Global stability of an epidemic model with carrier state in heterogeneous networks

IMA Journal of Applied Mathematics (Institute of Mathematics and Its Applications)

Volumn 80, Issue 4, 2014, Pages 1025-1048

  Cao, Jinde ^a   Wang, Yi ^a   Alofi, Abdulaziz ^b   Al Mazrooei, Abdullah ^b   Elaiw, Ahmed ^b  

^a SOUTHEAST UNIVERSITY   (China)

^b KING ABDULAZIZ UNIVERSITY   (Saudi Arabia)

Author keywords

basic reproduction number; carrier state; complex networks; final size formula; global stability; immunization strategies

Indexed keywords

CELL PROLIFERATION; COMPLEX NETWORKS; COMPUTER VIRUSES; HETEROGENEOUS NETWORKS; IMMUNIZATION; LYAPUNOV FUNCTIONS; TREES (MATHEMATICS); VIRUSES;

BASIC REPRODUCTION NUMBER; CARRIER STATE; DISEASE PROPAGATION; DISEASE-FREE EQUILIBRIUM; FINAL SIZE FORMULA; GLOBAL STABILITY; GLOBALLY ASYMPTOTICALLY STABLE; KIRCHHOFF'S MATRIX-TREE THEOREMS;

EPIDEMIOLOGY;

EID: 84939606788     PISSN: 02724960     EISSN: 14643634     Source Type: Journal    
DOI: 10.1093/imamat/hxu040     Document Type: Article

References (65)

