Traveling wave solutions in a two-group SIR epidemic model with constant recruitment



Host heterogeneity can be modeled by using multi-group structures in the population. In this paper we investigate the existence and nonexistence of traveling waves of a two-group SIR epidemic model with time delay and constant recruitment and show that the existence of traveling waves is determined by the basic reproduction number \(R_{0}.\) More specifically, we prove that (i) when the basic reproduction number \(R_{0}>1,\) there exists a minimal wave speed \(c^*>0,\) such that for each \(c \ge c^*\) the system admits a nontrivial traveling wave solution with wave speed c and for \(c<c^*\) there exists no nontrivial traveling wave satisfying the system; (ii) when \(R_{0} \le 1,\) the system admits no nontrivial traveling waves. Finally, we present some numerical simulations to show the existence of traveling waves of the system.


Two-group epidemic model Basic reproduction number Time delay Constant recruitment Traveling wave solutions 

Mathematics Subject Classification

35C07 35B40 35K57 92D30 



The authors are grateful to the two anonymous reviewers and the handling editor (Professor Klaus Dietz) for their helpful comments and suggestions which helped us in improving the paper.


  1. Ai S (2010) Traveling waves for a model of a fungal disease over a vineyard. SIAM J Math Anal 42:833–856MathSciNetCrossRefMATHGoogle Scholar
  2. Andersson H, Britton T (1998) Heterogeneity in epidemic models and its effect on the spread of infection. J Appl Probab 35:651–661MathSciNetCrossRefMATHGoogle Scholar
  3. Bai Z, Wu S-L (2015) Traveling waves in a delayed SIR epidemic model with nonlinear incidence. Appl Math Comput 263:221–232MathSciNetGoogle Scholar
  4. Bonzi B, Fall AA, Iggidr A, Sallet G (2011) Stability of differential susceptibility and infectivity epidemic models. J Math Biol 62:39–64MathSciNetCrossRefMATHGoogle Scholar
  5. Burie JB, Calonnec A, Ducrot A (2006) Singular perturbation analysis of travelling waves for a model in phytopathology. Math Model Nat Phenom 1:49–63MathSciNetCrossRefMATHGoogle Scholar
  6. Cai L, Xiang J, Li X (2012) A two-strain epidemic model with mutant strain and vaccination. J Appl Math Comput 40:125–142MathSciNetCrossRefMATHGoogle Scholar
  7. Clancy D, Pearce CJ (2013) The effect of population heterogeneities upon spread of infection. J Math Biol 67:963–987MathSciNetCrossRefMATHGoogle Scholar
  8. Demasse RD, Ducrot A (2013) An age\(-\)structured within-host model for multistrain malaria infections. SIAM J Appl Math 73:572–593MathSciNetCrossRefMATHGoogle Scholar
  9. Ducrot A, Magal P (2009) Travelling wave solutions for an infection-age structured model with diffusion. Proc R Soc Edinb 139A:459–482MathSciNetCrossRefMATHGoogle Scholar
  10. Ducrot A, Magal P (2011) Travelling wave solutions for an infection-age structured model with external supplies. Nonlinearity 24:2891–2911MathSciNetCrossRefMATHGoogle Scholar
  11. Ducrot A, Magal P, Ruan S (2010) Travelling wave solutions in multigroup age-structured epidemic models. Arch Ration Mech Anal 195:311–331MathSciNetCrossRefMATHGoogle Scholar
  12. Dwyer G, Elkinton JS, Buonaccorsi J (1997) Host heterogeneity in susceptibility and disease dynamics: tests of a mathematical model. Am Nat 150:685–770CrossRefGoogle Scholar
  13. Fitzgibbon WE, Langlais M, Parrott ME, Webb GF (1995a) A diffusive system with age dependency modeling FIV. Nonlinear Anal 25:975–989MathSciNetCrossRefMATHGoogle Scholar
