# American Institute of Mathematical Sciences

December  2011, 15(3): 867-892. doi: 10.3934/dcdsb.2011.15.867

## Asymptotic speed of propagation and traveling wavefronts for a SIR epidemic model

 1 School of Mathematics, South China Normal University, Guangzhou 510631, China

Received  January 2010 Revised  August 2010 Published  February 2011

In this article, the well-posedness of the initial value problem, the existence of traveling wavefronts and the asymptotic speed of propagation for a SIR epidemic model with stage structure and nonlocal response are studied. We further show that the minimum wave speed in fact coincides with the asymptotic speed of propagation.
Citation: Chufen Wu, Peixuan Weng. Asymptotic speed of propagation and traveling wavefronts for a SIR epidemic model. Discrete & Continuous Dynamical Systems - B, 2011, 15 (3) : 867-892. doi: 10.3934/dcdsb.2011.15.867
##### References:
 [1] D. G. Aronson, The asymptotic speed of propagation of a simple epidemic, in "Nonlinear Diffusion" (W. E. Fitzgibbon, H. F. Walker, eds.), Pitman, London, (1977), 1-23.  Google Scholar [2] D. G. Aronson and H. F. Weinberger, Nonlinear diffusion in population genetics, combustion, and nerve pulse propagation, in "Partial Differential Equations and Related Topics" (J. A. Goldstein, ed.), Leture Notes in Mathematics, 446, Springer, Berlin, (1975), 5-49.  Google Scholar [3] D. G. Aronson and H. F. Weinberger, Multidimensional nonlinear diffusion arising in population genetics, Adv. in Math., 30 (1978), 33-76. doi: 10.1016/0001-8708(78)90130-5.  Google Scholar [4] J. Al-Omari, S. A. Gourley, Monotone travelling fronts in an age-stuctured reaction-diffusion model for a single species, J. Math. Biol., 45 (2002), 294-312. doi: 10.1007/s002850200159.  Google Scholar [5] H. Berestycki, F. Hamel and L. Roques, Analisis of the periodically fragmented environment model: I-Species persistence, J. Math. Biol., 51 (2005), 75-113. doi: 10.1007/s00285-004-0313-3.  Google Scholar [6] H. Berestycki, F. Hamel and L. Roques, Analisis of the periodically fragmented environment model: II-Biological invasions and pulsating travelling fronts, J. Math. Pures Appl., 84 (2005), 1101-1146. doi: 10.1016/j.matpur.2004.10.006.  Google Scholar [7] V. Capasso and S. L. Paveri-Fontana, A mathematical model for the 1973 cholera epidemic in the European Mediterranean region, Revue d'Epidemiologic et de Sant$\acutee$ Publique, 27 (1979), 121-132. Google Scholar [8] O. Diekmann, Run for your life. A note on the asymptotic speed of propagation of an epidemic, J. Diff. Eqns., 33 (1979), 58-73. doi: 10.1016/0022-0396(79)90080-9.  Google Scholar [9] J. Fang, J. Wei and X.-Q. Zhao, Spatial dynamics of a nonlocal and time-delayed reaction-diffusion system, J. Diff. Eqns., 245 (2008), 2749-2770. doi: 10.1016/j.jde.2008.09.001.  Google Scholar [10] J. Fang and X.-Q. Zhao, Existence and uniqueness of traveling waves for non-monotone integral equations with applications, J. Diff. Eqns., 248 (2010), 2199-2226. doi: 10.1016/j.jde.2010.01.009.  Google Scholar [11] R. A. Fisher, The advance of advantageous genes, Ann. Eugenics., 7 (1937), 355-369. Google Scholar [12] S. A. Gourley and Y. Kuang, Wavefront and global stability in a time-delayed population model with stage structure, Proc. R. Soc. Lond. Ser. A., 459 (2003), 1563-1579. doi: 10.1098/rspa.2002.1094.  Google Scholar [13] S. A. Gourley and J. H. Wu, Delayed non-local diffusive sysrems in biological invasion and disease spread, Nonlinear dynamics and evolution equations, Fields Inst. Commun., Amer. Math. Soc., Providence, RI, 48 (2006), 137-200.  Google Scholar [14] A. N. Kolmogorov, I. G. Petrowsky and N. S. Piscounov, Etude de l'equation de la diffusion avec croissance de la quantité de matiere et son application a un probleme biologique, Univ. Bull. Moscow. Serie. Intern. Sec. A., 1 (1937), 1-26. Google Scholar [15] X. Liang and X.-Q. Zhao, Asymptotic speeds of spread and traveling waves for monotone semiflows with application, Commun. Pure Appl. Math., 60 (2007), 1-40. Erratum: 61 (2008), 137-138.  