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
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.

[1]

Jinping Fang, Guang Lin, Hui Wan. Analysis of a stage-structured dengue model. Discrete and Continuous Dynamical Systems - B, 2018, 23 (9) : 4045-4061. doi: 10.3934/dcdsb.2018125
[2]

Qing Zhu, Huaqin Peng, Xiaoxiao Zheng, Huafeng Xiao. Bifurcation analysis of a stage-structured predator-prey model with prey refuge. Discrete and Continuous Dynamical Systems - S, 2019, 12 (7) : 2195-2209. doi: 10.3934/dcdss.2019141
[3]

Shangzhi Li, Shangjiang Guo. Dynamics of a stage-structured population model with a state-dependent delay. Discrete and Continuous Dynamical Systems - B, 2020, 25 (9) : 3523-3551. doi: 10.3934/dcdsb.2020071
[4]

Ruijun Zhao, Jemal Mohammed-Awel. A mathematical model studying mosquito-stage transmission-blocking vaccines. Mathematical Biosciences & Engineering, 2014, 11 (5) : 1229-1245. doi: 10.3934/mbe.2014.11.1229
[5]

Mugen Huang, Moxun Tang, Jianshe Yu, Bo Zheng. A stage structured model of delay differential equations for Aedes mosquito population suppression. Discrete and Continuous Dynamical Systems, 2020, 40 (6) : 3467-3484. doi: 10.3934/dcds.2020042
[6]

Shangzhi Li, Shangjiang Guo. Dynamics of a two-species stage-structured model incorporating state-dependent maturation delays. Discrete and Continuous Dynamical Systems - B, 2017, 22 (4) : 1393-1423. doi: 10.3934/dcdsb.2017067
[7]

Thazin Aye, Guanyu Shang, Ying Su. On a stage-structured population model in discrete periodic habitat: III. unimodal growth and delay effect. Discrete and Continuous Dynamical Systems - B, 2021, 26 (4) : 1763-1781. doi: 10.3934/dcdsb.2021005
[8]

Hui Wan, Jing-An Cui. A model for the transmission of malaria. Discrete and Continuous Dynamical Systems - B, 2009, 11 (2) : 479-496. doi: 10.3934/dcdsb.2009.11.479
[9]

Guirong Jiang, Qishao Lu, Linping Peng. Impulsive Ecological Control Of A Stage-Structured Pest Management System. Mathematical Biosciences & Engineering, 2005, 2 (2) : 329-344. doi: 10.3934/mbe.2005.2.329
[10]

Bruno Buonomo, Deborah Lacitignola. On the stabilizing effect of cannibalism in stage-structured population models. Mathematical Biosciences & Engineering, 2006, 3 (4) : 717-731. doi: 10.3934/mbe.2006.3.717
[11]

Wei Feng, Michael T. Cowen, Xin Lu. Coexistence and asymptotic stability in stage-structured predator-prey models. Mathematical Biosciences & Engineering, 2014, 11 (4) : 823-839. doi: 10.3934/mbe.2014.11.823
[12]

Timothy C. Reluga, Jan Medlock, Alison Galvani. The discounted reproductive number for epidemiology. Mathematical Biosciences & Engineering, 2009, 6 (2) : 377-393. doi: 10.3934/mbe.2009.6.377
[13]

Manoj Atolia, Prakash Loungani, Helmut Maurer, Willi Semmler. Optimal control of a global model of climate change with adaptation and mitigation. Mathematical Control and Related Fields, 2022  doi: 10.3934/mcrf.2022009
[14]

Shangbing Ai, Jia Li, Jianshe Yu, Bo Zheng. Stage-structured models for interactive wild and periodically and impulsively released sterile mosquitoes. Discrete and Continuous Dynamical Systems - B, 2022, 27 (6) : 3039-3052. doi: 10.3934/dcdsb.2021172
[15]

Sabri Bensid, Jesús Ildefonso Díaz. On the exact number of monotone solutions of a simplified Budyko climate model and their different stability. Discrete and Continuous Dynamical Systems - B, 2019, 24 (3) : 1033-1047. doi: 10.3934/dcdsb.2019005
[16]

G. Buffoni, S. Pasquali, G. Gilioli. A stochastic model for the dynamics of a stage structured population. Discrete and Continuous Dynamical Systems - B, 2004, 4 (3) : 517-525. doi: 10.3934/dcdsb.2004.4.517
[17]

Liming Cai, Jicai Huang, Xinyu Song, Yuyue Zhang. Bifurcation analysis of a mosquito population model for proportional releasing sterile mosquitoes. Discrete and Continuous Dynamical Systems - B, 2019, 24 (11) : 6279-6295. doi: 10.3934/dcdsb.2019139
[18]

Lizhong Qiang, Bin-Guo Wang. An almost periodic malaria transmission model with time-delayed input of vector. Discrete and Continuous Dynamical Systems - B, 2017, 22 (4) : 1525-1546. doi: 10.3934/dcdsb.2017073
[19]

Jemal Mohammed-Awel, Ruijun Zhao, Eric Numfor, Suzanne Lenhart. Management strategies in a malaria model combining human and transmission-blocking vaccines. Discrete and Continuous Dynamical Systems - B, 2017, 22 (3) : 977-1000. doi: 10.3934/dcdsb.2017049
[20]

Yunfei Lv, Rong Yuan, Yuan He. Wavefronts of a stage structured model with state--dependent delay. Discrete and Continuous Dynamical Systems, 2015, 35 (10) : 4931-4954. doi: 10.3934/dcds.2015.35.4931

2018 Impact Factor: 1.313

Tools

Metrics

  • PDF downloads (65)
  • HTML views (0)
  • Cited by (0)

Other articles
by authors

[Back to Top]

Copyright © 2022 American Institute of Mathematical Sciences | Privacy Policy