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Periodically forced discrete-time SIS epidemic model with disease induced mortality
1. | Department of Mathematics, North Carolina State University, Raleigh, NC 27695-8205, United States |
2. | Department of Mathematics, Howard University, Washington, DC 20059, United States |
References:
[1] |
L. J. S. Allen and A. M. Burgin, Comparison of deterministic and stochastic SIS and SIR models in discrete-time, Math. Biosci., 163 (2000), 1-33.
doi: 10.1016/S0025-5564(99)00047-4. |
[2] |
L. J. S. Allen, Some discrete-time SI, SIR and SIS epidemic models, Math. Biosci., 124 (1994), 83-105.
doi: 10.1016/0025-5564(94)90025-6. |
[3] |
R. M. Anderson and R. M. May, "Infectious Diseases of Humans: Dynamics and Control," Oxford University Press, Oxford, UK, 1992. |
[4] |
N. T. J. Bailey, "The Mathematical Theory of Infectious Diseases and Its Applications," Griffin, London, 1975. |
[5] |
M. Begon, J. L. Harper and C. R. Townsend, "Ecology: Individuals, Populations and Communities," Blackwell Science, Williston, VT, 1996. |
[6] |
F. Berezovsky, C. Karev, B. Song and C. Castillo-Chavez, A simple model with surprising dynamics, Mathematical Biosciences and Engineering, 2 (2005), 133-152. |
[7] |
F. Berezovsky, S. Novozhilov and G. Karev, Population models with singular equilbrium, Math. Biosci., 208 (2007), 270-299.
doi: 10.1016/j.mbs.2006.10.006. |
[8] |
R. J. H. Beverton and S. J. Holt, "On the Dynamics of Exploited Fish Populations," Fish. Invest. Ser. II, H. M. Stationery Office, London, 1957. |
[9] |
C. Castillo-Chavez and A. Yakubu, Dispersal, disease and life-history evolution, Math. Biosci., 173 (2001), 35-53.
doi: 10.1016/S0025-5564(01)00065-7. |
[10] |
C. Castillo-Chavez and A. Yakubu, Discrete-time S-I-S models with complex dynamics, Nonlinear Anal., 47 (2001), 4753-4762.
doi: 10.1016/S0362-546X(01)00587-9. |
[11] |
C. Castillo-Chavez and A. A. Yakubu, Intraspecific competition, dispersal and disease dynamics in discrete-time patchy environments, in "Mathematical Approaches for Emerging and Reemerging Infectious Diseases: An Introduction to Models, Methods and Theory," C. Castillo-Chavez with S. Blower, P. van den Driessche, D. Kirschner,and A.-A. Yakubu, eds., Springer-Verlag, New York, 2002, 165-181. |
[12] |
B. D. Coleman, On the growth of populations with narrow spread in reproductive age. I. General theory and examples, J. Math. Biol., 6 (1978), 1-19.
doi: 10.1007/BF02478513. |
[13] |
C. S. Coleman and J. C. Frauenthal, Satiable egg eating predators, Math. Biosci., 63 (1983), 99-119.
doi: 10.1016/0025-5564(83)90053-6. |
[14] |
K. L. Cooke and J. A. Yorke, Some equations modelling growth processes of gonorrhea epidemics, Math. Biosci., 58 (1973), 93-109. |
[15] |
R. F. Costantino, J. M. Cushing, B. Dennis, R., A. Desharnais and S. M. Henson, Resonant population cycles in temporarily fluctuating habitats, Bull. Math. Biol., 60 (1998), 247-273.
doi: 10.1006/bulm.1997.0017. |
[16] |
J. M. Cushing and S. M. Henson, Global dynamics of some periodically forced, monotone difference equations, J. Differential Equations Appl., 7 (2001), 859-872.
doi: 10.1080/10236190108808308. |
[17] |
S. N. Elaydi, "Discrete Chaos," Chapman & Hall/CRC, Boca Raton, FL, 2000. |
[18] |
S. N. Elaydi, Periodicity and stability of linear Volterra difference equations, J. Math. Anal. Appl., 181 (1994), 483-492.
