# 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 and Continuous Dynamical Systems - B, 2017, 22 (2) : 383-405. doi: 10.3934/dcdsb.2017018
##### References:
 [1] P. Bantay and M. Janosi, Self-organization and anomalous diffusion, Phys. A, 185 (1992), 11-18. [2] V. Barbu, Nonlinear Differential Equations of Monotone Types in Banach Spaces New York, Springer, 2010. [3] V. Barbu, Self-organized criticality of cellular automata model; absorbtion in finite-time of supercritical region into the critical one, Math. Methods Appl. Sci., 36 (2013), 1726-1733.  doi: 10.1002/mma.2718. [4] C. Beauchemin, J. Samuel and J. Tuszynski, A simple cellular automaton models for influenza: A viral infections, J. Theor. Biol., 232 (2005), 223-234.  doi: 10.1016/j.jtbi.2004.08.001. [5] N. Boccara and K. Cheong, Critical behaviour of a probablistic automata network SIS model for the spread of an infectious disease in a population of moving individuals, J. Phys. A-Math. Gen., 26 (1993), 3707-3717. [6] D. Breda, O. Diekmann, W. F. de Graaf, A. Pugliese and A. R. Vermiglio, On the formulation of epidemic models (an appraisal of Kermack and McKendrick), J Biol Dyn., 6 (2012), 103-117.  doi: 10.1080/17513758.2012.716454. [7] H. Brezis, Opérateurs Maximaux Monotones et Semi-groupes de Contractions dans les Espaces de Hilbert North Holland, 1973. [8] S. Busenberg, K. Cooke and M. Iannelli, Endemic thresholds and stability in a class of age-structured epidemics, SIAM J. Appl. Math., 48 (1988), 1379-1395.  doi: 10.1137/0148085. [9] K. Chen, P. Bak and S. P. Obukhov, Self-organized criticality in a crack-propagation model of earthquakes, Physical Review, 43 (1991), 625-630. [10] J. T. Cox and R. Durrett, Limit theorems for the spread of epidemics and forest fires, Stochastic Processes and their Applications, 30 (1998), 171-191.  doi: 10.1016/0304-4149(88)90083-X. [11] O. Diekmann and J. A. P. Heesterbeek, Mathematical Epidemiology of Infectious Diseases Wiley, New York, 2000. [12] R. Eymard, Th. Gallouet and R. Herbin, Finite volume method, in Handbook of Numerial Analysis, (eds. G. Ciarlet and J. L. Lions), North Holland, 7 (2000), 713–1020. [13] M. A. Fuentes and M. N. Kuperman, Cellular automata and epidemiological models with spatial dependence, Phys. A, 267 (1999), p471. [14] A. Gandolfi, A. Pugliese and C. Sinisgalli, Epidemic dynamics and host immune response: A nested approach, J. Math. Biol., 70 (2015), 399-435.  doi: 10.1007/s00285-014-0769-8. [15] P. Grassberger, On the critical Behavior of the general epidemic process and dynamical percolation, Math. Biosci., 63 (1983), 151-172. [16] H. W. Hethcote, The mathematics of infectious diseases, SIAM Review, 42 (2000), 599-653.  doi: 10.1137/S0036144500371907. [17] J. L. Lions, Quelques Méthodes de Ré solution des Problémes aux Limites non Linéaires Dunod, Paris, 1969. [18] L. Lübeck and K. D. Usadel, Numerical determination of the avalanche exponents of the Bak-Tang-Wisenfeld model, Phys. Rev. E, 55 (1997), 4095-4099. [19] G. Marinoschi, A duality approach to nonlinear diffusion equations, Set-Valued Var. Anal., 22 (2014), 783-807.  doi: 10.1007/s11228-014-0288-1. [20] M. J. F. Martínez, E. G. Sánchez, J. E. G. Sánchez, A. M. Del Rey and G. R. Sánchez, A model based on cellular automata to simulate a SIS epidemic disease, J. Pure Appl. Math.: Adv. Appl., 5 (2011), 125-139. [21] C. J. Rhodes and R. M. Anderson, A scaling analysis of measles epidemics in a small population, Phil. Trans. R. Soc. Lond. B, 351 (1996), 1679-1688. [22] S. Steacy, J. McCloskey, C. J. Bean and J. Ren, Heterogeneity in a self-organised critical earthquake model, Geophysical Research Letters, 23 (1996), 383-386. [23] R. V. Sole, S. C. Manrubia, M. Benton and P. Bak, Self-similarity of extinction statistics in the fossil record, Nature, 388 (1997), 764-767. [24] Z. Xu and Y. Zhao, A reaction-diffusion model of dengue transmission, Discrete Contin. Dyn. Syst. Ser. B, 19 (2014), 2993-3018.  doi: 10.3934/dcdsb.2014.19.2993. [25] S. White, A. del Rey and G. Sánchez, Modeling epidemics using cellular automata, Appl. Math. Comput., 186 (2007), 193-202.  doi: 10.1016/j.amc.2006.06.126. [26] K. Wiesenfeld, C. Tang and P. Bak, A physicist's sandbox, Journal of Statistical Physics, 54 (1989), 1441-1458.  doi: 10.1007/BF01044728.

