Dynamic noise, chaos and parameter estimation in population biology

We revisit the parameter estimation framework for population biological dynamical systems, and apply it to calibrate various models in epidemiology with empirical time series, namely influenza and dengue fever. When it comes to more complex models such as multi-strain dynamics to describe the virus–host interaction in dengue fever, even the most recently developed parameter estimation techniques, such as maximum likelihood iterated filtering, reach their computational limits. However, the first results of parameter estimation with data on dengue fever from Thailand indicate a subtle interplay between stochasticity and the deterministic skeleton. The deterministic system on its own already displays complex dynamics up to deterministic chaos and coexistence of multiple attractors.

[1]  Y. Kuznetsov.  Elements of applied bifurcation theory (2nd ed.) , 1998 .

[2]  Michael P. H. Stumpf,et al.  Simulation-based model selection for dynamical systems in systems and population biology , 2009, Bioinform..

[3]  A. Doucet,et al.  A Tutorial on Particle Filtering and Smoothing: Fifteen years later , 2008 .

[4]  Vincent A. A. Jansen,et al.  Population Biology And Criticality: From Critical Birth-Death Processes To Self-Organized Criticality In Mutation Pathogen Systems , 2010 .

[5]  Tina Toni,et al.  Designing attractive models via automated identification of chaotic and oscillatory dynamical regimes , 2011, Nature communications.

[6]  W. Ebeling.  Stochastic Processes in Physics and Chemistry , 1995 .

[7]  E. Ott.  Chaos in Dynamical Systems: Contents , 2002 .

[8]  Nico Stollenwerk,et al.  The phase transition lines in pair approximation for the basic reinfection model SIRI , 2007 .

[9]  D. Sherrington.  Stochastic Processes in Physics and Chemistry , 1983 .

[10]  Aaron A. King,et al.  Time series analysis via mechanistic models , 2008, 0802.0021.

[11]  On the series expansion of the spatial SIS evolution operator , 2011 .

[12]  Bob W. Kooi,et al.  Scaling of Stochasticity in Dengue Hemorrhagic Fever Epidemics , 2012 .

[13]  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..

[14]  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.

[15]  N. Stollenwerk,et al.  A spatially stochastic epidemic model with partial immunization shows in mean field approximation the reinfection threshold , 2010, Journal of biological dynamics.

[16]  Simon J. Godsill,et al.  On sequential Monte Carlo sampling methods for Bayesian filtering , 2000, Stat. Comput..

[17]  Daniel T. Gillespie,et al.  Monte Carlo simulation of random walks with residence time dependent transition probability rates , 1978 .

[18]  N. Stollenwerk,et al.  Master equation solution of a plant disease model , 2000 .

[19]  David Welch,et al.  Approximate Bayesian computation scheme for parameter inference and model selection in dynamical systems , 2009, Journal of The Royal Society Interface.

[20]  S. Wood.  Statistical inference for noisy nonlinear ecological dynamic systems , 2010, Nature.

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

[22]  Yves F. Atchad'e,et al.  Iterated filtering , 2009, 0902.0347.

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

[24]  D. Gillespie.  A General Method for Numerically Simulating the Stochastic Time Evolution of Coupled Chemical Reactions , 1976 .

[25]  Nico Stollenwerk,et al.  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. , 2011, Journal of theoretical biology.

[26]  N. Stollenwerk,et al.  Irregularity in dengue fever epidemics: difference between first and secondary infections drives the rich dynamics more than the detailed number of strains , 2011, 1111.3844.

[27]  T. Başar,et al.  A New Approach to Linear Filtering and Prediction Problems , 2001 .

[28]  Nico Stollenwerk,et al.  Testing nonlinear stochastic models on phytoplankton biomass time series , 2001 .

[29]  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 .

[30]  Nando de Freitas,et al.  Sequential Monte Carlo Methods in Practice , 2001, Statistics for Engineering and Information Science.

[31]  R. E. Kalman,et al.  New Results in Linear Filtering and Prediction Theory , 1961 .

[32]  C. Gardiner.  Handbook of Stochastic Methods , 1983 .

[33]  N. Kampen,et al.  Stochastic processes in physics and chemistry , 1981 .

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

[35]  R. L. Stratonovich.  CONDITIONAL MARKOV PROCESSES , 1960 .

[36]  D. Ruelle.  Chaotic evolution and strange attractors , 1989 .