zbMATH — the first resource for mathematics

Examples
Geometry Search for the term Geometry in any field. Queries are case-independent.
Funct* Wildcard queries are specified by * (e.g. functions, functorial, etc.). Otherwise the search is exact.
"Topological group" Phrases (multi-words) should be set in "straight quotation marks".
au: Bourbaki & ti: Algebra Search for author and title. The and-operator & is default and can be omitted.
Chebyshev | Tschebyscheff The or-operator | allows to search for Chebyshev or Tschebyscheff.
"Quasi* map*" py: 1989 The resulting documents have publication year 1989.
so: Eur* J* Mat* Soc* cc: 14 Search for publications in a particular source with a Mathematics Subject Classification code (cc) in 14.
"Partial diff* eq*" ! elliptic The not-operator ! eliminates all results containing the word elliptic.
dt: b & au: Hilbert The document type is set to books; alternatively: j for journal articles, a for book articles.
py: 2000-2015 cc: (94A | 11T) Number ranges are accepted. Terms can be grouped within (parentheses).
la: chinese Find documents in a given language. ISO 639-1 language codes can also be used.

Operators
a & b logic and
a | b logic or
!ab logic not
abc* right wildcard
"ab c" phrase
(ab c) parentheses
Fields
any anywhere an internal document identifier
au author, editor ai internal author identifier
ti title la language
so source ab review, abstract
py publication year rv reviewer
cc MSC code ut uncontrolled term
dt document type (j: journal article; b: book; a: book article)
On the delayed Ross-Macdonald model for malaria transmission. (English) Zbl 1142.92040
Summary: The feedback dynamics from mosquitos to humans and back to mosquitos 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 humans and the mosquitos, we first propose a delayed Ross-Macdonald model [R. Ross, The prevention of malaria. 2nd ed., Murray, London (1911); G. Macdonald, The epidemiology and control of malaria. Oxford Univ. Press (1957)]. 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.
MSC:
92D30Epidemiology
34K60Qualitative investigation and simulation of models
References:
[1]Anderson, R.M., May, R.M., 1991. Infectious Diseases of Humans: Dynamics and Control. Oxford University Press, Oxford.
[2]Aron, J.L., 1988. Mathematical modeling of immunity to malaria. Math. Biosci. 90, 385–396. · Zbl 0651.92018 · doi:10.1016/0025-5564(88)90076-4
[3]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.
[4]Bailey, N.T.J., 1982. The Biomathematics of Malaria. Charles Griffin, London.
[5]Beier, J.C., 1998. Malaria parasite development in mosquitoes. Annu. Rev. Entomol. 43, 519–543. · doi:10.1146/annurev.ento.43.1.519
[6]Bray, R.S., Granham, P.C.C., 1982. The life cycle of primate malaria parasites. Br. Med. Bull. 38, 117–122.
[7]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.
[8]Chitnis, N., Cushing, J.M., Hymas, J.M., 2006. Bifurcation analysis of mathematical model for malaria transmission. SIAM J. Appl. Math. 67, 24–45. · Zbl 1107.92047 · doi:10.1137/050638941
[9]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.
[10]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. · doi:10.1016/S0169-4758(99)01396-4
[11]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.
[12]Dietz, K., Molineaux, L., Thomas, A., 1974. A malaria model tested in the African savannah. Bull. World Health Organ. 50, 347–357.
[13]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. · doi:10.1016/0035-9203(86)90199-9
[14]Garrett, L., 1996. The return of infectious disease. Foreign Aff. 75, 66–79. · doi:10.2307/20047468
[15]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.
[16]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. · doi:10.1016/S0035-9203(03)90018-6
[17]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. · doi:10.1016/S0001-706X(03)00020-2
[18]Gupta, S., Hill, A.V.S., 1995. Dynamic interactions in malaria: host heterogeneity meets parasite polymorphism. Proc. Roy. Soc. Lond. B 261, 271–277. · doi:10.1098/rspb.1995.0147
[19]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. · doi:10.1098/rspb.1994.0075
[20]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.
