A multipatch malaria model with logistic growth populations

SIAM Journal on Applied Mathematics

Volumn 72, Issue 3, 2012, Pages 819-841

  Gao, Daozhou ^a   Ruan, Shigui ^a  

^a University of Miami   (United States)

Author keywords

Basic reproduction number; Disease free equilibrium; Human movement; Malaria; Monotonicity; Patch model; Travel rate

Indexed keywords

BASIC REPRODUCTION NUMBER; DISEASE-FREE EQUILIBRIUM; HUMAN MOVEMENTS; MALARIA; MONOTONICITY; PATCH MODELS; TRAVEL RATE;

DISEASES; EPIDEMIOLOGY;

POPULATION STATISTICS;

EID: 84865687346     PISSN: 00361399     EISSN: None     Source Type: Journal    
DOI: 10.1137/110850761     Document Type: Article

References (50)

1. L. J. S. Allen, B. M. Bolker, Y. Lou, and A. L. Nevai, Asymptotic profiles of the steady states for an SIS epidemic patch model, SIAM J. Appl. Math., 67 (2007), pp. 1283-1309. (Pubitemid 47613207)
2. J. Arino, Diseases in metapopulations, in Modeling and Dynamics of Infectious Diseases, Ser. Contemp. Appl. Math. 11, Z. Ma, Y. Zhou, and J. Wu, eds., World Scientific, Singapore, 2009, pp. 65-123.
3. J. Arino, A. Ducrot, and P. Zongo, A metapopulation model for malaria with transmissionblocking partial immunity in hosts, J. Math. Biol., 64 (2012), pp. 423-448.
4. J. L. Aron and R. M. May, The population dynamics of malaria, in The Population Dynamics of Infectious Diseases: Theory and Applications, R. M. Anderson, ed., Chapman and Hall, London, 1982, pp. 139-179.
5. P. Auger, E. Kouokam, G. Sallet, M. Tchuente, and B. Tsanou, The Ross-Macdonald model in a patchy environment, Math. Biosci., 216 (2008), pp. 123-131.
6. R. S. Cantrell and C. Cosner, Spatial Ecology via Reaction-Diffusion Equations, JohnWiley & Sons, Chichester, UK, 2003.
7. N. Chitnis, J. M. Cushing, and J. M. Hyman, Bifurcation analysis of a mathematical model for malaria transmission, SIAM J. Appl. Math., 67 (2006), pp. 24-45. (Pubitemid 46384494)
8. N. Chitnis, J. M. Hyman, and J. M. Cushing, Determining important parameters in the spread of malaria through the sensitivity analysis of a mathematical model, Bull. Math. Biol., 70 (2008), pp. 1272-1296.
9. C. Cosner, J. C. Beier, R. S. Cantrell, D. Impoinvil, L. Kapitanski, M. D. Potts, A. Troyo, and S. Ruan, The effects of human movement on the persistence of vector-borne diseases, J. Theoret. Biol., 258 (2009), pp. 550-560.
10. J. Cui and L. Chen, The effect of diffusion on the time varying logistic population growth, Comput. Math. Appl., 36 (1998), pp. 1-9. (Pubitemid 128401477)
11. O. Diekmann, J. A. P. Heesterbeek, and J. A. J. Metz, On the definition and the computation of the basic reproduction ratio R0 in models for infectious diseases in heterogeneous populations, J. Math. Biol., 28 (1990), pp. 365-382.
12. K. Dietz, L. Molineaux, and A. Thomas, A malaria model tested in the African savannah, Bull. World Health Organ., 50 (1974), pp. 347-357.
13. C. Dye and G. Hasibeder, Population dynamics of mosquito-borne disease: effects of flies which bite some people more frequently than others, Trans. R. Soc. Trop. Med. Hyg., 80 (1986), pp. 69-77.
14. H. I. Freedman, S. G. Ruan, and M. X. Tang, Uniform persistence and flows near a closed positively invariant set, J. Dynam. Differential Equations, 6 (1994), pp. 583-600.
15. H. Guo, M. Y. Li, and Z. Shuai, Global stability of the endemic equilibrium of multigroup SIR epidemic models, Canad. Appl. Math. Quart., 14 (2006), pp. 259-284.
16. S. Gupta, J. Swinton, and R. M. Anderson, Theoretical studies of the effects of heterogeneity in the parasite population on the transmission dynamics of malaria, Proc. Roy. Soc. Lond. B, 256 (1994), pp. 231-238.
17. G. Hasibeder and C. Dye, Population dynamics of mosquito-borne disease: persistence in a completely heterogeneous environments, Theoret. Population Biol., 33 (1988), pp. 31-53.
18. M. W. Hirsch, The dynamical systems approach to differential equations, Bull. Amer. Math. Soc. (N.S.), 11 (1984), pp. 1-64.
19. R. A. Horn and C. R. Johnson, Topics in Matrix Analysis, Cambridge University Press, Cambridge, UK, 1991.
20. Y.-H. Hsieh, P. van den Driessche, and L. Wang, Impact of travel between patches for spatial spread of disease, Bull. Math. Biol., 69 (2007), pp. 1355-1375. (Pubitemid 46776471)
21. V. Hutson and K. Schmitt, Permanence and the dynamics of biological systems, Math. Biosci., 111 (1992), pp. 1-71.
