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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

John E. Franke 1, and  Abdul-Aziz Yakubu 2,

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.
Keywords: Epidemics, infectives, periodic environments, susceptibles..
Mathematics Subject Classification: Primary: 37G15, 37G35; Secondary: 39A11, 92B0.
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.  doi: 10.1016/S0025-5564(99)00047-4.  Google Scholar

[2]

L. J. S. Allen, Some discrete-time SI, SIR and SIS epidemic models,, Math. Biosci., 124 (1994), 83.  doi: 10.1016/0025-5564(94)90025-6.  Google Scholar

[3]

R. M. Anderson and R. M. May, "Infectious Diseases of Humans: Dynamics and Control,", Oxford University Press, (1992).   Google Scholar

[4]

N. T. J. Bailey, "The Mathematical Theory of Infectious Diseases and Its Applications,", Griffin, (1975).   Google Scholar

[5]

M. Begon, J. L. Harper and C. R. Townsend, "Ecology: Individuals, Populations and Communities,", Blackwell Science, (1996).   Google Scholar

[6]

F. Berezovsky, C. Karev, B. Song and C. Castillo-Chavez, A simple model with surprising dynamics,, Mathematical Biosciences and Engineering, 2 (2005), 133.   Google Scholar

[7]

F. Berezovsky, S. Novozhilov and G. Karev, Population models with singular equilbrium,, Math. Biosci., 208 (2007), 270.  doi: 10.1016/j.mbs.2006.10.006.  Google Scholar

[8]

R. J. H. Beverton and S. J. Holt, "On the Dynamics of Exploited Fish Populations,", Fish. Invest. Ser. II, (1957).   Google Scholar

[9]

C. Castillo-Chavez and A. Yakubu, Dispersal, disease and life-history evolution,, Math. Biosci., 173 (2001), 35.  doi: 10.1016/S0025-5564(01)00065-7.  Google Scholar

[10]

C. Castillo-Chavez and A. Yakubu, Discrete-time S-I-S models with complex dynamics,, Nonlinear Anal., 47 (2001), 4753.  doi: 10.1016/S0362-546X(01)00587-9.  Google Scholar

[11]

C. Castillo-Chavez and A. A. Yakubu, Intraspecific competition, dispersal and disease dynamics in discrete-time patchy environments,, in, (2002), 165.   Google Scholar

[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.  doi: 10.1007/BF02478513.  Google Scholar

[13]

C. S. Coleman and J. C. Frauenthal, Satiable egg eating predators,, Math. Biosci., 63 (1983), 99.  doi: 10.1016/0025-5564(83)90053-6.  Google Scholar

[14]

K. L. Cooke and J. A. Yorke, Some equations modelling growth processes of gonorrhea epidemics,, Math. Biosci., 58 (1973), 93.   Google Scholar

[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.  doi: 10.1006/bulm.1997.0017.  Google Scholar

[16]

J. M. Cushing and S. M. Henson, Global dynamics of some periodically forced, monotone difference equations,, J. Differential Equations Appl., 7 (2001), 859.  doi: 10.1080/10236190108808308.  Google Scholar

[17]

S. N. Elaydi, "Discrete Chaos,", Chapman & Hall/CRC, (2000).   Google Scholar

[18]

S. N. Elaydi, Periodicity and stability of linear Volterra difference equations,, J. Math. Anal. Appl., 181 (1994), 483.  doi: 10.1006/jmaa.1994.1037.  Google Scholar

[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.  doi: 10.1016/j.jde.2003.10.024.  Google Scholar

[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, (2005), 113.  doi: 10.1201/9781420034905.  Google Scholar

[21]

S. N. Elaydi and R. J. Sacker, Nonautonomous Beverton-Holt equations and the Cushing-Henson conjectures,, J. Differential Equations Appl., 11 (2005), 336.  doi: 10.1080/10236190412331335418.  Google Scholar

[22]

S. N. Elaydi and R. J. Sacker, Periodic Difference Equations, Populations Biology and the Cushing-Henson Conjectures,, Trinity University, ().   Google Scholar

[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.  doi: 10.1080/10236190290027666.  Google Scholar

[24]

