Figure 1.
Evolution of the infected nodes for different values of β and δ: a) β = δ = 0; b) β = 1, δ = 0; c) β = δ = 1; d) β = 0, δ = 1
-
Previous Article
Risk assessment for enterprise merger and acquisition via multiple classifier fusion
- DCDS-S Home
- This Issue
-
Next Article
Design of one type of linear network prediction controller for multi-agent system
August & September
2019, 12(4&5): 735-746.
doi: 10.3934/dcdss.2019048
A SIR-based model for contact-based messaging applications supported by permanent infrastructure
| 1. | Instituto Universitario de Matemática Pura y Aplicada, Universitat Politècnica de València, Spain |
| 2. | Departamento de Informática de Sistemas y Computadores, Universitat Politècnica de València, Spain |
| 3. | Institut Universitari de Matemàtiques i Aplicacions de Castelló (IMAC), Escuela Superior de Tecnología y Ciencias Experimentales, Universitat Jaume I, Spain |
- Abstract
- Full Text(HTML)
- Figure(4)
- Related Papers
Under a Creative Commons license
In this paper we focus on the study of coupled systems of ordinary differential equations (ODE's) describing the diffusion of messages between mobile devices. Communications in mobile opportunistic networks take place upon the establishment of ephemeral contacts among mobile nodes using direct communication. SIR (Sane, Infected, Recovered) models permit to represent the diffusion of messages using an epidemiological based approach.
The question we analyse in this work is whether the coexistence of a fixed infrastructure can improve the diffusion of messages and thus justify the additional costs. We analyse this case from the point of view of dynamical systems, finding and characterising the admissible equilibrium of this scenario. We show that a centralised diffusion is not efficient when people density reaches a sufficient value.
This result supports the interest in developing opportunistic networks for occasionally crowded places to avoid the cost of additional infrastructure.
Keywords:
SIR models,
PDEs in connection with computer science,
mobile networking in proximity,
contact-based messaging,
dynamical systems,
epidemic diffusion.
Mathematics Subject Classification:
Primary: 94C99; Secondary: 37N25.
Citation:
J. Alberto Conejero, Enrique Hernández-Orallo, Pietro Manzoni, Marina Murillo-Arcila. A SIR-based model for contact-based messaging applications supported by permanent infrastructure. Discrete and Continuous Dynamical Systems - S,
2019, 12
(4&5)
: 735-746.
doi: 10.3934/dcdss.2019048
![]()
References:
| [1] |
L. J. S. Allen, Mathematical epidemiology: Lecture notes in mathematics, An Introduction to Stochastic Epidemic Models, Springer Verlag, 1945 (2008), 81-130.
doi: 10.1007/978-3-540-78911-6_3. |
| [2] |
Roy M. Anderson (ed.), The population dynamics of infectious diseases: Theory and applications, Chapman and Hall, 1982. |
| [3] |
F. Brauer, Compartmental models in epidemiology, in Mathematical epidemiology, vol. 1945 of Lecture Notes in Math., Springer, Berlin, 2008, 19-79.
doi: 10.1007/978-3-540-78911-6_2. |
| [4] |
M. Brin and G. Stuck, Introduction to Dynamical Systems, Cambridge University Press, Cambridge, 2002.
doi: 10.1017/CBO9780511755316. |
| [5] |
C. S. De Abreu and R. M. Salles,
Modeling message diffusion in epidemical DTN, Ad Hoc Networks, 16 (2014), 197-209.
|
| [6] |
A. Dénes and G. Röst,
Global stability for SIR and SIRS models with nonlinear incidence and removal terms via Dulac functions, Discrete Contin. Dyn. Syst. Ser. B, 21 (2016), 1101-1117.
doi: 10.3934/dcdsb.2016.21.1101. |
| [7] |
R. Groenevelt, P. Nain and G. Koole,
The message delay in mobile ad hoc networks, Performance Evaluation, 62 (2005), 210-228.
|
| [8] |
Z. J. Haas and T. Small,
A new networking model for biological applications of ad hoc sensor networks, Networking, IEEE/ACM Transactions on, 14 (2006), 27-40.
