2011, 8(3): 753-768. doi: 10.3934/mbe.2011.8.753

Malaria model with stage-structured mosquitoes

1. 

Department of Mathematical Sciences, University of Alabama in Huntsville, Huntsville, AL 35899

Received  November 2010 Revised  November 2010 Published  June 2011

A simple SEIR model for malaria transmission dynamics is formulated as our baseline model. The metamorphic stages in the mosquito population are then included and a simple stage-structured mosquito population model is introduced, where the mosquito population is divided into two classes, with all three aquatic stages in one class and all adults in the other class, to keep the model tractable in mathematical analysis. After a brief investigation of this simple stage-structured mosquito model, it is incorporated into the baseline model to formulate a stage-structured malaria model. A basic analysis for the stage-structured malaria model is provided and it is shown that a theoretical framework can be built up for further studies on the impact of environmental or climate change on the malaria transmission. It is also shown that both the baseline and the stage-structured malaria models undergo backward bifurcations.
Citation: Jia Li. Malaria model with stage-structured mosquitoes. Mathematical Biosciences & Engineering, 2011, 8 (3) : 753-768. doi: 10.3934/mbe.2011.8.753
References:
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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.

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J. L. Aron, Mathematical modeling of immunity to malaria, Math. Biosci., 90 (1988), 385-396. doi: 10.1016/0025-5564(88)90076-4.

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N. Becker, "Mosquitoes and Their Control," Kluwer Academic/Plenum, New York, 2003.

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A. Berman and R. J. Plemmons, "Nonnegative Matrices in the Mathematical Sciences," Computer Science and Applied Mathematics, Acad. Press, New York-London, 1979.

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F. Brauer, Backward bifurcations in simple vacicnation models, J. Math. Anal. Appl., 298 (2004), 418-431. doi: 10.1016/j.jmaa.2004.05.045.

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CDC, Malaria Fact Sheet, 2010. Available from: http://www.cdc.gov/malaria/about/facts.html.

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A. N. Clements, "Development, Nutrition and Reproduction," The Biology of Mosquitoes, 1, CABI Publishing, 2000.

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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.

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K. Dietz, L. Molineaux and A. Thomas, A malaria model tested in the African savannah, Bull. World Health Org., 50 (1974), 347-357.

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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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