SIS and SIR Epidemic Models Under Virtual Dispersal

Bulletin of Mathematical Biology

Volumn 77, Issue 11, 2015, Pages 2004-2034

  Bichara, Derdei ^a   Kang, Yun ^b   Castillo Chavez, Carlos ^a   Horan, Richard ^c   Perrings, Charles ^d  

^a ARIZONA STATE UNIVERSITY   (United States)

^b ARIZONA STATE UNIVERSITY   (United States)

^c MICHIGAN STATE UNIVERSITY   (United States)

^d ARIZONA STATE UNIVERSITY   (United States)

Author keywords

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

Indexed keywords

BASIC REPRODUCTION NUMBER; BIOLOGICAL MODEL; COMMUNICABLE DISEASES; DISEASE TRANSMISSION; EPIDEMIC; HUMAN; MATHEMATICAL PHENOMENA; RISK FACTOR; STATISTICAL MODEL; STATISTICS AND NUMERICAL DATA; TRANSMISSION;

BASIC REPRODUCTION NUMBER; COMMUNICABLE DISEASES; DISEASE TRANSMISSION, INFECTIOUS; EPIDEMICS; HUMANS; MATHEMATICAL CONCEPTS; MODELS, BIOLOGICAL; MODELS, STATISTICAL; RISK FACTORS;

EID: 84948662106     PISSN: 00928240     EISSN: 15229602     Source Type: Journal    
DOI: 10.1007/s11538-015-0113-5     Document Type: Article

References (71)