|
| [9] |
E. Hernández-Orallo, M. Murillo-Arcila, C. T. Calafate, J. C. Cano, J. A. Conejero and P. Manzoni,
Analytical evaluation of the performance of contact-based messaging applications, Computer Networks, 111 (2016), 45-54.
|
| [10] |
H. Huang and M. Wang,
The reaction-diffusion system for an SIR epidemic model with a free boundary, Discrete Contin. Dyn. Syst. Ser. B, 20 (2015), 2039-2050.
doi: 10.3934/dcdsb.2015.20.2039. |
| [11] |
F. Jian and S. Dandan,
Complex network theory and its application research on p2p networks, Applied Mathematics and Nonlinear Sciences, 1 (2016), 45-52.
|
| [12] |
T. G. Kurtz, Approximation of Population Processes, SIAM, 1981. |
| [13] |
E. Toledano, D. Sawada, A. Lippman, H. Holtzman and F. Casalegno, Cocam: Real-time photo sharing based on opportunistic p2p networking, in Consumer Communications and Networking Conference (CCNC), 2013 IEEE, 2013, 877-878. |
| [14] |
X. Wang, An SIRS epidemic model with vital dynamics and a ratio-dependent saturation incidence rate Discrete Dyn. Nat. Soc., 2015 (2015), Art. ID 720682, 9pp.
doi: 10.1155/2015/720682. |
| [15] |
H. Wu and B. Zhao,
Overview of current techniques in remote data auditing, Applied Mathematics and Nonlinear Sciences, 1 (2016), 145-158.
|
| [16] |
Q. Xu, Z. Su, K. Zhang, P. Ren and X. S. Shen,
Epidemic information dissemination in mobile social networks with opportunistic links, Emerging Topics in Computing, IEEE Transactions on, 3 (2015), 399-409.
|
| [17] |
X. Zhang, G. Neglia, J. Kurose and D. Towsley,
Performance modeling of epidemic routing, Computer Networks, 51 (2007), 2867-2891.
|
show all references
References:
| [1] |
L. J. S. Allen, Mathematical epidemiology: Lecture notes in mathematics, An Introduction to Stochastic Epidemic Models, Springer Verlag, 1945 (2008), 81-130.
doi: 10.1007/978-3-540-78911-6_3. |
| [2] |
Roy M. Anderson (ed.), The population dynamics of infectious diseases: Theory and applications, Chapman and Hall, 1982. |
| [3] |
F. Brauer, Compartmental models in epidemiology, in Mathematical epidemiology, vol. 1945 of Lecture Notes in Math., Springer, Berlin, 2008, 19-79.
doi: 10.1007/978-3-540-78911-6_2. |
| [4] |
M. Brin and G. Stuck, Introduction to Dynamical Systems, Cambridge University Press, Cambridge, 2002.
doi: 10.1017/CBO9780511755316. |
| [5] |
C. S. De Abreu and R. M. Salles,
Modeling message diffusion in epidemical DTN, Ad Hoc Networks, 16 (2014), 197-209.
|
| [6] |
A. Dénes and G. Röst,
Global stability for SIR and SIRS models with nonlinear incidence and removal terms via Dulac functions, Discrete Contin. Dyn. Syst. Ser. B, 21 (2016), 1101-1117.
doi: 10.3934/dcdsb.2016.21.1101. |
| [7] |
R. Groenevelt, P. Nain and G. Koole,
The message delay in mobile ad hoc networks, Performance Evaluation, 62 (2005), 210-228.
|
| [8] |
Z. J. Haas and T. Small,
A new networking model for biological applications of ad hoc sensor networks, Networking, IEEE/ACM Transactions on, 14 (2006), 27-40.
|
| [9] |
E. Hernández-Orallo, M. Murillo-Arcila, C. T. Calafate, J. C. Cano, J. A. Conejero and P. Manzoni,
Analytical evaluation of the performance of contact-based messaging applications, Computer Networks, 111 (2016), 45-54.