doi: 10.1006/jmaa.1994.1037. |
[19] |
S. N. Elaydi and R. J. Sacker, Global stability of periodic orbits of nonautonomous difference equations and population biology, J. Differential Equations, 208 (2005), 258-273.
doi: 10.1016/j.jde.2003.10.024. |
[20] |
S. N. Elaydi and R. J. Sacker, Global stability of periodic orbits of nonautonomous difference equations in population biology and Cushing-Henson conjectures, in "Proceedings of the 8th International Conference on Difference Equations and Applications," Chapman & Hall/CRC, Boca Raton, FL, 2005, 113-126.
doi: 10.1201/9781420034905. |
[21] |
S. N. Elaydi and R. J. Sacker, Nonautonomous Beverton-Holt equations and the Cushing-Henson conjectures, J. Differential Equations Appl., 11 (2005), 336-346.
doi: 10.1080/10236190412331335418. |
[22] |
S. N. Elaydi and R. J. Sacker, Periodic Difference Equations, Populations Biology and the Cushing-Henson Conjectures,, Trinity University, ().
|
[23] |
S. N. Elaydi and A. A. Yakubu, Global stability of cycles: Lotka-Volterra competition model with stocking, J. Differential Equations Appl., 8 (2002), 537-549.
doi: 10.1080/10236190290027666. |
[24] |
J. E. Franke and J. F. Selgrade, Attractor for periodic dynamical systems, J. Math. Anal. Appl., 286 (2003), 64-79.
doi: 10.1016/S0022-247X(03)00417-7. |
[25] |
J. E. Franke and A. A. Yakubu, Disease-induced mortality in density dependent discrete-time SIS epidemic model, J. Math. Biology, 57 (2008), 755-790.
doi: 10.1007/s00285-008-0188-9. |
[26] |
J. E. Franke and A. A. Yakubu, Periodic dynamical systems in unidirectional metapopulation models, J. Differential Equations Appl., 11 (2005), 687-700.
doi: 10.1080/10236190412331334563. |
[27] |
J. E. Franke and A. A. Yakubu, Multiple attractors via cusp bifurcation in periodically varying environments, J. Differential Equations Appl., 11 (2005), 365-377.
doi: 10.1080/10236190412331335436. |
[28] |
J. E. Franke and A. A. Yakubu, Population models with periodic recruitment functions and survival rates, J. Differential Equations Appl., 11 (2005), 1169-1184.
doi: 10.1080/10236190500386275. |
[29] |
S. D. Fretwell, "Populations in a Seasonal Environment," Princeton University Press, Princeton, NJ, 1972. |
[30] |
M. P. Hassell, The Dynamics of Competition and Predation, in "Studies in Biol. 72," The Camelot Press, Southampton, UK, 1976. |
[31] |
S. M. Henson, The effect of periodicity in maps, J. Differential Equations Appl., 5 (1999), 31-56.
doi: 10.1080/10236199908808169. |
[32] |
S. M. Henson, R. F. Costantino, J. M. Cushing, B. Dennis and R. A. Desharnais, Multiple attractors, saddles, and population dynamics in periodic habitats, Bull. Math. Biol., 61 (1999), 1121-1149.
doi: 10.1006/bulm.1999.0136. |
[33] |
S. M. Henson and J. M. Cushing, The effect of periodic habitat fluctuations on a nonlinear insect population model, J. Math. Biol., 36 (1997), 201-226.
doi: 10.1007/s002850050098. |
[34] |
T. W. Hwang and Y. Kuang, Deterministic extinction effect in parasites on host populations, J. Math. Biology, 46 (2003), 17-30.
doi: 10.1007/s00285-002-0165-7. |
[35] |
D. Jillson, Insect populations respond to fluctuating environments, Nature, 288 (1980), 699-700.
doi: 10.1038/288699a0. |
[36] |
V. L. Kocic, A note on nonautonomous Beverton-Holt model, J. Differential Equations Appl., 11 (2005), 415-422.
doi: 10.1080/10236190412331335463. |
[37] |
V. L. Kocic and G. Ladas, Global Behavior of Nonlinear Difference Equations of Higher Order with Applications, in "Math. Appl. 256," Kluwer Academic Publishers, Dordrecht, The Netherlands, 1993. |
[38] |
R. Kon, Attenuant cycles of population models with periodic carrying capacity, J. Differential Equations Appl., 11 (2005), 423-430.