J. E. Franke and J. F. Selgrade, Attractor for periodic dynamical systems,, J. Math. Anal. Appl., 286 (2003), 64.  doi: 10.1016/S0022-247X(03)00417-7.  Google Scholar

[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.  doi: 10.1007/s00285-008-0188-9.  Google Scholar

[26]

J. E. Franke and A. A. Yakubu, Periodic dynamical systems in unidirectional metapopulation models,, J. Differential Equations Appl., 11 (2005), 687.  doi: 10.1080/10236190412331334563.  Google Scholar

[27]

J. E. Franke and A. A. Yakubu, Multiple attractors via cusp bifurcation in periodically varying environments,, J. Differential Equations Appl., 11 (2005), 365.  doi: 10.1080/10236190412331335436.  Google Scholar

[28]

J. E. Franke and A. A. Yakubu, Population models with periodic recruitment functions and survival rates,, J. Differential Equations Appl., 11 (2005), 1169.  doi: 10.1080/10236190500386275.  Google Scholar

[29]

S. D. Fretwell, "Populations in a Seasonal Environment,", Princeton University Press, (1972).   Google Scholar

[30]

M. P. Hassell, The Dynamics of Competition and Predation,, in, (1976).   Google Scholar

[31]

S. M. Henson, The effect of periodicity in maps,, J. Differential Equations Appl., 5 (1999), 31.  doi: 10.1080/10236199908808169.  Google Scholar

[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.  doi: 10.1006/bulm.1999.0136.  Google Scholar

[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.  doi: 10.1007/s002850050098.  Google Scholar

[34]

T. W. Hwang and Y. Kuang, Deterministic extinction effect in parasites on host populations,, J. Math. Biology, 46 (2003), 17.  doi: 10.1007/s00285-002-0165-7.  Google Scholar

[35]

D. Jillson, Insect populations respond to fluctuating environments,, Nature, 288 (1980), 699.  doi: 10.1038/288699a0.  Google Scholar

[36]

V. L. Kocic, A note on nonautonomous Beverton-Holt model,, J. Differential Equations Appl., 11 (2005), 415.  doi: 10.1080/10236190412331335463.  Google Scholar

[37]

V. L. Kocic and G. Ladas, Global Behavior of Nonlinear Difference Equations of Higher Order with Applications,, in, (1993).   Google Scholar

[38]

R. Kon, Attenuant cycles of population models with periodic carrying capacity,, J. Differential Equations Appl., 11 (2005), 423.  doi: 10.1080/10236190412331335472.  Google Scholar

[39]

Y. Kuang and E. Beretta, Global qualitative analysis of a ratio-dependent predator-prey system,, J. Math. Biology, 36 (1998), 389.  doi: 10.1007/s002850050105.  Google Scholar

[40]

J. Li, Periodic solutions of population models in a periodically fluctuating environment,, Math. Biosci., 110 (1992), 17.  doi: 10.1016/0025-5564(92)90012-L.  Google Scholar

[41]

R. M. May and G. F. Oster, Bifurcations and dynamic complexity in simple ecological models,, Amer. Naturalist, 110 (1976), 573.  doi: 10.1086/283092.  Google Scholar

[42]

R. M. May, Simple mathematical models with very complicated dynamics,, Nature, 261 (1977), 459.  doi: 10.1038/261459a0.  Google Scholar

[43]

R. M. May, "Stability and Complexity in Model Ecosystems,", Princeton University Press, (1974).   Google Scholar

[44]

A. J. Nicholson, Compensatory reactions of populations to stresses, and their evolutionary significance,, Aust. J. Zool., 2 (1954), 1.  doi: 10.1071/ZO9540001.  Google Scholar

[45]

R. M. Nisbet and W. S. C. Gurney, "Modelling Fluctuating Populations,", Wiley & Sons, (1982).   Google Scholar

[46]

S. Rosenblat, Population models in a periodically fluctuating environment,, J. Math. Biol., 9 (1980), 23.  doi: 10.1007/BF00276033.  Google Scholar

[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.  doi: 10.1016/S0167-2789(01)00324-4.  Google Scholar

[48]

