Irregularity in dengue fever epidemics: difference between first and secondary infections drives the rich dynamics more than the detailed number of strains Ma´ ıra Aguiar, 1, 2, ∗ Nico Stollenwerk, 1 and Bob W. Kooi 3 1 Centro de Matem´atica e Aplica¸ c˜oes Fundamentais, Lisbon University, Av. Prof. Gama Pinto, 2, 1649-003 Lisbon, Portugal 2 Funda¸ c˜ao Ezequiel Dias, Servi¸co de Virologia e Riquetisioses, Laborat´orio de dengue e febre amarela, Rua Conde Pereira Carneiro 80, 30510-010 Belo Horizonte-MG, Brazil 3 Vrije Universiteit, Faculty of Earth and Life Sciences, Department of Theoretical Biology, De Boelelaan 1087, NL 1081 HV Amsterdam, The Netherlands (Dated: September 30, 2011) PACS numbers: 87.19.xd, 87.85.Tu, 07.05.Tp, 02.30.Hq, S 05.45.-a, 05.70.Ln Dengue is a viral mosquito-borne infection, a leading cause of illness and death in the tropics and subtropics. There are four antigenically distinct but closely related dengue viruses, designated DEN-1, DEN-2, DEN-3, and DEN-4. Infection by one serotype confers life-long im- munity to that serotype and a short period of temporary cross-immunity to other serotypes. Two forms of the dis- ease exist: (DF) dengue fever, and (DHF) dengue hem- orrhagic fever which has been associated with secondary dengue infection due to the (ADE) antibody-dependent enhancement process, where the pre-existing antibodies to previous dengue infection cannot neutralize but rather enhance the new infection. In the first dengue infection virus particles are captured and processed by so-called antigen presenting cells. T-cells become activated, like- wise B-cells that produce antibodies used to inactivate the viruses. In a secondary infection the antibodies from the first infection attaches to the virus particles but does not inactivate them. The antibody-virus complex sup- presses innate immune responses, increasing intracellu- lar infection and generating inflammatory citokines and chemokines that, collectively, result in enhanced disease [1–3]. Dengue fever dynamics is well known to be particularly complex with large fluctuations of disease incidences. Several mathematical models found in the literature have been formulated to describe the transmission of dengue fever. Multi-strain dengue dynamics are generally mod- elled with extended SIR-type models, and have demon- strated to show critical fluctuations with power law dis- tributions of disease cases [4] and deterministic chaos via ADE, but without temporary cross-immunity, [5–7], when strong infectivity on secondary infection was as- sumed. In these models, the recovered individuals could be immediately infected with another strain. The combination of biological aspects such as tempo- rary cross-immunity and ADE have been studied by sev- eral authors [8–10] where four strains are involved, but again limiting the effect of ADE to increase the contribu- * Electronic address: maira@ptmat.fc.ul.pt tion of secondary cases to the force of infection. Aguiar et al. [11] have investigated a two-strain dengue model, ini- tially suggested and preliminarily analyzed in [5], where deterministic chaos was found in a wider parameter re- gions when including temporary cross-immunity [11–13], not needing to restrict the infectivity on secondary infec- tion to one or another region in parameter space. In this manuscript we compare the two-strain dengue model, which already captures differences between pri- mary and secondary infections, with the one that in- troduces the idea of competition of multiple strains in dengue epidemics, the four-strain dengue model, and show that the difference between first and secondary in- fections and temporary cross-immunity drives the rich dynamics more than the detailed number of strains. The parameter description and respective values for dengue fever epidemiology are given in Table I, if not otherwise explicitly stated. TABLE I: Parameter set for the basic multi-strain models. Par. Description Values Ref μ new born susceptible rate 1/65y [15] γ recovery rate 52y -1 [16, 17] β infection rate 2γ [5] α temporary cross-immunity rate ∈ [2, 52]y -1 [18] φ ratio of contrib. to force of inf. ∈ [0, 3] [11–13] In the basic two-strain model with temporary cross- immunity the population N is divided into ten classes. For two different strains, 1 and 2, we label the SIR classes for the hosts that have seen the individual strains, with epidemiological symmetry between strains, i.e. infections with strain one followed by strain two or vice versa con- tribute in the same way to the force of infection. Sus- ceptible individuals without a previous dengue infection S can possible get the primary infection with strain one (I 1 ) or strain two (I 2 ), with two different infection rates, depending on who (individual on his primary or sec- ondary infection) is transmitting the infection. The rele- vant difference concerning disease transmissibility is that the force of infection varies accordingly to the number of previous infections the hosts have experienced. In a pri-