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On the Delayed Ross–Macdonald Model for Malaria Transmission

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Abstract

The feedback dynamics from mosquito to human and back to mosquito involve considerable time delays due to the incubation periods of the parasites. In this paper, taking explicit account of the incubation periods of parasites within the human and the mosquito, we first propose a delayed Ross–Macdonald model. Then we calculate the basic reproduction number R 0 and carry out some sensitivity analysis of R 0 on the incubation periods, that is, to study the effect of time delays on the basic reproduction number. It is shown that the basic reproduction number is a decreasing function of both time delays. Thus, prolonging the incubation periods in either humans or mosquitos (via medicine or control measures) could reduce the prevalence of infection.

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References

  • Anderson, R.M., May, R.M., 1991. Infectious Diseases of Humans: Dynamics and Control. Oxford University Press, Oxford.

    Google Scholar 

  • Aron, J.L., 1988. Mathematical modeling of immunity to malaria. Math. Biosci. 90, 385–396.

    Article  MathSciNet  Google Scholar 

  • Aron, J.L., May, R.M., 1982. The population dynamics of malaria. In: Anderson, R.M. (Ed.), Population Dynamics of Infectious Diseases: Theory and Applications, pp. 139–179. Chapman & Hall, London.

    Google Scholar 

  • Bailey, N.T.J., 1982. The Biomathematics of Malaria. Charles Griffin, London.

    MATH  Google Scholar 

  • Beier, J.C., 1998. Malaria parasite development in mosquitoes. Annu. Rev. Entomol. 43, 519–543.

    Article  Google Scholar 

  • Bray, R.S., Granham, P.C.C., 1982. The life cycle of primate malaria parasites. Br. Med. Bull. 38, 117–122.

    Google Scholar 

  • Burkot, T.R., Graves, P.M., Paru, R., Battistuta, D., Barnes, A., Saul, A., 1990. Variations in malaria transmission rates are not related to anopheline survivorship per feeding cycle. Am. J. Trop. Med. Hyg. 43, 321–327.

    Google Scholar 

  • Chitnis, N., Cushing, J.M., Hymas, J.M., 2006. Bifurcation analysis of mathematical model for malaria transmission. SIAM J. Appl. Math. 67, 24–45.

    Article  MATH  MathSciNet  Google Scholar 

  • Claborn, D.M., Masuoka, P.M., Klein, T.A., Hooper, T., Lee, A., Andre, R.G., 2002. A cost comparison of two malaria control methods in Kyunggi Province, Republic of Korea, using remote sensing and geographic information systems. Am. J. Trop. Med. Hyg. 66, 680–685.

    Google Scholar 

  • Craig, M.H., Snow, R.W., Le Sueur, D., 1999. Climate-based distribution model of malaria transmission in sub-Saharan Africa. Parasitol. Today 15, 105–111.

    Article  Google Scholar 

  • Dietz, K., 1988. Mathematical models for transmission and control of malaria. In: Wernsdorfer, W., McGregor, Y. (Eds.), Principles and Practice of Malariology, pp. 1091–1133. Churchill Livingstone, Edinburgh.

    Google Scholar 

  • Dietz, K., Molineaux, L., Thomas, A., 1974. A malaria model tested in the African savannah. Bull. World Health Organ. 50, 347–357.

    Google Scholar 

  • Dye, C., Hasibeder, G., 1986. Population dynamics of mosquito-borne disease: effects of flies which bite some people more frequently than others. Trans. Roy. Soc. Trop. Med. Hyg. 80, 69–77.

    Article  Google Scholar 

  • Garrett, L., 1996. The return of infectious disease. Foreign Aff. 75, 66–79.

    Article  Google Scholar 

  • Gu, W., Novak, R.J., 2005. Habitat-based modeling of impacts of mosquito larval interventions on entomological inoculation rates, incidence, and prevalence of malaria. Am. J. Trop. Med. Hyg. 73, 546–552.

    Google Scholar 

  • Gu, W., Killeen, G.F., Mbogo, C.M., Regens, J.L., Githure, J.I., Beier, J.C., 2003a. An individual-based model of Plasmodium falciparum malaria transmission on the coast of Kenya. Trans. Roy. Soc. Trop. Med. Hyg. 97, 43–50.

    Article  Google Scholar 

  • Gu, W., Mbogo, C.M., Githure, J.I., Regens, J.L., Killeen, G.F., Swalm, C.M., Yan, G., Beier, J.C., 2003b. Low recovery rates stabilize malaria endemicity in areas of low transmission in coastal Kenya. Acta Trop. 86, 71–81.

    Article  Google Scholar 

  • Gupta, S., Hill, A.V.S., 1995. Dynamic interactions in malaria: host heterogeneity meets parasite polymorphism. Proc. Roy. Soc. Lond. B 261, 271–277.

    Article  Google Scholar 

  • 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. Roy. Soc. Lond. B 256, 231–238.

