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A periodic Ross-Macdonald model in a patchy environment
1. | Francis I. Proctor Foundation for Research in Ophthalmology, University of California, San Francisco, San Francisco, CA 94143, United States |
2. | Department of Applied Mathematics, The Hong Kong Polytechnic University, Hung Hom, Kowloon, Hong Kong |
3. | Department of Mathematics, University of Miami, Coral Gables, FL 33124 |
References:
[1] |
S. Altizer, A. Dobson, P. Hosseini, P. Hudson, M. Pascual and P. Rohani, Seasonality and the dynamics of infectious diseases,, Ecol. Lett., 9 (2006), 467.
doi: 10.1111/j.1461-0248.2005.00879.x. |
[2] |
G. Aronsson and R. B. Kellogg, On a differential equation arising from compartmental analysis,, Math. Biosci., 38 (1973), 113.
doi: 10.1016/0025-5564(78)90021-4. |
[3] |
P. Auger, E. Kouokam, G. Sallet, M. Tchuente and B. Tsanou, The Ross-Macdonald model in a patchy environment,, Math. Biosci., 216 (2008), 123.
doi: 10.1016/j.mbs.2008.08.010. |
[4] |
N. Bacaër and S. Guernaoui, The epidemic threshold of vector-borne diseases with seasonality,, J. Math. Biol., 53 (2006), 421.
doi: 10.1007/s00285-006-0015-0. |
[5] |
N. Bacaër, Approximation of the basic reproduction number $R_0$ for vector-borne diseases with a periodic vector population,, Bull. Math. Biol., 69 (2007), 1067.
doi: 10.1007/s11538-006-9166-9. |
[6] |
Z. Bai and Y. Zhou, Threshold dynamics of a Bacillary Dysentery model with seasonal fluctuation,, Discrete Contin. Dyn. Syst., 15 (2011), 1.
doi: 10.3934/dcdsb.2011.15.1. |
[7] |
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), 550.
doi: 10.1016/j.jtbi.2009.02.016. |
[8] |
C. Costantini, S. G. Li, A. D. Torre, N. Sagnon, M. Coluzzi and C. E. Taylor, Density, survival and dispersal of Anopheles gambiae complex mosquitoes in a West African Sudan savanna village,, Med. Vet. Entomol., 10 (1996), 203.
doi: 10.1111/j.1365-2915.1996.tb00733.x. |
[9] |
M. H. Craig, I. Kleinschmidt, J. B. Nawn, D. Le Sueur and B. L. Sharp, Exploring 30 years of malaria case data in KwaZulu-Natal, South Africa: Part I. The impact of climatic factors,, Trop. Med. Int. Health, 9 (2004), 1247.
doi: 10.1111/j.1365-3156.2004.01340.x. |
[10] |
B. Dembele, A. Friedman and A.-A. Yakubu, Malaria model with periodic mosquito birth and death rates,, J. Biol. Dynam., 3 (2009), 430.
doi: 10.1080/17513750802495816. |
[11] |
D. Gao and S. Ruan, A multi-patch malaria model with logistic growth populations,, SIAM J. Appl. Math., 72 (2012), 819.
doi: 10.1137/110850761. |
[12] |
H. Gao, L. Wang, S. Liang, Y. Liu, S. Tong, J. Wang, Y. Li, X. Wang, H. Yang, J. Ma, L. Fang and W. Cao, Change in rainfall drives malaria re-emergence in Anhui province, China,, PLoS ONE, 7 (2012).
doi: 10.1371/journal.pone.0043686. |
[13] |
N. C. Grassly and C. Fraser, Seasonal infectious disease epidemiology,, Proc. R. Soc. B., 273 (2006), 2541.
doi: 10.1098/rspb.2006.3604. |
[14] |
J. K. Hale, Ordinary Differential Equations,, Wiley-Interscience, (1980).
|
[15] |
M. W. Hirsch, H. L. Smith and X.-Q. Zhao, Chain transitivity, attractivity, and strong repellors for semidynamical systems,, J. Dyn. Differ. Equ., 13 (2001), 107.
doi: 10.1023/A:1009044515567. |
[16] |
K. D. Lafferty, The ecology of climate change and infectious diseases,, Ecology, 90 (2009), 888.
doi: 10.1890/08-0079.1. |
[17] |
L. Liu, X.-Q. Zhao and Y. Zhou, A tuberculosis model with seasonality,, Bull. Math. Biol., 72 (2010), 931.
doi: 10.1007/s11538-009-9477-8. |
[18] |
X. Liu and X.-Q. Zhao, A periodic epidemic model with age structure in a patchy environment,, SIAM J. Appl. Math., 71 (2011), 1896.
