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The basic reproduction number of discrete SIR and SEIS models with periodic parameters
1. | School of Science, Shaanxi University of Science & Technology, Xi'an, 710021, China |
2. | Department of Mathematics, Xi'an Jiaotong University, Xi'an, 710049 |
References:
[1] |
O. Diekmann, J. Heesterbeek and J. Metz, On the definition and the computation of the basic reproduction ratio $R_{0}$ in models for infectious disease in heterogeneous populations, J. Math. Biol., 28 (1990), 365-382.
doi: 10.1007/BF00178324. |
[2] |
P. van den Driessche and J. Watmough, Reproduction numbers and sub-threshold endemic equilibria for compartmental models of disease transmission, Math. Biosci., 180 (2002), 29-48.
doi: 10.1016/S0025-5564(02)00108-6. |
[3] |
W. Wang and X. Zhao, Threshold dynamics for compartmental epidemic models in periodic environments, J. Dyn. Diff. Equat., 20 (2008), 699-717.
doi: 10.1007/s10884-008-9111-8. |
[4] |
L. Allen and P. van den Driessche, The basic reproduction number in some discrete-time epidemic models, J. Difference Equations and Applications, 14 (2008), 1127-1147.
doi: 10.1080/10236190802332308. |
[5] |
N. Bacaër, Periodic matrix populaiton models: growth rate, basic reproduction number, and entropy, Bull. Math. Biol., 71 (2009), 1781-1792.
doi: 10.1007/s11538-009-9426-6. |
[6] |
M. Keeling and P. Rohani, "Modeling Infectious Diseases in Humans and Animals," Princeton University Press, 2008. |
[7] |
Y. Zhou and H. Cao, Discrete tuberculosis transmission models and their application, in "Fields Communications Series: New Perspectives in Mathematical Biology," (ed.S. Sivaloganathan), A co-publication of the AMS and Fields Institute, Canada, 57 (2010), 83-112. |
[8] |
I. Schwartz and H. Smith, Infinite subharmonic bifurcation in an SIER epidemic model, J. Math. Biol., 18 (1983), 233-253.
doi: 10.1007/BF00276090. |
[9] |
I. Schwartz, Small amplitude, long periodic outbreaks in seasonally driven epidemics, J. Math. Biol., 30 (1992), 473-491.
doi: 10.1007/BF00160532. |
[10] |
H. Smith, Multiple stable subharmonics for a periodic epidemic model, J. Math. Biol., 17 (1983), 179-190.
doi: 10.1007/BF00305758. |
[11] |
X. Zhao, "Dynamical Sytems in Population Biology," Springer-Verlag, New York, 2003. |
[12] |
J. M. Cushing, A juvenile-adult model with periodic vital rates, J. Math. Biol., 53 (2006), 520-539.
doi: 10.1007/s00285-006-0382-6. |
[13] |
J. Ma and Z. Ma, Epidemic threshold conditions for seasonally forced SEIR models, Math. Biosci. Eng., 3 (2006), 161-172. |
[14] |
N. Bacaër, Approximation of the basic reproduction number $R_0$ for vector-borne diseases with a periodic vector population, Bull. Math. Biol., 69 (2007), 1067-1091.
doi: 10.1007/s11538-006-9166-9. |
[15] |
N. Bacaër and M. G. M. Gomes, On the final size of epidemics with seasonality, Bull. Math. Biol., 71 (2009), 1954-1966.
doi: 10.1007/s11538-009-9433-7. |
[16] |
N. Bacaër and S. Guernaoui, The epidemic threshold of vector-borne diseases with seasonality, J. Math. Biol., 53 (2006), 421-436.
doi: 10.1007/s00285-006-0015-0. |
[17] |
F. Zhang and X. Zhao, A periodic epidemic model in a patchy enviroment, J. Math. Anal. Appl., 325 (2007), 496-516.
doi: 10.1016/j.jmaa.2006.01.085. |
[18] |
B. G. Williams and C. Dye, Infectious disease persistence when transmission varies seasonally, Math. Biosci., 145 (1997), 77-88.
