Bulletin of Mathematical Biology

, Volume 77, Issue 11, pp 2004–2034 | Cite as

SIS and SIR Epidemic Models Under Virtual Dispersal

  • Derdei Bichara
  • Yun Kang
  • Carlos Castillo-Chavez
  • Richard Horan
  • Charles Perrings
Original Article


We develop a multi-group epidemic framework via virtual dispersal where the risk of infection is a function of the residence time and local environmental risk. This novel approach eliminates the need to define and measure contact rates that are used in the traditional multi-group epidemic models with heterogeneous mixing. We apply this approach to a general n-patch SIS model whose basic reproduction number \({\mathcal {R}}_0 \) is computed as a function of a patch residence-time matrix \({\mathbb {P}}\). Our analysis implies that the resulting n-patch SIS model has robust dynamics when patches are strongly connected: There is a unique globally stable endemic equilibrium when \({\mathcal {R}}_0>1 \), while the disease-free equilibrium is globally stable when \({\mathcal {R}}_0\le 1 \). Our further analysis indicates that the dispersal behavior described by the residence-time matrix \({\mathbb {P}}\) has profound effects on the disease dynamics at the single patch level with consequences that proper dispersal behavior along with the local environmental risk can either promote or eliminate the endemic in particular patches. Our work highlights the impact of residence-time matrix if the patches are not strongly connected. Our framework can be generalized in other endemic and disease outbreak models. As an illustration, we apply our framework to a two-patch SIR single-outbreak epidemic model where the process of disease invasion is connected to the final epidemic size relationship. We also explore the impact of disease-prevalence-driven decision using a phenomenological modeling approach in order to contrast the role of constant versus state-dependent \({\mathbb {P}}\) on disease dynamics.


Epidemiology SIS–SIR models Dispersal Residence times Global stability Adaptive behavior Final size relationship 

Mathematics Subject Classfication

Primary 34D23 92D25 60K35 



These studies were made possible by grant #1R01GM100471-01 from the National Institute of General Medical Sciences (NIGMS) at the National Institutes of Health. The contents of this manuscript are solely the responsibility of the authors and do not necessarily represent the official views of DHS or NIGMS. Research of Y.K. is partially supported by NSF-DMS (1313312). The funders had no role in study design, data collection and analysis, decision to publish or preparation of the manuscript. The authors are grateful to two anonymous referees for helpful comments and suggestions which led to an improvement of this paper.