Google Scholar [16] X. Liang, Y. Yi and X.-Q. Zhao, Spreading speeds and traveling waves for periodic evolution systems, J. Diff. Eqns., 231 (2006), 57-77. doi: 10.1016/j.jde.2006.04.010.  Google Scholar [17] R. Lui, Biological growth and spread modeled by systems of recursions. I. Mathematical theory, Math. Biosci., 93 (1989), 269-295. doi: 10.1016/0025-5564(89)90026-6.  Google Scholar [18] R. Lui, Biological growth and spread modeled by systems of recursions. II. Biological theory, Math. Biosci., 93 (1989), 297-312. doi: 10.1016/0025-5564(89)90027-8.  Google Scholar [19] S. W. Ma, Traveling wavefronts for delayed reaction-diffusion systems via a fixed point theorem, J. Diff. Eqns., 171 (2001), 294-314. doi: 10.1006/jdeq.2000.3846.  Google Scholar [20] J. Radcliffe and L. Rass, The asymptotic speed of propagation of the deterministic non-reducible n-type epidemic, J. Math. Biol., 23 (1986), 341-359. doi: 10.1007/BF00275253.  Google Scholar [21] L. Rass and J. Radcliffe, "Spatial Deterministic Epidemics, Mathematical Surveys and Monographs," AMS, Rhode Island, USA, 2003.  Google Scholar [22] K. W. Schaaf, Asymptotic behavior and traveling wave solutions for parabolic functional differential equations, Trans. Amer. Math. Soc., 302 (1987), 587-615.  Google Scholar [23] H. R. Thieme, Asymptotic estimates of the solutions of nonlinear integral equatons and asymptotic speeds for the spread of populatins, J. Reine Angew. Math., 306 (1979), 94-121. doi: 10.1515/crll.1979.306.94.  Google Scholar [24] H. R. Thieme, Density-dependent regulation of spatially distributed populations and their asymptotic speed of spread, J. Math. Biol., 8 (1979), 173-187. doi: 10.1007/BF00279720.  Google Scholar [25] H. R. Thieme and X.-Q. Zhao, Asymptotic speeds of spread and traveling waves for integral equations and delayed reaction-diffusion model, J. Diff. Eqns., 195 (2003), 430-470. doi: 10.1016/S0022-0396(03)00175-X.  Google Scholar [26] X. H. Tian and R. Xu, Stability analysis of a delayed SIR epidemic with stage structure and nonlinear incidence, Discrete Dynamics in Nature and Society, (2009), Art. ID 979217, 17pp.  Google Scholar [27] H. F. Weinberger, Long-time behavior of a class of biological models, SIAM J. Math. Anal., 13 (1982), 353-396. doi: 10.1137/0513028.  Google Scholar [28] H. F. Weinberger, On spreading speeds and traveling waves for growth and migration models in a periodic habitat, J. Math. Biol., 45 (2002), 511-548. doi: 10.1007/s00285-002-0169-3.  Google Scholar [29] H. F. Weinberger, M. A. Lewis and B. Li, Analysis of linear determinacy for spread in cooperative models, J. Math. Biol., 45 (2002), 183-218. doi: 10.1007/s002850200145.  Google Scholar [30] P. X. Weng, H. X. Huang and J. H. Wu, Asymptotic speed of propagation of wave fronts in a lattice delay differential equation with global interaction, IMA J. Appl. Maths., 68 (2003), 409-439. doi: 10.1093/imamat/68.4.409.  Google Scholar [31] P. X. Weng and X.-Q. Zhao, Spreading speed and traveling waves for a multi-type SIS epidemic model, J. Diff. Eqns., 229 (2006), 270-296. doi: 10.1016/j.jde.2006.01.020.  Google Scholar [32] P. X. Weng and Z. T. Xu, Survey on progress for asymptotic speed of propagation and traveling wave solutions of some types of evolution equations(Chinese), Advance in Mathematics, 39 (2009), 1-22.  Google Scholar [33] C. F. Wu and P. X. Weng, Traveling wavefronts for a SIS epidemic model with stage structure, Dynamic Systems and Applications, 19 (2010), 125-146.  Google Scholar [34] X.-Q. Zhao and W. D. Wang, Fisher waves in an epidemic model, Discrete Continuous Dynam. Systems - B, 4 (2004), 1117-1128.  Google Scholar [35] X.-Q. Zhao, Spatial dynamics of some evolution system in biology, "Recent Progress on Reaction-Diffusion Systems and Viscosity Solutions," World Scientific Publishing Co. Pte. Ltd. Singapore, (2009), 332-363.  Google Scholar [36] X.-Q. Zhao and D. M. Xiao, The asymptotic speed of spread and traveling waves for a vector disease model, J. Dynam. Diff. Eqns., 18 (2006), 1001-1019; Erratum: 20 (2008), 277-279.  Google Scholar

