Models of infectious diseases in spatially heterogeneous environments. (English) Zbl 1323.92210
Summary: Most models of dynamics of infectious diseases have assumed homogeneous mixing in the host population. However, it is increasingly recognized that heterogeneity can arise through many processes. It is then important to consider the existence of subpopulations of hosts, and that the contact rate within subpopulations is different than that between subpopulations. We study models with hosts distributed in subpopulations as a consequence of spatial partitioning. Two types of models are considered. In the first one there is direct transmission. The second one is a model of dynamics of a mosquito-borne disease, with indirect transmission, and applicable to malaria. The contact between subpopulations is achieved through the visits of hosts. Two types of visit are considered: a first one in which the visit time is independent of the distance travelled, and a second one in which visit time decreases with distance. There are two types of spatial arrangement: one dimensional, and two dimensional. Conditions for the establishment of the disease are obtained. Results indicate that the disease becomes established with greater difficulty when the degree of spatial partition increases, and when visit time decreases. In addition, when visit time decreases with distance, the establishment of the disease is more difficult when the spatial arrangement is one dimensional than when it is two dimensional. The results indicate the importance of knowing the spatial distribution and mobility patterns to understand the dynamics of infectious diseases. The consequences of these results for the design of public health policies are discussed.
MSC:
| 92D30 | Epidemiology |
References:
| [1] | Anderson, R. M.; May, R. M., Infectious Diseases of Humans (1992), Oxford: Oxford University Press, Oxford |
| [2] | Andreasen, V.; Christiansen, F. B., Persistence of an infectious disease in a subdivided population, Math. Biosci., 96, 239-253 (1989) · Zbl 0679.92016 · doi:10.1016/0025-5564(89)90061-8 |
| [3] | Aron, J. L.; May, R. M.; Anderson, R. M., The population dynamics of malaria, Population Dynamics of Infectious Diseases: Theory and Applications, 139-179 (1982), London: Chapman & Hall, London |
| [4] | Bailey, N. T. J., The Biomathematics of Malaria (1982), London: Charles Griffin, London · Zbl 0494.92018 |
| [5] | Ball, F., Stochastic and deterministic models for SIS epidemics among a population partitioned into households, Math. Biosci., 156, 41-67 (1999) · Zbl 0979.92033 · doi:10.1016/S0025-5564(98)10060-3 |
| [6] | Bascompté, J.; Sole, R. V., Modelling Spatiotemporal Dynamics in Ecology (1998), New York: Springer, New York |
| [7] | Becker, N. G.; Bahrampour, A.; Dietz, K., Threshold parameters for epidemics in different community settings, Math. Biosci., 129, 189-208 (1995) · Zbl 0828.92027 · doi:10.1016/0025-5564(94)00061-4 |
| [8] | Becker, N. G.; Dietz, K., The effect of household distribution on transmission and control of highly infectious diseases, Math. Biosci., 127, 207-219 (1995) · Zbl 0824.92025 · doi:10.1016/0025-5564(94)00055-5 |
| [9] | Becker, N. G.; Hall, R., Immunization levels for preventing epidemics in a community of households made up of individuals of various types, Math. Biosci., 132, 205-216 (1996) · Zbl 0844.92020 · doi:10.1016/0025-5564(95)00080-1 |
| [10] | Becker, N. G.; Starczak, D. N., Optimal vaccination strategies for a community of households, Math. Biosci., 139, 117-132 (1997) · Zbl 0881.92028 · doi:10.1016/S0025-5564(96)00139-3 |
| [11] | Begon, M.; Harper, J. L.; Townsend, C. R., Ecology (1996), Oxford: Blackwell, Oxford |
| [12] | Berlin, T. H.; Kac, M., The spherical model of a ferromagnet, Phys. Rev., 86, 821-835 (1952) · Zbl 0047.45703 · doi:10.1103/PhysRev.86.821 |