    Article  Google Scholar 

  • Harada, M., Ikeshoji, T., Suguri, S., 1998. Studies on vector control by ‘Mosquito Candle’. In: Ishii, A., Nihei, N., Sasa, M. (Eds.), Malaria research in the Solomon Islands, pp. 120–125. Inter Group Co., Tokyo.

    Google Scholar 

  • Hasibeder, G., Dye, C., 1988. Population dynamics of mosquito-borne disease: persistence in a completely heterogeneous environments. Theor. Popul. Biol. 33, 31–53.

    Article  MATH  MathSciNet  Google Scholar 

  • Hay, S.I., Myers, M.F., Burke, D.S., Vaughn, D.W., Endy, T., Ananda, N., Shanks, G.D., Snow, R.W., Rogers, D.J., 2000. Etiology of interepidemic periods of mosquito-borne disease. PNAS 97, 9335–9339.

    Article  Google Scholar 

  • Hoshen, M.B., Morse, A.P., 2004. A weather-driven model of malaria transmission. Malaria J. 3, 32.

    Article  Google Scholar 

  • Ishikawa, H., Ishii, A., Nagai, N., Ohmae, H., Harada, M., Suguri, S., Leafasia, J., 2003. A mathematical model for the transmission of Plasmodium vivax malaria. Parasitol. Int. 52, 81–93.

    Article  Google Scholar 

  • Killeen, G.F., McKenzie, F.E., Foy, B.D., Schieffelin, C., Billingsley, P.F., Beier, J.C., 2000. A simplified model for predicting malaria entomological inoculation rates based on entomologic and parasitologic parameters relevant to control. Am. J. Trop. Med. Hyg. 62, 535–544.

    Google Scholar 

  • Koella, J.C., 1991. On the use of mathematical models of malaria transmission. Acta Trop. 49, 1–25.

    Article  Google Scholar 

  • Koella, J.C., Antia, R., 2003. Epidemiological models for the spread of anti-malarial resistance. Malaria J. 2, 3.

    Article  Google Scholar 

  • Koella, J.C., Boëte, C., 2003. A model for the coevolution of immunity and immune evasion in vector-borne diseases with implications for the epidemiology of malaria. Am. Nat. 161, 698–707.

    Article  Google Scholar 

  • Kreier, J.P., 1980. Malaria. Epidemiology, Chemotherapy, Morphology, and Metabolism, vol. 1. Academic, New York.

    Google Scholar 

  • Le Menach, A., McKenzie, F.E., Flahault, A., Smith, D.L., 2005. The unexpected importance of mosquito oviposition behaviour for malaria: non-productive larval habitats can be sources for malaria transmission. Malaria J. 4, 23.

    Article  Google Scholar 

  • Lopez, L.F., Coutinho, F.A.B., Burattini, M.N., Massad, E., 2002. Threshold conditions for infection persistence in complex host-vectors interactions. C.R. Biol. 325, 1073–1084.

    Article  Google Scholar 

  • Lotka, A.J., 1923. Contribution of the analysis of malaria epidemiology. Am. J. Hyg. 3(suppl. 1), 1–21.

    Google Scholar 

  • Lysenko, A.J., Beljaev, A.E., Rybalka, V.M., 1977. Population studies of Plasmodium vivax. 1. The theory of polymorphism of sporozoites and epidemiological phenomena of tertian malaria. Bull. World Health Organ. 55, 541–549.

    Google Scholar 

  • Macdonald, G., 1952. The analysis of sporozoite rate. Trop. Dis. Bull. 49, 569–585.

    Google Scholar 

  • Macdonald, G., 1956. Epidemiological basis of malaria control. Bull. World Health Organ. 15, 613–626.

    Google Scholar 

  • Macdonald, G., 1957. The Epidemiology and Control of Malaria. Oxford University Press, London.

    Google Scholar 

  • Martens, W.J.M., Niessen, L.W., Rotmans, J., Mcmichael, A.J., 1995. Potential impacts of global climate change on malaria risk. Environ. Health Perspect. 103, 458–464.

    Article  Google Scholar 

  • McKenzie, F.E., 2000. Why model malaria? Parasitol. Today 16, 511–516.

    Article  Google Scholar 

  • McKenzie, F.E., Bossert, W.H., 2005. An integrated model of Plasmodium falciparum dynamics. J. Theor. Biol. 232, 411–426.

    MathSciNet  Google Scholar 

  • McKenzie, F.E., Killeen, G.F., Beier, J.C., Bossert, W.H., 2001. Seasonality parasite diversity, and local extinctions in Plasmodium falciparum malaria. Ecology 82, 2673–2681.

    Google Scholar 

  • Molineaux, L., Gramiccia, G., 1980. The Garki Project. WHO, Geneva.

    Google Scholar 

  • Newman, R.D., Parise, M.E., Barber, A.M., Steketee, R.W., 2004. Malaria-related deaths among U.S. travelers, 1963–2001. Ann. Int. Med. 141, 547–555.