doi: 10.1137/100813610. |
[19] |
J. Lou, Y. Lou and J. Wu, Threshold virus dynamics with impulsive antiretroviral drug effects,, J. Math. Biol., 65 (2012), 623.
doi: 10.1007/s00285-011-0474-9. |
[20] |
Y. Lou and X.-Q. Zhao, The periodic Ross-Macdonald model with diffusion and advection,, Appl. Anal., 89 (2010), 1067.
doi: 10.1080/00036810903437804. |
[21] |
Y. Lou and X.-Q. Zhao, A climate-based malaria transmission model with structured vector population,, SIAM J. Appl. Math., 70 (2010), 2023.
doi: 10.1137/080744438. |
[22] |
Y. Lou and X.-Q. Zhao, Modelling malaria control by introduction of larvivorous fish,, Bull. Math. Biol., 73 (2011), 2384.
doi: 10.1007/s11538-011-9628-6. |
[23] |
G. Macdonald, The analysis of sporozoite rate,, Trop. Dis. Bull., 49 (1952), 569. Google Scholar |
[24] |
G. Macdonald, Epidemiological basis of malaria control,, Bull. World Health Organ., 15 (1956), 613. Google Scholar |
[25] |
G. Macdonald, The Epidemiology and Control of Malaria,, Oxford University Press, (1957). Google Scholar |
[26] |
J. T. Midega, C. M. Mbogo, H. Mwambi, M. D. Wilson, G. Ojwang, J. M. Mwangangi, J. G. Nzovu, J. I. Githure, G. Yan and J. C. Beier, Estimating dispersal and survival of Anopheles gambiae and Anopheles funestus along the Kenyan coast by using mark-release-recapture methods,, J. Med. Entomol., 44 (2007), 923.
doi: {10.1603/0022-2585(2007)44[923:EDASOA]2.0.CO;2}. |
[27] |
R. S. Ostfeld, Climate change and the distribution and intensity of infectious diseases,, Ecology, 90 (2009), 903.
doi: 10.1890/08-0659.1. |
[28] |
P. E. Parham and E. Michael, Modeling the effects of weather and climate change on malaria transmission,, Environ. Health Perspect., 118 (2010), 620.
doi: 10.1289/ehp.0901256. |
[29] |
D. Rain, Eaters of the Dry Season: Circular Labor Migration in the West African Sahel,, Westview Press, (1999). Google Scholar |
[30] |
A. Roca-Feltrer, J. R. Armstrong Schellenberg, L. Smith and I. Carneiro, A simple method for defining malaria seasonality,, Malaria J., 8 (2009).
doi: 10.1186/1475-2875-8-276. |
[31] |
R. Ross, The Prevention of Malaria,, 2nd edn., (1911). Google Scholar |
[32] |
S. Ruan, D. Xiao and J. C. Beier, On the delayed Ross-Macdonald model for malaria transmission,, Bull. Math. Biol., 70 (2008), 1098.
doi: 10.1007/s11538-007-9292-z. |
[33] |
D. L. Smith, J. Dushoff and F. E. McKenzie, The risk of a mosquito-borne infection in a heterogeneous environment,, PLoS Biology, 2 (2004), 1957. Google Scholar |
[34] |
H. L. Smith, Monotone Dynamical Systems: An Introduction to the Theory of Competitive and Cooperative Systems,, Mathematical Surveys and Monographs, (1995).
|
[35] |
H. L. Smith and P. Waltman, The Theory of the Chemostat,, Cambridge University Press, (1995).
doi: 10.1017/CBO9780511530043. |
[36] |
W. Wang and X.-Q. Zhao, Threshold dynamics for compartmental epidemic models in periodic environments,, J. Dyn. Differ. Equ., 20 (2008), 699.
doi: 10.1007/s10884-008-9111-8. |
[37] |
P. Weng and X.-Q. Zhao, Spatial dynamics of a nonlocal and delayed population model in a periodic habitat,, Discrete Contin. Dyn. Syst. Ser. A, 29 (2011), 343.
doi: 10.3934/dcds.2011.29.343. |
[38] |
J. Zhang, Z. Jin, G. Sun, X. Sun and S. Ruan, Modeling seasonal Rabies epidemics in China,, Bull. Math. Biol., 74 (2012), 1226.
doi: 10.1007/s11538-012-9720-6. |
[39] |
T. Zhang and Z. Teng, On a nonautonomous SEIRS model in epidemiology,, Bull. Math. Biol., 69 (2007), 2537.
doi: 10.1007/s11538-007-9231-z. |
[40] |
X.-Q. Zhao, Dynamical Systems in Population Biology,, Springer-Verlag, (2003).
