# American Institute of Mathematical Sciences

January  2013, 18(1): 37-56. doi: 10.3934/dcdsb.2013.18.37

## 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

Received  April 2011 Revised  May 2012 Published  September 2012

Seasonal fluctuations have been observed in many infectious diseases. Discrete epidemic models with periodic epidemiological parameters are formulated and studied to take into account seasonal variations of infectious diseases. The definition and the formula of the basic reproduction number $R_0$ are given by following the framework in [1,2,3,4,5]. Threshold results for a general model are obtained which show that the magnitude of $R_0$ determines whether the disease will go extinct (when $R_0<1$) or not (when $R_0>1$) in the population. Applications of these general results to discrete periodic SIR and SEIS models are demonstrated. The disease persistence and the existence of the positive periodic solution are established. Numerical explorations of the model properties are also presented via several examples including the calculations of the basic reproduction number, conditions for the disease extinction or persistence, and the existence of periodic solutions as well as its stability.
Citation: Hui Cao, Yicang Zhou. The basic reproduction number of discrete SIR and SEIS models with periodic parameters. Discrete and Continuous Dynamical Systems - B, 2013, 18 (1) : 37-56. doi: 10.3934/dcdsb.2013.18.37
##### 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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##### 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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