# 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 & Continuous Dynamical Systems - B, 2013, 18 (1) : 37-56. doi: 10.3934/dcdsb.2013.18.37
##### References:
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Bacaër, Periodic matrix populaiton models: growth rate, basic reproduction number, and entropy,, Bull. Math. Biol., 71 (2009), 1781. doi: 10.1007/s11538-009-9426-6. Google Scholar [6] M. Keeling and P. Rohani, "Modeling Infectious Diseases in Humans and Animals,", Princeton University Press, (2008). Google Scholar [7] Y. Zhou and H. Cao, Discrete tuberculosis transmission models and their application,, in, 57 (2010), 83. Google Scholar [8] I. Schwartz and H. Smith, Infinite subharmonic bifurcation in an SIER epidemic model,, J. Math. Biol., 18 (1983), 233. doi: 10.1007/BF00276090. Google Scholar [9] I. Schwartz, Small amplitude, long periodic outbreaks in seasonally driven epidemics,, J. Math. Biol., 30 (1992), 473. doi: 10.1007/BF00160532. Google Scholar [10] H. Smith, Multiple stable subharmonics for a periodic epidemic model,, J. Math. Biol., 17 (1983), 179. doi: 10.1007/BF00305758. Google Scholar [11] X. Zhao, "Dynamical Sytems in Population Biology,", Springer-Verlag, (2003). Google Scholar [12] J. M. Cushing, A juvenile-adult model with periodic vital rates,, J. Math. Biol., 53 (2006), 520. doi: 10.1007/s00285-006-0382-6. Google Scholar [13] J. Ma and Z. Ma, Epidemic threshold conditions for seasonally forced SEIR models,, Math. Biosci. Eng., 3 (2006), 161. Google Scholar [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. doi: 10.1007/s11538-006-9166-9. Google Scholar [15] N. Bacaër and M. G. M. Gomes, On the final size of epidemics with seasonality,, Bull. Math. Biol., 71 (2009), 1954. doi: 10.1007/s11538-009-9433-7. Google Scholar [16] N. Bacaër and S. Guernaoui, The epidemic threshold of vector-borne diseases with seasonality,, J. Math. Biol., 53 (2006), 421. doi: 10.1007/s00285-006-0015-0. Google Scholar [17] F. Zhang and X. Zhao, A periodic epidemic model in a patchy enviroment,, J. Math. Anal. Appl., 325 (2007), 496. doi: 10.1016/j.jmaa.2006.01.085. Google Scholar [18] B. G. Williams and C. Dye, Infectious disease persistence when transmission varies seasonally,, Math. Biosci., 145 (1997), 77. doi: 10.1016/S0025-5564(97)00039-4. Google Scholar [19] H. R. Thieme, Renewal theorems for linear periodic Volterra integral equations,, J. Integral Equations, 7 (1984), 253. Google Scholar [20] L. 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 [21] L. Allen and A. 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 [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. doi: 10.1016/S0096-3003(01)00192-8. Google Scholar [23] C. Castillo-Chavez and A. A. Yakubu, Discrete-time SIS models with complex dynamics,, Nonliear Anal., 47 (2001), 4753. doi: 10.1016/S0362-546X(01)00587-9. Google Scholar [24] C. Castillo-Chavez and A. A. Yakubu, Dispersal, disease and life-history evolution,, Math. Biosci., 173 (2001), 35. doi: 10.1016/S0025-5564(01)00065-7. Google Scholar [25] C. Castillo-Chavez and A. A. Yakubu, Discrete-time SIS models with simple and complex population dynamics,, in, (2002), 153. Google Scholar [26] Y. Zhou and P. Fergola, Dynamic of a discrete age-structured SIS models,, Discrete Contin. Dyn. Syst. Ser. B., 4 (2004), 843. Google Scholar [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. Google Scholar [28] X. Li and W. Wang, A discrete epidemic model with stage structure,, Chaos, 26 (2005), 947. Google Scholar [29] J. E. Franke and A. A. Yakubu, Discrete-time SIS epidemic model in a seasonal environment,, SIAM J. Appl. Math., 66 (2006), 1563. doi: 10.1137/050638345. Google Scholar [30] Ira M. Longini, Jr., The generalized discrete-time epidemic model with immunity: Asynthesis,, Math. Biosci., 82 (1986), 19. doi: 10.1016/0025-5564(86)90003-9. Google Scholar [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. doi: 10.1016/j.mbs.2007.07.005. Google Scholar [32] M. I. Gil, "Difference Equations in Normed Spaces Stability and Oscillations,", Elsevier Science, (2007). Google Scholar [33] R. A. Horn and C. A. Johnson, "Matrix Analysis,", Cambridge University press, (1985). Google Scholar [34] H. Smith and P. Waltman, "Theory of the Chemostat,", Cambridge University Press, (1995). Google Scholar [35] P. Hess, "Periodic-Parabolic Boundary Value Problems and Positivity,", Pitman Research Notes in Mathematics, (1991). Google Scholar [36] H. R. Thieme, Convergence results and a Poincare-Bendixson trichotomy for asymptotically autonomous differential equations,, J. Math. Biol., 30 (1992), 755. Google Scholar [37] X. Q. Zhao, Asymptotic behavior for asymptotically periodic semiflows with applications,, Commun. Appl. Nonlinear Anal., 3 (1996), 43. Google Scholar [38] P. Salceanu and H. Smith, Persistence in a discrete-time, stage-structured epidemic model,, J. Difference Equa. Appl., 16 (2010), 73. doi: 10.1080/10236190802400733. Google Scholar [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). Google Scholar

