Geodesic
Epidemiology of Dengue Fever: A Model with Temporary Cross-Immunity and Possible Secondary Infection Shows Bifurcations and Chaotic Behaviour in Wide Parameter Regions
Maíra Aguiar1, 2, 3 ; Bob Kooi4 ; Nico Stollenwerk1, 2
1 Centro de Matemática e Aplicações Fundamentais da Universidade de Lisboa, Avenida Prof. Gama Pinto 2, 1649-003 Lisboa, Portugal
2 Instituto Gulbenkian de Ciência, Apartado 14, 2781-901 Oeiras, Portugal
3 Laboratório de dengue e febre amarela, Fundação Ezequiel Dias, Rua Conde Pereira Carneiro 80, 30510-010 Belo Horizonte-MG, Brasil
4 Department of Theoretical Biology, Vrije Universiteit, De Boelelaan 1087, NL 1081 HV Amsterdam, The Netherlands
Mathematical modelling of natural phenomena, Tome 3 (2008) no. 4, pp. 48-70.

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Basic models suitable to explain the epidemiology of dengue fever have previously shown the possibility of deterministically chaotic attractors, which might explain the observed fluctuations found in empiric outbreak data. However, the region of bifurcations and chaos require strong enhanced infectivity on secondary infection, motivated by experimental findings of antibody-dependent-enhancement. Including temporary cross-immunity in such models, which is common knowledge among field researchers in dengue, we find bifurcations up to chaotic attractors in much wider and also unexpected parameter regions of reduced infectivity on secondary infection, realistically describing more likely hospitalization on secondary infection when the viral load becomes high. The model shows Hopf bifurcations, symmetry breaking bifurcations of limit cycles, coexisting isolas, and two different possible routes to chaos, via the Feigenbaum period doubling and via torus bifurcations.
DOI : 10.1051/mmnp:2008070

Maíra Aguiar 1, 2, 3 ; Bob Kooi 4 ; Nico Stollenwerk 1, 2

1 Centro de Matemática e Aplicações Fundamentais da Universidade de Lisboa, Avenida Prof. Gama Pinto 2, 1649-003 Lisboa, Portugal
2 Instituto Gulbenkian de Ciência, Apartado 14, 2781-901 Oeiras, Portugal
3 Laboratório de dengue e febre amarela, Fundação Ezequiel Dias, Rua Conde Pereira Carneiro 80, 30510-010 Belo Horizonte-MG, Brasil
4 Department of Theoretical Biology, Vrije Universiteit, De Boelelaan 1087, NL 1081 HV Amsterdam, The Netherlands 
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Maíra Aguiar; Bob Kooi; Nico Stollenwerk. Epidemiology of Dengue Fever: A Model with Temporary Cross-Immunity and Possible Secondary Infection Shows Bifurcations and Chaotic Behaviour in Wide Parameter Regions. Mathematical modelling of natural phenomena, Tome 3 (2008) no. 4, pp. 48-70. doi : 10.1051/mmnp:2008070. http://geodesic.mathdoc.fr/articles/10.1051/mmnp:2008070/

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