Journal of Mathematical Biology

, Volume 77, Issue 1, pp 107–134 | Cite as

Multi-patch and multi-group epidemic models: a new framework

  • Derdei Bichara
  • Abderrahman Iggidr


We develop a multi-patch and multi-group model that captures the dynamics of an infectious disease when the host is structured into an arbitrary number of groups and interacts into an arbitrary number of patches where the infection takes place. In this framework, we model host mobility that depends on its epidemiological status, by a Lagrangian approach. This framework is applied to a general SEIRS model and the basic reproduction number \({\mathcal {R}}_0\) is derived. The effects of heterogeneity in groups, patches and mobility patterns on \({\mathcal {R}}_0\) and disease prevalence are explored. Our results show that for a fixed number of groups, the basic reproduction number increases with respect to the number of patches and the host mobility patterns. Moreover, when the mobility matrix of susceptible individuals is of rank one, the basic reproduction number is explicitly determined and was found to be independent of the latter if the matrix is also stochastic. The cases where mobility matrices are of rank one capture important modeling scenarios. Additionally, we study the global analysis of equilibria for some special cases. Numerical simulations are carried out to showcase the ramifications of mobility pattern matrices on disease prevalence and basic reproduction number.


Multi-patch Multi-group Mobility Heterogeneity Residence times Global stability 

Mathematics Subject Classification

92D25 92D30 



We are grateful to two anonymous referees and the handling editor Dr. Gabriela Gomes for valuable comments and suggestions that led to an improvement of this paper. We also thank Bridget K. Druken for the careful reading and constructive comments. A. Iggidr acknowledges the partial support of Inria in the framework of the program STIC AmSud (project MOSTICAW).


