https://orcid.org/0000-0003-2399-1137
Abstract
This paper analyzes an age-group susceptible-infected-recovered (SIR) model. Theoretical results concerning the conservation of the total population, the positivity of the analytical solution, and the final size of the epidemic are derived. Since the model is a nonlinear system of ordinary differential equations (ODEs), a numerical approximation is considered, based on Standard and non-Standard Finite Difference methods, and on a Modified Patankar-Runge–Kutta (MPRK) method. The numerical preservation of the qualitative properties of the analytical solution is studied. The obtained results are applied to the diffusion of information in social networks, and the effectiveness of the different numerical approaches is shown through several numerical tests on real data.
References
1.
Rodrigues
,
H.
, 2016
, “
Application of SIR Epidemiological Model: New Trends
,” arXiv:1611.02565.2.
Fumanelli
,
L.
,
Ajelli
,
M.
,
Manfredi
,
P.
,
Vespignani
,
A.
, and
Merler
,
S.
, 2012
, “
Inferring the Structure of Social Contacts From Demographic Data in the Analysis of Infectious Diseases Spread
,” PLoS Comput. Biol.
,
8
(9
), p. e1002673
.10.1371/journal.pcbi.10026733.
Cardone
,
A.
,
Díaz de Alba
,
P.
, and
Paternoster
,
B.
, 2022
, “
Influence of Age Group in the Spreading of Fake News: Contact Matrices in Social Media
,” 16th International Conference on Signal-Image Technology & Internet-Based Systems (SITIS
),
Dijon, France
, Oct. 19–21, pp. 515
–521
.10.1109/SITIS57111.2022.000834.
Kermack
,
W. O.
, and
McKendrick
,
A. G.
, 1927
, “
A Contribution to the Mathematical Theory of Epidemics
,” Proc. R. Soc. London, Ser. A
,
115
, pp. 700
–721
.10.1098/rspa.1927.01185.
Brauer
,
F.
,
Castillo-Chavez
,
C.
, and
Feng
,
Z.
, 2019
, Mathematical Models in Epidemiology
, Vol.
32
,
Springer
, New York, p. xvii+619.6.
Kaddar
,
A.
, 2009
, “
On the Dynamics of a Delayed SIR Epidemic Model With a Modified Saturated Incidence Rate
,” Electron. J. Differ. Eq.
,
2009
(133
), pp. 1
–7
.https://ejde.math.txstate.edu/Volumes/2009/133/kaddar.pdf7.
Lin
,
Y.
,
Jiang
,
D.
, and
Xia
,
P.
, 2014
, “
Long-Time Behavior of a Stochastic SIR Model
,” Appl. Math. Comput.
,
236
, pp. 1
–9
.10.1016/j.amc.2014.03.0358.
Razaque
,
A.
,
Rizvi
,
S.
,
Khan
,
M. J.
,
Almiani
,
M.
, and
Rahayfeh
,
A. A.
, 2022
, “
State-of-Art Review of Information Diffusion Models and Their Impact on Social Network Vulnerabilities
,” J. King Saud Univ. - Comput. Inf. Sci.
,
34
(1
), pp. 1275
–1294
.10.1016/j.jksuci.2019.08.0089.
Rui
,
X.
,
Meng
,
F.
,
Wang
,
Z.
,
Yuan
,
G.
, and
Du
,
C.
, 2018
, “
SPIR: The Potential Spreaders Involved SIR Model for Information Diffusion in Social Networks
,” Phys. A
,
506
, pp. 254
–269
.10.1016/j.physa.2018.04.06210.
Woo
,
J.
, and
Chen
,
H.
, 2012
, “
An Event-Driven Sir Model for Topic Diffusion in Web Forums
,” ISI 2012 IEEE International Conference on Intelligence and Security Informatics: Cyberspace, Border, and Immigration Securities
, Washington, DC, June 11–14, pp. 108
–113
.10.1109/ISI.2012.628410111.
Woo
,
J.
, and
Chen
,
H.
, 2016
, “
Epidemic Model for Information Diffusion in Web Forums: Experiments in Marketing Exchange and Political Dialog
,” SpringerPlus
,
5
(1
), pp. 1
–19
.10.1186/s40064-016-1675-x12.
Li
,
Y.
,
Zhao
,
H.
, and
Chen
,
Y.
, 2022
, “
An Epidemic Model for Correlated Information Diffusion in Crowd Intelligence Networks
,” Int. J. Crowd Sci.