  1. Anderson RM, May RM (1982) Directly transmitted infections diseases: control by vaccination. Science 215:1053–1060zbMATHMathSciNetCrossRefGoogle Scholar
  2. Anderson RM, May RM (1991) Infectious diseases of humans. Dynamics and control. Oxford Science Publications, New YorkGoogle Scholar
  3. Arino J (2009) Diseases in metapopulations. In: Ma Z, Zhou Y, Wu J (eds) Modeling and dynamics of infectious diseases, vol 11. World Scientific, SingaporeGoogle Scholar
  4. Arino J, Davis J, Hartley D, Jordan R, Miller J, van den Driessche P (2005) A multi-species epidemic model with spatial dynamics. Math Med Biol 22:129–142zbMATHCrossRefGoogle Scholar
  5. Arino J, van den Driessche P (2003) The basic reproduction number in a multi-city compartmental model. Lecture notes in control and information science, vol 294, pp 135–142Google Scholar
  6. Arino J, van den Driessche P (2006) Disease spread in metapopulations. In: Zhao X-O, Zou X (eds) Nonlinear dynamics and evolution equations, vol 48. Fields Institute Communications, AMS, Providence, pp 1–13Google Scholar
  7. Berman A, Plemmons RJ (1994) Nonnegative matrices in the mathematical sciences, vol 9 of classics in applied mathematics. Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA. Revised reprint of the 1979 originalGoogle Scholar
  8. Bernoulli D (1766) Essai d’une nouvelle analyse de la mortalité causée par la petite vérole, Mem. Math. Phys. Acad. R. Sci. Paris, pp. 1–45Google Scholar
  9. Blythe SP, Castillo-Chavez C (1989) Like-with-like preference and sexual mixing models. Math Biosci 96:221–238zbMATHCrossRefGoogle Scholar
  10. Brauer F (2008) Epidemic models with heterogeneous mixing and treatment. Bull Math Biol 70:1869–1885zbMATHMathSciNetCrossRefGoogle Scholar
  11. Brauer F, Castillo-Chavez C (1994) Basic models in epidemiology. In: Steele J, Powell T (eds) Ecological time series. Raven Press, New York, pp 410–477Google Scholar
  12. Brauer F, Castillo-Chávez C (2012) Mathematical models in population biology and epidemiology. In: Marsden JE, Sirovich L, Golubitski M (eds) Applied mathematics, vol 40. Springer, New YorkGoogle Scholar
  13. Brauer F, Castillo-Chavez C, Velasco-Herná ndez JX (1996) Recruitment effects in heterosexually transmitted disease models. In: Kirschner D (ed) Advances in mathematical modeling of biological processes, vol 3:1. Int J Appl Sci Comput, pp 78–90Google Scholar
  14. Brauer F, Feng Z, Castillo-Chavez C (2010) Discrete epidemic models. Math Biosci Eng 7:1–15zbMATHMathSciNetCrossRefGoogle Scholar
  15. Brauer F, van den Driessche P (2001) Models for transmission of disease with immigration of infectives. Math Biosci 171:143–154Google Scholar
  16. Brauer F, van den Driessche P, Wang L (2008) Oscillations in a patchy environment disease model. Math Biosci 215:1–10zbMATHMathSciNetCrossRefGoogle Scholar
  17. Brauer F, Watmough J (2009) Age of infection epidemic models with heterogeneous mixing. J Biol Dyn 3:324–330MathSciNetCrossRefGoogle Scholar
  18. Castillo-Chavez C, Busenberg S (1991) A general solution of the problem of mixing of subpopulations and its application to risk-and age-structured epidemic models for the spread of AIDS. Math Med Biol 8:1–29zbMATHMathSciNetCrossRefGoogle Scholar
  19. Castillo-Chavez C, Cooke K, Huang W, Levin SA (1989) Results on the dynamics for models for the sexual transmission of the human immunodeficiency virus. Appl Math Lett 2:327–331zbMATHCrossRefGoogle Scholar
  20. Castillo-Chavez C, Hethcote H, Andreasen V, Levin S, Liu W (1989) Epidemiological models with age structure, proportionate mixing, and cross-immunity. J Math Biol 27:233–258zbMATHMathSciNetCrossRefGoogle Scholar
  21. Castillo-Chavez C, Huang W (1999) Age-structured core group modeland its impact on STD dynamics. In Mathematical approaches for emerging and reemerging infectious diseases: models, methods, and theory (Minneapolis, MN, 1999), vol. 126 of IMA, Math Appl, Springer, New York, 2002, pp. 261–273Google Scholar
  22. Castillo-Chavez C, Huang W, Li J (1996) Competitive exclusion in gonorrhea models and other sexually transmitted diseases. SIAM J Appl Math 56:494–508zbMATHMathSciNetCrossRefGoogle Scholar
  23. Castillo-Chavez C, Huang W, Li J (1999) Competitive exclusion and coexistence of multiple strains in an SIS STD model. SIAM J Appl Math 59:1790–1811 (electronic)zbMATHMathSciNetCrossRefGoogle Scholar
  24. Castillo-Chavez C, Thieme HR (1995) Asymptotically autonomous epidemic models. In: Arino ADE, Kimmel O, Kimmel M (eds) Mathematical population dynamics: analysis of heterogeneity, volume one: theory of epidemics. Wuerz, WinnipegGoogle Scholar
