zbMATH — the first resource for mathematics

Analysis of a vector-bias model on malaria transmission. (English) Zbl 1225.92030
Summary: We incorporate a vector-bias term into a malaria-transmission model to account for the greater attractiveness of infectious humans to mosquitoes in terms of differing probabilities that a mosquito arriving at a human at random picks that human depending on whether he is infectious or susceptible. We prove that transcritical bifurcation occurs at the basic reproductive ratio equalling 1 by projecting the flow onto the extended centre manifold. We next study the dynamics of the system when the incubation time of malaria parasites in mosquitoes is included, and find that the longer incubation time reduces the prevalence of malaria. Also, we incorporate a random movement of mosquitoes as a diffusion term and a chemically directed movement of mosquitoes to humans expressed in terms of sweat and body odour as a chemotaxis term to study the propagation of infected populations to uninfected populations. We find that a travelling wave occurs; its speed is calculated numerically and estimated for the lower bound analytically.

92C60 Medical epidemiology
34C60 Qualitative investigation and simulation of ordinary differential equation models
35Q92 PDEs in connection with biology, chemistry and other natural sciences
37N25 Dynamical systems in biology
Full Text: DOI
[1] Anderson, R. M.; May, R. M., Infectious diseases of humans: dynamics and control (1992), Oxford: Oxford University Press, Oxford
[2] Aron, J. L., Mathematical modeling of immunity to malaria, Math. Biosci., 90, 385-396 (1988) · Zbl 0651.92018
[3] Bailey, N. J. T., The mathematical theory of infectious diseases and its application (1975), London: Griffin, London · Zbl 0334.92024
[4] Beier, J. C., Malaria parasites development in mosquitoes, Ann. Rev. Entomol., 43, 519-543 (1998)
[5] Costantini, C.; Gibson, G.; Sagnon, N.; Torre, A. D.; Brady, J.; Coluzzi, M., Mosquito responses to carbon dioxide in a West African Sudan savanna, Med. Vet. Entomol., 10, 220-227 (1996)
[6] Dietz, K.; Molineaux, L.; Thomas, A., A malaria model tested in the African savannah, Bull. WHO, 50, 347-357 (1974)
[7] Doolan, D. L.; Dobano, C.; Baird, J. K., Acquired immunity to malaria, Clin. Microbiol. Rev., 22, 13-36 (2009)
[8] Drakeley, C.; Sutherland, C.; Bouserna, J. T.; Sauerwein, R. W.; Targett, G. A. T., The epidemiology of Plasmodium falciparum gametocytes: Weapons of mass dispersion, Trends Parasitol., 22, 424-430 (2006)
[9] Fenton, A.; Rands, S. A., The impact of parasite manipulation and predator foraging behavior on predator-prey communities, Ecology, 87, 2832-2841 (2006)
[10] Fillipe, J. A. N.; Riley, E. M.; Drakeley, C. J.; Sutherland, C. J.; Ghani, A. C., Determination of the processes driving the acquisition of immunity to malaria using a mathematical transmission, PLOS Comput. Biol., 3, e255 (2007)
[11] Glendinning, P., Stability, instability and chaos: an introduction to the theory of nonlinear differential equations (1999), Cambridge: Cambridge University Press, Cambridge
[12] Guerra, C. A.; Snow, R. W.; Hay, S. I., Mapping the global extent of malaria in 2005, Trends Parasitol., 22, 353-358 (2006)
[13] Gupta, S.; Swinton, J.; Anderson, R. M., Theoretical studies of the effects of heterogeneity in the parasite population on the transmission dynamics of malaria, Proc. R. Soc. Lond. B, 256, 231-238 (1994)
[14] Hosack, G. R.; Rossignol, P. A.; van den Driessche, P., The control of vector-borne disease epidemics, J. Theor. Biol., 255, 16-25 (2008) · Zbl 1400.92488
[15] Kingsolver, J. G., Mosquito host choice and the epidemiology of malaria, Am. Nat., 130, 811-827 (1987)
[16] Lacroix, R.; Mukabana, W. R.; Gouagna, L. C.; Koella, J. C., Malaria infection increases attractiveness of humans to mosquitoes, PLOS Biol., 3, e298 (2005)
[17] Lewis, M.; Renclawowicz, J.; van den Driessche, P., Travelling waves and spread rates for a West Nile virus model, Bull. Math. Biol., 68, 3-23 (2006) · Zbl 1334.92414
[18] Macdonald, G., The analysis of equilibrium in malaria, Trop. Dis. Bull, 49, 813-829 (1952)
[19] Macdonald, G., The epidemiology and control of malaria (1957), London: Oxford University Press, London
[20] Maidana, N. A.; Yang, H. M., Describing the geographic spread of dengue disease by travelling waves, Math. Biosci., 215, 64-77 (2008) · Zbl 1156.92037
[21] Mukabana, W. R.; Takken, W.; Killeen, G. I.; Knols, B. G. J., Allomonal effect of breath contributes to differential attractiveness of humans to the African malaria vector Anopheles gambiae, Malar. J., 3, 1-8 (2004)
[22] Murray, J. D., Mathematical biology: I. An introduction (2002), New York: Springer, New York
[23] Ross, R., An application of the theory of probabilities to the study of a priori pathometry, Proc. R. Soc. Lond. A, 92, 204-230 (1916) · JFM 46.0789.01
[24] Ruan, S.; Xiao, D.; Beier, J. C., On the delayed Ross-Macdonald model for malaria transmission, Bull. Math. Biol., 70, 1098-1114 (2008) · Zbl 1142.92040
[25] Skinner, W.; Tong, H.; Pearson, T.; Strauss, W.; Maibach, H., Human sweat components attractive to mosquitoes, Nature, 207, 661-662 (1965)
[26] Snow, R. W.; Guerra, C. A.; Noor, A. M.; Myint, H. Y.; Hay, S. I., The global distribution of clinical episodes of Plasmodium falciparum malaria, Nature, 434, 214-217 (2005)
[27] Takken, W.; Knols, B. G. J., Odour-mediated behaviour of Afrotropical malaria mosquitoes, Ann. Rev. Entomol., 44, 131-157 (1999)
[28] Wei, H. M.; Li, X. Z.; Martcheva, M., An epidemic model of a vector-borne disease with direct transmission and time delay, J. Math. Anal. Appl., 342, 895-908 (2008) · Zbl 1146.34059
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.