A self-organizing criticality mathematical model for contamination and epidemic spreading

Stelian Ion; Gabriela Marinoschi

doi:10.3934/dcdsb.2017018

Article Contents

2017, Volume 22, Issue 2: 383-405. Doi: 10.3934/dcdsb.2017018

This issue Previous Article Stability of equilibria of randomly perturbed maps Next Article Asymptotic behavior of the solution of a diffusion equation with nonlocal boundary conditions

A self-organizing criticality mathematical model for contamination and epidemic spreading

Stelian Ion and
Gabriela Marinoschi^,

Institute of Mathematical Statistics and Applied Mathematics of the Romanian Academy, Calea 13 Septembrie 13, Bucharest, Romania

* Corresponding author: Gabriela Marinoschi
* Corresponding author: Gabriela Marinoschi

Received: January 31, 2016

Revised: February 29, 2016

Published: December 2017

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Abstract

We introduce a new model to predict the spread of an epidemic, focusing on the contamination process and simulating the disease propagation by the means of a unique function viewed as a measure of the local infective energy. The model is intended to illustrate a map of the epidemic spread and not to compute the densities of various populations related to an epidemic, as in the classical models. First, the model is constructed as a cellular automaton exhibiting a self-organizing-type criticality process with two thresholds. This induces the consideration of an associate continuous model described by a nonlinear equation with two singularities, for whose solution we prove existence, uniqueness and certain properties. We provide numerical simulations to put into evidence the effect of some model parameters in various scenarios of the epidemic spread.

Keywords:

Mathematics Subject Classification: Primary:35K61, 35K67, 35Q92;Secondary:35K65, 37B15, 68Q80.

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Figure 1. Scenario 1 for $p_{R}=3: \log $-$\log $ plot of the statistics $S$ (left), $T$ (middle), $A$ (right)

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Figure 2. Scenario 1 for $p_{R}=4$: $\log $-$\log $ plot of the statistics $S $ (left), $T$ (middle), $A$ (right)

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Figure 3. Scenario 2 for $p_{R}=3$: $\log $-$\log $ plot of the statistics $S $ (left), $T$ (middle), $A$ (right)

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Figure 4. Scenario 2 for $p_{R}=4$: $\log $-$\log $ plot of the statistics $S$ (left), $T$ (middle), $A$ (right)

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Figure 5. Snapshots of the area contamination

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Figure 6. Epidemic evolution for $m_{p}=1: z(t,x,0,5)$ at various $t$ (left); $z(t,0.5,0.5)$ (middle); infected area $a(t)$ (right)

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Figure 7. Epidemic evolution for $m_{p}=100: z(t,x,0,5)$ at va-rious $t$ (left); $z(t,0.5,0.5)$ (middle); infected area $a(t)$ (right)

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Table 1. Power-laws exponents

Model $\tau _{s}$ $\tau _{t}$ $\tau _{a}$

BTW 1.293 1.480 1.183

S2, $p_{R}=3$ 1.5 1.7 1.25

S2, $p_{R}=4$ 3 3.8 3

S1, $p_{R}=3$ 1.25 1.7 1.25

S1, $p_{R}=4$ 2.2 3.5 2.8

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