doi: 10.1007/978-0-387-21761-1. |
show all references
References:
[1] |
S. Altizer, A. Dobson, P. Hosseini, P. Hudson, M. Pascual and P. Rohani, Seasonality and the dynamics of infectious diseases,, Ecol. Lett., 9 (2006), 467.
doi: 10.1111/j.1461-0248.2005.00879.x. |
[2] |
G. Aronsson and R. B. Kellogg, On a differential equation arising from compartmental analysis,, Math. Biosci., 38 (1973), 113.
doi: 10.1016/0025-5564(78)90021-4. |
[3] |
P. Auger, E. Kouokam, G. Sallet, M. Tchuente and B. Tsanou, The Ross-Macdonald model in a patchy environment,, Math. Biosci., 216 (2008), 123.
doi: 10.1016/j.mbs.2008.08.010. |
[4] |
N. Bacaër and S. Guernaoui, The epidemic threshold of vector-borne diseases with seasonality,, J. Math. Biol., 53 (2006), 421.
doi: 10.1007/s00285-006-0015-0. |
[5] |
N. Bacaër, Approximation of the basic reproduction number $R_0$ for vector-borne diseases with a periodic vector population,, Bull. Math. Biol., 69 (2007), 1067.
doi: 10.1007/s11538-006-9166-9. |
[6] |
Z. Bai and Y. Zhou, Threshold dynamics of a Bacillary Dysentery model with seasonal fluctuation,, Discrete Contin. Dyn. Syst., 15 (2011), 1.
doi: 10.3934/dcdsb.2011.15.1. |
[7] |
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), 550.
doi: 10.1016/j.jtbi.2009.02.016. |
[8] |
C. Costantini, S. G. Li, A. D. Torre, N. Sagnon, M. Coluzzi and C. E. Taylor, Density, survival and dispersal of Anopheles gambiae complex mosquitoes in a West African Sudan savanna village,, Med. Vet. Entomol., 10 (1996), 203.
doi: 10.1111/j.1365-2915.1996.tb00733.x. |
[9] |
M. H. Craig, I. Kleinschmidt, J. B. Nawn, D. Le Sueur and B. L. Sharp, Exploring 30 years of malaria case data in KwaZulu-Natal, South Africa: Part I. The impact of climatic factors,, Trop. Med. Int. Health, 9 (2004), 1247.
doi: 10.1111/j.1365-3156.2004.01340.x. |
[10] |
B. Dembele, A. Friedman and A.-A. Yakubu, Malaria model with periodic mosquito birth and death rates,, J. Biol. Dynam., 3 (2009), 430.
doi: 10.1080/17513750802495816. |
[11] |
D. Gao and S. Ruan, A multi-patch malaria model with logistic growth populations,, SIAM J. Appl. Math., 72 (2012), 819.
doi: 10.1137/110850761. |
[12] |
H. Gao, L. Wang, S. Liang, Y. Liu, S. Tong, J. Wang, Y. Li, X. Wang, H. Yang, J. Ma, L. Fang and W. Cao, Change in rainfall drives malaria re-emergence in Anhui province, China,, PLoS ONE, 7 (2012).
doi: 10.1371/journal.pone.0043686. |
[13] |
N. C. Grassly and C. Fraser, Seasonal infectious disease epidemiology,, Proc. R. Soc. B., 273 (2006), 2541.
doi: 10.1098/rspb.2006.3604. |
[14] |
J. K. Hale, Ordinary Differential Equations,, Wiley-Interscience, (1980).
|
[15] |
M. W. Hirsch, H. L. Smith and X.-Q. Zhao, Chain transitivity, attractivity, and strong repellors for semidynamical systems,, J. Dyn. Differ. Equ., 13 (2001), 107.
doi: 10.1023/A:1009044515567. |
[16] |
K. D. Lafferty, The ecology of climate change and infectious diseases,, Ecology, 90 (2009), 888.
doi: 10.1890/08-0079.1. |
[17] |
L. Liu, X.-Q. Zhao and Y. Zhou, A tuberculosis model with seasonality,, Bull. Math. Biol., 72 (2010), 931.
doi: 10.1007/s11538-009-9477-8. |
[18] |
X. Liu and X.-Q. Zhao, A periodic epidemic model with age structure in a patchy environment,, SIAM J. Appl. Math., 71 (2011), 1896.
doi: 10.1137/100813610. |
[19] |
J. Lou, Y. Lou and J. Wu, Threshold virus dynamics with impulsive antiretroviral drug effects,, J. Math. Biol., 65 (2012), 623.