  14. Fitzgibbon WE, Parrott ME, Webb GF (1995b) Diffusion epidemic models with incubation and crisscross dynamics. Math Biosci 128:131–155MathSciNetCrossRefMATHGoogle Scholar
  15. Földes J, Polác̆ik P (2009) On cooperative parabolic systems: Harnack inequalities and asymptotic symmetry. Discrete Contin Dyn Syst 25:133–157MathSciNetCrossRefMATHGoogle Scholar
  16. Friedman A (1964) Partial differential equations of parabolic type. Prentice-Hall, Englewood Cliffs, NJMATHGoogle Scholar
  17. Fu S-C (2016) Traveling waves for a diffusive SIR model with delay. J Math Anal Appl 435:20–37MathSciNetCrossRefMATHGoogle Scholar
  18. Gilbarg G, Trudinger N (2001) Elliptic partial differential equations of second order. Springer, BerlinMATHGoogle Scholar
  19. Guo H, Li MY, Shuai Z (2006) Global stability of the endemic equilibrium of multigroup SIR epidemic models. Can Appl Math Q 14:259–284MathSciNetMATHGoogle Scholar
  20. Guo H, Li MY, Shuai Z (2012) Global dynamics of a general class of multistage models for infectious diseases. SIAM J Appl Math 72:261–279MathSciNetCrossRefMATHGoogle Scholar
  21. Hadeler KP (1988) Hyperbolic travelling fronts. Proc Edinb Math Soc 31A:89–97MathSciNetCrossRefMATHGoogle Scholar
  22. Hadeler KP (1994) Travelling fronts for correlated random walks. Can Appl Math Q 2:27–43MathSciNetMATHGoogle Scholar
  23. Hadeler KP (2016) Stefan problem, traveling fronts, and epidemic spread. Discrete Contin Dyn Syst Ser B 2(1):417–436MathSciNetMATHGoogle Scholar
  24. Hadeler KP, Castillo-Chavez C (1995) A core group model for disease transmission. Math Biosci 128:41–55CrossRefMATHGoogle Scholar
  25. Hadeler KP, Ruan S (2007) Interaction of diffusion and delay. Discrete Contin Dyn Syst Ser B 8:95–105MathSciNetCrossRefMATHGoogle Scholar
  26. Hyman JM, Li J (2005) Differential susceptibility epidemic models. J Math Biol 50:626–644MathSciNetCrossRefMATHGoogle Scholar
  27. Hyman JM, Li J (2006) Differential susceptibility and infectivity epidemic models. Math Biosci Eng 3:89–100MathSciNetCrossRefMATHGoogle Scholar
  28. Katriel G (2012) The size of epidemics in populations with heterogeneous susceptibility. J Math Biol 65:237–262MathSciNetCrossRefMATHGoogle Scholar
  29. Li W-T, Yang F-Y (2014) Traveling waves for a nonlocal dispersal SIR model with standard incidence. J Integral Equ Appl 26:243–273MathSciNetCrossRefMATHGoogle Scholar
  30. Li J, Zou X (2009) Modeling spatial spread of infections diseases with a fixed latent period in a spatially continous domain. Bull Math Biol 71:2048–2079MathSciNetCrossRefMATHGoogle Scholar
  31. Li Y, Li W-T, Yang F-Y (2014) Traveling waves for a nonlocal disperal SIR model with delay and external supplies. Appl Math Comput 247:723–740MathSciNetMATHGoogle Scholar
  32. Li Y, Li W-T, Lin G (2015a) Traveling waves of a delayed diffusive SIR epidemic model. Commun Pure Appl Anal 14:1001–1022MathSciNetCrossRefMATHGoogle Scholar
  33. Li Y, Li W-T, Lin G (2015b) Damped oscillating traveling waves of a diffusive SIR epidemic model. Appl Math Lett 46:89–93MathSciNetCrossRefMATHGoogle Scholar
  34. Murray JD (1989) Mathematical biology. Springer, BerlinCrossRefMATHGoogle Scholar
  35. Novozhilov AS (2008) On the spread of epidemics in a closed heterogeneous population. Math Biosci 215:177–185MathSciNetCrossRefMATHGoogle Scholar