| [13] | Collins, F. H.; Paskewitz, S. M., Malaria: current and future prospects, Ann. Rev. Entomol., 40, 195-219 (1995) · doi:10.1146/annurev.en.40.010195.001211 |
| [14] | De Jong, M. C. M.; Diekmann, O.; Heesterbeck, J. A. P., The computation of R_0 for discrete-time epidemic models with dynamic heterogeneity, Math. Biosci., 119, 97-114 (1994) · Zbl 0791.92019 · doi:10.1016/0025-5564(94)90006-X |
| [15] | Diekmann, O.; Hesterbeck, J. A. P.; Metz, J. A. J., On the definition and the computation of the basic reproduction rate ratio R_0 in models for infectious diseases in heterogeneous populations, J. Math. Biol., 28, 365-382 (1990) · Zbl 0726.92018 · doi:10.1007/BF00178324 |
| [16] | Dietz, K.; Wernsdorfer, W.; McGregor, Y., Mathematical models for transmission and control of malaria, Principles and Practice of Malariology, 1091-1133 (1988), Edinburgh: Churchill Livingstone, Edinburgh |
| [17] | Dye, C.; Hasibeder, G., Population dynamics of mosquito-borne disease: effects of flies which bite some people more frequently than others, Trans. R. Soc. Trop. Med. Hyg., 80, 69-77 (1986) · doi:10.1016/0035-9203(86)90199-9 |
| [18] | Hanski, I.; Gilpin, M., Metapopulation Dynamics: Empirical and Theoretical Investigations (1991), New York: Academic Press, New York |
| [19] | Gratz, N. G., Emerging and resurging vector-borne diseases, Ann. Rev. Entomol., 44, 51-75 (1999) · doi:10.1146/annurev.ento.44.1.51 |
| [20] | Dobson, A. P.; Grenfell, B. T., Ecology of Infectious Diseases in Natural Populations (1995), Cambridge: Cambridge University Press, Cambridge · Zbl 0829.00038 |
| [21] | Grenfell, B. T.; Harwood, J., (Meta)populations dynamics of infectious diseases, Trends Ecol. Evol., 12, 395-399 (1997) · doi:10.1016/S0169-5347(97)01174-9 |
| [22] | Hanski, I., Metapopulation Ecology (1999), Oxford: Oxford University Press, Oxford |
| [23] | Hanski, I.; Gilpin, M. E., Metapopulation Biology. Ecology, Genetics, and Evolution (1997), New York: Academic Press, New York · Zbl 0913.92025 |
| [24] | Hasibeder, G.; Dye, C., Population dynamics of mosquito-borne disease: persistence in a completely heterogeneous environment, Theor. Popul. Biol., 33, 31-53 (1988) · Zbl 0647.92015 · doi:10.1016/0040-5809(88)90003-2 |
| [25] | Hess, G., Disease in metapopulation models: implications for conservation, Ecology, 77, 1617-1632 (1996) · doi:10.2307/2265556 |
| [26] | Hethcote, H. W., An immunization model for a heterogeneous population, Theor. Popul. Biol., 14, 338-349 (1978) · Zbl 0392.92009 · doi:10.1016/0040-5809(78)90011-4 |
| [27] | Hethcote, H. W.; Thieme, H. R., Stability of the endemic equilibrium in epidemic models with subpopulations, Math. Biosci., 75, 205-277 (1985) · Zbl 0582.92024 · doi:10.1016/0025-5564(85)90038-0 |
| [28] | Hethcote, H. W.; Van Ark, J. W., Epidemiological models for heterogeneous populations: proportionate mixing, parameter estimation, and immunization programs, Math. Biosci., 84, 85-118 (1987) · Zbl 0619.92006 · doi:10.1016/0025-5564(87)90044-7 |
| [29] | Lajmanovich, A.; Yorke, J. A., A deterministic model for gonorrhea in a nonhomogeneous population, Math. Biosci., 28, 221-236 (1976) · Zbl 0344.92016 · doi:10.1016/0025-5564(76)90125-5 |
| [30] | Levins, R.; Awerbuch, T.; Brinkman, U.; Eckardt, I.; Epstein, P.; Makhoul, N.; Albuquerque de Posas, C.; Puccia, C.; Spielman, A.; Wilson, M., The emergence of new diseases, Am. Sci., 82, 52-60 (1994) |
| [31] | Longini, I. M. Jr, A mathematical model for predicting the geographical spread of new infectious agents, Math. Biosci., 90, 367-383 (1988) · Zbl 0651.92016 · doi:10.1016/0025-5564(88)90075-2 |
| [32] | Longini, I. M. Jr; Fine, P. E. M.; Thacker, S. B., Predicting the global spread of new infectious agents, Am. J. Epidemiol., 123, 383-391 (1986) |
| [33] | Macdonald, G., The Epidemiology and Control of Malaria (1957), London: Oxford University Press, London |