,
3
(2
), pp. 168
–183
.10.1108/IJCS-01-2019-000513.
Sivaraman
,
N.
,
Baijal
,
S.
, and
Muthiah
,
S.
, 2022
, “
On the Usage of Epidemiological Models for Information Diffusion Over Twitter
,” SSRN Electron. J.
,
13
(1
), p. 133
.10.2139/ssrn.425102414.
Castiello
,
M.
,
Conte
,
D.
, and
Iscaro
,
S.
, 2023
, “
Using Epidemiological Models to Predict the Spread of Information on Twitter
,” Algorithms
,
16
(8
), p. 391
.10.3390/a1608039115.
Wang
,
P.
,
Yu
,
G.
,
Wu
,
X.
,
Wang
,
Y.
, and
He
,
X.
, 2019
, “
Spreading Patterns of Malicious Information on Single-Lane Platooned Traffic in a Connected Environment
,” Comput.-Aided Civ. Infrastruct. Eng.
,
34
(3
), pp. 248
–265
.10.1111/mice.1241616.
Dong
,
S.
,
Deng
,
Y.
, and
Huang
,
Y.
, 2017
, “
SEIR Model of Rumor Spreading in Online Social Network With Varying Total Population Size
,” Commun. Theor. Phys.
,
68
(4
), pp. 545
–552
.10.1088/0253-6102/68/4/54517.
Franceschi
,
J.
, and
Pareschi
,
L.
, 2022
, “
Spreading of Fake News, Competence and Learning: Kinetic Modelling and Numerical Approximation
,” Philos. Trans. R. Soc. A
,
380
(2224
), p. 20210159
.10.1098/rsta.2021.015918.
Lotito
,
Q.
,
Zanella
,
D.
, and
Casari
,
P.
, 2021
, “
Realistic Aspects of Simulation Models for Fake News Epidemics Over Social Networks
,” Future Internet
,
13
(3
), p. 76
.10.3390/fi1303007619.
D’Ambrosio
,
R.
,
Díaz de Alba
,
P.
,
Giordano
,
G.
, and
Paternoster
,
B.
, 2022
, “
A Modified SEIR Model: Stiffness Analysis and Application to the Diffusion of Fake News
,” Lect. Notes Comput. Sci.
,
13375
, pp. 90
–103
.10.1007/978-3-031-10522-7_720.
D’Ambrosio
,
R.
,
Giordano
,
G.
,
Mottola
,
S.
, and
Paternoster
,
B.
, 2021
, “
Stiffness Analysis to Predict the Spread Out of Fake News
,” Future Internet
,
13
(9
), p. 222
.10.3390/fi1309022221.
Deepak
,
P.
,
Chakraborty
,
T.
,
Long
,
C.
, and
Santhosh Kumar
,
G.
, 2021
, Data Science for Fake News
,
Springer
, Switzerland.22.
Giordano
,
G.
,
Mottola
,
S.
, and
Paternoster
,
B.
, 2020
, “
A Short Review of Some Mathematical Methods to Detect Fake News
,” Int. J. Circuits, Syst. Signal Process.
,
14
, pp. 255
–265
.10.46300/9106.2020.14.3723.
Mottola
,
S.
, 2020
, “
Las Fake News Como Fenómeno Social. análisis Lingüístico y Poder Persuasivo de Bulos en Italiano y Español
,” Discurso Sociedad
,
14
(3
), pp. 683
–706
.https://www.researchgate.net/publication/336059083_Las_fake_news_como_fenomeno_social_estructura_linguistica_y_poder_persuasivo_de_los_bulos_en_italiano_y_espanol24.
Mottola
,
S.
, 2021
, “
Las Fake News Como Expresión de Ideologías. entre Bulos, Posverdad y Creencias
,” Les Idéologies Linguistiques: Débats, Purismes et Stratégies Discursives
,
C.
Marimón Llorca
,
W.
Remysen
, and
F.
Rossi
, eds.,
Peter Lang Verlag
,
Berlin, Germany
, pp. 469
–494
.25.
Tandoc
,
J.
, and
Edson
,
C.
, 2019
, “
The Facts of Fake News: A Research Review
,” Sociol. Compass
,
13
(9
), p. e12724
.10.1111/soc4.1272426.
Mickens
,
R.