|
| [10] |
H. Huang and M. Wang,
The reaction-diffusion system for an SIR epidemic model with a free boundary, Discrete Contin. Dyn. Syst. Ser. B, 20 (2015), 2039-2050.
doi: 10.3934/dcdsb.2015.20.2039. |
| [11] |
F. Jian and S. Dandan,
Complex network theory and its application research on p2p networks, Applied Mathematics and Nonlinear Sciences, 1 (2016), 45-52.
|
| [12] |
T. G. Kurtz, Approximation of Population Processes, SIAM, 1981. |
| [13] |
E. Toledano, D. Sawada, A. Lippman, H. Holtzman and F. Casalegno, Cocam: Real-time photo sharing based on opportunistic p2p networking, in Consumer Communications and Networking Conference (CCNC), 2013 IEEE, 2013, 877-878. |
| [14] |
X. Wang, An SIRS epidemic model with vital dynamics and a ratio-dependent saturation incidence rate Discrete Dyn. Nat. Soc., 2015 (2015), Art. ID 720682, 9pp.
doi: 10.1155/2015/720682. |
| [15] |
H. Wu and B. Zhao,
Overview of current techniques in remote data auditing, Applied Mathematics and Nonlinear Sciences, 1 (2016), 145-158.
|
| [16] |
Q. Xu, Z. Su, K. Zhang, P. Ren and X. S. Shen,
Epidemic information dissemination in mobile social networks with opportunistic links, Emerging Topics in Computing, IEEE Transactions on, 3 (2015), 399-409.
|
| [17] |
X. Zhang, G. Neglia, J. Kurose and D. Towsley,
Performance modeling of epidemic routing, Computer Networks, 51 (2007), 2867-2891.
|
Figure 2.
Evolution of the infected nodes in the open model with fixed nodes. In all cases β = δ = 1. a) ρ = 0:5; b) ρ = 1; c) ρ = 2; d) ρ = 4;
Figure 3.
Message coverage depending on people density and renewal percentages. a) contact-based only diffusion; b) contactbased and fixed nodes diffusion for ρ = 1.
Figure 4.
Delivery time depending on the people density and with different renewal rates. The label with FN, refers to diffusion with Fixed-nodes. a) Delivery time to 95% of the nodes; b) Delivery time to 75% of nodes.
| [1] |
Kazuyuki Yagasaki. Optimal control of the SIR epidemic model based on dynamical systems theory. Discrete and Continuous Dynamical Systems - B, 2022, 27 (5) : 2501-2513. doi: 10.3934/dcdsb.2021144 |
| [2] |
Yoichi Enatsu, Yukihiko Nakata, Yoshiaki Muroya. Global stability for a class of discrete SIR epidemic models. Mathematical Biosciences & Engineering, 2010, 7 (2) : 347-361. doi: 10.3934/mbe.2010.7.347 |
| [3] |
Jean-Philippe Cointet, David Chavalarias. Multi-level science mapping with asymmetrical paradigmatic proximity. Networks and Heterogeneous Media, 2008, 3 (2) : 267-276. doi: 10.3934/nhm.2008.3.267 |
| [4] |
Jing Hui, Lansun Chen. Impulsive vaccination of sir epidemic models with nonlinear incidence rates. Discrete and Continuous Dynamical Systems - B, 2004, 4 (3) : 595-605. doi: 10.3934/dcdsb.2004.4.595 |
| [5] |
Andrea Franceschetti, Andrea Pugliese, Dimitri Breda. Multiple endemic states in age-structured $SIR$ epidemic models. Mathematical Biosciences & Engineering, 2012, 9 (3) : 577-599. doi: 10.3934/mbe.2012.9.577 |
| [6] |
Haomin Huang, Mingxin Wang. The reaction-diffusion system for an SIR epidemic model with a free boundary. Discrete and Continuous Dynamical Systems - B, 2015, 20 (7) : 2039-2050. doi: 10.3934/dcdsb.2015.20.2039 |
| [7] |