1. Anderson, R. M. & May, R. M. (1991) Infectious Diseases of Humans: Dynamics and Control. Oxford: Oxford University Press.
2. Anderson, R. & Watson, R. (1980) On the spread of a disease with gamma distributed latent and infectious periods. Biometrika, 67, 191-198.
3. Barabási, A.-L. & Albert, R. (1999) Emergence of scaling in random networks. Science, 286, 509-512.
4. Barthelemy, M., Barrat, A., Pastor-satorras, R. & Vespignani, A. (2004) Velocity and hierarchical spread of epidemic outbreaks in scale-free networks. Phys. Rev. Lett., 92, 178701-178704.
5. Beretta, E. & Takeuchi, Y. (1995) Global stability of an SIR epidemic model with time delays. J. Math. Biol., 33, 250-260.
6. Ca llaway, D. S., Newman, M. E. J., Strogatz, S. H. & Watts, D. J. (2000) Network robustness and fragility: percolation on random graphs. Phys. Rev. Lett., 85, 5468-5471.
7. Cohen, R., Havlin, S. & Ben-Avraham, D. (2003) Efficient immunization strategies for computer networks and populations. Phys. Rev. Lett., 91, 247901.
8. Danon, L., Ford, A. P., House, T., Jewell, C. P. & Keeling, M. J. (2011) Networks and the epidemiology of infectious disease. Interdiscip. Perspect. Infect. Dis., 2011, 1-28.
9. Diekmann, O. & Heesterbeek, J. A. P. (2000) Mathematical Epidemiology of Infectious Diseases: Model Building, Analysis and Interpretation. New York: John Wiley Sons.
10. nDiekmann, O., Heesterbeek, J. A. P. & Metz, J. A. J. (1990) On the definition and the computation of the basic reproduction ratio R0 in models for infectious diseases in heterogeneous populations. J. Math. Biol., 28, 365-382.
11. D'Onofrio, A. (2008) A note on the global behaviour of the network-based SIS epidemic model. Nonlinear Anal.: RWA, 9, 1567-1572.
12. Dorogovtsev, S. N., Goltsev, A. V. & Mendes, J. F. F. (2008) Criticla phenomena in complex networks. Rev. Mod. Phys., 80, 1275-1335.
13. Diekmann O.,Heesterbeek J. A. P.,Metz J. A. J. 2002 Reproduction numbers and subthreshold endemic equilibri a for compartmental models of disease transmission. Math. Biosci. 180 29-48.
14. Edmunds, W. J., Medley, G. F. & Nokes, D. J. (1996) The transmission dynamics and control of hepatitis B virus in the Gambia. Stat. Med., 15, 2215-2233.
15. Freedman, H. I., Ruan, S. & Tang, M. (1994) Uniform persistence and flows near a closed positively invariant set. J. Dyn. Differ. Equa., 6, 583-600.
16. Fu, X.C., Small, M., Walker, D. M. & Zhang, H. F. (2008) Epidemic dynamics on scale-free networks with piecewise linear infectivity and immunization. Phys. Rev. E, 77, 036113.
17. Guo, H., Li, M. Y. & Shuai, Z. (2008) A graph-theoretic approach to the method of global Lyapunov functions. Proc. Amer. Math. Soc., 136, 2793-2802.
18. Hayashi, Y., Minoura, M. & Matsukubo, J. (2004) Oscillatory epidemic prevalence in growing scale-free networks. Phys. Rev. E, 69, 016112.
19. Hethcote, H. W. (2000) The mathematics of infectious diseases. SIAM Rev., 42, 599-653.
20. Hethcote, H. W. & Driessche, P. (1995) An SIS epidemic model with variable population size and a delay. J. Math. Biol., 34, 177-194.
21. Hethcote, H. W., Stech, H. W. & Driessche, P. (1981) Nonlinear Oscillations in Epidemic Models. SIAM J. Appl. Math., 40, 1-9.
22. Joo, J. & Lebowitz, J. L. (2004) Behavior of susceptible-infected-susceptible epidemics on heterogeneous networks with saturation. Phys. Rev. E, 69, 066105.
23. Keeling, M. J. & Rohani, P. (2008) Modeling Infectious Disease in Humans and Animals. Princeton and Oxford: Princeton University Press.
24. Kendall, D. G. (1948) On the generalized " Birth-And-Death" process. Annal. Math. Stat., 19, 1-15.
25. Kermack, W. O. & McKendrick, A. G. (1927) A contribution to the mathematical theory of epidemics. Proc. R. Soc. A, 115, 700-721.
26. Kiss, I. Z., Green, D. M. & Kao, R. R. (2006) The effect of contact heterogeneity and multiple routes of transmission on final epidemic size. Math. Biosci., 203, 124-136.
27. Klovdahl, A. S. (1985) Social networks and the spread of infectious diseases: the AIDS example. Soc. Sci. Med., 21, 1203-1216.
28. Lajmanovich, A. & Yorke, J. A. (1976) A deterministic model for gonorrhea in a nonhomogeneous population. Math. Biosci., 28, 221-236.
29. LaSalle, J. P. (1976) The stability of dynamical systems. Philadelphia: CBMS-NSF Regional Conference Series in Applied Mathematics. SIAM, Philadelphia, PA.
30. Li, M.Y., Graef, J. R., Wang, L. & Karsai, J. (1999) Global dynamics of a SEIR model with varying total population size.Math. Biosci., 160, 191-213.
31. Li, M. Y. & Shuai, Z. S. (2010) Global-stability problem for coupled systems of differential equations on networks. J. Differential Equations, 248, 1-20.
