Abstract
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 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 population to uninfected population. We find that a travelling wave occurs; its speed is calculated numerically and estimated for the lower bound analytically.
Article PDF
Similar content being viewed by others
References
Anderson, R. M., & May, R. M. (1992). Infectious diseases of humans: dynamics and control (1st ed.). Oxford: Oxford University Press.
Aron, J. L. (1988). Mathematical modeling of immunity to malaria. Math. Biosci., 90, 385–396.
Bailey, N. J. T. (1975). The mathematical theory of infectious diseases and its application (1st ed.). London: Griffin.
Beier, J. C. (1998). Malaria parasites development in mosquitoes. Ann. Rev. Entomol., 43, 519–543.
Costantini, C., Gibson, G., Sagnon, N., Torre, A. D., Brady, J., & Coluzzi, M. (1996). Mosquito responses to carbon dioxide in a West African Sudan savanna. Med. Vet. Entomol., 10, 220–227.
Dietz, K., Molineaux, L., & Thomas, A. (1974). A malaria model tested in the African savannah. Bull. WHO, 50, 347–357.
Doolan, D. L., Dobano, C., & Baird, J. K. (2009). Acquired immunity to malaria. Clin. Microbiol. Rev., 22, 13–36.
Drakeley, C., Sutherland, C., Bouserna, J. T., Sauerwein, R. W., & Targett, G. A. T. (2006). The epidemiology of Plasmodium falciparum gametocytes: Weapons of mass dispersion. Trends Parasitol., 22, 424–430.
Fenton, A., & Rands, S. A. (2006). The impact of parasite manipulation and predator foraging behavior on predator-prey communities. Ecology, 87, 2832–2841.
Fillipe, J. A. N., Riley, E. M., Drakeley, C. J., Sutherland, C. J., & Ghani, A. C. (2007). Determination of the processes driving the acquisition of immunity to malaria using a mathematical transmission. PLOS Comput. Biol., 3, e255.
Glendinning, P. (1999). Stability, instability and chaos: an introduction to the theory of nonlinear differential equations (1st ed.). Cambridge: Cambridge University Press.
Guerra, C. A., Snow, R. W., & Hay, S. I. (2006). Mapping the global extent of malaria in 2005. Trends Parasitol., 22, 353–358.
Gupta, S., Swinton, J., & Anderson, R. M. (1994). 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.
Hosack, G. R., Rossignol, P. A., & van den Driessche, P. (2008). The control of vector-borne disease epidemics. J. Theor. Biol., 255, 16–25.
Kingsolver, J. G. (1987). Mosquito host choice and the epidemiology of malaria. Am. Nat., 130, 811–827.
Lacroix, R., Mukabana, W. R., Gouagna, L. C., & Koella, J. C. (2005). Malaria infection increases attractiveness of humans to mosquitoes. PLOS Biol., 3, e298.
Lewis, M., Renclawowicz, J., & van den Driessche, P. (2006). Travelling waves and spread rates for a West Nile virus model. Bull. Math. Biol., 68, 3–23.
Macdonald, G. (1952). The analysis of equilibrium in malaria. Trop. Dis. Bull, 49, 813–829.
Macdonald, G. (1957). The epidemiology and control of malaria (1st ed.). London: Oxford University Press.
Maidana, N. A., & Yang, H. M. (2008). Describing the geographic spread of dengue disease by travelling waves. Math. Biosci., 215, 64–77.
Mukabana, W. R., Takken, W., Killeen, G. I., & Knols, B. G. J. (2004). Allomonal effect of breath contributes to differential attractiveness of humans to the African malaria vector Anopheles gambiae. Malar. J., 3, 1–8.
Murray, J. D. (2002). Mathematical biology: I. An introduction (3rd ed.). New York: Springer.
Ross, R. (1916). An application of the theory of probabilities to the study of a priori pathometry. Proc. R. Soc. Lond. A, 92, 204–230.
Ruan, S., Xiao, D., & Beier, J. C. (2008). On the delayed Ross-Macdonald model for malaria transmission. Bull. Math. Biol., 70, 1098–1114.
Skinner, W., Tong, H., Pearson, T., Strauss, W., & Maibach, H. (1965). Human sweat components attractive to mosquitoes. Nature, 207, 661–662.
Snow, R. W., Guerra, C. A., Noor, A. M., Myint, H. Y., & Hay, S. I. (2005). The global distribution of clinical episodes of Plasmodium falciparum malaria. Nature, 434, 214–217.
Takken, W., & Knols, B. G. J. (1999). Odour-mediated behaviour of Afrotropical malaria mosquitoes. Ann. Rev. Entomol., 44, 131–157.
Wei, H. M., Li, X. Z., & Martcheva, M. (2008). An epidemic model of a vector-borne disease with direct transmission and time delay. J. Math. Anal. Appl., 342, 895–908.
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Chamchod, F., Britton, N.F. Analysis of a Vector-Bias Model on Malaria Transmission. Bull Math Biol 73, 639–657 (2011). https://doi.org/10.1007/s11538-010-9545-0
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s11538-010-9545-0