  25. Castillo-Chavez C, Velasco-Hernández JX, Fridman S (1994) Modeling contact structures in biology. In: Levin SA (ed) Frontiers in mathematical biology, vol 100. Springer, Berlin ch. 454–491CrossRefGoogle Scholar
  26. Chowell D, Castillo-Chavez C, Krishna S, Qiu X, Anderson KS (2015) Modelling the effect of early detection of Ebola. Lancet 15:148–149CrossRefGoogle Scholar
  27. Diekmann O, Heesterbeek JAP, Metz JAJ (1990) On the definition and the computation of the basic reproduction ratio \(R_0\) in models for infectious diseases in heterogeneous populations. J Math Biol 28:365–382zbMATHMathSciNetCrossRefGoogle Scholar
  28. Dietz K, Heesterbeek J (2002) Daniel Bernoulli’s epidemiological model revisited. Math Biosci 180:1–21zbMATHMathSciNetCrossRefGoogle Scholar
  29. Dietz K, Schenzle D (1985) Mathematical models for infectious disease statistics. In: Atkinson, Anthony, Fienberg, Stephen E (eds) A celebration of statistics. Springer, New York, pp 167–204Google Scholar
  30. Fall A, Iggidr A, Sallet G, Tewa J-J (2007) Epidemiological models and Lyapunov functions. Math Model Nat Phenom 2:62–68MathSciNetCrossRefGoogle Scholar
  31. Fenichel E, Castillo-Chavez C, Ceddia MG, Chowell G,Gonzalez Parra P, Hickling GJ, Holloway G, Horan R, Morin B, Perrings C, Springborn M, Valazquez L, Villalobos C (2011) Adaptive human behavior in epidemiological models. PNAS 208(15):6306–6311Google Scholar
  32. Guo H, Li M, Shuai Z (2006) Global stability of the endemic equilibrium of multigroup models. Can Appl Math Q 14:259–284zbMATHMathSciNetGoogle Scholar
  33. Guo H, Li M, Shuai Z (2008) A graph-theoretic approach to the method of global Lyapunov functions. Proc Am Math Soc 136(8):2793–2802Google Scholar
  34. Hadeler K, Castillo-Chavez C (1995) A core group model for disease transmission. Math Biosci 128:41–55zbMATHCrossRefGoogle Scholar
  35. Heiderich KR, Huang W, Castillo-Chavez C (2002) Nonlocal response in a simple epidemiological model. In: Appli IVM (ed) Mathematical approaches for emerging and reemerging infectious diseases: an introduction, vol 125. Springer, New York, pp 129–151CrossRefGoogle Scholar
  36. Hernandez-Ceron N, Feng Z, Castillo-Chavez C (2013) Discrete epidemic models with arbitrary stage distributions and applications to disease control. Bull Math Biol 75:1716–1746zbMATHMathSciNetCrossRefGoogle Scholar
  37. Hethcote HW (1976) Qualitative analyses of communicable disease models. Math Biosci 28:335–356zbMATHMathSciNetCrossRefGoogle Scholar
  38. Hethcote HW (2000) The mathematics of infectious diseases. SIAM Rev 42:599–653 (electronic)zbMATHMathSciNetCrossRefGoogle Scholar
  39. Hethcote HW, Thieme HR (1985) Stability of the endemic equilibrium in epidemic models with subpopulations. Math Biosci 75:205–227zbMATHMathSciNetCrossRefGoogle Scholar
  40. Hethcote HW, Yorke J (1984) Gonorrhea: transmission dynamics and control, vol 56. Lecture notes in biomathematics. SpringerGoogle Scholar
  41. Hirsch M (1984) The dynamical system approach to differential equations. Bull AMS 11:1–64zbMATHCrossRefGoogle Scholar
  42. Horan DR, Fenichel EP (2007) Economics and ecology of managing emerging infectious animal diseases. Am J Agric Econ 89:1232–1238CrossRefGoogle Scholar
  43. Horan DR, Fenichel EP, Melstrom RT (2011) Wildlife disease bioeconomics. Int Rev Environ Resour Econ 5:23–61CrossRefGoogle Scholar
  44. Horan DR, Fenichel EP, Wolf CA, Graming BM (2010) Managing infectious animal disease systems. Annu Rev Resour Econ 2:101–124CrossRefGoogle Scholar
  45. Hsu Schmitz S-F (2000a) Effect of treatment or/and vaccination on HIV transmission in homosexual with genetic heterogeneity. Math Biosci 167:1–18zbMATHCrossRefGoogle Scholar
  46. Hsu Schmitz S-F (2000b) A mathematical model of HIV transmission in homosexuals with genetic heterogeneity. J Theor Med 2:285–296Google Scholar
  47. Hsu Schmitz S-F (2007) The influence of treatment and vaccination induced changes in the risky contact rate on HIV transmisssion. Math Popul Stud 14:57–76zbMATHMathSciNetCrossRefGoogle Scholar
  48. Huang W, Cooke K, Castillo-Chavez C (1992) Stability and bifurcation for a multiple-group model for the dynamics of HIV/AIDS transmission. SIAM J Appl Math 52:835–854zbMATHMathSciNetCrossRefGoogle Scholar
  49. Huang W, Cooke KL, Castillo-Chavez C (1992) Stability and bifurcation for a multiple-group model for the dynamics of HIV/AIDS transmission. SIAM J Appl Math 52:835–854zbMATHMathSciNetCrossRefGoogle Scholar
  50. Hutson V (1984) A theorem on average Lyapunov functions. Monatshefte für Mathematik 98:267–275zbMATHMathSciNetCrossRefGoogle Scholar