show all references

##### References:
 [1] D. G. Aronson, The asymptotic speed of propagation of a simple epidemic, in "Nonlinear Diffusion" (W. E. Fitzgibbon, H. F. Walker, eds.), Pitman, London, (1977), 1-23.  Google Scholar [2] D. G. Aronson and H. F. Weinberger, Nonlinear diffusion in population genetics, combustion, and nerve pulse propagation, in "Partial Differential Equations and Related Topics" (J. A. Goldstein, ed.), Leture Notes in Mathematics, 446, Springer, Berlin, (1975), 5-49.  Google Scholar [3] D. G. Aronson and H. F. Weinberger, Multidimensional nonlinear diffusion arising in population genetics, Adv. in Math., 30 (1978), 33-76. doi: 10.1016/0001-8708(78)90130-5.  Google Scholar [4] J. Al-Omari, S. A. Gourley, Monotone travelling fronts in an age-stuctured reaction-diffusion model for a single species, J. Math. Biol., 45 (2002), 294-312. doi: 10.1007/s002850200159.  Google Scholar [5] H. Berestycki, F. Hamel and L. Roques, Analisis of the periodically fragmented environment model: I-Species persistence, J. Math. Biol., 51 (2005), 75-113. doi: 10.1007/s00285-004-0313-3.  Google Scholar [6] H. Berestycki, F. Hamel and L. Roques, Analisis of the periodically fragmented environment model: II-Biological invasions and pulsating travelling fronts, J. Math. Pures Appl., 84 (2005), 1101-1146. doi: 10.1016/j.matpur.2004.10.006.  Google Scholar [7] V. Capasso and S. L. Paveri-Fontana, A mathematical model for the 1973 cholera epidemic in the European Mediterranean region, Revue d'Epidemiologic et de Sant$\acutee$ Publique, 27 (1979), 121-132. Google Scholar [8] O. Diekmann, Run for your life. A note on the asymptotic speed of propagation of an epidemic, J. Diff. Eqns., 33 (1979), 58-73. doi: 10.1016/0022-0396(79)90080-9.  Google Scholar [9] J. Fang, J. Wei and X.-Q. Zhao, Spatial dynamics of a nonlocal and time-delayed reaction-diffusion system, J. Diff. Eqns., 245 (2008), 2749-2770. doi: 10.1016/j.jde.2008.09.001.  Google Scholar [10] J. Fang and X.-Q. Zhao, Existence and uniqueness of traveling waves for non-monotone integral equations with applications, J. Diff. Eqns., 248 (2010), 2199-2226. doi: 10.1016/j.jde.2010.01.009.  Google Scholar [11] R. A. Fisher, The advance of advantageous genes, Ann. Eugenics., 7 (1937), 355-369. Google Scholar [12] S. A. Gourley and Y. Kuang, Wavefront and global stability in a time-delayed population model with stage structure, Proc. R. Soc. Lond. Ser. A., 459 (2003), 1563-1579. doi: 10.1098/rspa.2002.1094.  Google Scholar [13] S. A. Gourley and J. H. Wu, Delayed non-local diffusive sysrems in biological invasion and disease spread, Nonlinear dynamics and evolution equations, Fields Inst. Commun., Amer. Math. Soc., Providence, RI, 48 (2006), 137-200.  Google Scholar [14] A. N. Kolmogorov, I. G. Petrowsky and N. S. Piscounov, Etude de l'equation de la diffusion avec croissance de la quantité de matiere et son application a un probleme biologique, Univ. Bull. Moscow. Serie. Intern. Sec. A., 1 (1937), 1-26. Google Scholar [15] X. Liang and X.