doi: 10.1016/S0025-5564(97)00039-4. |
[19] |
H. R. Thieme, Renewal theorems for linear periodic Volterra integral equations, J. Integral Equations, 7 (1984), 253-277. |
[20] |
L. Allen, Some discrete-time SI, SIR, and SIS epidemic models, Math. Biosci., 124 (1994), 83-105.
doi: 10.1016/0025-5564(94)90025-6. |
[21] |
L. Allen and A. 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. |
[22] |
L. Allen, D. Flores, R. Ratnayake and J. Herbold, Discrete-time deterministic and stochastic models for the spread of rabies, Appl. Math. Comput., 132 (2002), 271-292.
doi: 10.1016/S0096-3003(01)00192-8. |
[23] |
C. Castillo-Chavez and A. A. Yakubu, Discrete-time SIS models with complex dynamics, Nonliear Anal., 47 (2001), 4753-4762.
doi: 10.1016/S0362-546X(01)00587-9. |
[24] |
C. Castillo-Chavez and A. A. Yakubu, Dispersal, disease and life-history evolution, Math. Biosci., 173 (2001), 35-53.
doi: 10.1016/S0025-5564(01)00065-7. |
[25] |
C. Castillo-Chavez and A. A. Yakubu, Discrete-time SIS models with simple and complex population dynamics, in "Mathematical Approaches for Emerging and Reemerging Infectious Diseases: A introduction," (ed. C. Castillo-Chavez with S. Blower, P. van den Driessche, D. Kirschner, and A. A. Yakubu), Springger-Verlag, New York, (2002), 153-163. |
[26] |
Y. Zhou and P. Fergola, Dynamic of a discrete age-structured SIS models, Discrete Contin. Dyn. Syst. Ser. B., 4 (2004), 843-852. |
[27] |
Y. Zhou and Z. Ma, Global stability of a class of discrete age-structured SIS models with immigration, Math. Biosci. Eng., 6 (2009), 409-425. |
[28] |
X. Li and W. Wang, A discrete epidemic model with stage structure, Chaos, Solitions and Fractals., 26 (2005), 947-958. |
[29] |
J. E. Franke and A. A. Yakubu, Discrete-time SIS epidemic model in a seasonal environment, SIAM J. Appl. Math., 66 (2006), 1563-1587.
doi: 10.1137/050638345. |
[30] |
Ira M. Longini, Jr., The generalized discrete-time epidemic model with immunity: Asynthesis, Math. Biosci., 82 (1986), 19-41.
doi: 10.1016/0025-5564(86)90003-9. |
[31] |
N. Bacaër and R. Ouifki, Growth rate and basic reproduction number for population models with a simple periodic factor, Math. Biosci., 210 (2007), 647-58.
doi: 10.1016/j.mbs.2007.07.005. |
[32] |
M. I. Gil, "Difference Equations in Normed Spaces Stability and Oscillations," Elsevier Science, 2007. |
[33] |
R. A. Horn and C. A. Johnson, "Matrix Analysis," Cambridge University press, Cambridge, 1985. |
[34] |
H. Smith and P. Waltman, "Theory of the Chemostat," Cambridge University Press, Cambridge, 1995. |
[35] |
P. Hess, "Periodic-Parabolic Boundary Value Problems and Positivity," Pitman Research Notes in Mathematics,Series 247, Longman Scientific and Technical, 1991. |
[36] |
H. R. Thieme, Convergence results and a Poincare-Bendixson trichotomy for asymptotically autonomous differential equations, J. Math. Biol., 30 (1992), 755-763. |
[37] |
X. Q. Zhao, Asymptotic behavior for asymptotically periodic semiflows with applications, Commun. Appl. Nonlinear Anal., 3 (1996), 43-66. |
[38] |
P. Salceanu and H. Smith, Persistence in a discrete-time, stage-structured epidemic model, J. Difference Equa. Appl., 16 (2010), 73-103.
