h

(t), h = 1, . . . , n, (20)

where

Φ

h

(t) =

−

∑

4

j=2

λ

7(h−1)+ j

Ω

7(h−1)+ j,h

µ

h

= −

1

µ

h

−λ

7(h−1)+2

ρ

h

E

h

S

h

+ E

h

+ A

h

+ R

Ah

+λ

7(h−1)+3

ρ

h

E

h

+ τ

h

A

h

S

h

+ E

h

+ A

h

+ R

Ah

− λ

7(h−1)+4

τ

h

A

h

S

h

+ E

h

+ A

h

+ R

Ah

(21)

whenever Φ

h

(t) is in [u

m

h

, u

M

h

]. Otherwise, if Φ

h

(t) ≥

u

M

h

or Φ

h

(t) ≤ u

m

h

the optimal control is given by u

M

h

or, respectively, u

m

i

.

Note that, as usual, the switching functions Φ

h

(t)

depend on the costate evolution, given by Eq. (17)

solved backward starting from the ﬁnal condition

Λ(t

f

) = 0.

ICINCO 2022 - 19th International Conference on Informatics in Control, Automation and Robotics

620

3 NUMERICAL RESULTS

In this section it is analysed the epidemic trend of

a simulated population in which, for its peculiari-

ties, four different groups can be identiﬁed; these

groups may be differentiated based on their weak-

ness and susceptibility with respect to the virus in-

fection and on different healing capability. It may

correspond to age-class subdivision or activity-related

differences; the four groups, referred for simplicity

as G

i

, i = 1, 2, 3, 4, are deﬁned by the initial condi-

tions given in Table 1 and by the model parameters

given in Tables 2, 3. Each control u

h

is limited by the

box constraint [0.05,0.99]. Fig.2 shows the network

regarding the infection spread within the population

with the 4 selected groups. The scenario described in

Figure 2: Virus spread in a network of 4 connected groups.

this paper assumes that group G

1

has a higher infec-

tion capability with respect to the others; this means

that the contact rates β

i1

for i = 1, ..., 4 are higher with

respect to the other values of the matrix β collecting

the contact rates β

i j

, i, j = 1, ..., 4

β = 10

−7

9.5 7 5 2

9.4 8 7 4

9.4 5 8 7

9.8 5 7 8

(22)

Among the groups G

i

, i = 1, ..., 4, it is assumed a de-

creasing capability of healing from the infection, as

well as an increasing probability of hospitalization

after the infection, as i increases (see Tables 2, 3).

The described situation could correspond to a possi-

ble splitting of the population into 4 groups depending

on age; therefore, as the age increases the subjects of

each group become more susceptible to the infection,

weaker and with increased probability of hospitaliza-

tion. The rate of new incomers in each S

h

is estimated

as S

h

(0)·d

Sh

·100, that is 100 times the corresponding

death rate.

The choice of the weights in the cost index de-

pends on the speciﬁc goal to be pursued. In the

considered scenario, the aim is to minimize the in-

fected subjects allocating optimally the available re-

Table 1: Initial conditions.

G

1

G

2

G

3

G

4

S

i

(0) 10

6

3 · 10

6

4.5 · 10

6

10

6

E

i

(0) 10

2

4 · 10

2

3 · 10

3

2 · 10

2

I

i

(0) 50 10 20 4

A

i

(0) 10

2

3 5 10

H

i

(0) 10 10 10 5 · 10

2

R

i

(0) 10

2

4 · 10

3

10

3

2 · 10

2

R

Ai

(0) 5 · 10

2

10

2

8 · 10

2

10

2

Table 2: Death rated parameters for each group i = 1..., 4;

note that the values in the table must be multiplied by 10

−5

.

G

1

G

2

G

3

G

4

d

Si

· 10

5

1 1.5 1.5 2

d

Ei

· 10

5

1 1.5 1.5 2.3

d

Ii

· 10

5

1.1 1.7 1.8 3

d

Ai

· 10

5

1.02 1.5 1.6 3.4

d

Hi

· 10

5

1.2 1.8 1.8 4

d

Ri

· 10

5

1 1.5 1.5 1.5

d

RAi

· 10

5

1 1.5 1.5 2

sources, that is deﬁning the optimal planning of the

swab test campaign, which is very useful in case of

resource limitation, especially at the beginning of a

sanitary emergency but also for a prolonged surveil-

lance, during and after the pandemic and endemic pe-

riod. As said, group G

1

includes subjects with higher

infectious capability and lower probability of showing

symptoms; therefore, a possible goal is to decrease

the number of patients in A

1

; at the same time, it is

mandatory to avoid the infection and hospitalization

of the weak subjects. With this framework in mind,

the values in Table 4 are proposed.

