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Threshold conditions for a non-autonomous epidemic system describing the population dynamics of dengue. (English) Zbl 1296.92226
Summary: A non-autonomous dynamical system, in which the seasonal variation of a mosquito vector population is modeled, is proposed to investigate dengue overwintering. A time-dependent threshold, \(R(t)\), is deduced such that when its yearly average, denoted by \(\overline{R}\), is less than 1, the disease does not invade the populations and when \(\overline{R}\) is greater than 1 it does. By not invading the population we mean that the number of infected individuals always decrease in subsequent seasons of transmission. Using the same threshold, all the qualitative features of the resulting epidemic can be understood. Our model suggests that trans-ovarial infection in the mosquitoes facilitates dengue overwintering. We also explain the delay between the peak in the mosquitoes population and the peak in dengue cases.

92D30 Epidemiology
92D25 Population dynamics (general)
Full Text: DOI
[1] Bancroft, T. L., On the aetiology of dengue fever, Aust. Med. Gazette, 25, 17-18 (1906)
[2] Beaty, B. J.; Marquardt, W. C., The Biology of Disease Vectors (1996), Niwot, Colorado: University Press of Colorado, Niwot, Colorado
[3] Crochu, S.; Cook, S.; Attoui, H.; Charrel, R. N.; Chesse, R. D.; Belhouchet, M.; Lemasson, J. J.; Micco, P.; Lamballerie, X., J. Gen. Virol., 85, 1971-1980 (2004)
[4] El’sgol’ts, E. L., Introduction to the Theory of Differential Equations with Deviating Arguments (1966), San Francisco: Holden-Day Inc., San Francisco · Zbl 0133.33502
[5] Forattini, O.P., 2002. Medical Culicidology EDUSP. São Paulo (in Portuguese).
[6] Forattini, O. P.; Kakitani, I.; Massad, E.; Marucci, D., Studies mosquitoes (Diptera: Culicidae) and anthropic environment. 9-Synanthropic and epidemiological vector role of aedes scapularis in South-Eastern Brazil, Revis. Saúde Públ., 29, 3, 199-207 (1995)
[7] Gubler, D. J.; Kuno, G., Dengue and Dengue Hemorrhagic Fever (1997), New York, Wallingford: CABI Publishing, New York, Wallingford
[8] Gubler, D. J.; Kuno, G., Dengue and Dengue Hemorrhagic Fever (1997), Wallingford: CABI Publishing, Wallingford
[9] Hauck Center for the Albert B., 2005. Sabin Archives, Box 12, file 5. http://sabin.uc.edu/dengue. ucm.
[10] Joshi, V.; Mourya, D. T.; Sharma, R. C., Persistence of dengue-3 virus through transovarial transmission passage in successive generation of Aedes aegypti mosquitoes, Am. J. Trop. Med. Hyg., 67, 2, 158-161 (2002)
[11] Lopez, L. F.; Coutinho, F. A.B.; Burattini, M. N.; Massad, E., Threshold conditions for infection persistence in complex host-vectors interactions, Comptes Rendus Biol. Acad. Sci. Paris, 325, 1073-1084 (2002)
[12] Luz, P. M.; Codeço, P. T.; Massad, E.; Struchiner, C. J., Uncertainties regarding dengue modelling in Rio de Janeiro, Brazil, Mem. Inst. Oswaldo Cruz, 98, 7, 871-878 (2003)
[13] Macdonald, G., The analysis of equilibrium in malaria, Trop. Dis. Bull., 49, 813-828 (1952)
[14] Massad, E.; Coutinho, F. A.B.; Burattini, M. N.; Lopez, L. F., The risk of yellow fever in a dengue infested area, Trans. R. Soc. Trop. Med., 95, 4, 370-374 (2001)
[15] Monath, T.; Heinz, F. X.; Fields, B. N.; Howley, P. M.; Griffin, D. E.; Lamb, R. A.; Martin, M. A.; Roizman, B.; Straus, S. E.; Knipe, D. M., Flaviviruses, Virology, 961-1034 (1996), Philadelphia: Lippincott-Raven, Philadelphia
[16] Rodahin, F.; Rosen, L.; Gubler, D. J.; Kuno, G., Mosquito vectors and dengue virus-vector relationships, Dengue and Dengue Hemorrhagic Fever (1997), New York: CABI Publishing, New York
[17] Shroyer, D. A., Vertical maintenance of dengue-1 virus in sequential generations of Aedes albopictus, J. Am. Mosq. Control. Assoc., 6, 2, 312-314 (1990)
[18] SEADE Foundation of São Paulo State. Demographic data. http://www.seade.gov.br/.
[19] Beaty, B. J.; Marquardt, W. C., The Biology of Disease Vector (1996), Niwot, Colorado.: University Press of Colorado, Niwot, Colorado.
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