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##### References:
 [1] P. Bantay and M. Janosi, Self-organization and anomalous diffusion, Phys. A, 185 (1992), 11-18. [2] V. Barbu, Nonlinear Differential Equations of Monotone Types in Banach Spaces New York, Springer, 2010. [3] V. Barbu, Self-organized criticality of cellular automata model; absorbtion in finite-time of supercritical region into the critical one, Math. Methods Appl. Sci., 36 (2013), 1726-1733.  doi: 10.1002/mma.2718. [4] C. Beauchemin, J. Samuel and J. Tuszynski, A simple cellular automaton models for influenza: A viral infections, J. Theor. Biol., 232 (2005), 223-234.  doi: 10.1016/j.jtbi.2004.08.001. [5] N. Boccara and K. Cheong, Critical behaviour of a probablistic automata network SIS model for the spread of an infectious disease in a population of moving individuals, J. Phys. A-Math. Gen., 26 (1993), 3707-3717. [6] D. Breda, O. Diekmann, W. F. de Graaf, A. Pugliese and A. R. Vermiglio, On the formulation of epidemic models (an appraisal of Kermack and McKendrick), J Biol Dyn., 6 (2012), 103-117.  doi: 10.1080/17513758.2012.716454. [7] H. Brezis, Opérateurs Maximaux Monotones et Semi-groupes de Contractions dans les Espaces de Hilbert North Holland, 1973. [8] S. Busenberg, K. Cooke and M. Iannelli, Endemic thresholds and stability in a class of age-structured epidemics, SIAM J. Appl. Math., 48 (1988), 1379-1395.  doi: 10.1137/0148085. [9] K. Chen, P. Bak and S. P. Obukhov, Self-organized criticality in a crack-propagation model of earthquakes, Physical Review, 43 (1991), 625-630. [10] J. T. Cox and R. Durrett, Limit theorems for the spread of epidemics and forest fires, Stochastic Processes and their Applications, 30 (1998), 171-191.  doi: 10.1016/0304-4149(88)90083-X. [11] O. Diekmann and J. A. P. Heesterbeek, Mathematical Epidemiology of Infectious Diseases Wiley, New York, 2000. [12] R. Eymard, Th. Gallouet and R. Herbin, Finite volume method, in Handbook of Numerial Analysis, (eds. G. Ciarlet and J. L. Lions), North Holland, 7 (2000), 713–1020. [13] M. A. Fuentes and M. N. Kuperman, Cellular automata and epidemiological models with spatial dependence, Phys. A, 267 (1999), p471. [14] A. Gandolfi, A. Pugliese and C. Sinisgalli, Epidemic dynamics and host immune response: A nested approach, J. Math. Biol., 70 (2015), 399-435.  doi: 10.1007/s00285-014-0769-8. [15] P. Grassberger, On the critical Behavior of the general epidemic process and dynamical percolation, Math. Biosci., 63 (1983), 151-172. [16] H. W. Hethcote, The mathematics of infectious diseases, SIAM Review, 42 (2000), 599-653.  doi: 10.1137/S0036144500371907. [17] J. L. Lions, Quelques Méthodes de Ré solution des Problémes aux Limites non Linéaires Dunod, Paris, 1969. [18] L. Lübeck and K. D. Usadel, Numerical determination of the avalanche exponents of the Bak-Tang-Wisenfeld model, Phys. Rev. E, 55 (1997), 4095-4099. [19] G. Marinoschi, A duality approach to nonlinear diffusion equations, Set-Valued Var. Anal., 22 (2014), 783-807.  doi: 10.1007/s11228-014-0288-1. [20] M. J. F. Martínez, E. G. Sánchez, J. E. G. Sánchez, A. M. Del Rey and G. R. Sánchez, A model based on cellular automata to simulate a SIS epidemic disease, J. Pure Appl. Math.: Adv. Appl., 5 (2011), 125-139. [21] C. J. Rhodes and R. M. Anderson, A scaling analysis of measles epidemics in a small population, Phil. Trans. R. Soc. Lond. B, 351 (1996), 1679-1688. [22] S. Steacy, J. McCloskey, C. J. Bean and J. Ren, Heterogeneity in a self-organised critical earthquake model, Geophysical Research Letters, 23 (1996), 383-386. [23] R. V. Sole, S. C. Manrubia, M. Benton and P. Bak, Self-similarity of extinction statistics in the fossil record, Nature, 388 (1997), 764-767. [24] Z. Xu and Y. Zhao, A reaction-diffusion model of dengue transmission, Discrete Contin. Dyn. Syst. Ser. B, 19 (2014), 2993-3018.  doi: 10.3934/dcdsb.2014.19.2993. [25] S. White, A. del Rey and G. Sánchez, Modeling epidemics using cellular automata, Appl. Math. Comput., 186 (2007), 193-202.  doi: 10.1016/j.amc.2006.06.126. [26] K. Wiesenfeld, C. Tang and P. Bak, A physicist's sandbox, Journal of Statistical Physics, 54 (1989), 1441-1458.  doi: 10.1007/BF01044728.
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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