[21]Hasibeder, G., Dye, C., 1988. Population dynamics of mosquito-borne disease: persistence in a completely heterogeneous environments. Theor. Popul. Biol. 33, 31–53. · Zbl 0647.92015 · doi:10.1016/0040-5809(88)90003-2
[22]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. · doi:10.1073/pnas.97.16.9335
[23]Hoshen, M.B., Morse, A.P., 2004. A weather-driven model of malaria transmission. Malaria J. 3, 32. · doi:10.1186/1475-2875-3-32
[24]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. · doi:10.1016/S1383-5769(02)00084-3
[25]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.
[26]Koella, J.C., 1991. On the use of mathematical models of malaria transmission. Acta Trop. 49, 1–25. · doi:10.1016/0001-706X(91)90026-G
[27]Koella, J.C., Antia, R., 2003. Epidemiological models for the spread of anti-malarial resistance. Malaria J. 2, 3. · doi:10.1186/1475-2875-2-3
[28]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. · doi:10.1086/374202
[29]Kreier, J.P., 1980. Malaria. Epidemiology, Chemotherapy, Morphology, and Metabolism, vol. 1. Academic, New York.
[30]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. · doi:10.1186/1475-2875-4-23
[31]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. · doi:10.1016/S1631-0691(02)01534-2
[32]Lotka, A.J., 1923. Contribution of the analysis of malaria epidemiology. Am. J. Hyg. 3(suppl. 1), 1–21.
[33]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.
[34]Macdonald, G., 1952. The analysis of sporozoite rate. Trop. Dis. Bull. 49, 569–585.
[35]Macdonald, G., 1956. Epidemiological basis of malaria control. Bull. World Health Organ. 15, 613–626.
[36]Macdonald, G., 1957. The Epidemiology and Control of Malaria. Oxford University Press, London.
[37]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. · doi:10.2307/3432584
[38]McKenzie, F.E., 2000. Why model malaria? Parasitol. Today 16, 511–516. · doi:10.1016/S0169-4758(00)01789-0
[39]McKenzie, F.E., Bossert, W.H., 2005. An integrated model of Plasmodium falciparum dynamics. J. Theor. Biol. 232, 411–426.
[40]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.
[41]Molineaux, L., Gramiccia, G., 1980. The Garki Project. WHO, Geneva.
[42]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.
[43]Ngwa, G.A., 2006. On the population dynamics of the malaria vector. Bull. Math. Biol. 68, 2161–2189. · doi:10.1007/s11538-006-9104-x
[44]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.
[45]Rodriguez, D.J., Torres-Sorando, L., 2001. Models of infectious diseases in spatially heterogeneous environments. Bull. Math. Biol. 63, 547–571. · doi:10.1006/bulm.2001.0231
[46]Rogers, D.J., Randolph, S.E., 2000. The global spread of malaria in a future, warmer world. Science 289, 1763–1766. · doi:10.1126/science.289.5478.391b
[47]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. · doi:10.1038/415710a
[48]Ross, R., 1911. The Prevention of Malaria, 2nd edn. Murray, London.
[49]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.
[50]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.
[51]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. · Zbl 1065.35059 · doi:10.1017/S0308210500003590
[52]Smith, H.L., 1995. Monotone Dynamical Systems, An Introduction to the Theory of Competitive and Cooperative Systems. Am. Math. Soc., Providence.
[53]Smith, D.L., McKenzie, F.E., 2004. Statics and dynamics of malaria infection in Anopheles mosquitoes. Malaria J. 3, 13. · doi:10.1186/1475-2875-3-13
[54]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. · doi:10.1371/journal.pbio.0020368
[55]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. · doi:10.1038/nature04024
[56]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. · doi:10.1186/1475-2875-3-44
[57]Torres-Sorando, L., Rodriguez, D.J., 1997. Models of spatio-temporal dynamics in malaria. Ecol. Model. 104, 231–240. · doi:10.1016/S0304-3800(97)00135-X
[58]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.
[59]World Health Organization, 2005. Roll back malaria: what is malaria? http://mosquito.who.int/cmc_upload/0/000/015/372/RBMInfosheet_1.htm .