As far as the control weights, without speciﬁc rea-

sons for privileging the action over one category, it

was decided to weigh the controls u

h

, h = 1, ...4 in

the same way, assuming µ

k

= 0.5 · 10

5

for k = 1, ..., 4.

Note the different order of magnitude of the weights

of the state variables and of the controls; this is due to

make these quantities comparable and effective in the

cost index. The simulations are obtained over a period

of 60 days, with a sampling time of 0.5. With the cho-

sen parameters, it can be noted the suggested strategy,

Fig. 3; by using the maximum allowed effort for the

swab test campaign in G

1

for about 93% of the control

period, it is possible to apply the control on groups

G

2

, G

3

and G

4

for about 80%, 70% and, respectively,

65% of the control period. These choices produce a

reduced increase in the trends of asymptomatic, in-

fected and hospitalized patients, as shown in Figs.4, 5,

6. It can be noted, especially for the subjects in G

2

and G

3

, an increase in the number of asymptomatic

subjects after the end of the swab test campaign, but a

Optimal Resource Allocation for Fast Epidemic Monitoring in Networked Populations

621

Table 3: Model parameters for each group i = 1..., 4.

G

1

G

2

G

3

G

4

k

i

1

6

1

6

1

6

1

6

ρ

i

10

6

10

6

10

6

10

6

p

i

0.1 0.5 0.7 0.85

ν

i

1

5

1

4

1

3

1

2

τ

i

10

4

10

4

10

4

10

4

o

i

0.1 0.4 0.6 0.8

γ

A

i

1

7

1

8

1

9

1

14

γ

H

i

1

7

1

8

1

9

1

14

γ

I

i

1

7

1

8

1

9

1

14

Table 4: Choice of the weights in the cost index.

G

1

G

2

G

3

G

4

α

A

k

0.99 0.08 0.02 0.008

α

I

k

0.04 0.05 0.08 0.1

α

H

k

0.4 0.5 0.7 0.99

general decrease, or at least not increasing trend, for

the infected patients in all the groups. To check the

effects of the application of the optimal controls u

o

i

,

i = 1, ..., 4 to the overall infectious situation, in Fig.

7 the total number of infected and infectious subjects

(that is all the infected, asymptomatic and hospital-

ized patients) is shown, both with the application of

the obtained optimal control (continuous line) and in

free evolution (dashed line). It can be noted that the

controlled spread is delayed with respect to the free

evolution; this is a positive consequence, since it al-

lows a simpler managing of the sanitary emergency.

Moreover, it can be evaluated the total number of dead

subjects in the population, both with and without the

application of the proposed surveillance control strat-

egy; the total number of deaths is equal to 8.85 × 10

4

without the determined control, whereas is 1.94 × 10

4

with the determined action that actually produces a

decrease of more than 78%. These results show the

strong sanitary impact of a suitably planned test cam-

Figure 3: Optimal control strategy.

Figure 4: Trend of the asymptomatic subjects in the 4

groups.

Figure 5: Trend of the infected subjects in the 4 groups.

paign. Finally, it is interesting to study the inﬂuence

of the total number of asymptomatic subjects on the

infected isolated and hospitalized patients. It can be

noted from the state trajectory of Fig. 8 that, in the ab-

sence of an optimal swab test campaign, the number

of infected patients (isolated and hospitalized) keeps

Figure 6: Trend of the hospitalized subjects in the 4 groups.

ICINCO 2022 - 19th International Conference on Informatics in Control, Automation and Robotics

622

on increasing even when the number of asymptomatic

individuals has started decreasing.

Figure 7: Comparison between the trend of total number of

infected patients in the controlled and non controlled con-

ditions.

Figure 8: Asymptomatic subjects in the population (that is

∑

4

h=1

A

h

) versus the sum of infected and hospitalized pa-

tients (that is

∑

4

h=1

(I

h

+ H

h

)); in continuous line the con-

trolled case, in dashed line the non controlled one.

