# American Institute of Mathematical Sciences

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

* 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, 2019, 12 (4&5) : 735-746. doi: 10.3934/dcdss.2019048
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##### References:
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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