A. A. Yakubu and M. Fogarty, Spatially discrete metapopulation models with directional dispersal,, Math. Biosci., 204 (2006), 68.  doi: 10.1016/j.mbs.2006.05.007.  Google Scholar

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.  doi: 10.1016/S0025-5564(99)00047-4.  Google Scholar

[2]

L. J. S. Allen, Some discrete-time SI, SIR and SIS epidemic models,, Math. Biosci., 124 (1994), 83.  doi: 10.1016/0025-5564(94)90025-6.  Google Scholar

[3]

R. M. Anderson and R. M. May, "Infectious Diseases of Humans: Dynamics and Control,", Oxford University Press, (1992).   Google Scholar

[4]

N. T. J. Bailey, "The Mathematical Theory of Infectious Diseases and Its Applications,", Griffin, (1975).   Google Scholar

[5]

M. Begon, J. L. Harper and C. R. Townsend, "Ecology: Individuals, Populations and Communities,", Blackwell Science, (1996).   Google Scholar

[6]

F. Berezovsky, C. Karev, B. Song and C. Castillo-Chavez, A simple model with surprising dynamics,, Mathematical Biosciences and Engineering, 2 (2005), 133.   Google Scholar

[7]

F. Berezovsky, S. Novozhilov and G. Karev, Population models with singular equilbrium,, Math. Biosci., 208 (2007), 270.  doi: 10.1016/j.mbs.2006.10.006.  Google Scholar

[8]

R. J. H. Beverton and S. J. Holt, "On the Dynamics of Exploited Fish Populations,", Fish. Invest. Ser. II, (1957).   Google Scholar

[9]

C. Castillo-Chavez and A. Yakubu, Dispersal, disease and life-history evolution,, Math. Biosci., 173 (2001), 35.  doi: 10.1016/S0025-5564(01)00065-7.  Google Scholar

[10]

C. Castillo-Chavez and A. Yakubu, Discrete-time S-I-S models with complex dynamics,, Nonlinear Anal., 47 (2001), 4753.  doi: 10.1016/S0362-546X(01)00587-9.  Google Scholar

[11]

C. Castillo-Chavez and A. A. Yakubu, Intraspecific competition, dispersal and disease dynamics in discrete-time patchy environments,, in, (2002), 165.   Google Scholar

[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.  doi: 10.1007/BF02478513.  Google Scholar

[13]

C. S. Coleman and J. C. Frauenthal, Satiable egg eating predators,, Math. Biosci., 63 (1983), 99.  doi: 10.1016/0025-5564(83)90053-6.  Google Scholar

[14]

K. L. Cooke and J. A. Yorke, Some equations modelling growth processes of gonorrhea epidemics,, Math. Biosci., 58 (1973), 93.   Google Scholar

[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.  doi: 10.1006/bulm.1997.0017.  Google Scholar

[16]

J. M. Cushing and S. M. Henson, Global dynamics of some periodically forced, monotone difference equations,, J. Differential Equations Appl., 7 (2001), 859.  doi: 10.1080/10236190108808308.  Google Scholar

[17]

S. N. Elaydi, "Discrete Chaos,", Chapman & Hall/CRC, (2000).   Google Scholar

[18]

S. N. Elaydi, Periodicity and stability of linear Volterra difference equations,, J. Math. Anal. Appl., 181 (1994), 483.  doi: 10.1006/jmaa.1994.1037.  Google Scholar

[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.  doi: 10.1016/j.jde.2003.10.024.  Google Scholar

[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, (2005), 113.  doi: 10.1201/9781420034905.  Google Scholar

[21]

S. N. Elaydi and R. J. Sacker, Nonautonomous Beverton-Holt equations and the Cushing-Henson conjectures,, J. Differential Equations Appl., 11 (2005), 336.  doi: 10.1080/10236190412331335418.  Google Scholar

[22]

S. N. Elaydi and R. J. Sacker, Periodic Difference Equations, Populations Biology and the Cushing-Henson Conjectures,, Trinity University, ().   Google Scholar

[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.  doi: 10.1080/10236190290027666.  Google Scholar

[24]

J. E. Franke and J. F. Selgrade, Attractor for periodic dynamical systems,, J. Math. Anal. Appl., 286 (2003), 64.  doi: 10.1016/S0022-247X(03)00417-7.  Google Scholar

