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Backward bifurcations in dengue transmission dynamics. (English) Zbl 1156.92036
Summary: A deterministic model for the transmission dynamics of a strain of dengue disease, which allows transmission by exposed humans and mosquitoes, is developed and rigorously analysed. The model, consisting of seven mutually-exclusive compartments representing the human and vector dynamics, has a locally-asymptotically stable disease-free equilibrium (DFE) whenever a certain epidemiological threshold, known as the basic reproduction number $$\mathcal R_0$$, is less than unity. Further, the model exhibits the phenomenon of backward bifurcation, where the stable DFE coexists with a stable endemic equilibrium. The epidemiological consequence of this phenomenon is that the classical epidemiological requirement of making $$\mathcal R_0$$ less than unity is no longer sufficient, although necessary, for effectively controlling the spread of dengue in a community.
The model is extended to incorporate an imperfect vaccine against the strain of dengue. Using the theory of centre manifolds, the extended model is also shown to undergo backward bifurcations. In both the original and the extended models, it is shown, using Lyapunov function theory and the LaSalle invariance principle, that the backward bifurcation phenomenon can be removed by substituting the associated standard incidence function with a mass action incidence. In other words, in addition to establishing the presence of backward bifurcation in models of dengue transmission, this study shows that the use of standard incidence in modelling dengue disease causes the backward bifurcation phenomenon of dengue disease.

##### MSC:
 92D30 Epidemiology 34C60 Qualitative investigation and simulation of ordinary differential equation models 37N25 Dynamical systems in biology 92C60 Medical epidemiology 93A30 Mathematical modelling of systems (MSC2010)
##### Keywords:
dengue; mosquitoes; equilibria; stability; bifurcation; vaccine
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##### References:
 [1] Anderson, R.M.; May, R.M., Infectious diseases of humans: dynamics and control, (1991), Oxford University Press [2] Anderson, R.M.; May, R.M., Population biology of infectious diseases, (1982), Springer-Verlag Berlin, Heilderberg, New York [3] Arino, J.; McCluskey, C.C.; van den Driessche, P., Global results for an epidemic model with vaccination that exhibits backward bifurcation, SIAM journal on applied mathematics, 64, 260-276, (2003) · Zbl 1034.92025 [4] Bancroft, T.L., On the etiology of dengue fever, Australian medical gazette, 25, 17-18, (1906) [5] Bartley, L.M.; Donnelly, C.A.; Garnett, G.P., The seasonal pattern of dengue in endemic areas: mathematical models of mechanisms, Transaction of the royal society of tropical medicine and hygiene, 96, 4, 387-397, (2002) [6] Blaney Jr, J.E.; Sathe, N.S.; Hanson, C.T.; Firestone, C.Y.; Murphy, B.R.; Whitehead, S.S., Vaccine candidates for dengue virus type 1 (den 1) generated by replacement of the structural genes of rden 4 and rden4$$\operatorname{\Delta}$$30 with those of den 1, Virology journal, 4, 23, 1-11, (2007) [7] Blaney Jr, J.E.; Matro, J.M.; Murphy, B.R.; Whitehead, S.S., Recombinant, live-attenuated tetravalent dengue virus vaccine formulations induce a balanced, broad, and protective neutralizing anitibody response against each of the four serotypes in rhesus monkeys, Journal of virology, 79, 9, 5516-5528, (2005) [8] Blower, S.M.; McLean, A.R., Prophylactic vaccines, risk behaviour change, and the probability of eradicating HIV in San Francisco, Science, 265, 1451-1454, (1994) [9] Bowman, C.; Gumel, A.B.; van den Drissche, P.; Wu, J.; Zhu, H., Mathematical model for assessing control strategies against west nile virus, Bulletin of mathematical biology, 67, 1107-1133, (2005) · Zbl 1334.92392 [10] Brauer, F., Backward bifurcation