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
References:
[1]

P. Bantay and M. Janosi, Self-organization and anomalous diffusion, Phys. A, 185 (1992), 11-18. Google Scholar

[2]

V. Barbu, Nonlinear Differential Equations of Monotone Types in Banach Spaces New York, Springer, 2010.Google Scholar

[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. Google Scholar

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C. BeaucheminJ. 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. Google Scholar

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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. Google Scholar

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D. BredaO. DiekmannW. F. de GraafA. 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. Google Scholar

[7]

H. Brezis, Opérateurs Maximaux Monotones et Semi-groupes de Contractions dans les Espaces de Hilbert North Holland, 1973.Google Scholar

[8]

S. BusenbergK. 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. Google Scholar

[9]

K. ChenP. Bak and S. P. Obukhov, Self-organized criticality in a crack-propagation model of earthquakes, Physical Review, 43 (1991), 625-630. Google Scholar

[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. Google Scholar

[11]

O. Diekmann and J. A. P. Heesterbeek, Mathematical Epidemiology of Infectious Diseases Wiley, New York, 2000.Google Scholar

[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.Google Scholar

[13]

M. A. Fuentes and M. N. Kuperman, Cellular automata and epidemiological models with spatial dependence, Phys. A, 267 (1999), p471.Google Scholar

[14]

A. GandolfiA. 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. Google Scholar

[15]

P. Grassberger, On the critical Behavior of the general epidemic process and dynamical percolation, Math. Biosci., 63 (1983), 151-172. Google Scholar

[16]

H. W. Hethcote, The mathematics of infectious diseases, SIAM Review, 42 (2000), 599-653. doi: 10.1137/S0036144500371907. Google Scholar

[17]

J. L. Lions, Quelques Méthodes de Ré solution des Problémes aux Limites non Linéaires Dunod, Paris, 1969.Google Scholar

[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. Google Scholar

[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. Google Scholar

[20]

M. J. F. MartínezE. G. SánchezJ. E. G. SánchezA. 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. Google Scholar

[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. Google Scholar

[22]

S. SteacyJ. McCloskeyC. J. Bean and J. Ren, Heterogeneity in a self-organised critical earthquake model, Geophysical Research Letters, 23 (1996), 383-386. Google Scholar

[23]

R. V. SoleS. C. ManrubiaM. Benton and P. Bak, Self-similarity of extinction statistics in the fossil record, Nature, 388 (1997), 764-767. Google Scholar

[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. Google Scholar

[25]

S. WhiteA. 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. Google Scholar

[26]

K. WiesenfeldC. Tang and P. Bak, A physicist's sandbox, Journal of Statistical Physics, 54 (1989), 1441-1458. doi: 10.1007/BF01044728. Google Scholar

show all references

References:
[1]

P. Bantay and M. Janosi, Self-organization and anomalous diffusion, Phys. A, 185 (1992), 11-18. Google Scholar

[2]

V. Barbu, Nonlinear Differential Equations of Monotone Types in Banach Spaces New York, Springer, 2010.Google Scholar

[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. Google Scholar

[4]

C. BeaucheminJ. 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. Google Scholar

[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. Google Scholar

[6]

D. BredaO. DiekmannW. F. de GraafA. 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. Google Scholar

[7]

H. Brezis, Opérateurs Maximaux Monotones et Semi-groupes de Contractions dans les Espaces de Hilbert North Holland, 1973.Google Scholar

[8]

S. BusenbergK. 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. Google Scholar

[9]

K. ChenP. Bak and S. P. Obukhov, Self-organized criticality in a crack-propagation model of earthquakes, Physical Review, 43 (1991), 625-630. Google Scholar

[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. Google Scholar

[11]

O. Diekmann and J. A. P. Heesterbeek, Mathematical Epidemiology of Infectious Diseases Wiley, New York, 2000.Google Scholar

[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.Google Scholar

[13]

M. A. Fuentes and M. N. Kuperman, Cellular automata and epidemiological models with spatial dependence, Phys. A, 267 (1999), p471.Google Scholar

[14]

A. GandolfiA. 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. Google Scholar

[15]

P. Grassberger, On the critical Behavior of the general epidemic process and dynamical percolation, Math. Biosci., 63 (1983), 151-172. Google Scholar

[16]

H. W. Hethcote, The mathematics of infectious diseases, SIAM Review, 42 (2000), 599-653. doi: 10.1137/S0036144500371907. Google Scholar

[17]

J. L. Lions, Quelques Méthodes de Ré solution des Problémes aux Limites non Linéaires Dunod, Paris, 1969.Google Scholar

[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. Google Scholar

[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. Google Scholar

[20]

M. J. F. MartínezE. G. SánchezJ. E. G. SánchezA. 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. Google Scholar

[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. Google Scholar

[22]

S. SteacyJ. McCloskeyC. J. Bean and J. Ren, Heterogeneity in a self-organised critical earthquake model, Geophysical Research Letters, 23 (1996), 383-386. Google Scholar

[23]

R. V. SoleS. C. ManrubiaM. Benton and P. Bak, Self-similarity of extinction statistics in the fossil record, Nature, 388 (1997), 764-767. Google Scholar

[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. Google Scholar

[25]

S. WhiteA. 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. Google Scholar

[26]

K. WiesenfeldC. Tang and P. Bak, A physicist's sandbox, Journal of Statistical Physics, 54 (1989), 1441-1458. doi: 10.1007/BF01044728. Google Scholar

Figure 1.  Scenario 1 for $p_{R}=3: \log $-$\log $ plot of the statistics $S$ (left), $T$ (middle), $A$ (right)
Figure 2.  Scenario 1 for $p_{R}=4$: $\log $-$\log $ plot of the statistics $S $ (left), $T$ (middle), $A$ (right)
Figure 3.  Scenario 2 for $p_{R}=3$: $\log $-$\log $ plot of the statistics $S $ (left), $T$ (middle), $A$ (right)
Figure 4.  Scenario 2 for $p_{R}=4$: $\log $-$\log $ plot of the statistics $S$ (left), $T$ (middle), $A$ (right)
Figure 5.  Snapshots of the area contamination
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)
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)
Table 1.  Power-laws exponents
Model$\tau _{s}$$\tau _{t}$$\tau _{a}$
BTW1.2931.4801.183
S2, $p_{R}=3$1.51.71.25
S2, $p_{R}=4$33.83
S1, $p_{R}=3$1.251.71.25
S1, $p_{R}=4$2.23.52.8
Model$\tau _{s}$$\tau _{t}$$\tau _{a}$
BTW1.2931.4801.183
S2, $p_{R}=3$1.51.71.25
S2, $p_{R}=4$33.83
S1, $p_{R}=3$1.251.71.25
S1, $p_{R}=4$2.23.52.8
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