doi: 10.1080/10236190412331335472. |
[39] |
Y. Kuang and E. Beretta, Global qualitative analysis of a ratio-dependent predator-prey system, J. Math. Biology, 36 (1998), 389-406.
doi: 10.1007/s002850050105. |
[40] |
J. Li, Periodic solutions of population models in a periodically fluctuating environment, Math. Biosci., 110 (1992), 17-25.
doi: 10.1016/0025-5564(92)90012-L. |
[41] |
R. M. May and G. F. Oster, Bifurcations and dynamic complexity in simple ecological models, Amer. Naturalist, 110 (1976), 573-579.
doi: 10.1086/283092. |
[42] |
R. M. May, Simple mathematical models with very complicated dynamics, Nature, 261 (1977), 459-469.
doi: 10.1038/261459a0. |
[43] |
R. M. May, "Stability and Complexity in Model Ecosystems," Princeton University Press, Princeton, NJ, 1974. |
[44] |
A. J. Nicholson, Compensatory reactions of populations to stresses, and their evolutionary significance, Aust. J. Zool., 2 (1954), 1-65.
doi: 10.1071/ZO9540001. |
[45] |
R. M. Nisbet and W. S. C. Gurney, "Modelling Fluctuating Populations," Wiley & Sons, New York, 1982. |
[46] |
S. Rosenblat, Population models in a periodically fluctuating environment, J. Math. Biol., 9 (1980), 23-36.
doi: 10.1007/BF00276033. |
[47] |
J. F. Selgrade and H. D. Roberds, On the structure of attractors for discrete, periodically forced systems with applications to population models, Phys. D, 158 (2001), 69-82.
doi: 10.1016/S0167-2789(01)00324-4. |
[48] |
A. A. Yakubu and M. Fogarty, Spatially discrete metapopulation models with directional dispersal, Math. Biosci., 204 (2006), 68-101.
doi: 10.1016/j.mbs.2006.05.007. |
show all references
References:
[1] |
L. J. S. Allen and A. M. Burgin, Comparison of deterministic and stochastic SIS and SIR models in discrete-time, Math. Biosci., 163 (2000), 1-33.
doi: 10.1016/S0025-5564(99)00047-4. |
[2] |
L. J. S. Allen, Some discrete-time SI, SIR and SIS epidemic models, Math. Biosci., 124 (1994), 83-105.
doi: 10.1016/0025-5564(94)90025-6. |
[3] |
R. M. Anderson and R. M. May, "Infectious Diseases of Humans: Dynamics and Control," Oxford University Press, Oxford, UK, 1992. |
[4] |
N. T. J. Bailey, "The Mathematical Theory of Infectious Diseases and Its Applications," Griffin, London, 1975. |
[5] |
M. Begon, J. L. Harper and C. R. Townsend, "Ecology: Individuals, Populations and Communities," Blackwell Science, Williston, VT, 1996. |
[6] |
F. Berezovsky, C. Karev, B. Song and C. Castillo-Chavez, A simple model with surprising dynamics, Mathematical Biosciences and Engineering, 2 (2005), 133-152. |
[7] |
F. Berezovsky, S. Novozhilov and G. Karev, Population models with singular equilbrium, Math. Biosci., 208 (2007), 270-299.
doi: 10.1016/j.mbs.2006.10.006. |
[8] |
R. J. H. Beverton and S. J. Holt, "On the Dynamics of Exploited Fish Populations," Fish. Invest. Ser. II, H. M. Stationery Office, London, 1957. |
[9] |
C. Castillo-Chavez and A. Yakubu, Dispersal, disease and life-history evolution, Math. Biosci., 173 (2001), 35-53.
doi: 10.1016/S0025-5564(01)00065-7. |
[10] |
C. Castillo-Chavez and A. Yakubu, Discrete-time S-I-S models with complex dynamics, Nonlinear Anal., 47 (2001), 4753-4762.