in simple vaccination models, Journal of mathematical analysis and application, 298, 2, 418-431, (2004) · Zbl 1063.92037 [11] Carr, J., Applications of centre manifold theory, (1981), Springer-Verlag New York · Zbl 0464.58001 [12] Castillo-Chavez, C.; Song, B., Dynamical model of tuberclosis and their applications, Mathematical biosciences and engineering, 1, 2, 361-404, (2004) · Zbl 1060.92041 [13] Center For Disease Control, Dengue Fact Sheet, (2007). www.cdc.gov/ncidod/dvbid/dengue/resources/DengueFactSheet.pdf. (Accessed August 2007). [14] Center For Vaccine Development (2007). Live attenuated tetravalent DEN vaccine. www2.mahidol.ac.th/spectrum/page3vol9no3.htm. (Accessed August 2007). [15] Chitnis, N.; Cushing, J.M.; Hyman, J.M., Bifurcation analysis of a mathematical model for malaria transmission, SIAM journal on applied mathematics, 67, 24-45, (2006) · Zbl 1107.92047 [16] Chiyaka, C.; Garira, W.; Dube, S., Transmission model of endemic human malaria in a partially immune population, Mathematical and computer modelling, 46, 806-822, (2007) · Zbl 1126.92043 [17] Chiyaka, C.; Tchuenche, J.M.; Garira, W.; Dube, S., A mathematical analysis of the effects of control strategies on the transmission dynamics of malaria, Applied mathematics and computation, 195, 2, 641-662, (2008) · Zbl 1128.92022 [18] Chowell, G.; Diaz-Duenas, P.; Miller, J.C.; Alcazar-Velazco, A.; Hyman, J.M.; Fenimore, P.W.; Castillo Chavez, C., Estimation of the reproduction number of dengue fever from spatial epidemic data, Mathematical biosciences, 208, 2, 571-589, (2007) · Zbl 1119.92055 [19] Coutinho, F.A.B.; Burattini, M.N.; Lopez, L.F.; Massad, E., Threshold conditions for a non-autonomous epidemic system describing the population dynamics of dengue, Bulletin of mathematical biology, 68, 2263-2282, (2006) · Zbl 1296.92226 [20] Derouich, M.; Boutayeb, A., Dengue fever: mathematical modelling and computer simulation, Applied mathematics and computation, 177, 2, 528-544, (2006) · Zbl 1121.92056 [21] Dung, N.M., Double-blind comparison of four intravenous-fluid regimens, Clinical infectious diseases, 29, 4, 795-796, (1999) [22] Eckels, K.H.; Putnak, R., Formalin-inactivated whole virus and recombinant subunit flavivirus vaccines, Advances in virus research, 61, 395-418, (2003) [23] Elbasha, E.H.; Gumel, A.B., Theoretical assessment of public health impact of imperfect prophylactic HIV-1 vaccines with therapeutic benefits, Bulletin of mathematical biology, 68, 577-614, (2006) · Zbl 1334.91060 [24] Esteva, L.; Vargas, C., Coexistence of different serotypes of dengue virus, Journal of mathematical biology, 46, 31-47, (2003) · Zbl 1015.92023 [25] Esteva, L.; Vargas, C., Analysis of a dengue disease transmission model, Mathematical biosciences, 150, 131-151, (1998) · Zbl 0930.92020 [26] Esteva, L.; Vargas, C., A model for dengue disease with variable human population, Journal of mathematical biology, 38, 220-240, (1999) · Zbl 0981.92016 [27] Esteva, L.; Vargas, C., Influence of vertical and mechanical transmission on the dynamics of dengue disease, Mathematical biosciences, 167, 51-64, (2000) · Zbl 0970.92011 [28] Y. Eshita, T. Takasaki, I. Takashima, N. Komalamisra, H. Ushijima and I. Kurane, Vector competence of Japanese mosquitoes for dengue and West Nile viruses, Pesticide Chemistry (2007) doi:10.1002/9783527611249.ch23. [29] Feng, Z.; Jorge Velasco-Hernandez, X., Competitive exclusion in a vector-host model for the dengue fever, Journal of mathematical biology, 35, 523-544, (1997) · Zbl 0878.92025 [30] Ferguson, N.M.; Donnelly, C.A.; Anderson, R.M., Transmission dynamics and epidemiology of dengue: insights from age-stratified sero-prevalence surveys, Philosophical transactions of the royal society of London B, 354, 757-768, (1999) [31] Gubler, D.J.; Kuno, G., Dengue and dengue hemorrhagic fever, (1997), CAB International London [32] Cruz-Pacheco, G.; Esteva, L.; Monta-Hirose, J.; Vargas, C., Modelling