-Q. Zhao, Asymptotic speeds of spread and traveling waves for monotone semiflows with application, Commun. Pure Appl. Math., 60 (2007), 1-40. Erratum: 61 (2008), 137-138.  Google Scholar [16] X. Liang, Y. Yi and X.-Q. Zhao, Spreading speeds and traveling waves for periodic evolution systems, J. Diff. Eqns., 231 (2006), 57-77. doi: 10.1016/j.jde.2006.04.010.  Google Scholar [17] R. Lui, Biological growth and spread modeled by systems of recursions. I. Mathematical theory, Math. Biosci., 93 (1989), 269-295. doi: 10.1016/0025-5564(89)90026-6.  Google Scholar [18] R. Lui, Biological growth and spread modeled by systems of recursions. II. Biological theory, Math. Biosci., 93 (1989), 297-312. doi: 10.1016/0025-5564(89)90027-8.  Google Scholar [19] S. W. Ma, Traveling wavefronts for delayed reaction-diffusion systems via a fixed point theorem, J. Diff. Eqns., 171 (2001), 294-314. doi: 10.1006/jdeq.2000.3846.  Google Scholar [20] J. Radcliffe and L. Rass, The asymptotic speed of propagation of the deterministic non-reducible n-type epidemic, J. Math. Biol., 23 (1986), 341-359. doi: 10.1007/BF00275253.  Google Scholar [21] L. Rass and J. Radcliffe, "Spatial Deterministic Epidemics, Mathematical Surveys and Monographs," AMS, Rhode Island, USA, 2003.  Google Scholar [22] K. W. Schaaf, Asymptotic behavior and traveling wave solutions for parabolic functional differential equations, Trans. Amer. Math. Soc., 302 (1987), 587-615.  Google Scholar [23] H. R. Thieme, Asymptotic estimates of the solutions of nonlinear integral equatons and asymptotic speeds for the spread of populatins, J. Reine Angew. Math., 306 (1979), 94-121. doi: 10.1515/crll.1979.306.94.  Google Scholar [24] H. R. Thieme, Density-dependent regulation of spatially distributed populations and their asymptotic speed of spread, J. Math. Biol., 8 (1979), 173-187. doi: 10.1007/BF00279720.  Google Scholar [25] H. R. Thieme and X.-Q. Zhao, Asymptotic speeds of spread and traveling waves for integral equations and delayed reaction-diffusion model, J. Diff. Eqns., 195 (2003), 430-470. doi: 10.1016/S0022-0396(03)00175-X.  Google Scholar [26] X. H. Tian and R. Xu, Stability analysis of a delayed SIR epidemic with stage structure and nonlinear incidence, Discrete Dynamics in Nature and Society, (2009), Art. ID 979217, 17pp.  Google Scholar [27] H. F. Weinberger, Long-time behavior of a class of biological models, SIAM J. Math. Anal., 13 (1982), 353-396. doi: 10.1137/0513028.  Google Scholar [28] H. F. Weinberger, On spreading speeds and traveling waves for growth and migration models in a periodic habitat, J. Math. Biol., 45 (2002), 511-548. doi: 10.1007/s00285-002-0169-3.  Google Scholar [29] H. F. Weinberger, M. A. Lewis and B. Li, Analysis of linear determinacy for spread in cooperative models, J. Math. Biol., 45 (2002), 183-218. doi: 10.1007/s002850200145.  