1. Anderson RM, May RM (1982) Directly transmitted infections diseases: control by vaccination. Science 215:1053–1060
2. Anderson RM, May RM (1991) Infectious diseases of humans. Dynamics and control. Oxford Science Publications, New York
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, Singapore
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 22:129–142
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 135–142
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 1–13
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 original
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. 1–45
9. Blythe SP, Castillo-Chavez C (1989) Like-with-like preference and sexual mixing models. Math Biosci 96:221–238
10. Brauer F (2008) Epidemic models with heterogeneous mixing and treatment. Bull Math Biol 70:1869–1885
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 410–477
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 York
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 78–90
14. Brauer F, Feng Z, Castillo-Chavez C (2010) Discrete epidemic models. Math Biosci Eng 7:1–15
15. Brauer F, van den Driessche P (2001) Models for transmission of disease with immigration of infectives. Math Biosci 171:143–154
16. Brauer F, van den Driessche P, Wang L (2008) Oscillations in a patchy environment disease model. Math Biosci 215:1–10
17. Brauer F, Watmough J (2009) Age of infection epidemic models with heterogeneous mixing. J Biol Dyn 3:324–330
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 8:1–29
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 2:327–331
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 27:233–258
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. 261–273
22. Castillo-Chavez C, Huang W, Li J (1996) Competitive exclusion in gonorrhea models and other sexually transmitted diseases. SIAM J Appl Math 56:494–508
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)
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, Winnipeg
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. 454–491
26. Chowell D, Castillo-Chavez C, Krishna S, Qiu X, Anderson KS (2015) Modelling the effect of early detection of Ebola. Lancet 15:148–149
27. Diekmann O, Heesterbeek JAP, Metz JAJ (1990) On the definition and the computation of the basic reproduction ratio $$R_0$$R0 in models for infectious diseases in heterogeneous populations. J Math Biol 28:365–382
28. Dietz K, Heesterbeek J (2002) Daniel Bernoulli’s epidemiological model revisited. Math Biosci 180:1–21
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 167–204
30. Fall A, Iggidr A, Sallet G, Tewa J-J (2007) Epidemiological models and Lyapunov functions. Math Model Nat Phenom 2:62–68
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):6306–6311
32. Guo H, Li M, Shuai Z (2006) Global stability of the endemic equilibrium of multigroup models. Can Appl Math Q 14:259–284
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):2793–2802
34. Hadeler K, Castillo-Chavez C (1995) A core group model for disease transmission. Math Biosci 128:41–55
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 129–151
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 75:1716–1746
37. Hethcote HW (1976) Qualitative analyses of communicable disease models. Math Biosci 28:335–356
38. Hethcote HW (2000) The mathematics of infectious diseases. SIAM Rev 42:599–653 (electronic)
39. Hethcote HW, Thieme HR (1985) Stability of the endemic equilibrium in epidemic models with subpopulations. Math Biosci 75:205–227
40. Hethcote HW, Yorke J (1984) Gonorrhea: transmission dynamics and control, vol 56. Lecture notes in biomathematics. Springer
41. Hirsch M (1984) The dynamical system approach to differential equations. Bull AMS 11:1–64
42. Horan DR, Fenichel EP (2007) Economics and ecology of managing emerging infectious animal diseases. Am J Agric Econ 89:1232–1238
43. Horan DR, Fenichel EP, Melstrom RT (2011) Wildlife disease bioeconomics. Int Rev Environ Resour Econ 5:23–61
44. Horan DR, Fenichel EP, Wolf CA, Graming BM (2010) Managing infectious animal disease systems. Annu Rev Resour Econ 2:101–124
45. Hsu Schmitz S-F (2000a) Effect of treatment or/and vaccination on HIV transmission in homosexual with genetic heterogeneity. Math Biosci 167:1–18
46. Hsu Schmitz S-F (2000b) A mathematical model of HIV transmission in homosexuals with genetic heterogeneity. J Theor Med 2:285–296
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 14:57–76
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 52:835–854
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 52:835–854
50. Hutson V (1984) A theorem on average Lyapunov functions. Monatshefte für Mathematik 98:267–275
51. Iggidr A, Sallet G, Tsanou B (2012) Global stability analysis of a metapopulation SIS epidemic model. Math Popul Stud 19:115–129
52. Jacquez JA, Simon CP (1993) The stochastic SI model with recruitment and deaths I. Comparison with the closed SIS model. Math Biosci 117:77–125
53. Jacquez JA, Simon CP, Koopman J (1991) The reproduction number in deterministic models of contagious diseases. Comment Theor Biol 2:159–209
54. Jacquez JA, Simon CP, Koopman J, Sattenspiel L, Perry T (1988) Modeling and analyzing HIV transmission: the effect of contact patterns. Math Biosci 92:119–199
55. Kuniya T, Muroya Y (2014) Global stability of a multi-group SIS epidemic model for population migration. DCDS Ser B 19(4):1105–1118
56. Lajmanovich A, Yorke J (1976) A deterministic model for gonorrhea in a nonhomogeneous population. Math Biosci 28:221–236
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 34:282–295
58. Morin B, Castillo-Chavez C (2003) SIR dynamics with economically driven contact rates. Nat Resour Model 26:505–525
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 5:381–391
60. Nold A (1980) Heterogeneity in disease-transmission modeling. Math Biosci 52:227
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):464–475
62. Ross R (1911) The prevention of malaria. John Murray, London
63. Rushton S, Mautner A (1955) The deterministic model of a simple epidemic for more than one community. Biometrika 42(1/2):126–132
64. Sattenspiel L, Dietz K (1995) A structured epidemic model incorporating geographic mobility among regions. Math Biosci 128:71–91
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)]
66. Shuai Z, van den Driessche P (2013) Global stability of infectious disease models using lyapunov functions. SIAM J Appl Math 73:1513–1532
67. Simon CP, Jacquez JA (1992) Reproduction numbers and the stability of equilibria of SI models for heterogeneous populations. SIAM J Appl Math 52:541–576
68. van den Driessche P, Watmough J (2002) Reproduction numbers and sub-threshold endemic equilibria for compartmental models of disease transmission. Math Biosci 180:29–48
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 13:175–192
70. Vidyasagar M (1980) Decomposition techniques for large-scale systems with nonadditive interactions: stability and stabilizability. IEEE Trans Autom Control 25:773–779
71. Yorke JA, Hethcote HW, Nold A (1978) Dynamics and control of the transmission of gonorrhea. Sex Transm Dis 5:51–56