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Malaria model with stage-structured mosquitoes
1. | Department of Mathematical Sciences, University of Alabama in Huntsville, Huntsville, AL 35899 |
References:
[1] |
R. M. Anderson and R. M. May, "Infectious Diseases of Humans," Oxford Univ. Press, Oxford, 1991. |
[2] |
J. Arino, C. C. McCluskey and P. van den Driessche, Global results for an epidemic model with vaccination that exhibits backward bifurcation, SIAM J. Appl. Math., 64 (2003), 260-276.
doi: 10.1137/S0036139902413829. |
[3] |
J. L. Aron, Mathematical modeling of immunity to malaria, Math. Biosci., 90 (1988), 385-396.
doi: 10.1016/0025-5564(88)90076-4. |
[4] |
N. Becker, "Mosquitoes and Their Control," Kluwer Academic/Plenum, New York, 2003. |
[5] |
A. Berman and R. J. Plemmons, "Nonnegative Matrices in the Mathematical Sciences," Computer Science and Applied Mathematics, Acad. Press, New York-London, 1979. |
[6] |
F. Brauer, Backward bifurcations in simple vacicnation models, J. Math. Anal. Appl., 298 (2004), 418-431.
doi: 10.1016/j.jmaa.2004.05.045. |
[7] |
CDC, Malaria Fact Sheet, 2010. Available from: http://www.cdc.gov/malaria/about/facts.html. |
[8] |
A. N. Clements, "Development, Nutrition and Reproduction," The Biology of Mosquitoes, 1, CABI Publishing, 2000. |
[9] |
R. C. Dhiman, S. Pahwa and A. P. Dash, Climate change and malaria in India: Interplay between temperatures ans mosquitoes, Regional Health Forum, 12 (2008), 27-31. |
[10] |
K. Dietz, Mathematical models for transmission and control of malaria, in "Malaria: Principles and Practice of Malariology," II (eds. W. H. Wernsdorfer and Sir I. McGregor), Churchill Livingstoe, Edinburgh, (1988), 1091-1133. |
[11] |
K. Dietz, L. Molineaux and A. Thomas, A malaria model tested in the African savannah, Bull. World Health Org., 50 (1974), 347-357. |
[12] |
F. Dumortier, J. Llibre and J. C. Artés, "Qualitative Theory of Plannar Differential Systems," Springer-Verlag, Berlin, 2006. |
[13] |
J. Dushoff, W. Huang and C. Castillo-Chavez, Backwards bifurcations and catastrophe in simple models of fatal diseases, J. Math. Biol., 36 (1998), 227-248.
doi: 10.1007/s002850050099. |
[14] |
C. Dye, Intraspecific competition amongst larval Aedes aegypti: Food exploitation or chemical interference, Ecol. Entom., 7 (1982), 39-46.
doi: 10.1111/j.1365-2311.1982.tb00642.x. |
[15] |
D. A. Focks, D. G. Haile, E. Daniels and G. A. Mount, Dynamics life table model for Aedes aegypti (L.) (Diptera: Culicidae), J. Med. Entomol., 30 (1993), 1003-1017. |
[16] |
S. M. Garba, A. B. Gumel and M. R. Abu Bakar, Backward bifurcations in dengue transmission dynamics, Math. Biosci., 215 (2008), 11-25.
doi: 10.1016/j.mbs.2008.05.002. |
[17] |
R. M. Gleiser, J. Urrutia and D. E. Gorla, Effects of crowding on populations of Aedes albifasciatus larvae under laboratory conditions, Entomologia Experimentalis et Applicata, 95 (2000), 135-140.
doi: 10.1046/j.1570-7458.2000.00651.x. |
[18] |
E. A. Gould and S. Higgs, Impact of climate change and other factors on emerging arbovirus diseases, Transactions of the Royal Society of Tropical Medicine and Hygience, 103 (2009), 109-121.
doi: 10.1016/j.trstmh.2008.07.025. |
[19] |
K. P. Hadeler and P. van den Driessche, Backward bifurcation in epidemic control, Math. Biosci., 146 (1997), 15-35.
doi: 10.1016/S0025-5564(97)00027-8. |
[20] |
A. Hainesa, R. S. Kovatsa, D. Campbell-Lendrumb and C. Corvalan, Climate change and human health: Impacts, vulnerability and public health, Public Health, 120 (2006), 585-596.
doi: 10.1016/j.puhe.2006.01.002. |
[21] |
M. B. Hoshen and A. P. Morse, A weather-driven model of malaria transmission, Malaria Journal, 3 (2004), 32.
doi: 10.1186/1475-2875-3-32. |
[22] |
J. M. Hyman and Jia Li, The Reproductive number for an HIV model with differential infectivity and staged progression, Lin. Al. Appl., 398 (2005), 101-116.
