×

Discrete-time deterministic and stochastic models for the spread of rabies. (English) Zbl 1017.92027

Summary: Discrete spatial and temporal models for the spread and control of rabies are developed, analyzed and simulated. First, a deterministic model is formulated, then an analogous stochastic model. The models are structured with respect to space \((m\) patches), age (juveniles and adults) and disease state. For each patch there are six state variables corresponding to either juveniles or adults and their disease state: susceptible, infected, or vaccinated. The models have seven stages which repeat every year.
The impact of different vaccination strategies on the dynamics of the deterministic and stochastic models are compared. In particular, the relationships among the vaccination proportion, the width of the vaccination barrier, the initial number infected, and the transmissibility of the disease are examined. An estimate for the probability of disease elimination is given for the stochastic model. It is shown that in some cases where the deterministic model predicts disease persistence, the stochastic model predicts a high probability of disease elimination.

MSC:

92D30 Epidemiology
65C20 Probabilistic models, generic numerical methods in probability and statistics
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] Rupprecht, J.; Smith, S.; Fekadu, M.; Childs, J. E., The ascension of wildlife rabies: a cause for public health concern or intervention?, Emerging Infect. Dis., 1, 107-114 (1995)
[2] Meehan, S. K., Rabies epizootic in coyotes combated with oral vaccination program, JAVMA, 206, 1097-1099 (1995)
[3] Texas Department of Health, Zoonosis Control Division, Oral Rabies Vaccination Program Information 2000, available at: http://www.tdh.state.tx.us/zoonosis/orvp/information/; Texas Department of Health, Zoonosis Control Division, Oral Rabies Vaccination Program Information 2000, available at: http://www.tdh.state.tx.us/zoonosis/orvp/information/
[4] Murray, J. D., Mathematical Biology (1993), Springer: Springer Berlin · Zbl 0779.92001
[5] Anderson, R. M.; Jackson, H. C.; May, R. M.; Smith, A. M., Population dynamics of fox rabies in Europe, Nature, 289, 765-771 (1981)
[6] Artois, M.; Langlais, M.; Suppo, C., Simulation of rabies control within an increasing fox population, Ecol. Modelling, 97, 23-34 (1997)
[7] Bacon, P. J., Discrete time temporal models of rabies, (Bacon, P. J., Population Dynamics of Rabies in Wildlife (1985), Academic Press: Academic Press London), 148-196
[8] Bacon, P. J.; MacDonald, D. W., To control rabies: vaccinate foxes, New Scientist, 87, 640-645 (1980)
[9] Ball, F. G., Spatial models for the spread and control of rabies incorporating group size, (Bacon, P. J., Population Dynamics of Rabies in Wildlife (1985), Academic Press: Academic Press London), 197-222
[10] Garnerin, Ph.; Hazout, S.; Valerin, A.-J., Estimation of two epidemiological parameters of fox rabies: the length of the incubation period and the dispersion distance of cubs, Ecol. Modelling, 33, 123-135 (1986)
[11] Källén, A.; Arcuri, P.; Murray, J. D., A simple model for the spatial spread and control of rabies, J. Theoret. Biol., 116, 377-393 (1985)
[12] Mollison, D.; Kuulasmaa, K., Spatial epidemic models theory and simulations, (Bacon, P. J., Population Dynamics of Rabies in Wildlife (1985), Academic Press: Academic Press London), 291-309
[13] Murray, J. D.; Seward, W. L., On the spatial spread of rabies among foxes with immunity, J. Theoret. Biol., 156, 327-348 (1992)
[14] Murray, J. D.; Stanley, E. A.; Brown, D. L., On the spatial spread of rabies among foxes, Proc. Roy. Soc. Lond. B, 229, 111-150 (1986)
[15] Smith, G. C.; Harris, S., Rabies in urban foxes Vulpes vulpes in Britain: the use of a spatial stochastic simulation model to examine the pattern of spread and evaluate the efficacy of different control regimes, Philos. Trans. Roy. Soc. Lond. B, 334, 450-479 (1991)
