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2011, 8(2): 385-408. doi: 10.3934/mbe.2011.8.385

## 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

Received  February 2010 Revised  August 2010 Published  April 2011

We use a periodically forced SIS epidemic model with disease induced mortality to study the combined effects of seasonal trends and death on the extinction and persistence of discretely reproducing populations. We introduce the epidemic threshold parameter, $R_0$, for predicting disease dynamics in periodic environments. Typically, $R_0<1$ implies disease extinction. However, in the presence of disease induced mortality, we extend the results of Franke and Yakubu to periodic environments and show that a small number of infectives can drive an otherwise persistent population with $R_0>1$ to extinction. Furthermore, we obtain conditions for the persistence of the total population. In addition, we use the Beverton-Holt recruitment function to show that the infective population exhibits period-doubling bifurcations route to chaos where the disease-free susceptible population lives on a 2-cycle (non-chaotic) attractor.
Citation: John E. Franke, Abdul-Aziz Yakubu. Periodically forced discrete-time SIS epidemic model with disease induced mortality. Mathematical Biosciences & Engineering, 2011, 8 (2) : 385-408. doi: 10.3934/mbe.2011.8.385
##### 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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