the dynamics of west nile virus, Bulletin of mathematical biology, 67, 1157-1172, (2005) · Zbl 1334.92397 [33] Halstead, S.B.; Nimmannitya, S.; Cohen, S.N., Observations related to pathogenesis of dengue hemorrhagic fever IV. relation of disease severity to antibody response and virus recovered, Yale journal of biology and medicine, 42, 311-322, (1970) [34] Hethcote, H.W., The mathematics of infectious diseases, SIAM review, 42, 599-653, (2000) · Zbl 0993.92033 [35] International Vaccine Institute (2007). Pediatric dengue vaccine initiative. http://www.pdvi.org/. (Accessed August 2007). [36] Kawaguchi, I.; Sasaki, A.; Boots, M., Why are dengue virus serotypes so distantly related? enhancement and limiting serotype similarity between dengue virus strains, Proceedings of the royal society of London B, 270, 2241-2247, (2003) [37] Koraka, P.; Benton, S.; van Amerongen, G.; Stitelaar, K.J.; Osterhaus, A.D.M.E., Efficacy of a live attenuated tetravalent candidate dengue vaccine in naive and previously infected cynomolgus macaques, Vaccine, 25, 5409-5416, (2007) [38] Lakshmikantham, V.; Leela, S.; Martynyuk, A.A., Stability analysis of nonlinear systems, (1989), Marcel Dekker Inc. New York and Basel · Zbl 0676.34003 [39] LaSalle, J.P., The stability of dynamical systems, Regional conference series in applied mathematics, (1976), SIAM Philadelphia · Zbl 0364.93002 [40] Lewis, M.A.; Renclawowicz, J.; van den Driessche, P.; Wonham, M., A comparison of continuous and discrete time west nile virus models, Bulletin of mathematical biology, 68, 491-509, (2006) · Zbl 1334.92413 [41] Pongsumpun, P.; Tang, I.M., Transmission of dengue hemorrhagic fever in an age structured population, Mathamatical and computer modelling, 37, 949-961, (2003) · Zbl 1045.92040 [42] Saluzzo, J.F., Empirically derived live attenuated vaccines against dengue and Japanese encephalitis, Advances in virus research, 61, 419-443, (2003) [43] Sharomi, O.; Podder, C.N.; Gumel, A.B.; Elbasha, E.H.; Watmough, J., Role of incidence function in vaccine-induced backward bifurcation in some HIV models, Mathematical biosciences, 210, 436-463, (2007) · Zbl 1134.92026 [44] Shekhar, C., Deadly dengue: new vaccines promise to tackle this escalating global menace, Chemistry and biology, 14, 871-872, (2007) [45] Struchiner, C.J.; Luz, P.M.; Codeco, C.T.; Coelho, F.C.; Massad, E., Current research issues in mosquito-borne diseases modelling, Contemporary mathematics, 410, 349-352, (2006) · Zbl 1152.92335 [46] Tewa, J.J.; Dimi, J.L.; Bowang, S., Lyapunov functions for a dengue disease transmission model, Chaos, solitons and fractals, (2007) [47] van den Driessche, P.; Watmough, J., Reproduction numbers and sub-threshold endemic equilibria for compartmental models of disease transmission, Mathematical biosciences, 180, 29-48, (2002) · Zbl 1015.92036 [48] Wonham, M.J.; de-Camino-Beck, T.; Lewis, M., An epidemiological model for west nile virus: invasion analysis and control applications, Proceeding of the royal society London B, 271, 1538, 501-507, (2004) [49] World Health Organization (2007), Dengue and dengue hemorrhagic fever. www.who.int/mediacentre/factsheets/fs117/en/. (Accessed August 2007). [50] World Health Organization, Immunological correlates of protection induced by dengue vaccines, Vaccine. 25 (2007) 4130-4139. [51] Wilder-Smith, A.; Foo, W.; Earnest, A.; Sremulanathan, S.; Paton, N.I., Seroepidemiology of dengue in the adult population of Singapore, Tropical medicine and international health, 9, 2, 305-308, (2004) [52] Yang, H.M.; Ferreira, C.P., Assessing the effects of vector control on dengue transmission, Applied mathematics and computation, 198, 401-413, (2008) · Zbl 1133.92015 [53] Yebakima, A., Genetic heterogeneity of the dengue vector aedes aegypti in Martinique, Tropical medicine and international health, 9, 5, 582-587, (2004)
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