# American Institute of Mathematical Sciences

## 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

* Corresponding author: J. Alberto Conejero

Received  November 2017 Revised  January 2018 Published  November 2018

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.

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 & Continuous Dynamical Systems - S, 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.

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##### 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.
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
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;
Message coverage depending on people density and renewal percentages. a) contact-based only diffusion; b) contactbased and fixed nodes diffusion for ρ = 1.
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.
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