Google Scholar [30] P. X. Weng, H. X. Huang and J. H. Wu, Asymptotic speed of propagation of wave fronts in a lattice delay differential equation with global interaction, IMA J. Appl. Maths., 68 (2003), 409-439. doi: 10.1093/imamat/68.4.409.  Google Scholar [31] P. X. Weng and X.-Q. Zhao, Spreading speed and traveling waves for a multi-type SIS epidemic model, J. Diff. Eqns., 229 (2006), 270-296. doi: 10.1016/j.jde.2006.01.020.  Google Scholar [32] P. X. Weng and Z. T. Xu, Survey on progress for asymptotic speed of propagation and traveling wave solutions of some types of evolution equations(Chinese), Advance in Mathematics, 39 (2009), 1-22.  Google Scholar [33] C. F. Wu and P. X. Weng, Traveling wavefronts for a SIS epidemic model with stage structure, Dynamic Systems and Applications, 19 (2010), 125-146.  Google Scholar [34] X.-Q. Zhao and W. D. Wang, Fisher waves in an epidemic model, Discrete Continuous Dynam. Systems - B, 4 (2004), 1117-1128.  Google Scholar [35] X.-Q. Zhao, Spatial dynamics of some evolution system in biology, "Recent Progress on Reaction-Diffusion Systems and Viscosity Solutions," World Scientific Publishing Co. Pte. Ltd. Singapore, (2009), 332-363.  Google Scholar [36] X.-Q. Zhao and D. M. Xiao, The asymptotic speed of spread and traveling waves for a vector disease model, J. Dynam. Diff. Eqns., 18 (2006), 1001-1019; Erratum: 20 (2008), 277-279.  Google Scholar
 [1] Dashun Xu, Xiao-Qiang Zhao. Asymptotic speed of spread and traveling waves for a nonlocal epidemic model. Discrete & Continuous Dynamical Systems - B, 2005, 5 (4) : 1043-1056. doi: 10.3934/dcdsb.2005.5.1043 [2] Shuang-Ming Wang, Zhaosheng Feng, Zhi-Cheng Wang, Liang Zhang. Spreading speed and periodic traveling waves of a time periodic and diffusive SI epidemic model with demographic structure. Communications on Pure & Applied Analysis, , () : -. doi: 10.3934/cpaa.2021145 [3] Yan Li, Wan-Tong Li, Guo Lin. Traveling waves of a delayed diffusive SIR epidemic model. Communications on Pure & Applied Analysis, 2015, 14 (3) : 1001-1022. doi: 10.3934/cpaa.2015.14.1001 [4] Cui-Ping Cheng, Wan-Tong Li, Zhi-Cheng Wang. Asymptotic stability of traveling wavefronts in a delayed population model with stage structure on a two-dimensional spatial lattice. Discrete & Continuous Dynamical Systems - B, 2010, 13 (3) : 559-575. doi: 10.3934/dcdsb.2010.13.559 [5] Lara Abi Rizk, Jean-Baptiste Burie, Arnaud Ducrot. Asymptotic speed of spread for a nonlocal evolutionary-epidemic system. Discrete & Continuous Dynamical Systems, 2021, 41 (10) : 4959-4985. doi: 10.3934/dcds.2021064 [6] Toshikazu Kuniya, Jinliang Wang, Hisashi Inaba. A multi-group SIR epidemic model with age structure. Discrete & Continuous Dynamical Systems - B, 2016, 21 (10) : 3515-3550. doi: 10.3934/dcdsb.2016109 [7] Liang Zhang, Zhi-Cheng Wang. Threshold dynamics of a reaction-diffusion epidemic model with stage structure. Discrete & Continuous Dynamical Systems - B, 2017, 22 (10) : 3797-3820. doi: 10.3934/dcdsb.2017191 [8] Wan-Tong Li, Guo Lin, Cong Ma, Fei-Ying Yang. Traveling wave solutions of a nonlocal delayed SIR model without outbreak threshold. Discrete & Continuous Dynamical Systems - B, 2014, 19 (2) : 467-484. doi: 10.3934/dcdsb.2014.19.467 [9] Yang Yang, Yun-Rui Yang, Xin-Jun Jiao. Traveling waves for a nonlocal dispersal SIR model equipped delay and generalized incidence. Electronic Research Archive, 2020, 28 (1) : 1-13. doi: 10.3934/era.2020001 [10] Wanbiao Ma, Yasuhiro Takeuchi. Asymptotic properties of a delayed SIR epidemic model with density dependent birth rate. Discrete & Continuous Dynamical Systems - B, 2004, 4 (3) : 671-678. doi: 10.3934/dcdsb.2004.4.671 [11] Chin-Chin Wu. Existence of traveling wavefront for discrete bistable competition model. Discrete & Continuous Dynamical Systems - B, 2011, 16 (3) : 973-984. doi: 10.3934/dcdsb.2011.16.973 [12] Fei-Ying Yang, Yan Li, Wan-Tong Li, Zhi-Cheng Wang. Traveling waves in a nonlocal dispersal Kermack-McKendrick epidemic model. Discrete & Continuous Dynamical Systems - B, 2013, 18 (7) : 1969-1993. doi: 10.3934/dcdsb.2013.18.1969 [13] Jingdong Wei, Jiangbo Zhou, Wenxia Chen, Zaili Zhen, Lixin Tian. Traveling waves in a nonlocal dispersal epidemic model with spatio-temporal delay. Communications on Pure & Applied Analysis, 2020, 19 (5) : 2853-2886. doi: 10.3934/cpaa.2020125 [14] Ran Zhang, Shengqiang Liu. On the asymptotic behaviour of traveling wave solution for a discrete diffusive epidemic model. Discrete & Continuous Dynamical Systems - B, 2021, 26 (2) : 1197-1204. doi: 10.3934/dcdsb.2020159 [15] Sun-Ho Choi, Hyowon Seo, Minha Yoo. A multi-stage SIR model for rumor spreading. Discrete & Continuous Dynamical Systems - B, 2020, 25 (6) : 2351-2372. doi: 10.3934/dcdsb.2020124 [16] Peixuan Weng. Spreading speed and traveling wavefront of an age-structured population diffusing in a 2D lattice strip. Discrete & Continuous Dynamical Systems - B, 2009, 12 (4) : 883-904. doi: 10.3934/dcdsb.2009.12.883 [17] Jinling Zhou, Yu Yang. Traveling waves for a nonlocal dispersal SIR model with general nonlinear incidence rate and spatio-temporal delay. Discrete & Continuous Dynamical Systems - B, 2017, 22 (4) : 1719-1741. doi: 10.3934/dcdsb.2017082 [18] Prabir Panja, Soovoojeet Jana, Shyamal kumar Mondal. Dynamics of a stage structure prey-predator model with ratio-dependent functional response and anti-predator behavior of adult prey. Numerical Algebra, Control & Optimization, 2021, 11 (3) : 391-405. doi: 10.3934/naco.2020033 [19] Qianqian Cui, Zhipeng Qiu, Ling Ding. An SIR epidemic model with vaccination in a patchy environment. Mathematical Biosciences & Engineering, 2017, 14 (5&6) : 1141-1157. doi: 10.3934/mbe.2017059 [20] Zhen Jin, Zhien Ma. The stability of an SIR epidemic model with time delays. Mathematical Biosciences & Engineering, 2006, 3 (1) : 101-109. doi: 10.3934/mbe.2006.3.101

2020 Impact Factor: 1.327

## Metrics

• HTML views (0)
• Cited by (6)

• on AIMS