[16] Suppo, Ch.; Naulin, J. M.; Langlais, M.; Artois, M., A modelling approach to vaccination and contraception programmes for rabies control in fox populations, Proc. R. Soc. Lond. B, 267, 1575-1582 (2000)
[17] Tischendorf, L.; Thulke, H.-H.; Staubach, C.; Müller, M. S.; Jeltsch, F.; Goretzki, J.; Selhorst, T.; Müller, T.; Schlüter, H.; Wissel, C., Chance and risk of controlling rabies in large-scale and long-term immunized fox populations, Proc. R. Soc. Lond. B, 265, 839-846 (1998)
[18] (Bacon, P. J., Population Dynamics of Rabies in Wildlife (1985), Academic Press: Academic Press London)
[19] Barlow, N. D., Critical evaluation of wildlife disease models, (Grenfell, B. T.; Dobson, A. P., Ecology of Infectious Diseases in Natural Populations (1995), Cambridge University Press: Cambridge University Press London), 230-259
[20] Durrett, R.; Levin, S. A., The importance of being discrete (and spatial), Theoret. Popul. Biol., 46, 363-394 (1994) · Zbl 0846.92027
[21] Durrett, R.; Levin, S. A., Stochastic spatial models: a user’s guide to ecological applications, Philos. Trans. Roy. Soc. Lond. B, 343, 329-350 (1994)
[22] Holmes, E. E., Basic epidemiological concepts in a spatial context, (Tilman, D.; Kareiva, P., Spatial Ecology. The Role of Space in Population Dynamics and Interspecific Interactions (1997), Princeton University Press: Princeton University Press Princeton, NJ), 111-136
[23] Keymer, J. E.; Marquet, P.; Johnson, A., Pattern formation in a patch occupancy metapopulation model: a cellular automata approach, J. Theoret. Biol., 194, 79-90 (1998)
[24] Allen, L. J.S.; Burgin, A. M., Comparison of deterministic and stochastic SIS and SIR models in discrete time, Math. Biosci., 163, 1-33 (2000) · Zbl 0978.92024
[25] H.W. Hethcote, A thousand and one epidemic models, in: S.A. Levin (Ed.), Frontiers in Mathematical Biology, Lecture Notes in Biomathematics, vol. 100, Springer, Berlin, 1994; H.W. Hethcote, A thousand and one epidemic models, in: S.A. Levin (Ed.), Frontiers in Mathematical Biology, Lecture Notes in Biomathematics, vol. 100, Springer, Berlin, 1994 · Zbl 0819.92020
[26] Clark, K. A.; Neill, S. U.; Smith, J. S.; Wilson, P. J.; Whadford, V. W.; McKirahan, G. W., Epizootic canine rabies transmitted by coyotes in South Texas, JAVMA, 204, 536-540 (1994)
[27] R.K. Ratnayake, Spread of disease in an age-structured model with application to rabies, M.S. Thesis, Texas Tech University, Lubbock, TX, USA, 1998; R.K. Ratnayake, Spread of disease in an age-structured model with application to rabies, M.S. Thesis, Texas Tech University, Lubbock, TX, USA, 1998
[28] Hethcote, H. W., The mathematics of infectious diseases, SIAM Rev., 42, 599-653 (2000) · Zbl 0993.92033
[29] Anderson, R. M.; May, R. M., Infectious Diseases of Humans: Dynamics and Control (1991), Oxford University Press: Oxford University Press Oxford, UK
[30] Dushoff, J.; Huang, W.; Castillo-Chavez, C., Backwards bifurcations and catastrophe in simple models of fatal diseases, J. Math. Biol., 36, 227-248 (1998) · Zbl 0917.92022
[31] Hadeler, K. P.; van den Driessche, P., Backward bifurcation in epidemic control, Math. Biosci., 146, 15-35 (1997) · Zbl 0904.92031
[32] Kribs-Zaleta, C. M.; Velasco-Hernandez, J. X., A simple vaccination model with a backward vaccination, Math. Biosci., 164, 183-201 (2000) · Zbl 0954.92023
[33] van den Driessche, P.; Watmough, J., A simple SIS model with a backward bifurcation, J. Math. Biol., 40, 525-540 (2000) · Zbl 0961.92029
[34] Bailey, N. T.J., The Elements of Stochastic Processes with Applications to the Natural Science (1964), Wiley: Wiley New York · Zbl 0127.11203
[35] Nåsell, I., The quasi-stationary distribution of the closed endemic SIS model, Adv. Appl. Prob., 28, 895-932 (1996) · Zbl 0854.92020
[36] Nåsell, I., On the quasi-stationary distribution of the stochastic logistic epidemic, Math. Biosci., 156, 21-40 (1999) · Zbl 0954.92024
[37] Nåsell, I., On the time to extinction in recurrent epidemics, J. R. Statist. Soc. B, 61, 309-330 (1999) · Zbl 0917.92023
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.