, 1994
, Nonstandard Finite Difference Models of Differential Equations
,
World Scientific Publishing
, Singapore.27.
Mickens
,
R.
, 2007
, “
Calculation of Denominator Functions for Nonstandard Finite Difference Schemes for Differential Equations Satisfying a Positivity Condition
,” Numer. Methods Partial Differ. Eq.
,
23
(3
), pp. 672
–691
.10.1002/num.2019828.
Conte
,
D.
,
Guarino
,
N.
,
Pagano
,
G.
, and
Paternoster
,
B.
, 2022
, “
On the Advantages of Nonstandard Finite Difference Discretizations for Differential Problems
,” Numer. Anal. Appl.
,
15
(3
), pp. 219
–235
.10.1134/S199542392203004129.
Conte
,
D.
,
Guarino
,
N.
,
Pagano
,
G.
, and
Paternoster
,
B.
, 2022
, “
Positivity-Preserving and Elementary Stable Nonstandard Method for a COVID-19 SIR Model
,” Dolomites Res. Notes Approximations
,
15
(5
), pp. 65
–77
.10.14658/PUPJ-DRNA-2022-5-730.
Conte
,
D.
,
Pagano
,
G.
, and
Paternoster
,
B.
, 2023
, “
Nonstandard Finite Differences Numerical Methods for a Vegetation Reaction-Diffusion Model
,” J. Comput. Appl. Math.
,
419
, p. 114790
.10.1016/j.cam.2022.11479031.
Burchard
,
H.
,
Deleersnijder
,
E.
, and
Meister
,
A.
, 2003
, “
A High-Order Conservative Patankar-Type Discretisation for Stiff Systems of Production-Destruction Equations
,” Appl. Numer. Math.
,
47
(1
), pp. 1
–30
.10.1016/S0168-9274(03)00101-632.
Conte
,
D.
, and
Frasca-Caccia
,
G.
, 2022
, “
Exponentially Fitted Methods That Preserve Conservation Laws
,” Commun. Nonlinear Sci. Numer. Simul.
,
109
, p. 106334
.10.1016/j.cnsns.2022.10633433.
Frasca-Caccia
,
G.
, and
Hydon
,
P. E.
, 2022
, “
A New Technique for Preserving Conservation Laws
,” Found. Comput. Math.
,
22
(2
), pp. 477
–506
.10.1007/s10208-021-09511-134.
Hairer
,
E.
,
Wanner
,
G.
, and
Lubich
,
C.
, 2006
, Geometric Numerical Integration Structure-Preserving Algorithms for Ordinary Differential Equations
,
Springer
, Berlin, Heidelberg.35.
Arregui
,
S.
,
Aleta
,
A.
,
Sanz
,
J.
, and
Moreno
,
Y.
, 2018
, “
Projecting Social Contact Matrices to Different Demographic Structures
,” PLOS Comput. Biol.
,
14
(12
), p. e1006638
.10.1371/journal.pcbi.100663836.
Diekmann
,
O.
,
Heesterbeek
,
J. A. P.
, and
Roberts
,
M. G.
, 2010
, “
The Construction of Next-Generation Matrices for Compartmental Epidemic Models
,” J. R. Soc. Interface
,
7
(47
), pp. 873
–885
.10.1098/rsif.2009.038637.
Diekmann
,
O.
,
Heesterbeek
,
J. A. P.
, and
Metz
,
J. A. J.
, 1990
, “
On the Definition and the Computation of the Basic Reproduction Ratio R0 in Models for Infectious Diseases in Heterogeneous Populations
,” J. Math. Biol.
,
28
(4
), pp. 365
–382
.10.1007/BF0017832438.
Martcheva
,
M.
, 2015
, An Introduction to Mathematical Epidemiology
,
Springer
, New York.39.
Magal
,
P.
,
Seydi
,
O.
, and
Webb
,
G.
, 2018
, “
Final Size of a Multi-Group Sir Epidemic Model: Irreducible and Non-Irreducible Modes of Transmission
,” Math Biosci.
,
301
, pp. 59
–67
.10.1016/j.mbs.2018.03.02040.
Burrage
,
K.
, 1995
, Parallel and Sequential Methods for Ordinary Differential Equations
,
Clarendon Press
, New York.41.
De Luca
,
P.
,
Di Luccio
,
D.
,
Galletti
,
A.
,
Giunta
,
G.
,
Marcellino
,
L.
