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Dynamics of an infectious diseases with media/psychology induced non-smooth incidence
On latencies in malaria infections and their impact on the disease dynamics
1. | Department of Applied Mathematics, University of Western Ontario, London, Ontario, N6A 5B7, Canada |
2. | Department of Applied Mathematics, University of Western Ontario, London, Ontario N6A 5B7 |
References:
[1] |
R. M. Anderson and R. M. May, "Infectious Diseases of Humans: Dynamics and Control," Oxford University Press, Oxford, 1991. |
[2] |
J. L. Aron and R. M. May, The population dynamics of malaria, in "Population Dynamics Of Infectious Diseases: Theory and Applications" (ed. R. M. Anderson), Chapman And Hall Press, (1982), 139-179. |
[3] |
F. Chanchod and N. F. Britton, Analysis of a vector-bias model on malaria, Bull. Math. Biol., 73 (2011), 639-657.
doi: 10.1007/s11538-010-9545-0. |
[4] |
C. Castillo-Chavez and H. R. Thieme, Asymptotically autonomous epidemic models, in "Mathematical Population Dynamics: Analysis of Heterogeneity, I. Theory of Epidemics" (eds. O. Arino et al.), Wuerz, Winnepeg, Canada, (1995), 33-50. |
[5] |
O. Diekmann, J. A. P Heesterbeek and J. A. J. Metz, On the definition and the computation of the basic reproduction ratio $\mathcalR_0$ in models for infectious diseases, J. Math. Biol., 35 (1990), 503-522. |
[6] |
O. Diekmann, J. A. P. Heesterbeek and M. G. Robert, The construction of next generation matrices for compartmental epidemic models, J. R. Soc. Interface, 7 (2011), 873-885. |
[7] |
H. Guo, M. Y. Li and Z. Shuai, Global stability of the endemic equilibrium of multigroup SIR epidemic models, Can. Appl. Math. Q., 14 (2006), 259-284. |
[8] |
H. Guo, M. Y. Li and Z. Shuai, A graph-theoretic approach to the method of global Lyapunov functions, Proc. Amer. Math. Soc., 136 (2008), 2793-2802.
doi: 10.1090/S0002-9939-08-09341-6. |
[9] |
J. K. Hale and S. M. Verduyn Lunel, "Introduction to Functional Differential Equations," Springer-Verlag, New York, 1993. |
[10] |
J. M. Heffernan, R. J. Smith and L. M. Wahl, Perspectives on the basic reproduction ratio, J. R. Soc. Interface, 2 (2005), 281-293. |
[11] |
W. M. Hirsch, H. Hanisch and J. P. Gabriel, Differential equation models of some parasitic infections: methods for the study of asymptotic behavior, Comm. Pure Appl. Math., 38 (1985), 733-753.
doi: 10.1002/cpa.3160380607. |
[12] |
A. Korobeinikov, Lyapunov function and global properties for SIR and SEIR epidemic models, Math. Med. Biol., 21 (2004), 75-83.
doi: 10.1007/s11538-008-9352-z. |
[13] |
A. Korobeinikov and P. K. Maini, A lyapunov function and global properties for SIR and SEIR epidemiological models with nonlinear incidence, Math. Biosci. Eng., 1 (2004), 57-60.
doi: 10.3934/mbe.2004.1.57. |
[14] |
Y. Lou and X-Q. Zhao, A reaction-diffusion malaria model with incubation period in the vector population, J. Math. Biol., 62 (2011), 543-568.
doi: 10.1007/s00285-010-0346-8. |
[15] |
R. K. Miller, "Nonlinear Volterra Integral Equations," Benjamin, Menlo Park, California, 1971. |
[16] |
G. Macdonald, The analysis of sporozoite rate, Trop. Dis. Bull., 49 (1952), 569-585. |
[17] |
G. Macdonald, Epidemiological basis of malaria control, Bull. WHO, 15 (1956), 613-626. |
[18] |
G. Macdonald, "The Epidemiology And Control Of Malaria," Oxford University Press, London, 1957. |
[19] |
R. Ross, "The Prevention Of Malaria," J. Murray, London, 1910. |
[20] |
S. Ruan, D. Xiao and J. C. Beier, On the delayed Ross-Macdonald model for Malaria transmission, Bull. Math. Biol., 70 (2008), 1098-1114.
doi: 10.1007/s11538-007-9292-z. |
[21] |
H. L. Smith, "Monotone Dynamical Systems. An Introduction to the Theory of Competitive and Cooperative Systems," 41. AMS, Providence, 1995. |
[22] |
A. M. Talman, O. Domarle, F. McKenzie, F. Ariey and V. Robert, Gametocytogenesis: the puberty of Plasmodium falciparum, Malaria Journal, 3 (2004), 24 pp. |
[23] |
H. R. Thieme, "Mathematics In Population Biology," Princeton University Press, Princeton, NJ, 2003. |
[24] |
H. R. Thieme, Persistence under relaxed point-dissipativity (with application to an endemic model), SIAM J. Math. Anal., 24 (1993), 407-435.
doi: 10.1137/0524026. |
[25] |
P. van den Driessche, L. Wang and X. Zou, Modeling diseases with latency and relapse, Math. Biosci. Eng., 4 (2007), 205-219.
doi: 10.3934/mbe.2007.4.205. |
[26] |
P. van den Drissche and J. Watmough, Reproduction numbers and sub-threshold endemic equilibria for compartmental models of disease transmission, Math. Biosci., 180 (2002), 29-48.