[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.  doi: 10.1007/s00285-008-0188-9.  Google Scholar

[26]

J. E. Franke and A. A. Yakubu, Periodic dynamical systems in unidirectional metapopulation models,, J. Differential Equations Appl., 11 (2005), 687.  doi: 10.1080/10236190412331334563.  Google Scholar

[27]

J. E. Franke and A. A. Yakubu, Multiple attractors via cusp bifurcation in periodically varying environments,, J. Differential Equations Appl., 11 (2005), 365.  doi: 10.1080/10236190412331335436.  Google Scholar

[28]

J. E. Franke and A. A. Yakubu, Population models with periodic recruitment functions and survival rates,, J. Differential Equations Appl., 11 (2005), 1169.  doi: 10.1080/10236190500386275.  Google Scholar

[29]

S. D. Fretwell, "Populations in a Seasonal Environment,", Princeton University Press, (1972).   Google Scholar

[30]

M. P. Hassell, The Dynamics of Competition and Predation,, in, (1976).   Google Scholar

[31]

S. M. Henson, The effect of periodicity in maps,, J. Differential Equations Appl., 5 (1999), 31.  doi: 10.1080/10236199908808169.  Google Scholar

[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.  doi: 10.1006/bulm.1999.0136.  Google Scholar

[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.  doi: 10.1007/s002850050098.  Google Scholar

[34]

T. W. Hwang and Y. Kuang, Deterministic extinction effect in parasites on host populations,, J. Math. Biology, 46 (2003), 17.  doi: 10.1007/s00285-002-0165-7.  Google Scholar

[35]

D. Jillson, Insect populations respond to fluctuating environments,, Nature, 288 (1980), 699.  doi: 10.1038/288699a0.  Google Scholar

[36]

V. L. Kocic, A note on nonautonomous Beverton-Holt model,, J. Differential Equations Appl., 11 (2005), 415.  doi: 10.1080/10236190412331335463.  Google Scholar

[37]

V. L. Kocic and G. Ladas, Global Behavior of Nonlinear Difference Equations of Higher Order with Applications,, in, (1993).   Google Scholar

[38]

R. Kon, Attenuant cycles of population models with periodic carrying capacity,, J. Differential Equations Appl., 11 (2005), 423.  doi: 10.1080/10236190412331335472.  Google Scholar

[39]

Y. Kuang and E. Beretta, Global qualitative analysis of a ratio-dependent predator-prey system,, J. Math. Biology, 36 (1998), 389.  doi: 10.1007/s002850050105.  Google Scholar

[40]

J. Li, Periodic solutions of population models in a periodically fluctuating environment,, Math. Biosci., 110 (1992), 17.  doi: 10.1016/0025-5564(92)90012-L.  Google Scholar

[41]

R. M. May and G. F. Oster, Bifurcations and dynamic complexity in simple ecological models,, Amer. Naturalist, 110 (1976), 573.  doi: 10.1086/283092.  Google Scholar

[42]

R. M. May, Simple mathematical models with very complicated dynamics,, Nature, 261 (1977), 459.  doi: 10.1038/261459a0.  Google Scholar

[43]

R. M. May, "Stability and Complexity in Model Ecosystems,", Princeton University Press, (1974).   Google Scholar

[44]

A. J. Nicholson, Compensatory reactions of populations to stresses, and their evolutionary significance,, Aust. J. Zool., 2 (1954), 1.  doi: 10.1071/ZO9540001.  Google Scholar

[45]

R. M. Nisbet and W. S. C. Gurney, "Modelling Fluctuating Populations,", Wiley & Sons, (1982).   Google Scholar

[46]

S. Rosenblat, Population models in a periodically fluctuating environment,, J. Math. Biol., 9 (1980), 23.  doi: 10.1007/BF00276033.  Google Scholar

[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.  doi: 10.1016/S0167-2789(01)00324-4.  Google Scholar

[48]

A. A. Yakubu and M. Fogarty, Spatially discrete metapopulation models with directional dispersal,, Math. Biosci., 204 (2006), 68.  doi: 10.1016/j.mbs.2006.05.007.  Google Scholar

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