, and
Montella
,
R.
, 2022
, “
Towards a GPU Parallel Software for Environmental Data Fitting
,” Proceedings of the 15th International Conference on PErvasive Technologies Related to Assistive Environments, PETRA ’22
,
Association for Computing Machinery
, Corfu Island, Greece, June 29–July 1, pp. 469
–472
.10.1145/3529190.353477642.
Montella
,
R.
,
Di Luccio
,
D.
,
De Vita
,
C. G.
,
Mellone
,
G.
,
Lapegna
,
M.
,
Ortega
,
G.
,
Marcellino
,
L.
,
Zambianchi
,
E.
, and
Giunta
,
G.
, 2023
, “
A Highly Scalable High-Performance Lagrangian Transport and Diffusion Model for Marine Pollutants Assessment
,” 2023 31st Euromicro International Conference on Parallel, Distributed and Network-Based Processing (PDP
),
Naples, Italy, Mar. 1–3, pp. 17
–26
.10.1109/PDP59025.2023.0001243.
Lambert
,
J.
, 1992
, Numerical Methods for Ordinary Differential Systems
,
Wiley
,
New York
.44.
Perko
,
L.
, 2001
, Differential Equations and Dynamical Systems
, 3rd ed., Vol.
7
,
Springer-Verlag
,
New York
.45.
Mickens
,
R. E.
, 2005
, Advances in the Application of Nonstandard Finite Difference Schemes
,
World Scientific
,
Singapore
.46.
Torlo
,
D.
,
Öffner
,
P.
, and
Ranocha
,
H.
, 2022
, “
Issues With Positivity-Preserving Patankar-Type Schemes
,” Appl. Numer. Math.
,
182
, pp. 117
–147
.10.1016/j.apnum.2022.07.01447.
Blanes
,
S.
,
Iserles
,
A.
, and
Macnamara
,
S.
, 2022
, “
Positivity-Preserving Methods for Ordinary Differential Equations
,” ESAIM: Math. Modell. Numer. Anal.
,
56
(6
), pp. 1843
–1870
.10.1051/m2an/202204248.
Patankar
,
S.
, 1980
, Numerical Heat Transfer and Fluid Flow
, Vol.
32
,
McGraw-Hill
,
New York
.49.
United Nations Development Programme
, “Human Development Data Center,” United Nations Development Programme, New York, accessed June 22, 2023, http://hdr.undp.org/en/data50.
Statista
, “Empowering People With Data,” Statista, Hamburg, Germany, accessed June 22, 2023, https://www.statista.com51.
The Global Economy
, “Business and Economic Data for 200 Countries,” accessed June 22, 2023, https://www.theglobaleconomy.com/52.
Brunner
,
H.
, 2004
, Collocation Methods for Volterra Integral and Related Functional Differential Equations
, Vol.
15
,
Cambridge University Press
,
Cambridge
.53.
Cardone
,
A.
,
Conte
,
D.
,
D’Ambrosio
,
R.
, and
Paternoster
,
B.
, 2018
, “
Collocation Methods for Volterra Integral and Integro-Differential Equations: A Review
,” Axioms
,
7
(3
), p. 45
.10.3390/axioms703004554.
Cardone
,
A.
,
Conte
,
D.
, and
Paternoster
,
B.
, 2022
, “
Stability of Two-Step Spline Collocation Methods for Initial Value Problems for Fractional Differential Equations
,” Commun. Nonlinear Sci. Numer. Simul
,
115
, p. 106726
.10.1016/j.cnsns.2022.10672655.
Cardone
,
A.
,
D'Ambrosio
,
R.
, and
Paternoster
,
B.
, 2017
, “
High Order Exponentially Fitted Methods for Volterra Integral Equations With Periodic Solution
,” Appl.Numer. Math.
,
114
, pp. 18
–29
.10.1016/j.apnum.2016.05.00356.
Hairer
,
E.
,
Nørsett
,
S. P.
, and
Wanner
,
G.
, 1993
, Solving Ordinary Differential Equations. I
, 2nd ed., Vol.
8
,
Springer-Verlag
,
Berlin
.57.
Moradi
,
A.
,
D'Ambrosio
,
R.
, and
Paternoster
,
B.
, 2023
, “
Variable Stepsize Multivalue Collocation Methods
,” Appl. Numer. Math.
,
190
, pp. 1
–14
.10.1016/j.apnum.2023.03.008
