zbMATH — the first resource for mathematics

Epidemic thresholds and vaccination in a lattice model of disease spread. (English) Zbl 0886.92027
Summary: We use a lattice-based epidemic model to study the spatial and temporal rates of disease spread in a spatially distributed host population. The prevalence of the disease in the population is studied as well as the spread of infection about a point source of infection. In particular, two distinct critical population densities are identified. The first relates to the minimum population density for an epidemic to occur, whilst the second is the minimum population density for long-term persistence to occur. Vaccination regimes are introduced that are used to measure the impact of spatially and nonspatially dependent intervention strategies. Specifically we show how a ring of vaccinated susceptibles, of sufficient thickness, can halt the spread of infection across space.

92D30 Epidemiology
Full Text: DOI
[1] Anderson, R.M.; May, R.M., Infectious diseases of humans, dynamics and control, (1991), Oxford Univ. Press Oxford
[2] Anderson, R.M.; Jackson, A.C.; May, R.M.; Smith, A.M., Population dynamics of fox rabies in Europe, Nature (lond.), 289, 765-771, (1981)
[3] Bailey, N.J.T, The mathematical theory of infectious diseases and its applications, (1975), Oxford University Press
[4] Bartlett, M.S, The critical community size for measles in the U.S, J. R. statist. soc. A, 123, 37-44, (1960)
[5] Black, F.L, Measles endemicity in insular populations: critical community size and its evolutionary implications, J. theo. biol., 11, 207-211, (1966)
[6] Boccara, N.; Cheong, K, Automata network models for the spread of infectious diseases in a population of moving individuals, J. phys. A, 25, 2447-2461, (1992) · Zbl 0752.92024
[7] Bolker, B.; Grenfell, B.T, Space, persistence and dynamics of measles epidemics, Phil. trans. R. soc. (lond.), 348, 309-320, (1995)
[8] Cox, J.T.; Durrett, R, Limit theorems for the spread of epidemics and forest fires, Stoc. proc. applics., 30, 171-191, (1988) · Zbl 0667.92016
[9] Durrett, R, Spatial epidemic models, Epidemic models: their structure and relation to data, (1995), Cambridge Univ. Press Cambridge · Zbl 0841.92024
[10] Durrett, R.; Neuhauser, C, Particle systems and reaction- diffusion equations, Ann. probab., 22, 289-333, (1994) · Zbl 0799.60093
[11] Durrett, R.; Neuhauser, C, Epidemics with recovery ind, Ann. appl. prob., 1, 189-206, (1991) · Zbl 0733.92022
[12] Durrett, R.; Levin, S, The importance of being discrete (and spatial), Theor. popul. biol., 46, 363-394, (1994) · Zbl 0846.92027
[13] Durrett, R.; Levin, S.A, Stochastic spatial models: a User’s guide to ecological applications, Phil. trans. R. soc. (lond.) B, 343, 329-350, (1994)
[14] Grenfell, B.T, Chance and chaos in measles dynamics, J. R. statist. soc. B, 54, 383-398, (1992)
[15] Hassell, M.P.; Comins, H.N.; May, R.M, Spatial structure and chaos in insect population dynamics, Nature (lond.), 353, 255-258, (1991)
[16] Johansen, A, Spatio-temporal self-organisation in a model of disease spreading, Physica D, 78, 186-193, (1994) · Zbl 0812.60102
[17] Johansen, A, A simple model of recurrent epidemics, J. theo. biol., 178, 45-51, (1966)
[18] May, R.M.; Anderson, R.M, Spatial heterogeneity and the design of immunisation programs, Math. biosci., 76, 1-16, (1984)
[19] May, R.M.; Nowak, M, Evolutionary games and spatial chaos, Nature (lond.), 359, 826-829, (1992)
[20] Mollison, D, Spatial contact models for ecological and epidemic spread,, J. R. statist. soc. B, 39, 283-326, (1977) · Zbl 0374.60110
[21] Mollison, D, Dependence of epidemic and population velocities on basic parameters, Math. biosci., 107, 255-287, (1991) · Zbl 0743.92029
[22] Murray, J.D.; Stanley, A.E.; Brown, D.L, On the spatial spread of rabies among foxes, Proc. roy. soc. (lond.) B, 229, 111-150, (1986)
[23] Noble, J.V, Geographic and temporal development of plagues, Nature (lond.), 250, 726-728, (1974)
[24] Olsen, L.F.; Schaffer, W.M, Chaos versus noisy periodicity: alternative hypotheses for childhood epidemics, Science (wash.), 249, 499-504, (1990)
[25] Olsen, L.F.; Truty, G.L.; Schaffer, W.M, Oscillations and chaos in epidemics: A nonlinear dynamic study of six childhood diseases in Copenhagen, Denmark, Theo. popul. biol., 33, 344-370, (1988) · Zbl 0639.92012
[26] Pacala, S.W.; Hassell, M.P.; May, R.M, Host-parasitoid associations in patchy environments, Nature (lond.), 344, 150-153, (1990)
[27] Rand, D.A.; Keeling, M.; Wilson, H.B, Invasion stability and evolution to criticality in spatially extended, artificial host-pathogen ecologies, Proc. roy. soc. (lond.) B, 259, 55-63, (1995)
[28] Rhodes, C.J.; Anderson, R.M, Dynamics in a lattice epidemic model, Phys. lett. A, 210, 183-188, (1996) · Zbl 1075.92520
[29] Rhodes, C.J.; Anderson, R.M, Persistence and dynamics in lattice models of epidemic spread, J. theor. biol., (1966)
[30] Rhodes, C.J.; Anderson, R.M, Power laws governing epidemics in isolated populations, Nature (lond.), 381, 600-602, (1996)
[31] Schenzle, D, An age-structured model of pre- and post-vaccination massles transmission, IMA J. math. appl. med. biol., 1, 169-191, (1984) · Zbl 0611.92021
[32] Schonfisch, B, Propagation of fronts in cellular automata, Physica D, 80, 433-450, (1995) · Zbl 0882.68098
[33] Smith, G.C.; Harris, S, Rabies in urban foxes (vulpes vulpes, Proc. roy. soc. (lond.) B, 334, 459-479, (1991)
[34] Viswanathan, G.M.; Afanasyev, V.; Buldyrev, S.V.; Murphy, E.J.; Prince, P.A.; Stanley, H.E, Levy flight search patterns of wandering albatrosses, Nature (lond.), 381, 413-415, (1996)
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. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.