22. J. C. Koella, On the use of mathematical models of malaria transmission, Acta Trop., 49 (1991), pp. 1-25.
23. Y. Lou and X.-Q. Zhao, A climate-based malaria transmission model with structured vector population, SIAM J. Appl. Math., 70 (2010), pp. 2023-2044.
24. G. Macdonald, The analysis of sporozoite rate, Trop. Dis. Bull., 49 (1952), pp. 569-585.
25. G. Macdonald, Epidemiological basis of malaria control, Bull.World Health Organ., 15 (1956), pp. 613-626.
26. G. Macdonald, The Epidemiology and Control of Malaria, Oxford University Press, London, 1957.
27. P. Martens and L. Hall, Malaria on the move: human population movement and malaria transmission, Emerg. Inf. Dis., 6 (2000), pp. 103-109. (Pubitemid 30226887)
28. W. J. Martens, L. W. Niessen, J. Rotmans, T. H. Jetten, and A. J. McMichael, Potential impact of global climate change on malaria risk, Environ. Health Perspect., 103 (1995), pp. 458-464.
29. J. Müller and K. P. Hadeler, Monotonicity of the number of passages in linear chains and of the basic reproduction number in epidemic models, Z. Anal. Anwendungen, 19 (2000), pp. 61-75.
30. J. Nedelman, Introductory review: some new thoughts about some old malaria models, Math. Biosci., 73 (1985), pp. 159-182. (Pubitemid 15144915)
31. R. D. Newman, M. E. Parise, A. M. Barber, and R. W. Steketee, Malaria-related deaths among U.S. travelers, 1963-2001, Ann. Intern. Med., 141 (2004), pp. 547-555. (Pubitemid 39350245)
32. G. A. Ngwa, Modelling the dynamics of endemic malaria in growing populations, Discrete Contin. Dyn. Syst. Ser. B, 4 (2004), pp. 1173-1202. (Pubitemid 39332908)
33. G. A. Ngwa and W. S. Shu, A mathematical model for endemic malaria with variable human and mosquito populations, Math. Comput. Modelling, 32 (2000), pp. 747-763.
34. D. J. Rodriguez and L. Torres-Sorando, Models of infectious diseases in spatially heterogeneous environments, Bull. Math. Biol., 63 (2001), pp. 547-571. (Pubitemid 32577704)
35. R. Ross, The Prevention of Malaria, 2nd ed., Murray, London, 1911.
36. S. Ruan, D. Xiao, and J. C. Beier, On the delayed Ross-Macdonald model for malaria transmission, Bull. Math. Biol., 70 (2008), pp. 1098-1114.
37. M. Salmani and P. van den Driessche, A model for disease transmission in a patchy environment, Discrete Contin. Dyn. Syst. Ser. B, 6 (2006), pp. 185-202. (Pubitemid 43294244)
38. D. L. Smith, J. Dushoff, and F. E. McKenzie, The risk of a mosquito-borne infection in a heterogeneous environment, PLoS Biol., 2 (2004), pp. 1957-1964.
39. H. L. Smith, Monotone Dynamical Systems: An Introduction to The theory of Competitive and Cooperative Systems, Math. Surveys Monogr. 41, AMS, Providence, RI, 1995.
40. H. L. Smith and H. R. Thieme, Dynamical Systems and Population Persistence, Grad. Stud. Math. 118, Amer. Math. Soc., Providence, RI, 2011.
41. T. J. Tatem, S. I. Hay, and D. J. Rogers, Global traffic and disease vector dispersal, Proc. Natl. Acad. Sci. USA, 103 (2006), pp. 6242-6247.
42. A. J. Tatem and D. L. Smith, International population movements and regional plasmodium falciparum malaria elimination strategies, Proc. Natl. Acad. Sci. USA, 107 (2010), pp. 12222-12227.
43. H. R. Thieme, Persistence under relaxed point-dissipativity (with application to an endemic model), SIAM J. Math. Anal., 24 (1993), pp. 407-435.
44. L. Torres-Sorando and D. J. Rodriguez, Models of spatio-temporal dynamics in malaria, Ecol. Model., 104 (1997), pp. 231-240. (Pubitemid 28091308)
45. P. van den Driessche and J. Watmough, Reproduction numbers and sub-threshold endemic equilibria for compartmental models of disease transmission, Math. Biosci., 180 (2002), pp. 29-48. (Pubitemid 35304346)
46. W. Wang, Epidemic models with population dispersal, in Mathematics for Life Sciences and Medicine, Y. Takeuchi, Y. Iwasa, and K. Sato, eds., Springer, Berlin, 2007, pp. 67-95.
47. W. Wang and X.-Q. Zhao, An epidemic model in a patchy environment, Math. Biosci., 190 (2004), pp. 97-112. (Pubitemid 38692826)
48. WHO, World Malaria Report 2010, World Health Organization, Geneva, 2010.
49. X.-Q. Zhao, Uniform persistence and periodic coexistence states in infinite-dimensional periodic semiflows with applications, Canad. Appl. Math. Quart., 3 (1995), pp. 473-495.
50. X.-Q. Zhao and Z.-J. Jing, Global asymptotic behavior in some cooperative systems of functional-differential equations, Canad. Appl. Math. Quart., 4 (1996), pp. 421-444.