The role of seasonality and import in a minimalistic multi-strain dengue model capturing differences between primary and secondary infections: complex dynamics and its implications for data analysis.

In many countries in Asia and South-America dengue fever (DF) and dengue hemorrhagic fever (DHF) has become a substantial public health concern leading to serious social-economic costs. Mathematical models describing the transmission of dengue viruses have focussed on the so-called antibody-dependent enhancement (ADE) effect and temporary cross-immunity trying to explain the irregular behavior of dengue epidemics by analyzing available data. However, no systematic investigation of the possible dynamical structures has been performed so far. Our study focuses on a seasonally forced (non-autonomous) model with temporary cross-immunity and possible secondary infection, motivated by dengue fever epidemiology. The notion of at least two different strains is needed in a minimalistic model to describe differences between primary infections, often asymptomatic, and secondary infection, associated with the severe form of the disease. We extend the previously studied non-seasonal (autonomous) model by adding seasonal forcing, mimicking the vectorial dynamics, and a low import of infected individuals, which is realistic in the dynamics of dengue fever epidemics. A comparative study between three different scenarios (non-seasonal, low seasonal and high seasonal with a low import of infected individuals) is performed. The extended models show complex dynamics and qualitatively a good agreement between empirical DHF monitoring data and the obtained model simulation. We discuss the role of seasonal forcing and the import of infected individuals in such systems, the biological relevance and its implications for the analysis of the available dengue data. At the moment only such minimalistic models have a chance to be qualitatively understood well and eventually tested against existing data. The simplicity of the model (low number of parameters and state variables) offer a promising perspective on parameter values inference from the DHF case notifications.

[1]  Y. Kuznetsov Elements of Applied Bifurcation Theory , 2023, Applied Mathematical Sciences.

[2]  A. Teeraratkul,et al.  Changing epidemiology of dengue hemorrhagic fever in Thailand , 1999, Epidemiology and Infection.

[3]  Parameter Estimation in Epidemiology: from Simple to Complex Dynamics , 2011, AIP conference proceedings.

[4]  Prida Malasit,et al.  Cross-Reacting Antibodies Enhance Dengue Virus Infection in Humans , 2010, Science.

[5]  Nico Stollenwerk,et al.  Diversity in pathogenicity can cause outbreaks of meningococcal disease. , 2004, Proceedings of the National Academy of Sciences of the United States of America.

[6]  D. Gubler,et al.  Viraemia in patients with naturally acquired dengue infection. , 1981, Bulletin of the World Health Organization.

[7]  Pejman Rohani,et al.  Ecological and immunological determinants of dengue epidemics. , 2006, Proceedings of the National Academy of Sciences of the United States of America.

[8]  Ira B Schwartz,et al.  Instabilities in multiserotype disease models with antibody-dependent enhancement. , 2007, Journal of theoretical biology.

[9]  Edward L. Ionides,et al.  Plug-and-play inference for disease dynamics: measles in large and small populations as a case study , 2009, Journal of The Royal Society Interface.

[10]  S. Halstead,et al.  Epidemiologic studies on Dengue in Santiago de Cuba, 1997. , 2000, American journal of epidemiology.

[11]  Xavier Deparis,et al.  Discrimination between Primary and Secondary Dengue Virus Infection by an Immunoglobulin G Avidity Test Using a Single Acute-Phase Serum Sample , 2005, Journal of Clinical Microbiology.

[12]  L. Stone,et al.  Seasonal dynamics of recurrent epidemics , 2007, Nature.

[13]  Uniqueness of Monotone Mono-stable Waves for Reaction-Diffusion Equations with Time Delay , 2009 .

[14]  S. Halstead,et al.  Immune enhancement of viral infection. , 1982, Progress in allergy.

[15]  Mario Recker,et al.  The Effects of Tertiary and Quaternary Infections on the Epidemiology of Dengue , 2010, PloS one.

[16]  Bob W. Kooi,et al.  Torus bifurcations, isolas and chaotic attractors in a simple dengue fever model with ADE and temporary cross immunity , 2008, Int. J. Comput. Math..

[17]  E. Wimmer Cellular receptors for animal viruses , 1994 .

[18]  S. Halstead 25 Antibody-dependent Enhancement of Infection: A Mechanism for Indirect Virus Entry into Cells , 1994 .

[19]  Thomas F. Fairgrieve,et al.  AUTO 2000 : CONTINUATION AND BIFURCATION SOFTWARE FOR ORDINARY DIFFERENTIAL EQUATIONS (with HomCont) , 1997 .