doi: 10.1080/10236190802400733. |
[39] |
P. A. Gonzalez, R. A. Saenz, B. N. Sanchez, C. Castillo-Chavez and A. A. Yakubu, "Dispersal Between Two Patches in a Discrete Time SEIS Model," MTBI technical Report, 2000. |
show all references
References:
[1] |
O. Diekmann, J. Heesterbeek and J. Metz, On the definition and the computation of the basic reproduction ratio $R_{0}$ in models for infectious disease in heterogeneous populations, J. Math. Biol., 28 (1990), 365-382.
doi: 10.1007/BF00178324. |
[2] |
P. van den Driessche and J. Watmough, Reproduction numbers and sub-threshold endemic equilibria for compartmental models of disease transmission, Math. Biosci., 180 (2002), 29-48.
doi: 10.1016/S0025-5564(02)00108-6. |
[3] |
W. Wang and X. Zhao, Threshold dynamics for compartmental epidemic models in periodic environments, J. Dyn. Diff. Equat., 20 (2008), 699-717.
doi: 10.1007/s10884-008-9111-8. |
[4] |
L. Allen and P. van den Driessche, The basic reproduction number in some discrete-time epidemic models, J. Difference Equations and Applications, 14 (2008), 1127-1147.
doi: 10.1080/10236190802332308. |
[5] |
N. Bacaër, Periodic matrix populaiton models: growth rate, basic reproduction number, and entropy, Bull. Math. Biol., 71 (2009), 1781-1792.
doi: 10.1007/s11538-009-9426-6. |
[6] |
M. Keeling and P. Rohani, "Modeling Infectious Diseases in Humans and Animals," Princeton University Press, 2008. |
[7] |
Y. Zhou and H. Cao, Discrete tuberculosis transmission models and their application, in "Fields Communications Series: New Perspectives in Mathematical Biology," (ed.S. Sivaloganathan), A co-publication of the AMS and Fields Institute, Canada, 57 (2010), 83-112. |
[8] |
I. Schwartz and H. Smith, Infinite subharmonic bifurcation in an SIER epidemic model, J. Math. Biol., 18 (1983), 233-253.
doi: 10.1007/BF00276090. |
[9] |
I. Schwartz, Small amplitude, long periodic outbreaks in seasonally driven epidemics, J. Math. Biol., 30 (1992), 473-491.
doi: 10.1007/BF00160532. |
[10] |
H. Smith, Multiple stable subharmonics for a periodic epidemic model, J. Math. Biol., 17 (1983), 179-190.
doi: 10.1007/BF00305758. |
[11] |
X. Zhao, "Dynamical Sytems in Population Biology," Springer-Verlag, New York, 2003. |
[12] |
J. M. Cushing, A juvenile-adult model with periodic vital rates, J. Math. Biol., 53 (2006), 520-539.
doi: 10.1007/s00285-006-0382-6. |
[13] |
J. Ma and Z. Ma, Epidemic threshold conditions for seasonally forced SEIR models, Math. Biosci. Eng., 3 (2006), 161-172. |
[14] |
N. Bacaër, Approximation of the basic reproduction number $R_0$ for vector-borne diseases with a periodic vector population, Bull. Math. Biol., 69 (2007), 1067-1091.
doi: 10.1007/s11538-006-9166-9. |
[15] |
N. Bacaër and M. G. M. Gomes, On the final size of epidemics with seasonality, Bull. Math. Biol., 71 (2009), 1954-1966.
doi: 10.1007/s11538-009-9433-7. |
[16] |
N. Bacaër and S. Guernaoui, The epidemic threshold of vector-borne diseases with seasonality, J. Math. Biol., 53 (2006), 421-436.
doi: 10.1007/s00285-006-0015-0. |
[17] |
F. Zhang and X. Zhao, A periodic epidemic model in a patchy enviroment, J. Math. Anal. Appl., 325 (2007), 496-516.
doi: 10.1016/j.jmaa.2006.01.085. |
[18] |
B. G. Williams and C. Dye, Infectious disease persistence when transmission varies seasonally, Math. Biosci., 145 (1997), 77-88.