  1. Anderson RM, May RM (1991) Infectious diseases of humans. Dynamics and control. Oxford science publications, OxfordGoogle Scholar
  2. Arino J (2009) Disease in metapopulations model. In: Ma Z, Zhou Y, Wu J (eds) Modeling and dynamics of infectious diseases. World Scientific Publishing, Singapore, pp 65–123Google Scholar
  3. Arino J, Portet S (2015) Epidemiological implications of mobility between a large urban centre and smaller satellite cities. J Math Biol 71:1243–1265MathSciNetCrossRefzbMATHGoogle Scholar
  4. 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
  5. Berman A, Plemmons RJ (1994) Nonnegative matrices in the mathematical sciences. SIAM, PhiladelphiaCrossRefzbMATHGoogle Scholar
  6. Bhatia NP, Szegö GP (1970) Stability theory of dynamical systems. Springer, BerlinCrossRefzbMATHGoogle Scholar
  7. Bichara D, Castillo-Chavez C (2016) Vector-borne diseases models with residence times—a Lagrangian perspective. Math Biosci 281:128–138MathSciNetCrossRefzbMATHGoogle Scholar
  8. Bichara D, Kang Y, Castillo-Chavez C, Horan R, Perrings C (2015) Sis and sir epidemic models under virtual dispersal. Bull Math Biol 77:2004–2034MathSciNetCrossRefzbMATHGoogle Scholar
  9. Bichara D, Holechek SA, Velázquez-Castro J, Murillo AL, Castillo-Chavez C (2016) On the dynamics of dengue virus type 2 with residence times and vertical transmission. Lett Biomath 3:140–160MathSciNetCrossRefGoogle Scholar
  10. Blythe SP, Castillo-Chavez C (1989) Like-with-like preference and sexual mixing models. Math Biosci 96:221–238CrossRefzbMATHGoogle Scholar
  11. Bonzi B, Fall A, Iggidr A, Sallet G (2011) Stability of differential susceptibility and infectivity epidemic models. J Math Biol 62(1):39–64MathSciNetCrossRefzbMATHGoogle Scholar
  12. 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 8:1–29MathSciNetCrossRefzbMATHGoogle Scholar
  13. Castillo-Chavez C, Thieme HR (1995) Asymptotically autonomous epidemic models. In: Arino O, Axelrod DE, Kimmel M (eds) Mathematical population dynamics: analysis of heterogeneity, volume one: theory of epidemics. Wuerz, WinnipegGoogle Scholar
  14. Castillo-Chavez C, Bichara D, Morin BR (2016) Perspectives on the role of mobility, behavior, and time scales in the spread of diseases. Proc Natl Acad Sci 113:14582–14588CrossRefGoogle Scholar
  15. Cosner C, Beier J, Cantrell R, Impoinvil D, Kapitanski L, Potts M, Troyo A, Ruan S (2009) The effects of human movement on the persistence of vector-borne diseases. J Theor Biol 258:550–560MathSciNetCrossRefGoogle Scholar
  16. 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 28:365–382MathSciNetCrossRefzbMATHGoogle Scholar
  17. Dushoff J, Levin S (1995) The effects of population heterogeneity on disease invasion. Math Biosci 128:25–40CrossRefzbMATHGoogle Scholar
  18. Eckenrode S, Bakullari A, Metersky ML, Wang Y, Pandolfi MM, Galusha D, Jaser L, Eldridge N (2014) The association between age, sex, and hospital-acquired infection rates: results from the 2009–2011 national medicare patient safety monitoring system. Infect Control Hosp Epidemiol 35:S3–S9CrossRefGoogle Scholar
  19. Falcón-Lezama JA, Martínez-Vega RA, Kuri-Morales PA, Ramos-Castañeda J, Adams B (2016) Day-to-day population movement and the management of dengue epidemics. Bull Math Biol 78:2011–2033MathSciNetCrossRefzbMATHGoogle Scholar
  20. Fall A, Iggidr A, Sallet G, Tewa J-J (2007) Epidemiological models and lyapunov functions. Math Model Nat Phenom 2:62–68MathSciNetCrossRefzbMATHGoogle Scholar
  21. 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 108:6306–6311CrossRefGoogle Scholar
  22. Hethcote HW, Thieme HR (1985) Stability of the endemic equilibrium in epidemic models with subpopulations. Math Biosci 75:205–227MathSciNetCrossRefzbMATHGoogle Scholar
  23. Hirsch M (1984) The dynamical system approach to differential equations. Bull AMS 11:1–64MathSciNetCrossRefzbMATHGoogle Scholar
  24. Horn RA, Johnson CR (1985) Matrix analysis. Cambridge University Press, New YorkCrossRefzbMATHGoogle Scholar
  25. 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 52:835–854MathSciNetCrossRefzbMATHGoogle Scholar
  26. Iggidr A, Sallet G, Tsanou B (2012) Global stability analysis of a metapopulation sis epidemic model. Math Popul Stud 19:115–129MathSciNetCrossRefzbMATHGoogle Scholar
  27. Iggidr A, Sallet G, Souza MO (2016) On the dynamics of a class of multi-group models for vector-borne diseases. J Math Anal Appl 2:723–743MathSciNetCrossRefzbMATHGoogle Scholar
  28. Jacquez JA, Simon CP, Koopman J, Sattenspiel L, Perry T (1988) Modeling and analyzing HIV transmission: the effect of contact patterns. Math Biosci 92:119–199MathSciNetCrossRefzbMATHGoogle Scholar
  29. Jacquez JA, Simon CP, Koopman J (1996) Core groups and the r0s for subgroups in heterogeneous SIS and SI models. In: Mollison D (ed) Epidemics models: their structure and relation to data. Cambridge University Press, Cambridge, pp 279–301Google Scholar
  30. Kaplan V, Angus DC, Griffin MF, Clermont G, Scott Watson R, Linde-zwirble WT (2002) Hospitalized community-acquired pneumonia in the elderly: age-and sex-related patterns of care and outcome in the united states. Am J Respir Crit Care Med 165:766–772CrossRefGoogle Scholar
  31. Kermack W, McKendrick A (1927) A contribution to the mathematical theory of epidemics. Proc R Soc A115:700–721CrossRefzbMATHGoogle Scholar
  32. Lajmanovich A, Yorke J (1976) A deterministic model for gonorrhea in a nonhomogeneous population. Math Biosci 28:221–236MathSciNetCrossRefzbMATHGoogle Scholar
  33. LaSalle JP, Lefschetz S (1961) Stability by Liapunov’s direct method. Academic Press, CambridgeGoogle Scholar
  34. Metz JA, Diekmann O (2014) The dynamics of physiologically structured populations, vol 68. Springer, BerlinzbMATHGoogle Scholar
  35. Nold A (1980) Heterogeneity in disease-transmission modeling. Math Biosci 52:227MathSciNetCrossRefzbMATHGoogle Scholar
  36. 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 11:464–475CrossRefGoogle Scholar
  37. Prothero RM (1977) Disease and mobility: a neglected factor in epidemiology. Int J Epidemiol 6:259–267CrossRefGoogle Scholar
  38. Rodríguez DJ, Torres-Sorando L (2001) Models of infectious diseases in spatially heterogeneous environments. Bull Math Biol 63:547–571CrossRefzbMATHGoogle Scholar
  39. Ruktanonchai NW, Smith DL, De Leenheer P (2016) Parasite sources and sinks in a patched ross-macdonald malaria model with human and mosquito movement: implications for control. Math Biosci 279:90–101MathSciNetCrossRefzbMATHGoogle Scholar
  40. Rushton S, Mautner A (1955) The deterministic model of a simple epidemic for more than one community. Biometrika 42:126–132MathSciNetCrossRefzbMATHGoogle Scholar
  41. Salmani M, van den Driessche P (2006) A model for disease transmission in a patchy environment. DCDS Ser B 6:185–202MathSciNetzbMATHGoogle Scholar
  42. Sattenspiel L, Dietz K (1995) A structured epidemic model incorporating geographic mobility among regions. Math Biosci 128:71–91CrossRefzbMATHGoogle Scholar
  43. 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)Google Scholar
  44. van den Driessche P, Watmough J (2002) reproduction numbers and sub-threshold endemic equilibria for compartmental models of disease transmission. Math Biosci 180:29–48MathSciNetCrossRefzbMATHGoogle Scholar
  45. Vidyasagar M (1980) Decomposition techniques for large-scale systems with nonadditive interactions: stability and stabilizability. IEEE Trans Autom Control 25:773–779MathSciNetCrossRefzbMATHGoogle Scholar
  46. Xiao Y, Zou X (2014) Transmission dynamics for vector-borne diseases in a patchy environment. J Math Biol 69:113–146MathSciNetCrossRefzbMATHGoogle Scholar
  47. Yorke JA, Hethcote HW, Nold A (1978) Dynamics and control of the transmission of gonorrhea. Sex Transm Dis 5:51–56CrossRefGoogle Scholar

Copyright information

© Springer-Verlag GmbH Germany, part of Springer Nature 2017

Authors and Affiliations

  1. 1.Department of Mathematics and Center for Computational and Applied MathematicsCalifornia State UniversityFullertonUSA
  2. 2.Inria, Université de Lorraine, CNRS, Institut Elie Cartan de Lorraine, UMR 7502Metz Cedex 01France

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