doi: 10.1016/j.laa.2004.07.017. |
[23] |
H. Kielhöfer, "Bifurcation Theory: An Introduction with Applications to PDEs," Applied Mathematical Sciences, 156, Springer-Verlag, New York, 2004. |
[24] |
Jia Li, Malaria models with partial immunity in humans, Math. Biol. Eng., 5 (2008), 789-801. |
[25] |
Jia Li, Simple stage-structured models for wild and transgenic mosquito populations, J. Diff. Eqns. Appl., 15 (2009), 327-347. |
[26] |
Y. Lou and X.-Q. Zhao, A climate-based malaria transmission model with structured vector population, SIAM J. Appl. Math., 70 (2010), 2023-2044.
doi: 10.1137/080744438. |
[27] |
W. J. M. Martens, L. Niessen, J. Rotmans, T. H. Jetten and J. McMichael, Climate change and vector-borne diseases: A global modelling perspective, Global Environ Change, 5 (1995), 195-209.
doi: 10.1016/0959-3780(95)00051-O. |
[28] |
G. MacDonald, "The Epidemiology and Control of Malaria," Oxford Univ. Press, London, 1957. |
[29] |
P. Martens, R. S. Kovats, S. Nijhof, P. Vries, M. T. J. Livermore, D. J. Bradley, J. Cox and A. J. McMichael, Climate change and future populations at risk of malaria, Global Environ Change, 9 (1999), S89-S107.
doi: 10.1016/S0959-3780(99)00020-5. |
[30] |
L. Molineaux, The pros and cons of modeling malaria transmission, Trans. R. Soc. Trop. Med. Hyg., 79 (1985), 743-747.
doi: 10.1016/0035-9203(85)90107-5. |
[31] |
Mosquito, 2010. Available from: http://www.enchantedlearning.com/subjects/insects/mosquito. |
[32] |
G. A. Ngwa, Modelling the dynamics of endemic malaria in growing populations, Discrete and Continuous Dynamical Systems-Series B, 4 (2004), 1173-1202.
doi: 10.3934/dcdsb.2004.4.1173. |
[33] |
G. A. Ngwa, On the population dynamics of the malaria vector, Bull Math Biol., 68 (2006), 2161-2189.
doi: 10.1007/s11538-006-9104-x. |
[34] |
G. A. Ngwa and W. S. Shu, A mathematical model for endemic malaria with variable human and mosquito populations, Math. Comp. Modelling, 32 (2000), 747-763.
doi: 10.1016/S0895-7177(00)00169-2. |
[35] |
M. Otero, H. G. Solari and N. Schweigmann, A stochastic population dynamics model for Aedes aegypti: Formulation and application to a city with temperate climate, Bull. Math. Biol., 68 (2006), 1945-1974.
doi: 10.1007/s11538-006-9067-y. |
[36] |
K. P. Paaijmans, A. F. Read and M. B. Thomas, Understanding the link between malaria risk and climate, Proc. Natl. Acad. Sci. 106 (2009), 13844-13849.
doi: 10.1073/pnas.0903423106. |
[37] |
R. Ross, "The Prevention of Malaria," John Murray, London, 1911. |
[38] |
D. Ruiz, G. Poveda, I. D. Velez, M. L. Quinones, G. L. Rua, L. E. Velasquesz and J. S. Zuluaga, Modelling entomological-climatic interactions of Plasmodium falciparum malaria transmission in two Colombian endemic-regions: Contributions to a national malaria early warning system, Malaria Journal, 5 (2006), 66.
doi: 10.1186/1475-2875-5-66. |
[39] |
M. Safan, H. Heesterbeek and K. Dietz, The minimum effort required to eradicate infections in models with backward bifurcation, J. Math. Biol., 53 (2006), 703-718.
doi: 10.1007/s00285-006-0028-8. |
[40] |
W. H. Wernsdorfer, The importance of malaria in the world, in "Malaria," 1, Epidemology, Chemotherapy, Morphology, and Metabolism (ed. J. P. Kreier), Academic Press, New York, 1980. |
[41] |
WHO, "Malaria Fact Sheets," 2010. Available from: http://www.who.int/mediacentre/factsheets/fs094/en/index.html. |
[42] |
H. M. Yang, Malaria transmission model for different levels of acquired immunity and temperature-dependent parameters (vector), Rev Saude Publica, 34 (2000), 223-231.
doi: 10.1590/S0034-89102000000300003. |
[43] |
H. M. Yang and M. U. Ferreira, Assessing the effects of global warming and local social economic conditions on the malaria transmission, Rev Saude Publica, 34 (2000), 214-222.