32. Li, M. Y., Smith, H. L. & Wang, L. (2001) Global dynamics of an SEIR epidemic model with vertical transmission. SIAM J. Appl. Math., 62, 58-69.
33. Li, C., Tsai, C. & Yang, S. (2014) Analysis of epidemic spreading of an SIRS model in complex heterogeneous networks. Commun. Nonlinear Sci. Numer. Simul., 19, 1042-1054.
34. Liu, G. T., Si, C. W. & Wang, Q. H. et al. (2002) Comments on the prevention and research of chronic hepatitis in China. Natl. Med. J. China, 82, 74-76.
35. Ma, J. L. & Earn, D. J. D. (2006) Generality of the final size formula for an epidemic of a newly invading infectious disease. Bull. Math. Biol., 68, 679-702.
36. Ma, Z. E., Zhou, Y. C. & Wu, J. H. (2009) Modeling and Dynamics of Infectious Diseases. Beijing: Higher Education Press.
37. May, R. M. & Anderson, R. M. (1987) Transmission dynamics of HIV infection. Nature, 326, 137-142.
38. May, R. M. & Lloyd, A. L. (2001) Infection dynamics on scale-free networks. Phys. Rev. E, 64, 066112.
39. Moon, J. W. (1970) Counting Labelled Trees. London: William Clowes and Sons.
40. Murray, J. D. (2003) Mathematical Biology, 3rd edn. New York: Springer.
41. Nian, F. Z. & Wang, X. Y. (2010) Efficient immunization strategies on complex networks. J. Theor. Biol., 264, 77-83.
42. Pastor-Satorras, R. & Vespignani, A. (2001a ) Epidemic dynamics and endemic states in complex networks. Phys. Rev. E, 63, 066117.
43. Pastor-Satorras, R. & Vespignani, A. (2001b ) Epidemic spreading in scale-free networks. Phys. Rev. Lett., 86, 3200-3203.
44. Pastor-Satorras, R. & Vespignani, A. (2002a ) Immunization of complex networks. Phys. Rev. E, 65, 036104.
45. Pastor-Satorras, R. & Vespignani, A. (2002b ) Epidemic dynamics in finite size scale-free networks. Phys. Rev. E, 65, 035108(R).
46. Ross, R. (1911) The Prevention of Malaria, 2nd edn. London: John Murray.
47. Ruan, S. G. & Wang, W. D. (2003) Dynamical behavior of an epidemic model with a nonlinear incidence rate. J. Differential Equations, 188, 135-163.
48. Shi, H. J., Duan, Z. S. & Ch en, G. R. (2008) An SIS model with infective medium on complex networks. Physica A, 387, 2133-2144.
49. Shulgin, B., Stone, L. & Agur, Z. (1998) Pulse vaccination strategy in the SIR epidemic model. Bull. Math. Biol., 60, 1-26.
50. Smith, H. L. (1983) Subharmonic bifurcation in an SIR epidemic model. J. Math. Biol., 17, 163-177.
51. Wang, W. D. (2006) Backward bifurcation of an epidemic model with treatment. Math. Biol., 201, 58-71.
52. Wang, Y., Cao, J. D., Jin, Z., Zhang, H. F. & Sun, G. Q. (2013) Impact of media coverage on epidemic spreading in complex networks. Physica A, 392, 5824-5838.
53. Wang, L. & Dai, G. Z. (2008) Global stability of virus spreading in complex heterogeneous networks. SIAM. J. Appl. Math., 68, 1495-1502.
54. Wang, Y., Jin, Z., Yang, Z., Zhang, Z. K., Zhou, T. & Sun, G. Q. (2012) Global analysis of an SIS model with an infective vector on complex network s. Nonlinear Anal.: RWA, 13, 543-557.
55. WHO: Hepatitis B. (2008) http://www.who.int/mediacentre/factsheets/fs204/en/index.html (revised August 2008). Accessed 2013.
56. Williams R. 2006 Global challenges in liver disease Hepatol. 44, 521-526.
57. Xia, C. Y., Wang, L., Sun, S. W. & Wang, J. (2012) An SIR model with infection delay and propagation vector in complex networks. Nonlinear Dyn., 69, 927-934.
58. Xu, X. J., Peng, H. Q., Wang, X. M. & Wang, Y. H. (2006) Epidemic spreading with time delay in complex networks. Physica A, 367, 525-530.
59. Yan, G., Zhou , T., Wang, J., Fu, Z. Q. & Wang, B. H. (2005) Epidemic spread in weighted scale-free networks. Chinese Phys. Lett., 22, 510-514.
60. Yang, Z. M. & Zhou, T. (2012) Epidemic spreading in weighted networks: an edge-based mean-field solution. Phys. Rev. E, 85, 056106.
61. Zhang, H. F. & Fu, X. C. (2009) Spreading of epidemics on scale-free networks with nonlinear infectivity. Nonlinear Anal.: TMA, 70, 3273-3278.
62. Zhou, T., Liu, J. G., Bai, W. J., Chen, G. R. & Wang, B. H. (2004) Behaviors of susceptible-infected epidemics on scale-free networks with identical infectivity. Phys. Rev. E, 74, 056109.
63. Zh u, G., Fu, X. & Chen, G. (2012) Global attractivity of a network-based epidemic SIS model with nonlinear infectivity. Commun. Nonlinear Sci. Numer. Simul., 17, 2588-2594.
64. Zou, S. F., Wu, J. H. & Chen , Y. M. (2011) Multiple epidemic waves in delayed susceptible-infected-recovered models on complex networks. Phys. Rev. E, 83, 056121.
65. Zou, L., Zhang, W. N. & Ruan, S. G. (2010) Modeling the transmission dynamics and control of hepatitis B virus in China. J. Theor. Biol., 262, 330-338.