doi: 10.1007/s00285-011-0474-9. |
[20] |
Y. Lou and X.-Q. Zhao, The periodic Ross-Macdonald model with diffusion and advection,, Appl. Anal., 89 (2010), 1067.
doi: 10.1080/00036810903437804. |
[21] |
Y. Lou and X.-Q. Zhao, A climate-based malaria transmission model with structured vector population,, SIAM J. Appl. Math., 70 (2010), 2023.
doi: 10.1137/080744438. |
[22] |
Y. Lou and X.-Q. Zhao, Modelling malaria control by introduction of larvivorous fish,, Bull. Math. Biol., 73 (2011), 2384.
doi: 10.1007/s11538-011-9628-6. |
[23] |
G. Macdonald, The analysis of sporozoite rate,, Trop. Dis. Bull., 49 (1952), 569. Google Scholar |
[24] |
G. Macdonald, Epidemiological basis of malaria control,, Bull. World Health Organ., 15 (1956), 613. Google Scholar |
[25] |
G. Macdonald, The Epidemiology and Control of Malaria,, Oxford University Press, (1957). Google Scholar |
[26] |
J. T. Midega, C. M. Mbogo, H. Mwambi, M. D. Wilson, G. Ojwang, J. M. Mwangangi, J. G. Nzovu, J. I. Githure, G. Yan and J. C. Beier, Estimating dispersal and survival of Anopheles gambiae and Anopheles funestus along the Kenyan coast by using mark-release-recapture methods,, J. Med. Entomol., 44 (2007), 923.
doi: {10.1603/0022-2585(2007)44[923:EDASOA]2.0.CO;2}. |
[27] |
R. S. Ostfeld, Climate change and the distribution and intensity of infectious diseases,, Ecology, 90 (2009), 903.
doi: 10.1890/08-0659.1. |
[28] |
P. E. Parham and E. Michael, Modeling the effects of weather and climate change on malaria transmission,, Environ. Health Perspect., 118 (2010), 620.
doi: 10.1289/ehp.0901256. |
[29] |
D. Rain, Eaters of the Dry Season: Circular Labor Migration in the West African Sahel,, Westview Press, (1999). Google Scholar |
[30] |
A. Roca-Feltrer, J. R. Armstrong Schellenberg, L. Smith and I. Carneiro, A simple method for defining malaria seasonality,, Malaria J., 8 (2009).
doi: 10.1186/1475-2875-8-276. |
[31] |
R. Ross, The Prevention of Malaria,, 2nd edn., (1911). Google Scholar |
[32] |
S. Ruan, D. Xiao and J. C. Beier, On the delayed Ross-Macdonald model for malaria transmission,, Bull. Math. Biol., 70 (2008), 1098.
doi: 10.1007/s11538-007-9292-z. |
[33] |
D. L. Smith, J. Dushoff and F. E. McKenzie, The risk of a mosquito-borne infection in a heterogeneous environment,, PLoS Biology, 2 (2004), 1957. Google Scholar |
[34] |
H. L. Smith, Monotone Dynamical Systems: An Introduction to the Theory of Competitive and Cooperative Systems,, Mathematical Surveys and Monographs, (1995).
|
[35] |
H. L. Smith and P. Waltman, The Theory of the Chemostat,, Cambridge University Press, (1995).
doi: 10.1017/CBO9780511530043. |
[36] |
W. Wang and X.-Q. Zhao, Threshold dynamics for compartmental epidemic models in periodic environments,, J. Dyn. Differ. Equ., 20 (2008), 699.
doi: 10.1007/s10884-008-9111-8. |
[37] |
P. Weng and X.-Q. Zhao, Spatial dynamics of a nonlocal and delayed population model in a periodic habitat,, Discrete Contin. Dyn. Syst. Ser. A, 29 (2011), 343.
doi: 10.3934/dcds.2011.29.343. |
[38] |
J. Zhang, Z. Jin, G. Sun, X. Sun and S. Ruan, Modeling seasonal Rabies epidemics in China,, Bull. Math. Biol., 74 (2012), 1226.
doi: 10.1007/s11538-012-9720-6. |
[39] |
T. Zhang and Z. Teng, On a nonautonomous SEIRS model in epidemiology,, Bull. Math. Biol., 69 (2007), 2537.
doi: 10.1007/s11538-007-9231-z. |
[40] |
X.-Q. Zhao, Dynamical Systems in Population Biology,, Springer-Verlag, (2003).
doi: 10.1007/978-0-387-21761-1. |
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