    Google Scholar 

  • Ngwa, G.A., 2006. On the population dynamics of the malaria vector. Bull. Math. Biol. 68, 2161–2189.

    Article  MathSciNet  Google Scholar 

  • Oaks, Jr. S.C., Mitchell, V.S., Pearson, G.M., Carpenter, C.C.J., 1991. Malaria: obstacles and opportunities. A report of the Committee for the Study on Malaria Prevention and Control: Status Review and Alternative Strategies, Division of International Health, Institute of Medicine, Washington, DC, National Academy Press.

  • Rodriguez, D.J., Torres-Sorando, L., 2001. Models of infectious diseases in spatially heterogeneous environments. Bull. Math. Biol. 63, 547–571.

    Article  Google Scholar 

  • Rogers, D.J., Randolph, S.E., 2000. The global spread of malaria in a future, warmer world. Science 289, 1763–1766.

    Article  Google Scholar 

  • Rogers, D.J., Randolph, S.E., Snow, R.W., Hay, S.I., 2002. Satellite imagery in the study and forecast of malaria. Nature 415, 710–715.

    Article  Google Scholar 

  • Ross, R., 1911. The Prevention of Malaria, 2nd edn. Murray, London.

    Google Scholar 

  • Ruan, S., 2006. Spatial-temporal dynamics in nonlocal epidemiological models. In: Iwasa, Y., Sato, K., Takeuchi, Y. (Eds.), Mathematics for Life Science and Medicine, vol. 2, pp. 99–122. Springer, New York.

    Google Scholar 

  • Ruan, S., Wei, J., 2003. On the zeros of transcendental functions with applications to stability delay differential equations with two delays. Dyn. Contin. Discrete Impuls. Syst. Ser. A 10, 863–874.

    MATH  MathSciNet  Google Scholar 

  • Ruan, S., Xiao, D., 2004. Stability of steady states and existence of traveling waves in a vector disease model. Proc. Roy. Soc. Edinb. Sect. A Math. 134, 991–1011.

    Article  MATH  MathSciNet  Google Scholar 

  • Smith, H.L., 1995. Monotone Dynamical Systems, An Introduction to the Theory of Competitive and Cooperative Systems. Am. Math. Soc., Providence.

    MATH  Google Scholar 

  • Smith, D.L., McKenzie, F.E., 2004. Statics and dynamics of malaria infection in Anopheles mosquitoes. Malaria J. 3, 13.

    Article  Google Scholar 

  • Smith, D.L., Dushoff, J., McKenzie, F.E., 2004. The risk of a mosquito-borne infection in a heterogeneous environment. PLoS Biol. 2, 1957–1964.

    Article  Google Scholar 

  • Smith, D.L., Dushoff, J., Snow, R.W., Hay, S.I., 2005. The entomological inoculation rate, Plasmodium falciparum infection in African children. Nature 438, 492–495.

    Article  Google Scholar 

  • Teklehaimanot, H.D., Schwartz, J., Teklehaimanot, A., Lipsitch, M., 2004. Weather-based prediction of Plasmodium falciparum malaria in epidemic-prone regions of Ethiopia II. Weather-based prediction systems perform comparably to early detection systems in identifying times for interventions. Malaria J. 3, 44.

    Article  Google Scholar 

  • Torres-Sorando, L., Rodriguez, D.J., 1997. Models of spatio-temporal dynamics in malaria. Ecol. Model. 104, 231–240.

    Article  Google Scholar 

  • Walter Reed Army Institute of Research, 1998. Addressing Emerging Infectious Disease Threats: A Strategic Plan for the Department of Defense, Washington DC, Division of Preventive Medicine.

  • World Health Organization, 2005. Roll back malaria: what is malaria? http://mosquito.who.int/cmc_upload/0/000/015/372/RBMInfosheet_1.htm.

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Authors and Affiliations

  1. Department of Mathematics, University of Miami, P.O. Box 249085, Coral Gables, FL, 33124-4250, USA

    Shigui Ruan

  2. Department of Mathematics, Shanghai Jiao Tong University, Shanghai, 200030, China

    Dongmei Xiao

  3. Department of Epidemiology and Public Health, University of Miami, Miami, FL, 33177, USA

    John C. Beier

Authors
  1. Shigui Ruan
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  2. Dongmei Xiao
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  3. John C. Beier
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Correspondence to Shigui Ruan.

Additional information

S. Ruan’s research was partially supported by NIH grants P20RR020770-02 and R01GM083607-01, NSF grant DMS-0715772. D. Xiao’s research was supported by the National Natural Science Fund (NNSF) of China.

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Ruan, S., Xiao, D. & Beier, J.C. On the Delayed Ross–Macdonald Model for Malaria Transmission. Bull. Math. Biol. 70, 1098–1114 (2008). https://doi.org/10.1007/s11538-007-9292-z

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  • DOI: https://doi.org/10.1007/s11538-007-9292-z

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