doi: 10.1016/S0025-5564(02)00108-6. |
show all references
References:
[1] |
R. M. Anderson and R. M. May, "Infectious Diseases of Humans: Dynamics and Control," Oxford University Press, Oxford, 1991. |
[2] |
J. L. Aron and R. M. May, The population dynamics of malaria, in "Population Dynamics Of Infectious Diseases: Theory and Applications" (ed. R. M. Anderson), Chapman And Hall Press, (1982), 139-179. |
[3] |
F. Chanchod and N. F. Britton, Analysis of a vector-bias model on malaria, Bull. Math. Biol., 73 (2011), 639-657.
doi: 10.1007/s11538-010-9545-0. |
[4] |
C. Castillo-Chavez and H. R. Thieme, Asymptotically autonomous epidemic models, in "Mathematical Population Dynamics: Analysis of Heterogeneity, I. Theory of Epidemics" (eds. O. Arino et al.), Wuerz, Winnepeg, Canada, (1995), 33-50. |
[5] |
O. Diekmann, J. A. P Heesterbeek and J. A. J. Metz, On the definition and the computation of the basic reproduction ratio $\mathcalR_0$ in models for infectious diseases, J. Math. Biol., 35 (1990), 503-522. |
[6] |
O. Diekmann, J. A. P. Heesterbeek and M. G. Robert, The construction of next generation matrices for compartmental epidemic models, J. R. Soc. Interface, 7 (2011), 873-885. |
[7] |
H. Guo, M. Y. Li and Z. Shuai, Global stability of the endemic equilibrium of multigroup SIR epidemic models, Can. Appl. Math. Q., 14 (2006), 259-284. |
[8] |
H. Guo, M. Y. Li and Z. Shuai, A graph-theoretic approach to the method of global Lyapunov functions, Proc. Amer. Math. Soc., 136 (2008), 2793-2802.
doi: 10.1090/S0002-9939-08-09341-6. |
[9] |
J. K. Hale and S. M. Verduyn Lunel, "Introduction to Functional Differential Equations," Springer-Verlag, New York, 1993. |
[10] |
J. M. Heffernan, R. J. Smith and L. M. Wahl, Perspectives on the basic reproduction ratio, J. R. Soc. Interface, 2 (2005), 281-293. |
[11] |
W. M. Hirsch, H. Hanisch and J. P. Gabriel, Differential equation models of some parasitic infections: methods for the study of asymptotic behavior, Comm. Pure Appl. Math., 38 (1985), 733-753.
doi: 10.1002/cpa.3160380607. |
[12] |
A. Korobeinikov, Lyapunov function and global properties for SIR and SEIR epidemic models, Math. Med. Biol., 21 (2004), 75-83.
doi: 10.1007/s11538-008-9352-z. |
[13] |
A. Korobeinikov and P. K. Maini, A lyapunov function and global properties for SIR and SEIR epidemiological models with nonlinear incidence, Math. Biosci. Eng., 1 (2004), 57-60.
doi: 10.3934/mbe.2004.1.57. |
[14] |
Y. Lou and X-Q. Zhao, A reaction-diffusion malaria model with incubation period in the vector population, J. Math. Biol., 62 (2011), 543-568.
doi: 10.1007/s00285-010-0346-8. |
[15] |
R. K. Miller, "Nonlinear Volterra Integral Equations," Benjamin, Menlo Park, California, 1971. |
[16] |
G. Macdonald, The analysis of sporozoite rate, Trop. Dis. Bull., 49 (1952), 569-585. |
[17] |
G. Macdonald, Epidemiological basis of malaria control, Bull. WHO, 15 (1956), 613-626. |
[18] |
G. Macdonald, "The Epidemiology And Control Of Malaria," Oxford University Press, London, 1957. |
[19] |
R. Ross, "The Prevention Of Malaria," J. Murray, London, 1910. |
[20] |
S. Ruan, D. Xiao and J. C. Beier, On the delayed Ross-Macdonald model for Malaria transmission, Bull. Math. Biol., 70 (2008), 1098-1114.
doi: 10.1007/s11538-007-9292-z. |
[21] |
H. L. Smith, "Monotone Dynamical Systems. An Introduction to the Theory of Competitive and Cooperative Systems," 41. AMS, Providence, 1995. |
[22] |
A. M. Talman, O. Domarle, F. McKenzie, F. Ariey and V. Robert, Gametocytogenesis: the puberty of Plasmodium falciparum, Malaria Journal, 3 (2004), 24 pp. |
[23] |
H. R. Thieme, "Mathematics In Population Biology," Princeton University Press, Princeton, NJ, 2003. |
[24] |
H. R. Thieme, Persistence under relaxed point-dissipativity (with application to an endemic model), SIAM J. Math. Anal., 24 (1993), 407-435.
doi: 10.1137/0524026. |
[25] |
P. van den Driessche, L. Wang and X. Zou, Modeling diseases with latency and relapse, Math. Biosci. Eng., 4 (2007), 205-219.
doi: 10.3934/mbe.2007.4.205. |
[26] |
P. van den Drissche and J. Watmough, Reproduction numbers and sub-threshold endemic equilibria for compartmental models of disease transmission, Math. Biosci., 180 (2002), 29-48.
doi: 10.1016/S0025-5564(02)00108-6. |
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