doi: 10.1016/S0362-546X(01)00587-9. |
[11] |
C. Castillo-Chavez and A. A. Yakubu, Intraspecific competition, dispersal and disease dynamics in discrete-time patchy environments, in "Mathematical Approaches for Emerging and Reemerging Infectious Diseases: An Introduction to Models, Methods and Theory," C. Castillo-Chavez with S. Blower, P. van den Driessche, D. Kirschner,and A.-A. Yakubu, eds., Springer-Verlag, New York, 2002, 165-181. |
[12] |
B. D. Coleman, On the growth of populations with narrow spread in reproductive age. I. General theory and examples, J. Math. Biol., 6 (1978), 1-19.
doi: 10.1007/BF02478513. |
[13] |
C. S. Coleman and J. C. Frauenthal, Satiable egg eating predators, Math. Biosci., 63 (1983), 99-119.
doi: 10.1016/0025-5564(83)90053-6. |
[14] |
K. L. Cooke and J. A. Yorke, Some equations modelling growth processes of gonorrhea epidemics, Math. Biosci., 58 (1973), 93-109. |
[15] |
R. F. Costantino, J. M. Cushing, B. Dennis, R., A. Desharnais and S. M. Henson, Resonant population cycles in temporarily fluctuating habitats, Bull. Math. Biol., 60 (1998), 247-273.
doi: 10.1006/bulm.1997.0017. |
[16] |
J. M. Cushing and S. M. Henson, Global dynamics of some periodically forced, monotone difference equations, J. Differential Equations Appl., 7 (2001), 859-872.
doi: 10.1080/10236190108808308. |
[17] |
S. N. Elaydi, "Discrete Chaos," Chapman & Hall/CRC, Boca Raton, FL, 2000. |
[18] |
S. N. Elaydi, Periodicity and stability of linear Volterra difference equations, J. Math. Anal. Appl., 181 (1994), 483-492.
doi: 10.1006/jmaa.1994.1037. |
[19] |
S. N. Elaydi and R. J. Sacker, Global stability of periodic orbits of nonautonomous difference equations and population biology, J. Differential Equations, 208 (2005), 258-273.
doi: 10.1016/j.jde.2003.10.024. |
[20] |
S. N. Elaydi and R. J. Sacker, Global stability of periodic orbits of nonautonomous difference equations in population biology and Cushing-Henson conjectures, in "Proceedings of the 8th International Conference on Difference Equations and Applications," Chapman & Hall/CRC, Boca Raton, FL, 2005, 113-126.
doi: 10.1201/9781420034905. |
[21] |
S. N. Elaydi and R. J. Sacker, Nonautonomous Beverton-Holt equations and the Cushing-Henson conjectures, J. Differential Equations Appl., 11 (2005), 336-346.
doi: 10.1080/10236190412331335418. |
[22] |
S. N. Elaydi and R. J. Sacker, Periodic Difference Equations, Populations Biology and the Cushing-Henson Conjectures,, Trinity University, ().
|
[23] |
S. N. Elaydi and A. A. Yakubu, Global stability of cycles: Lotka-Volterra competition model with stocking, J. Differential Equations Appl., 8 (2002), 537-549.
doi: 10.1080/10236190290027666. |
[24] |
J. E. Franke and J. F. Selgrade, Attractor for periodic dynamical systems, J. Math. Anal. Appl., 286 (2003), 64-79.
doi: 10.1016/S0022-247X(03)00417-7. |
[25] |
J. E. Franke and A. A. Yakubu, Disease-induced mortality in density dependent discrete-time SIS epidemic model, J. Math. Biology, 57 (2008), 755-790.
doi: 10.1007/s00285-008-0188-9. |
[26] |
J. E. Franke and A. A. Yakubu, Periodic dynamical systems in unidirectional metapopulation models, J. Differential Equations Appl., 11 (2005), 687-700.
doi: 10.1080/10236190412331334563. |
[27] |
J. E. Franke and A. A. Yakubu, Multiple attractors via cusp bifurcation in periodically varying environments, J. Differential Equations Appl., 11 (2005), 365-377.