doi: 10.1590/S0034-89102000000300002. |
show all references
References:
[1] |
R. M. Anderson and R. M. May, "Infectious Diseases of Humans," Oxford Univ. Press, Oxford, 1991. |
[2] |
J. Arino, C. C. McCluskey and P. van den Driessche, Global results for an epidemic model with vaccination that exhibits backward bifurcation, SIAM J. Appl. Math., 64 (2003), 260-276.
doi: 10.1137/S0036139902413829. |
[3] |
J. L. Aron, Mathematical modeling of immunity to malaria, Math. Biosci., 90 (1988), 385-396.
doi: 10.1016/0025-5564(88)90076-4. |
[4] |
N. Becker, "Mosquitoes and Their Control," Kluwer Academic/Plenum, New York, 2003. |
[5] |
A. Berman and R. J. Plemmons, "Nonnegative Matrices in the Mathematical Sciences," Computer Science and Applied Mathematics, Acad. Press, New York-London, 1979. |
[6] |
F. Brauer, Backward bifurcations in simple vacicnation models, J. Math. Anal. Appl., 298 (2004), 418-431.
doi: 10.1016/j.jmaa.2004.05.045. |
[7] |
CDC, Malaria Fact Sheet, 2010. Available from: http://www.cdc.gov/malaria/about/facts.html. |
[8] |
A. N. Clements, "Development, Nutrition and Reproduction," The Biology of Mosquitoes, 1, CABI Publishing, 2000. |
[9] |
R. C. Dhiman, S. Pahwa and A. P. Dash, Climate change and malaria in India: Interplay between temperatures ans mosquitoes, Regional Health Forum, 12 (2008), 27-31. |
[10] |
K. Dietz, Mathematical models for transmission and control of malaria, in "Malaria: Principles and Practice of Malariology," II (eds. W. H. Wernsdorfer and Sir I. McGregor), Churchill Livingstoe, Edinburgh, (1988), 1091-1133. |
[11] |
K. Dietz, L. Molineaux and A. Thomas, A malaria model tested in the African savannah, Bull. World Health Org., 50 (1974), 347-357. |
[12] |
F. Dumortier, J. Llibre and J. C. Artés, "Qualitative Theory of Plannar Differential Systems," Springer-Verlag, Berlin, 2006. |
[13] |
J. Dushoff, W. Huang and C. Castillo-Chavez, Backwards bifurcations and catastrophe in simple models of fatal diseases, J. Math. Biol., 36 (1998), 227-248.
doi: 10.1007/s002850050099. |
[14] |
C. Dye, Intraspecific competition amongst larval Aedes aegypti: Food exploitation or chemical interference, Ecol. Entom., 7 (1982), 39-46.
doi: 10.1111/j.1365-2311.1982.tb00642.x. |
[15] |
D. A. Focks, D. G. Haile, E. Daniels and G. A. Mount, Dynamics life table model for Aedes aegypti (L.) (Diptera: Culicidae), J. Med. Entomol., 30 (1993), 1003-1017. |
[16] |
S. M. Garba, A. B. Gumel and M. R. Abu Bakar, Backward bifurcations in dengue transmission dynamics, Math. Biosci., 215 (2008), 11-25.
doi: 10.1016/j.mbs.2008.05.002. |
[17] |
R. M. Gleiser, J. Urrutia and D. E. Gorla, Effects of crowding on populations of Aedes albifasciatus larvae under laboratory conditions, Entomologia Experimentalis et Applicata, 95 (2000), 135-140.
doi: 10.1046/j.1570-7458.2000.00651.x. |
[18] |
E. A. Gould and S. Higgs, Impact of climate change and other factors on emerging arbovirus diseases, Transactions of the Royal Society of Tropical Medicine and Hygience, 103 (2009), 109-121.
doi: 10.1016/j.trstmh.2008.07.025. |
[19] |
K. P. Hadeler and P. van den Driessche, Backward bifurcation in epidemic control, Math. Biosci., 146 (1997), 15-35.
doi: 10.1016/S0025-5564(97)00027-8. |
[20] |
A. Hainesa, R. S. Kovatsa, D. Campbell-Lendrumb and C. Corvalan, Climate change and human health: Impacts, vulnerability and public health, Public Health, 120 (2006), 585-596.
doi: 10.1016/j.puhe.2006.01.002. |
[21] |
M. B. Hoshen and A. P. Morse, A weather-driven model of malaria transmission, Malaria Journal, 3 (2004), 32.
doi: 10.1186/1475-2875-3-32. |
[22] |
J. M. Hyman and Jia Li, The Reproductive number for an HIV model with differential infectivity and staged progression, Lin. Al. Appl., 398 (2005), 101-116.
