American Institute of Mathematical Sciences

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

Malaria model with stage-structured mosquitoes

Jia Li 1,

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.
Keywords: backward bifurcation., mosquito metamorphic stages, reproductive number, climate change, Malaria transmission, stage-structured model.
Mathematics Subject Classification: 39A11, 39A99, 92D25, 92D4.
Citation: Jia Li. Malaria model with stage-structured mosquitoes. Mathematical Biosciences & Engineering, 2011, 8 (3) : 753-768. doi: 10.3934/mbe.2011.8.753
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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.  Google Scholar

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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.  Google Scholar

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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. Google Scholar

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F. Dumortier, J. Llibre and J. C. Artés, "Qualitative Theory of Plannar Differential Systems," Springer-Verlag, Berlin, 2006.  Google Scholar

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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.  Google Scholar

[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.  Google Scholar

[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. Google Scholar

[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.  Google Scholar

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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.  Google Scholar

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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.  Google Scholar

[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.  Google Scholar

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Jia Li, Malaria models with partial immunity in humans, Math. Biol. Eng., 5 (2008), 789-801.  Google Scholar

[25]

Jia Li, Simple stage-structured models for wild and transgenic mosquito populations, J. Diff. Eqns. Appl., 15 (2009), 327-347.  Google Scholar

[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.  Google Scholar

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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.  Google Scholar

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G. MacDonald, "The Epidemiology and Control of Malaria," Oxford Univ. Press, London, 1957. Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[31]

Mosquito, 2010., Available from: \url{http://www.enchantedlearning.com/subjects/insects/mosquito}., 2010 ().   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[37]

R. Ross, "The Prevention of Malaria," John Murray, London, 1911. Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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. Google Scholar

[41]

WHO, "Malaria Fact Sheets," 2010., Available from: \url{http://www.who.int/mediacentre/factsheets/fs094/en/index.html}., 2010 ().   Google Scholar

[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.  Google Scholar

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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.  Google Scholar

show all references

References:
[1]

R. M. Anderson and R. M. May, "Infectious Diseases of Humans," Oxford Univ. Press, Oxford, 1991. Google Scholar

[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.  Google Scholar

[3]

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

[4]

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

[5]

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

[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.  Google Scholar

[7]

CDC, Malaria Fact Sheet, 2010., Available from: \url{http://www.cdc.gov/malaria/about/facts.html}., ().   Google Scholar

[8]

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

[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. Google Scholar

[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. Google Scholar

[11]

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

[12]

F. Dumortier, J. Llibre and J. C. Artés, "Qualitative Theory of Plannar Differential Systems," Springer-Verlag, Berlin, 2006.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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. Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[23]

H. Kielhöfer, "Bifurcation Theory: An Introduction with Applications to PDEs," Applied Mathematical Sciences, 156, Springer-Verlag, New York, 2004.  Google Scholar

[24]

Jia Li, Malaria models with partial immunity in humans, Math. Biol. Eng., 5 (2008), 789-801.  Google Scholar

[25]

Jia Li, Simple stage-structured models for wild and transgenic mosquito populations, J. Diff. Eqns. Appl., 15 (2009), 327-347.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[28]

G. MacDonald, "The Epidemiology and Control of Malaria," Oxford Univ. Press, London, 1957. Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[31]

Mosquito, 2010., Available from: \url{http://www.enchantedlearning.com/subjects/insects/mosquito}., 2010 ().   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[37]

R. Ross, "The Prevention of Malaria," John Murray, London, 1911. Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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. Google Scholar

[41]

WHO, "Malaria Fact Sheets," 2010., Available from: \url{http://www.who.int/mediacentre/factsheets/fs094/en/index.html}., 2010 ().   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

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