  51. Iggidr A, Sallet G, Tsanou B (2012) Global stability analysis of a metapopulation SIS epidemic model. Math Popul Stud 19:115–129zbMATHMathSciNetCrossRefGoogle Scholar
  52. Jacquez JA, Simon CP (1993) The stochastic SI model with recruitment and deaths I. Comparison with the closed SIS model. Math Biosci 117:77–125zbMATHMathSciNetCrossRefGoogle Scholar
  53. Jacquez JA, Simon CP, Koopman J (1991) The reproduction number in deterministic models of contagious diseases. Comment Theor Biol 2:159–209Google Scholar
  54. Jacquez JA, Simon CP, Koopman J, Sattenspiel L, Perry T (1988) Modeling and analyzing HIV transmission: the effect of contact patterns. Math Biosci 92:119–199Google Scholar
  55. Kuniya T, Muroya Y (2014) Global stability of a multi-group SIS epidemic model for population migration. DCDS Ser B 19(4):1105–1118Google Scholar
  56. Lajmanovich A, Yorke J (1976) A deterministic model for gonorrhea in a nonhomogeneous population. Math Biosci 28:221–236zbMATHMathSciNetCrossRefGoogle Scholar
  57. Lin X, So JW-H (1993) Global stability of the endemic equilibrium and uniform persistence in epidemic models with subpopulations. J Aust Math Soc Ser B 34:282–295zbMATHMathSciNetCrossRefGoogle Scholar
  58. Morin B, Castillo-Chavez C (2003) SIR dynamics with economically driven contact rates. Nat Resour Model 26:505–525MathSciNetCrossRefGoogle Scholar
  59. Mossong J, Hens N, Jit M, Beutels P, Mikolajczyk R, Massari M, Salmaso S, Tomba GS, Wallinga J, Heijne J, Sadkowska-Todys M, Rosinska M, Edmunds WJ (2008) Social contacts and mixing patterns relevant to the spread of infectious diseases. PLoS Med 5:381–391CrossRefGoogle Scholar
  60. Nold A (1980) Heterogeneity in disease-transmission modeling. Math Biosci 52:227zbMATHMathSciNetCrossRefGoogle Scholar
  61. Perrings C, Castillo-Chavez C, Chowell G, Daszak P, Fenichel EP, Finnoff D, Horan RD, Kilpatrick AM, Kinzig AP, Kuminoff NV, Levin S, Morin B, Smith KF, Springborn M (2014) Merging economics and epidemiology to improve the prediction and management of infectious disease. Ecohealth 11(4):464–475Google Scholar
  62. Ross R (1911) The prevention of malaria. John Murray, LondonGoogle Scholar
  63. Rushton S, Mautner A (1955) The deterministic model of a simple epidemic for more than one community. Biometrika 42(1/2):126–132Google Scholar
  64. Sattenspiel L, Dietz K (1995) A structured epidemic model incorporating geographic mobility among regions. Math Biosci 128:71–91zbMATHCrossRefGoogle Scholar
  65. Sattenspiel L, Simon CP (1988) The spread and persistence of infectious diseases in structured populations. Math Biosci 90:341–366 [Nonlinearity in biology and medicine (Los Alamos, NM, 1987)]zbMATHMathSciNetCrossRefGoogle Scholar
  66. Shuai Z, van den Driessche P (2013) Global stability of infectious disease models using lyapunov functions. SIAM J Appl Math 73:1513–1532zbMATHMathSciNetCrossRefGoogle Scholar
  67. Simon CP, Jacquez JA (1992) Reproduction numbers and the stability of equilibria of SI models for heterogeneous populations. SIAM J Appl Math 52:541–576zbMATHMathSciNetCrossRefGoogle Scholar
  68. van den Driessche P, Watmough J (2002) Reproduction numbers and sub-threshold endemic equilibria for compartmental models of disease transmission. Math Biosci 180:29–48zbMATHMathSciNetCrossRefGoogle Scholar
  69. Velasco-Hernández JX, Brauer F, Castillo-Chavez C (1996) Effects of treatment and prevalence-dependent recruitment on the dynamics of a fatal disease. IMA J Math Appl Med Biol 13:175–192zbMATHCrossRefGoogle Scholar
  70. Vidyasagar M (1980) Decomposition techniques for large-scale systems with nonadditive interactions: stability and stabilizability. IEEE Trans Autom Control 25:773–779zbMATHMathSciNetCrossRefGoogle Scholar
  71. Yorke JA, Hethcote HW, Nold A (1978) Dynamics and control of the transmission of gonorrhea. Sex Transm Dis 5:51–56CrossRefGoogle Scholar

Copyright information

© Society for Mathematical Biology 2015

Authors and Affiliations

  • Derdei Bichara
    • 1
  • Yun Kang
    • 2
  • Carlos Castillo-Chavez
    • 1
  • Richard Horan
    • 3
  • Charles Perrings
    • 4
  1. 1.SAL Mathematical, Computational and Modeling Science CenterArizona State UniversityTempeUSA
  2. 2.Sciences and Mathematics Faculty, College of Letters and SciencesArizona State UniversityMesaUSA
  3. 3.Department of Agricultural, Food and Resource EconomicsMichigan State UniversityEast LansingUSA
  4. 4.School of Life SciencesArizona State UniversityTempeUSA

Personalised recommendations