doi: 10.1016/j.laa.2004.07.017. |
[23] |
H. Kielhöfer, "Bifurcation Theory: An Introduction with Applications to PDEs," Applied Mathematical Sciences, 156, Springer-Verlag, New York, 2004. |
[24] |
Jia Li, Malaria models with partial immunity in humans, Math. Biol. Eng., 5 (2008), 789-801. |
[25] |
Jia Li, Simple stage-structured models for wild and transgenic mosquito populations, J. Diff. Eqns. Appl., 15 (2009), 327-347. |
[26] |
Y. Lou and X.-Q. Zhao, A climate-based malaria transmission model with structured vector population, SIAM J. Appl. Math., 70 (2010), 2023-2044.
doi: 10.1137/080744438. |
[27] |
W. J. M. Martens, L. Niessen, J. Rotmans, T. H. Jetten and J. McMichael, Climate change and vector-borne diseases: A global modelling perspective, Global Environ Change, 5 (1995), 195-209.
doi: 10.1016/0959-3780(95)00051-O. |
[28] |
G. MacDonald, "The Epidemiology and Control of Malaria," Oxford Univ. Press, London, 1957. |
[29] |
P. Martens, R. S. Kovats, S. Nijhof, P. Vries, M. T. J. Livermore, D. J. Bradley, J. Cox and A. J. McMichael, Climate change and future populations at risk of malaria, Global Environ Change, 9 (1999), S89-S107.
doi: 10.1016/S0959-3780(99)00020-5. |
[30] |
L. Molineaux, The pros and cons of modeling malaria transmission, Trans. R. Soc. Trop. Med. Hyg., 79 (1985), 743-747.
doi: 10.1016/0035-9203(85)90107-5. |
[31] |
Mosquito, 2010. Available from: http://www.enchantedlearning.com/subjects/insects/mosquito. |
[32] |
G. A. Ngwa, Modelling the dynamics of endemic malaria in growing populations, Discrete and Continuous Dynamical Systems-Series B, 4 (2004), 1173-1202.
doi: 10.3934/dcdsb.2004.4.1173. |
[33] |
G. A. Ngwa, On the population dynamics of the malaria vector, Bull Math Biol., 68 (2006), 2161-2189.
doi: 10.1007/s11538-006-9104-x. |
[34] |
G. A. Ngwa and W. S. Shu, A mathematical model for endemic malaria with variable human and mosquito populations, Math. Comp. Modelling, 32 (2000), 747-763.
doi: 10.1016/S0895-7177(00)00169-2. |
[35] |
M. Otero, H. G. Solari and N. Schweigmann, A stochastic population dynamics model for Aedes aegypti: Formulation and application to a city with temperate climate, Bull. Math. Biol., 68 (2006), 1945-1974.
doi: 10.1007/s11538-006-9067-y. |
[36] |
K. P. Paaijmans, A. F. Read and M. B. Thomas, Understanding the link between malaria risk and climate, Proc. Natl. Acad. Sci. 106 (2009), 13844-13849.
doi: 10.1073/pnas.0903423106. |
[37] |
R. Ross, "The Prevention of Malaria," John Murray, London, 1911. |
[38] |
D. Ruiz, G. Poveda, I. D. Velez, M. L. Quinones, G. L. Rua, L. E. Velasquesz and J. S. Zuluaga, Modelling entomological-climatic interactions of Plasmodium falciparum malaria transmission in two Colombian endemic-regions: Contributions to a national malaria early warning system, Malaria Journal, 5 (2006), 66.
doi: 10.1186/1475-2875-5-66. |
[39] |
M. Safan, H. Heesterbeek and K. Dietz, The minimum effort required to eradicate infections in models with backward bifurcation, J. Math. Biol., 53 (2006), 703-718.
doi: 10.1007/s00285-006-0028-8. |
[40] |
W. H. Wernsdorfer, The importance of malaria in the world, in "Malaria," 1, Epidemology, Chemotherapy, Morphology, and Metabolism (ed. J. P. Kreier), Academic Press, New York, 1980. |
[41] |
WHO, "Malaria Fact Sheets," 2010. Available from: http://www.who.int/mediacentre/factsheets/fs094/en/index.html. |
[42] |
H. M. Yang, Malaria transmission model for different levels of acquired immunity and temperature-dependent parameters (vector), Rev Saude Publica, 34 (2000), 223-231.
doi: 10.1590/S0034-89102000000300003. |
[43] |
H. M. Yang and M. U. Ferreira, Assessing the effects of global warming and local social economic conditions on the malaria transmission, Rev Saude Publica, 34 (2000), 214-222.
doi: 10.1590/S0034-89102000000300002. |
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