doi: 10.1016/S0025-5564(97)00039-4. |
[19] |
H. R. Thieme, Renewal theorems for linear periodic Volterra integral equations, J. Integral Equations, 7 (1984), 253-277. |
[20] |
L. Allen, Some discrete-time SI, SIR, and SIS epidemic models, Math. Biosci., 124 (1994), 83-105.
doi: 10.1016/0025-5564(94)90025-6. |
[21] |
L. Allen and A. 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. |
[22] |
L. Allen, D. Flores, R. Ratnayake and J. Herbold, Discrete-time deterministic and stochastic models for the spread of rabies, Appl. Math. Comput., 132 (2002), 271-292.
doi: 10.1016/S0096-3003(01)00192-8. |
[23] |
C. Castillo-Chavez and A. A. Yakubu, Discrete-time SIS models with complex dynamics, Nonliear Anal., 47 (2001), 4753-4762.
doi: 10.1016/S0362-546X(01)00587-9. |
[24] |
C. Castillo-Chavez and A. A. Yakubu, Dispersal, disease and life-history evolution, Math. Biosci., 173 (2001), 35-53.
doi: 10.1016/S0025-5564(01)00065-7. |
[25] |
C. Castillo-Chavez and A. A. Yakubu, Discrete-time SIS models with simple and complex population dynamics, in "Mathematical Approaches for Emerging and Reemerging Infectious Diseases: A introduction," (ed. C. Castillo-Chavez with S. Blower, P. van den Driessche, D. Kirschner, and A. A. Yakubu), Springger-Verlag, New York, (2002), 153-163. |
[26] |
Y. Zhou and P. Fergola, Dynamic of a discrete age-structured SIS models, Discrete Contin. Dyn. Syst. Ser. B., 4 (2004), 843-852. |
[27] |
Y. Zhou and Z. Ma, Global stability of a class of discrete age-structured SIS models with immigration, Math. Biosci. Eng., 6 (2009), 409-425. |
[28] |
X. Li and W. Wang, A discrete epidemic model with stage structure, Chaos, Solitions and Fractals., 26 (2005), 947-958. |
[29] |
J. E. Franke and A. A. Yakubu, Discrete-time SIS epidemic model in a seasonal environment, SIAM J. Appl. Math., 66 (2006), 1563-1587.
doi: 10.1137/050638345. |
[30] |
Ira M. Longini, Jr., The generalized discrete-time epidemic model with immunity: Asynthesis, Math. Biosci., 82 (1986), 19-41.
doi: 10.1016/0025-5564(86)90003-9. |
[31] |
N. Bacaër and R. Ouifki, Growth rate and basic reproduction number for population models with a simple periodic factor, Math. Biosci., 210 (2007), 647-58.
doi: 10.1016/j.mbs.2007.07.005. |
[32] |
M. I. Gil, "Difference Equations in Normed Spaces Stability and Oscillations," Elsevier Science, 2007. |
[33] |
R. A. Horn and C. A. Johnson, "Matrix Analysis," Cambridge University press, Cambridge, 1985. |
[34] |
H. Smith and P. Waltman, "Theory of the Chemostat," Cambridge University Press, Cambridge, 1995. |
[35] |
P. Hess, "Periodic-Parabolic Boundary Value Problems and Positivity," Pitman Research Notes in Mathematics,Series 247, Longman Scientific and Technical, 1991. |
[36] |
H. R. Thieme, Convergence results and a Poincare-Bendixson trichotomy for asymptotically autonomous differential equations, J. Math. Biol., 30 (1992), 755-763. |
[37] |
X. Q. Zhao, Asymptotic behavior for asymptotically periodic semiflows with applications, Commun. Appl. Nonlinear Anal., 3 (1996), 43-66. |
[38] |
P. Salceanu and H. Smith, Persistence in a discrete-time, stage-structured epidemic model, J. Difference Equa. Appl., 16 (2010), 73-103.
doi: 10.1080/10236190802400733. |
[39] |
P. A. Gonzalez, R. A. Saenz, B. N. Sanchez, C. Castillo-Chavez and A. A. Yakubu, "Dispersal Between Two Patches in a Discrete Time SEIS Model," MTBI technical Report, 2000. |
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