Mudassar Imran, Mohamed Ben-Romdhane, Ali R. Ansari, Helmi Temimi. Numerical study of an influenza epidemic dynamical model with diffusion. Discrete and Continuous Dynamical Systems - S, 2020, 13 (10) : 2761-2787. doi: 10.3934/dcdss.2020168 |
| [8] |
Azmy S. Ackleh, Linda J. S. Allen. Competitive exclusion in SIS and SIR epidemic models with total cross immunity and density-dependent host mortality. Discrete and Continuous Dynamical Systems - B, 2005, 5 (2) : 175-188. doi: 10.3934/dcdsb.2005.5.175 |
| [9] |
E. Almaraz, A. Gómez-Corral. On SIR-models with Markov-modulated events: Length of an outbreak, total size of the epidemic and number of secondary infections. Discrete and Continuous Dynamical Systems - B, 2018, 23 (6) : 2153-2176. doi: 10.3934/dcdsb.2018229 |
| [10] |
Jinliang Wang, Xianning Liu, Toshikazu Kuniya, Jingmei Pang. Global stability for multi-group SIR and SEIR epidemic models with age-dependent susceptibility. Discrete and Continuous Dynamical Systems - B, 2017, 22 (7) : 2795-2812. doi: 10.3934/dcdsb.2017151 |
| [11] |
Yoichi Enatsu, Yukihiko Nakata, Yoshiaki Muroya. Global stability of SIR epidemic models with a wide class of nonlinear incidence rates and distributed delays. Discrete and Continuous Dynamical Systems - B, 2011, 15 (1) : 61-74. doi: 10.3934/dcdsb.2011.15.61 |
| [12] |
István Balázs, Jan Bouwe van den Berg, Julien Courtois, János Dudás, Jean-Philippe Lessard, Anett Vörös-Kiss, JF Williams, Xi Yuan Yin. Computer-assisted proofs for radially symmetric solutions of PDEs. Journal of Computational Dynamics, 2018, 5 (1&2) : 61-80. doi: 10.3934/jcd.2018003 |
| [13] |
Timothy C. Reluga, Jan Medlock. Resistance mechanisms matter in SIR models. Mathematical Biosciences & Engineering, 2007, 4 (3) : 553-563. doi: 10.3934/mbe.2007.4.553 |
| [14] |
Qianqian Cui, Zhipeng Qiu, Ling Ding. An SIR epidemic model with vaccination in a patchy environment. Mathematical Biosciences & Engineering, 2017, 14 (5&6) : 1141-1157. doi: 10.3934/mbe.2017059 |
| [15] |
Zhen Jin, Zhien Ma. The stability of an SIR epidemic model with time delays. Mathematical Biosciences & Engineering, 2006, 3 (1) : 101-109. doi: 10.3934/mbe.2006.3.101 |
| [16] |
Yan Li, Wan-Tong Li, Guo Lin. Traveling waves of a delayed diffusive SIR epidemic model. Communications on Pure and Applied Analysis, 2015, 14 (3) : 1001-1022. doi: 10.3934/cpaa.2015.14.1001 |
| [17] |
Weidong Bao, Haoran Ji, Xiaomin Zhu, Ji Wang, Wenhua Xiao, Jianhong Wu. ACO-based solution for computation offloading in mobile cloud computing. Big Data & Information Analytics, 2016, 1 (1) : 1-13. doi: 10.3934/bdia.2016.1.1 |
| [18] |
Fred Brauer. Some simple epidemic models. Mathematical Biosciences & Engineering, 2006, 3 (1) : 1-15. doi: 10.3934/mbe.2006.3.1 |
| [19] |
Fred Brauer, Zhilan Feng, Carlos Castillo-Chávez. Discrete epidemic models. Mathematical Biosciences & Engineering, 2010, 7 (1) : 1-15. doi: 10.3934/mbe.2010.7.1 |
| [20] |
Oanh Chau, R. Oujja, Mohamed Rochdi. A mathematical analysis of a dynamical frictional contact model in thermoviscoelasticity. Discrete and Continuous Dynamical Systems - S, 2008, 1 (1) : 61-70. doi: 10.3934/dcdss.2008.1.61 |
2021 Impact Factor: 1.865
Tools
Metrics
Other articles
by authors
[Back to Top]