doi: 10.1080/10236190412331335436. |
[28] |
J. E. Franke and A. A. Yakubu, Population models with periodic recruitment functions and survival rates, J. Differential Equations Appl., 11 (2005), 1169-1184.
doi: 10.1080/10236190500386275. |
[29] |
S. D. Fretwell, "Populations in a Seasonal Environment," Princeton University Press, Princeton, NJ, 1972. |
[30] |
M. P. Hassell, The Dynamics of Competition and Predation, in "Studies in Biol. 72," The Camelot Press, Southampton, UK, 1976. |
[31] |
S. M. Henson, The effect of periodicity in maps, J. Differential Equations Appl., 5 (1999), 31-56.
doi: 10.1080/10236199908808169. |
[32] |
S. M. Henson, R. F. Costantino, J. M. Cushing, B. Dennis and R. A. Desharnais, Multiple attractors, saddles, and population dynamics in periodic habitats, Bull. Math. Biol., 61 (1999), 1121-1149.
doi: 10.1006/bulm.1999.0136. |
[33] |
S. M. Henson and J. M. Cushing, The effect of periodic habitat fluctuations on a nonlinear insect population model, J. Math. Biol., 36 (1997), 201-226.
doi: 10.1007/s002850050098. |
[34] |
T. W. Hwang and Y. Kuang, Deterministic extinction effect in parasites on host populations, J. Math. Biology, 46 (2003), 17-30.
doi: 10.1007/s00285-002-0165-7. |
[35] |
D. Jillson, Insect populations respond to fluctuating environments, Nature, 288 (1980), 699-700.
doi: 10.1038/288699a0. |
[36] |
V. L. Kocic, A note on nonautonomous Beverton-Holt model, J. Differential Equations Appl., 11 (2005), 415-422.
doi: 10.1080/10236190412331335463. |
[37] |
V. L. Kocic and G. Ladas, Global Behavior of Nonlinear Difference Equations of Higher Order with Applications, in "Math. Appl. 256," Kluwer Academic Publishers, Dordrecht, The Netherlands, 1993. |
[38] |
R. Kon, Attenuant cycles of population models with periodic carrying capacity, J. Differential Equations Appl., 11 (2005), 423-430.
doi: 10.1080/10236190412331335472. |
[39] |
Y. Kuang and E. Beretta, Global qualitative analysis of a ratio-dependent predator-prey system, J. Math. Biology, 36 (1998), 389-406.
doi: 10.1007/s002850050105. |
[40] |
J. Li, Periodic solutions of population models in a periodically fluctuating environment, Math. Biosci., 110 (1992), 17-25.
doi: 10.1016/0025-5564(92)90012-L. |
[41] |
R. M. May and G. F. Oster, Bifurcations and dynamic complexity in simple ecological models, Amer. Naturalist, 110 (1976), 573-579.
doi: 10.1086/283092. |
[42] |
R. M. May, Simple mathematical models with very complicated dynamics, Nature, 261 (1977), 459-469.
doi: 10.1038/261459a0. |
[43] |
R. M. May, "Stability and Complexity in Model Ecosystems," Princeton University Press, Princeton, NJ, 1974. |
[44] |
A. J. Nicholson, Compensatory reactions of populations to stresses, and their evolutionary significance, Aust. J. Zool., 2 (1954), 1-65.
doi: 10.1071/ZO9540001. |
[45] |
R. M. Nisbet and W. S. C. Gurney, "Modelling Fluctuating Populations," Wiley & Sons, New York, 1982. |
[46] |
S. Rosenblat, Population models in a periodically fluctuating environment, J. Math. Biol., 9 (1980), 23-36.
doi: 10.1007/BF00276033. |
[47] |
J. F. Selgrade and H. D. Roberds, On the structure of attractors for discrete, periodically forced systems with applications to population models, Phys. D, 158 (2001), 69-82.
doi: 10.1016/S0167-2789(01)00324-4. |
[48] |
A. A. Yakubu and M. Fogarty, Spatially discrete metapopulation models with directional dispersal, Math. Biosci., 204 (2006), 68-101.
doi: 10.1016/j.mbs.2006.05.007. |
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