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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. doi: 10.1007/BF00178324. Google Scholar [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. doi: 10.1016/S0025-5564(02)00108-6. Google Scholar [3] W. Wang and X. Zhao, Threshold dynamics for compartmental epidemic models in periodic environments,, J. Dyn. Diff. Equat., 20 (2008), 699. doi: 10.1007/s10884-008-9111-8. Google Scholar [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. doi: 10.1080/10236190802332308. Google Scholar [5] N. Bacaër, Periodic matrix populaiton models: growth rate, basic reproduction number, and entropy,, Bull. Math. Biol., 71 (2009), 1781. doi: 10.1007/s11538-009-9426-6. Google Scholar [6] M. Keeling and P. Rohani, "Modeling Infectious Diseases in Humans and Animals,", Princeton University Press, (2008). Google Scholar [7] Y. Zhou and H. Cao, Discrete tuberculosis transmission models and their application,, in, 57 (2010), 83. Google Scholar [8] I. Schwartz and H. Smith, Infinite subharmonic bifurcation in an SIER epidemic model,, J. Math. Biol., 18 (1983), 233. doi: 10.1007/BF00276090. Google Scholar [9] I. Schwartz, Small amplitude, long periodic outbreaks in seasonally driven epidemics,, J. Math. Biol., 30 (1992), 473. doi: 10.1007/BF00160532. Google Scholar [10] H. Smith, Multiple stable subharmonics for a periodic epidemic model,, J. Math. Biol., 17 (1983), 179. doi: 10.1007/BF00305758. Google Scholar [11] X. Zhao, "Dynamical Sytems in Population Biology,", Springer-Verlag, (2003). Google Scholar [12] J. M. Cushing, A juvenile-adult model with periodic vital rates,, J. Math. Biol., 53 (2006), 520. doi: 10.1007/s00285-006-0382-6. Google Scholar [13] J. Ma and Z. Ma, Epidemic threshold conditions for seasonally forced SEIR models,, Math. Biosci. Eng., 3 (2006), 161. Google Scholar [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. doi: 10.1007/s11538-006-9166-9. Google Scholar [15] N. Bacaër and M. G. M. Gomes, On the final size of epidemics with seasonality,, Bull. Math. Biol., 71 (2009), 1954. doi: 10.1007/s11538-009-9433-7. Google Scholar [16] N. Bacaër and S. Guernaoui, The epidemic threshold of vector-borne diseases with seasonality,, J. Math. Biol., 53 (2006), 421. doi: 10.1007/s00285-006-0015-0. Google Scholar [17] F. Zhang and X. Zhao, A periodic epidemic model in a patchy enviroment,, J. Math. Anal. Appl., 325 (2007), 496. doi: 10.1016/j.jmaa.2006.01.085. Google Scholar [18] B. G. Williams and C. Dye, Infectious disease persistence when transmission varies seasonally,, Math. Biosci., 145 (1997), 77. doi: 10.1016/S0025-5564(97)00039-4. Google Scholar [19] H. R. Thieme, Renewal theorems for linear periodic Volterra integral equations,, J. Integral Equations, 7 (1984), 253. Google Scholar [20] L. 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 [21] L. Allen and A. 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 [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. doi: 10.1016/S0096-3003(01)00192-8. Google Scholar [23] C. Castillo-Chavez and A. A. Yakubu, Discrete-time SIS models with complex dynamics,, Nonliear Anal., 47 (2001), 4753. doi: 10.1016/S0362-546X(01)00587-9. Google Scholar [24] C. Castillo-Chavez and A. A. Yakubu, Dispersal, disease and life-history evolution,, Math. Biosci., 173 (2001), 35. doi: 10.1016/S0025-5564(01)00065-7. Google Scholar [25] C. Castillo-Chavez and A. A. Yakubu, Discrete-time SIS models with simple and complex population dynamics,, in, (2002), 153. Google Scholar [26] Y. Zhou and P. Fergola, Dynamic of a discrete age-structured SIS models,, Discrete Contin. Dyn. Syst. Ser. B., 4 (2004), 843. Google Scholar [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. Google Scholar [28] X. Li and W. Wang, A discrete epidemic model with stage structure,, Chaos, 26 (2005), 947. Google Scholar [29] J. E. Franke and A. A. Yakubu, Discrete-time SIS epidemic model in a seasonal environment,, SIAM J. Appl. Math., 66 (2006), 1563. doi: 10.1137/050638345. Google Scholar [30] Ira M. Longini, Jr., The generalized discrete-time epidemic model with immunity: Asynthesis,, Math. Biosci., 82 (1986), 19. doi: 10.1016/0025-5564(86)90003-9. Google Scholar [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. doi: 10.1016/j.mbs.2007.07.005. Google Scholar [32] M. I. Gil, "Difference Equations in Normed Spaces Stability and Oscillations,", Elsevier Science, (2007). Google Scholar [33] R. A. Horn and C. A. Johnson, "Matrix Analysis,", Cambridge University press, (1985). Google Scholar [34] H. Smith and P. Waltman, "Theory of the Chemostat,", Cambridge University Press, (1995). Google Scholar [35] P. Hess, "Periodic-Parabolic Boundary Value Problems and Positivity,", Pitman Research Notes in Mathematics, (1991). Google Scholar [36] H. R. Thieme, Convergence results and a Poincare-Bendixson trichotomy for asymptotically autonomous differential equations,, J. Math. Biol., 30 (1992), 755. Google Scholar [37] X. Q. Zhao, Asymptotic behavior for asymptotically periodic semiflows with applications,, Commun. Appl. Nonlinear Anal., 3 (1996), 43. Google Scholar [38] P. Salceanu and H. Smith, Persistence in a discrete-time, stage-structured epidemic model,, J. Difference Equa. Appl., 16 (2010), 73. doi: 10.1080/10236190802400733. Google Scholar [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). Google Scholar
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