[20]  Mario Recker,et al.  Immunological serotype interactions and their effect on the epidemiological pattern of dengue , 2009, Proceedings of the Royal Society B: Biological Sciences.

[21]  L. Esteva,et al.  Analysis of a dengue disease transmission model. , 1998, Mathematical biosciences.

[22]  Nico Stollenwerk,et al.  A new chaotic attractor in a basic multi-strain epidemiological model with temporary cross-immunity , 2007, 0704.3174.

[23]  Hermann Haken,et al.  At least one Lyapunov exponent vanishes if the trajectory of an attractor does not contain a fixed point , 1983 .

[24]  V. Deubel,et al.  Enzyme-Linked Immunosorbent Assay Specific to Dengue Virus Type 1 Nonstructural Protein NS1 Reveals Circulation of the Antigen in the Blood during the Acute Phase of Disease in Patients Experiencing Primary or Secondary Infections , 2002, Journal of Clinical Microbiology.

[25]  S. Halstead,et al.  Neutralization and antibody-dependent enhancement of dengue viruses. , 2003, Advances in virus research.

[26]  Alan L Rothman,et al.  Spatial and temporal circulation of dengue virus serotypes: a prospective study of primary school children in Kamphaeng Phet, Thailand. , 2002, American journal of epidemiology.

[27]  Fenguangzhai Song CD , 1992 .

[28]  Bob W. Kooi,et al.  Stabilization and complex dynamics in a predator-prey model with predator suffering from an infectious disease , 2011 .

[29]  E L Ionides,et al.  Inference for nonlinear dynamical systems , 2006, Proceedings of the National Academy of Sciences.

[30]  N. Ferguson,et al.  The effect of antibody-dependent enhancement on the transmission dynamics and persistence of multiple-strain pathogens. , 1999, Proceedings of the National Academy of Sciences of the United States of America.

[31]  Ira B Schwartz,et al.  Chaotic desynchronization of multistrain diseases. , 2005, Physical review. E, Statistical, nonlinear, and soft matter physics.

[32]  D. Vaughn Invited commentary: Dengue lessons from Cuba. , 2000, American journal of epidemiology.

[33]  Nico Stollenwerk,et al.  Meningitis, pathogenicity near criticality: the epidemiology of meningococcal disease as a model for accidental pathogens. , 2003, Journal of theoretical biology.

[34]  U. Parlitz,et al.  Lyapunov exponents from time series , 1991 .

[35]  Charles H. Hoke,et al.  SEROTYPE-SPECIFIC DENGUE VIRUS CIRCULATION AND DENGUE DISEASE IN BANGKOK, THAILAND FROM 1973 TO 1999 , 2003 .

[36]  Richard G Jarman,et al.  Analysis of repeat hospital admissions for dengue to estimate the frequency of third or fourth dengue infections resulting in admissions and dengue hemorrhagic fever, and serotype sequences. , 2007, The American journal of tropical medicine and hygiene.

[37]  Eckmann,et al.  Liapunov exponents from time series. , 1986, Physical review. A, General physics.

[38]  P. Holmes,et al.  Nonlinear Oscillations, Dynamical Systems, and Bifurcations of Vector Fields , 1983, Applied Mathematical Sciences.

[39]  S. Cohen,et al.  Observations related to pathogenesis of dengue hemorrhagic fever. IV. Relation of disease severity to antibody response and virus recovered. , 1970, The Yale journal of biology and medicine.

[40]  E. Ott Chaos in Dynamical Systems: Contents , 1993 .

[41]  Bob W. Kooi,et al.  Epidemiology of Dengue Fever: A Model with Temporary Cross-Immunity and Possible Secondary Infection Shows Bifurcations and Chaotic Behaviour in Wide Parameter Regions , 2008 .

[42]  Katia Koelle,et al.  Decreases in dengue transmission may act to increase the incidence of dengue hemorrhagic fever , 2008, Proceedings of the National Academy of Sciences.

[43]  Alan L Rothman,et al.  Dengue: defining protective versus pathologic immunity. , 2004, The Journal of clinical investigation.

[44]  Ezio Venturino,et al.  Evidence of chaos in eco-epidemic models. , 2009, Mathematical biosciences and engineering : MBE.

[45]  L. Esteva,et al.  Influence of vertical and mechanical transmission on the dynamics of dengue disease. , 2000, Mathematical biosciences.

[46]  Lauterborn,et al.  Liapunov exponents from a time series of acoustic chaos. , 1989, Physical review. A, General physics.

[47]  S. Petrovskii,et al.  Spatiotemporal patterns in ecology and epidemiology : theory, models, and simulation , 2007 .