# American Institute of Mathematical Sciences

March  2017, 22(2): 383-405. doi: 10.3934/dcdsb.2017018

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

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

* Corresponding author: Gabriela Marinoschi

Received  February 2016 Revised  March 2016 Published  December 2016

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.

Citation: Stelian Ion, Gabriela Marinoschi. A self-organizing criticality mathematical model for contamination and epidemic spreading. Discrete & Continuous Dynamical Systems - B, 2017, 22 (2) : 383-405. doi: 10.3934/dcdsb.2017018
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##### References:
Scenario 1 for $p_{R}=3: \log$-$\log$ plot of the statistics $S$ (left), $T$ (middle), $A$ (right)
Scenario 1 for $p_{R}=4$: $\log$-$\log$ plot of the statistics $S$ (left), $T$ (middle), $A$ (right)
Scenario 2 for $p_{R}=3$: $\log$-$\log$ plot of the statistics $S$ (left), $T$ (middle), $A$ (right)
Scenario 2 for $p_{R}=4$: $\log$-$\log$ plot of the statistics $S$ (left), $T$ (middle), $A$ (right)
Snapshots of the area contamination
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)
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)
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
 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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