| [34] | May, R. M., Stability and Complexity in Model Ecosystems. Monographs in Population Biology (1974), Princeton: Princeton University Press, Princeton |
| [35] | May, R. M.; Anderson, R. M., Spatial Heterogeneity and the design of immunization programs, Math. Biosci., 72, 83-111 (1984) · Zbl 0564.92016 · doi:10.1016/0025-5564(84)90063-4 |
| [36] | Nold, A., Heterogeneity in disease-tansmission modelling, Math. Biosci., 52, 227-240 (1980) · Zbl 0454.92020 · doi:10.1016/0025-5564(80)90069-3 |
| [37] | Post, W. M.; DeAngelis, D. L.; Travis, C. C., Endemic disease in environments with spatially heterogeneous host populations, Math. Biosci., 63, 289-302 (1983) · Zbl 0528.92018 · doi:10.1016/0025-5564(82)90044-X |
| [38] | Prothero, M.; Aragon, L. E., Resettlement and health: Amazonian and tropical perspective, A Desordem Ecologico na Amazonia, 161-182 (1991), Belem: Editora Universitaria, Belem |
| [39] | Ross, R., The Prevention of Malaria (1911), London: Murray, London |
| [40] | Rubio-Palis, Y.; Wirtz, R. A.; Curtis, C. F., Malaria entomological inoculation rates in western Venezuela, Acta Tropica, 52, 167-174 (1992) · doi:10.1016/0001-706X(92)90033-T |
| [41] | Rushton, S.; Mautner, A. J., The deterministic model of a simple epidemic for more than one community, Biometrika, 42, 126-132 (1955) · Zbl 0064.39102 · doi:10.2307/2333429 |
| [42] | Rvachev, L. A.; Longini, I. M. Jr, A mathematical model for the global spread of influenza, Math. Biosci., 75, 3-22 (1985) · Zbl 0567.92017 · doi:10.1016/0025-5564(85)90064-1 |
| [43] | Sandia-Mago, A., Venezuela: malaria y movilidad humana estacional de las comunidades indígenas del río Riecito del estado Apure, Fermentum, 3/4, 102-123 (1994) |
| [44] | Sattenspiel, L.; Dietz, K., A structured epidemic model incorporating geographic mobility among regions, Math. Biosci., 128, 71-91 (1995) · Zbl 0833.92020 · doi:10.1016/0025-5564(94)00068-B |
| [45] | Sattenspiel, L.; Herring, D. A., Structured epidemic models and the spread of influenza in the Central Canada Subarctic, Hum. Biol., 70, 91-115 (1998) |
| [46] | Sattenspiel, L.; Mobarry, A.; Herring, D. A., Modeling the influence of settlement structure on the spread of influenza among communities, Am. J. Hum. Biol., 12, 736-748 (2000) · doi:10.1002/1520-6300(200011/12)12:6<736::AID-AJHB3>3.0.CO;2-4 |
| [47] | Sattenspiel, L.; Simon, C. P., The spread and persistence of infectious diseases in structured populations, Math. Biosci., 90, 341-366 (1988) · Zbl 0659.92013 · doi:10.1016/0025-5564(88)90074-0 |
| [48] | Searle, S. R., Matrix Algebra Useful for Statistics (1982), New York: Wiley, New York · Zbl 0555.62002 |
| [49] | Sifontes, R., VenezuelaLa, Escuela de Malariología y el Saneamiento Ambiental y la Accíon Sanitaria en las Repúblicas Latinoamericanas, 519-559 (1985), Caracas: Fundación Bicentenario de Simón Bolívar, Caracas |
| [50] | Thrall, P. H.; Burdon, J. J., Host-pathogen dynamics in a metapopulation context: the ecological and evolutionary consequences of being spatial, J. Ecol., 85, 743-753 (1997) |
| [51] | Kareiva, P.; Tilman, D., Spatial Ecoloy. Monographs in Population Biology (1997), Princeton: Princeton University Press, Princeton · Zbl 0932.90001 |
| [52] | Torres-Sorando, L. J., Modelos Espacio-temporales y Estudio del Comportamiento Dinámico de la Incidencia de Malaria en Venezuela (1998), Caracas: Universidad Central de Venezuela, Caracas |
| [53] | Torres-Sorando, L. J.; Rodríguez, D. J., Models of spatio-temporal dynamics in malaria, Ecol. Modelling, 104, 231-240 (1997) · doi:10.1016/S0304-3800(97)00135-X |
| [54] | Travis, C. C.; Lenhart, S. M., Eradication of infectious diseases in heterogeneous populations, Math. Biosci., 83, 191-198 (1987) · Zbl 0613.92022 · doi:10.1016/0025-5564(87)90111-8 |
| [55] | Watson, R. K., On an epidemic in a stratified population, J. Appl. Prob., 9, 659-666 (1972) · Zbl 0245.92019 · doi:10.2307/3212334 |
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.
