Fluctuation effects in an epidemic model

C. P. Warren; G. Mikus; E. Somfai; L. M. Sander

doi:10.1103/PhysRevE.63.056103

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Fluctuation effects in an epidemic model

C. P. Warren¹, G. Mikus¹, E. Somfai², and L. M. Sander¹

¹Michigan Center for Theoretical Physics, Department of Physics, The University of Michigan, Ann Arbor, Michigan 48109-1120
²Instituut Lorentz, Leiden, NL-2333 CA, Netherlands

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Phys. Rev. E 63, 056103 – Published 11 April, 2001

DOI: https://doi.org/10.1103/PhysRevE.63.056103

Abstract

We study a discrete epidemic model $A + \vec{B} 2 A$ in one and two dimensions (1D and 2D). In 1D for low concentration θ, we find that a depletion zone exists ahead of the front and the average velocity of the front approaches $v = θ / 2.$ In the 1D high concentration limit, we find that the velocity approaches $v = 1 - e^{- θ / 2} .$ In 2D, for low concentration we also find a depletion zone, and the velocity scales as $v \sim θ^{0.6},$ which is different from the scaling expected from the mean field approximation, $v \sim θ^{0.5} .$ Analysis of the interface width scaling properties demonstrated that the front dynamics of this reaction are not governed by the Kardar-Parisi-Zhang equation.

References (16)

J. Mai, I. M. Sokolov, and A. Blumen, Phys. Rev. Lett. 77, 4462 (1996).
J. Mai, I. M. Sokolov, and A. Blumen, Europhys. Lett. 44, 7 (1998).
E. Brunet and B. Derrida, Phys. Rev. E 56, 2597 (1997).
D. A. Kessler, Z. Ner, and L. M. Sander, Phys. Rev. E 58, 107 (1998).
J. Riordan, C. Doering, and D. ben-Avraham, Phys. Rev. Lett. 75, 565 (1995). These authors model the reversible version of this reaction, $A + B \leftrightarrow 2 A$ , by looking at $A \leftrightarrow 2 A$ . This model differs from ours in that B is assumed to be in excess or otherwise maintained at a steady level. They thus suppress any variation in B, and this spatial variation in B is at the heart of our analysis and results, as we will see.
R. A. Fisher, Annals of Eugenics7, 355 (1937); A. Kolmogorov, I. Petrovsky, and N. Piscounov, Moscow Univ. Bull. Math. A1, 1 (1937).
J. D. Murray, Mathematical Biology (Springer, Berlin, 1993).
We used a version of periodic boundary conditions appropriate to this problem: $A$ ’s that walked off the left edge were added to the right edge as $B$ ’s and vice versa.
A preliminary version of these results was presented in C. Warren, E. Somfai, and L. Sander, Braz. J. Phys. 30, 157 (2000).
V. K. Rohatgi, An Introduction to Probability Theory and Mathematical Statistics (Wiley, New York, 1976).
In 2D, we used periodic boundary conditions on the top and bottom of the strip, and the same conditions on the ends of the strip as in 1D.
M. Plischke and Z. Racz, Phys. Rev. A 32, 3825 (1985); F. Family, and T. Vicsek, J. Phys. A 18, L75 (1985).
A. L. Barabasi and H. E. Stanley, Fractal Concepts in Surface Growth (Cambridge University Press, New York, 1995).
R. Goodman, D. Graff, L. M. Sander, P. Leroux-Hugon, and E. Clement, Phys. Rev. E 52, 5904 (1995); L. M. Sander and S. V. Ghaisas, Physica A 233, 629 (1996).
However, Riordan et al. [5] use an ensemble average in their definition of interface width, where we use a lattice average.
G. Tripathy and W. van Saarloos, Phys. Rev. Lett. 85, 3556 (2000).

[c1] J. Mai, I. M. Sokolov, and A. Blumen, Phys. Rev. Lett. 77, 4462 (1996).

[c2] J. Mai, I. M. Sokolov, and A. Blumen, Europhys. Lett. 44, 7 (1998).

[c3] E. Brunet and B. Derrida, Phys. Rev. E 56, 2597 (1997).

[c4] D. A. Kessler, Z. Ner, and L. M. Sander, Phys. Rev. E 58, 107 (1998).

[c5] J. Riordan, C. Doering, and D. ben-Avraham, Phys. Rev. Lett. 75, 565 (1995). These authors model the reversible version of this reaction, $A + B \leftrightarrow 2 A$ , by looking at $A \leftrightarrow 2 A$ . This model differs from ours in that B is assumed to be in excess or otherwise maintained at a steady level. They thus suppress any variation in B, and this spatial variation in B is at the heart of our analysis and results, as we will see.

[c6] R. A. Fisher, Annals of Eugenics7, 355 (1937); A. Kolmogorov, I. Petrovsky, and N. Piscounov, Moscow Univ. Bull. Math. A1, 1 (1937).

[c7] J. D. Murray, Mathematical Biology (Springer, Berlin, 1993).

[c8] We used a version of periodic boundary conditions appropriate to this problem: $A$ ’s that walked off the left edge were added to the right edge as $B$ ’s and vice versa.

[c9] A preliminary version of these results was presented in C. Warren, E. Somfai, and L. Sander, Braz. J. Phys. 30, 157 (2000).

[c10] V. K. Rohatgi, An Introduction to Probability Theory and Mathematical Statistics (Wiley, New York, 1976).

[c11] In 2D, we used periodic boundary conditions on the top and bottom of the strip, and the same conditions on the ends of the strip as in 1D.

[c12] M. Plischke and Z. Racz, Phys. Rev. A 32, 3825 (1985); F. Family, and T. Vicsek, J. Phys. A 18, L75 (1985).

[c13] A. L. Barabasi and H. E. Stanley, Fractal Concepts in Surface Growth (Cambridge University Press, New York, 1995).

[c14] R. Goodman, D. Graff, L. M. Sander, P. Leroux-Hugon, and E. Clement, Phys. Rev. E 52, 5904 (1995); L. M. Sander and S. V. Ghaisas, Physica A 233, 629 (1996).

[c15] However, Riordan et al. [5] use an ensemble average in their definition of interface width, where we use a lattice average.

[c16] G. Tripathy and W. van Saarloos, Phys. Rev. Lett. 85, 3556 (2000).

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Fluctuation effects in an epidemic model

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