  36. Protter MH, Weinberger HF (1983) Maximum principles in differential equations. Prentice-Hall, Englewood CliffsMATHGoogle Scholar
  37. Rass L, Radcliffe J (2003) Spatial deterministic epidemics, Mathematical surveys and monographs 102. American Mathematical Society, Providence, RICrossRefMATHGoogle Scholar
  38. Rodrigues P, Margheri A, Rebelo C, Gomes MGM (2009) Heterogeneity in susceptibility to infection can explain high reinfection rates. J Theor Biol 259:280–290MathSciNetCrossRefGoogle Scholar
  39. Ruan S (2007) Spatial-temporal dynamics in nonlocal epidemiological models. In: Takeuchi Y, Sato K, Iwasa Y (eds) Mathematics for life science and medicine. Springer, Berlin, pp 99–122Google Scholar
  40. Ruan S, Wu J (2009) Modeling spatial spread of communicable diseasesinvolving animal hosts. In: Cantrell SR, Cosner C, Ruan S (eds) Spatial ecology. Chapman & Hall/CRC, Boca Raton, FL, pp 293–316Google Scholar
  41. Shuai Z, van den Driessche P (2012) Impact of heterogeneity on the dynamics of an SEIR epidemic model. Math Biosci Eng 9:393–411MathSciNetCrossRefMATHGoogle Scholar
  42. Veliov VM (2005) On the effect of population heterogeneity on dynamics of epidemic diseases. J Math Biol 51:123–143MathSciNetCrossRefMATHGoogle Scholar
  43. Wang Z-C, Wu J (2010) Traveling waves of a diffusive Kermack–McKendrick epidemic model with nonlocal delayed transmission. Proc R Soc A 466:237–261CrossRefMATHGoogle Scholar
  44. Wang Z-C, Wu J, Liu R (2012) Traveling waves of the spread of avian influenza. Proc Am Math Soc 140:3931–3946MathSciNetCrossRefMATHGoogle Scholar
  45. Weng P, Zhao X-Q (2005) Spreading speed and traveling waves for a multi-type SIS epidemic model. J Differ Equ 229:270–296MathSciNetCrossRefMATHGoogle Scholar
  46. Wu C, Weng P (2011) Asymptotic speed of propagation and traveling wavefronts for a SIR epidemic model. Discrete Contin Dyn Syst Ser B 15:867–892MathSciNetCrossRefMATHGoogle Scholar
  47. Yang J, Liang S, Zhang Y (2011) Traveling waves of a delayed SIR epidemic model with nonlinear incidence rate and spatial diffusion. PLoS ONE 6:e21128CrossRefGoogle Scholar
  48. Yang F-Y, Li Y, Li W-T, Wang Z-C (2013) Traveling waves in a nonlocal anisotropic dispersal Kermack–McKendrick epidemic model. Discrete Contin Dyn Syst B 18:1969–1993MathSciNetCrossRefMATHGoogle Scholar
  49. Yuan Z, Zou X (2010) Global threshold property in an epidemic model for disease with latency spreading in a heterogeneous host population. Nonlinear Anal Real World Appl 11:3479–3490MathSciNetCrossRefMATHGoogle Scholar
  50. Zhang T, Wang W (2014) Existence of traveling waves for influenza with treatment. J Math Anal Appl 419:469–495MathSciNetCrossRefMATHGoogle Scholar
  51. Zhao L, Wang Z-C (2016) Traveling wave fronts in a diffusive epidemic model with multiple parallel infectious stages. IMA J Appl Math 81:795–823MathSciNetCrossRefGoogle Scholar
  52. Zhao L, Wang Z-C, Ruan S (2017) Traveling wave solutions of a two-group epidemic model with latent period. Nonlinearity 30:1287–1325MathSciNetCrossRefMATHGoogle Scholar

Copyright information

© Springer-Verlag GmbH Germany, part of Springer Nature 2018

Authors and Affiliations

  1. 1.School of Mathematics and StatisticsLanzhou UniversityLanzhouPeople’s Republic of China
  2. 2.Department of Applied MathematicsLanzhou University of TechnologyLanzhouPeople’s Republic of China
  3. 3.Department of MathematicsUniversity of MiamiCoral GablesUSA

Personalised recommendations