Cellular automaton modeling of epidemics. (English) Zbl 0709.92018

Summary: This study investigates random cellular automaton models with emphasis on their application to epidemiology. We conjecture that there may be a spatial factor involved in contagious disease. Random cellular automata would seem a natural mode for statistical exploration of this conjecture. We examine a random cellular automaton measle model proposed by D. Mollison [J. R. Stat. Soc., Ser. B 39, 283-326 (1977; Zbl 0374.60110)]; a homogeneous version of this model is shown to coincide with the macroscopic spatially invariant “basic stochastic” epidemic model discussed by N. T. J. Bailey [The mathematical theory of infectious diseases and its applications. (1975; Zbl 0334.92024)].
Some theory and proposed statistical methodology are suggested. Our experimental findings indicate that the course of an epidemic depends strongly on initial configuration of the infectives, all other parameters remaining fixed. This is consistent with our conjecture that the spatial distribution of the carriers contains significant information. Whether the random cellular automata are valid models for actual epidemis awaits statistical resolution.


92D30 Epidemiology
60J20 Applications of Markov chains and discrete-time Markov processes on general state spaces (social mobility, learning theory, industrial processes, etc.)
68Q45 Formal languages and automata
Full Text: DOI


[1] Anderson, R. M.; May, R. M., Spatial heterogeneity and the design of immunization programs, Math. Biosci., 72, 83-111 (1984) · Zbl 0564.92016
[2] Bailey, N. T.J., The Mathematical Theory of Infectious Diseases and Its Applications (1975), Hafner: Hafner New York · Zbl 0115.37202
[3] Berlekamp, E.; Conway, J.; Guy, R., Winning Ways for Your Mathematical Plays (1982), Academic: Academic New York · Zbl 0485.00025
[4] Billingsley, P., Statistical methods in Markov Chains, Ann. Math. Statist., 32, 12-40 (1961) · Zbl 0104.12802
[5] Denny, J. L., Markovian dependence, (Encyclopedia of Statistical Sciences, Vol. 3 (1985), Wiley: Wiley New York) · Zbl 0394.62026
[6] Denny, J. L.; Yakowitz, S., Admissible run-contingency type tests for independence, J. Amer. Statist. Assoc., 72, 177-181 (1978) · Zbl 0386.62042
[7] Denny, J. L.; Wright, A. L., On tests for Markov dependence, Z. Wahrsch., 43, 331-338 (1978) · Zbl 0394.62026
[8] Dewdney, A. K., The Armchair Universe (1988), Freeman: Freeman San Francisco
[9] Durrett, R., Lecture Notes on Particle Systems and Percolation (1988), Wadsworth: Wadsworth Monterey, Calif · Zbl 0659.60129
[10] Greenhalgh, D., Optimal control of an epidemic by ring vaccination, Stochastic Models, 5, 1, 131-159 (1989)
[11] Heyward, W. L.; Curran, J. H., The epidemology of AIDS in the U.S., Sci. Amer., 259, 4, 72-81 (1988)
[12] Kindermann, R.; Laurie Snell, J., Markov Random Fields and Their Applications, Amer. Math. Soc. (1980), Providence, R.I. · Zbl 1229.60003
[13] Mann, J. H.; Clin, J.; Piot, P.; Quinn, T., The international epidemology of AIDS, Sci. Amer., 259, 4, 82-89 (1988)
[14] Mollison, D., Spatial contact models for ecological and epidemic spread, J. Roy. Statist. Soc. Ser. B, 39, 283-326 (1977) · Zbl 0374.60110
[15] Mollison, D., Modelling biological invasion: Chance, explanation, prediction, Philos. Trans. Roy. Soc. London Ser. B, 314, 675-693 (1986)
[16] Mollison, D.; Kuulasmaa, K., Spatial epidemic models: Theory and simulation, (Bacon, P. J., The Population Dynamics of Rabies and Wildlife (1985), Academic: Academic London)
[17] Richardson, D., Random growth in a tessellation, Proc. Cambridge Philos. Soc., 74, 515-528 (1973) · Zbl 0295.62094
[18] Yakowitz, S., Small sample hypothesis tests of Markov order, with application to simulated and hydrologic chains, J. Amer. Statist. Assoc., 71, 132-136 (1976) · Zbl 0329.62068
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.