To stress the potentialities of the approach, it is

proposed a second case in which the subjects in G

1

are both more infectious and more susceptible to the

infection than the other groups,preserving their gen-

eral good health, as well as the weak probability of

having severe complications. The matrix β represent-

ing the new assumption on G

1

is given by

β = 10

−7

9.5 9 9 9

9.4 8 7 4

9.4 5 8 7

9.8 5 7 8

. (23)

In Fig. 9 it is shown the optimal control strategy for

the swab test campaign; all the actions start at their

maximal rate but only few days after the beginning of

the control period, that is after about 5 days, and they

maintain the maximum value for different periods: u

o

1

up to day 55, u

o

2

up to day 48, u

o

3

up to day 40, and

u

o

4

up to day 37. Using this control strategy that op-

timizes the swab test campaign, the total number of

dead people is reduced more than 79% w.r.t. the un-

controlled case, i.e. from 1.2 · 10

5

, in the absence of

any control, to 2.4 · 10

4

, with the application of the

optimal strategy.

Figure 9: Optimal control strategy for the scenario de-

scribed by the contact rate matrix (23).

In Fig. 10 it is shown a comparison of the num-

ber of infected in the four groups, with and without

the application of the swab test campaign; it is evi-

dent the strong reduction of the infected patients, es-

pecially for the subjects in groups G

2

, G

3

and G

4

.

Similar trends can be observed in Fig. 11 that shows

Figure 10: Comparison between I

o

i

, i = 1, ..., 4 (Continuous

line) and I

i

(dotted line) with the contact rate matrix (23);

ﬁrst row, from left to right: infected patients in G

1

, G

2

;

second row, from left to right: infected patients in G

3

, G

4

.

the comparison between the hospitalized patients with

and without the control. The improvements are

due to the resource allocation obtained by the optimal

swab test campaign that strongly reduces the number

Optimal Resource Allocation for Fast Epidemic Monitoring in Networked Populations

623

Figure 11: Comparison between H

o

i

, i = 1, ..., 4 (Continu-

ous line) and H

i

(dotted line) with the contact rate matrix

(23); ﬁrst row, from left to right: hospitalized patients in

G

1

, G

2

; second row, from left to right: hospitalized patients

in G

3

, G

4

.

Figure 12: Comparison between A

o

i

, i = 1, ..., 4 (Continuous

line) and A

i

(dotted line) with the contact rate matrix (23);

ﬁrst row, from left to right: asymptomatic subjects in G

1

,

G

2

; second row, from left to right: asymptomatic subjects

in G

3

, G

4

.

of asymptomatic subjects, see Fig.12. The results of

Figs. 10, 11, 12 can be summarized in Fig. 13 where

it is shown the trend of the infected and infectious

individuals both in the controlled and non controlled

case; note that there is one order of magnitude of dif-

ference in the peaks of the two curves (3.1·10

6

versus

2.7 · 10

7

patients). The effect of an effective swab test

campaign on the number of asymptomatic individu-

als, and therefore on the infected isolated patients and

on the hospitalized ones, is also highlighted by the

state trajectory reported in Fig. 14; note that in the

controlled case (continuous line), both asymptomatic

subjects and infected patients initially increase but,

after a while, they both decrease. Conversely, with-

Figure 13: Comparison between the trend of total number of

infected patients in the controlled and non controlled con-

ditions in the second case-study.

Figure 14: Asymptomatic subjects in the population (that

is

∑

4

h=1

A

h

) versus the sum of the infected and hospitalized

patients (that is

∑

4

h=1

(I

h

+H

h

)) in the second case-study; in

continuous line the controlled case, in dashed line the non

controlled one.

out any action (dashed line), the number of infected

isolated and hospitalized patients remains high even

when the number of asymptomatic subjects is actu-

ally decreasing.

4 CONCLUSIONS

The management of a pandemic, such as that due to

Sars-COV 2, must consider various aspects, as the al-

location of limited resources from different points of

view. The endemic phase also requires attention and

active surveillance. In this work, referring to COVID-

19, we propose a methodology for the deﬁnition of a

protocol for the management of swab tests in a pop-

ulation in which different groups can be identiﬁed

ICINCO 2022 - 19th International Conference on Informatics in Control, Automation and Robotics

624

based on some characteristics (age, work, social con-

dition, etc.). These groups are connected by a network

of contacts; their different susceptibility to the virus,

infectious capacity and speed of healing suggest an

optimal strategy for a swab test campaign. The anal-

ysis, supported by the numerical results, suggests a

control strategy that particularly focuses on the most

infectious individuals, allowing less surveillance on

the most fragile subjects. In future work different net-

work population characteristics could be included.

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