Stochastic models to estimate population dynamics
AbstractThe growth dynamics that a population follows is mainly due to births, deaths or migrations, each of thesephenomena is affected by other factors such as public health, birth control, work sources, economy, safety and conditions of quality of life in neighboring countries, among many others. In this paper is proposed two statistical models based on a system of stochastic differential equations (SDE) that model the dynamics of population growth, and three computational algorithms that allow the generation of probability distribution samples in high dimensions, in models that have non-linear structures and that are useful for making inferences. The algorithms allow to estimate simultaneously states solutions and parameters in SDE models. The interpretation of the parameters is important because they are related to the variables of growth, mortality, migration, physical-chemical conditions of the environment, among other factors. The algorithms are illustrated using real data from a sector of the population of the Republic of Ecuador, and are compared with the results obtained with the models used by theWorld Bank for the same data, which shows that stochastic models Proposals based on an SDE more adequately and reliably adjust the dynamics of demographic randomness, sampling errors and environmental randomness in comparison with the deterministic models used by the World Bank. It is observed that the population grows year by year and seems to have a definite tendency; that is, a clearly growing behavior is seen. To measure the relative success of the algorithms, the relative error was estimated, obtaining small percentage errors.
E.J. Allen, Stochastic Differential Equations and persistence time of two interacting populations, Dynamics of Continuous, Discrete,and Impulsive Systems, pp. 271–281, 1999.
E.J. Allen and H.D. Victory , Modelling and simulation of a schistosomiasis infection with biological control, Acta Tropical, vol.87, pp. 251–261, 2003.
E.J. Allen, L.J.S. Allen and H. Schurz, A comparison of persistence-time estimation for discrete and continuous population models that include demographic and environmental variability, Mathematical Biosciences, vol. 196, pp. 14–38, 2005.
L.J.S. Allen, An Introduction To Stochastic Processes With Applications to Biology, Pearson Education Inc., Upper Saddle River,New Jersey, 2003.
L.J.S. Allen and E.J. Allen, A comparison of three different stochastic population models with regard to persistence time, Theoretical Population Biology, vol. 68, pp. 439–449, 2003.
L.J.S. Allen, Modelling with It^o Stochastic Differential Equations, Mathematical Modelling: Theory and Applications, Springer,2007.
C. Andrieu, A. Doucet and R. Holenstein, Particle Markov Chain Monte Carlo methods, Journal of the Royal Statistical Society B,vol. 72, no. 3, pp. 269–342, 2010.
S. Ditlevsen and A. Samson , Introduction to stochastic models in biology, In Stochastic biomathematical models, Springer, 2013.
G. Evensen and P.van Leeuwen, Assimilation of geosat altimeter data for the agulhas current using the ensemble Kalman filter with a quasi-geostrophic model, Monthly Weather Review, vol. 24, pp. 85–96, 1996.
K.E. Emmertand and L.J.S. Allen, Population extinction in deterministic and stochastic discrete-time epidemic models with periodic coefficients with applications to amphibians, Natural Resource Modeling, vol. 19, pp. 117–164, 2006.
V. Fernandez and J. Rubio, Estimating dynamic equilibrium economies: Linear versus nonlinear likelihood, Journal of Applied Econometrics, vol. 20, pp. 891–910, 2005.
G. Gause, The struggle for existence, Baltimore, 1934.
J.B. Gelman, H.S. Stern, D.B. Dunson, A. Vehtari, and D.R. Rubin, Bayesian Data Analysis, Third Edition, Chapman and Hall/CRC, 2013.
A. Golightly and D. J. Wilkinson, Bayesian parameter inference for stochastic biochemical network models using particle Markov chain Monte Carlo, Interface Focus, vol. 1, pp. 807–820, 2012.
A. Harvey, Forecasting, Structural Time Series Models and the Kalman Filter, Cambridge University Press, 1989.
W. K. Hastings, Monte Carlo Sampling Methods Using Markov Chains and Their Applications, Biometrika, vol. 57, pp. 97–109,1970.
S. Heston, A Closed-Form Solution for Options with Stochastic Volatility with Applications to Bond and Currency Options, The Review of Financial Studies, vol. 6, pp. 327–343, 1993.
S. Infante, C. Luna, L. S´anchez and A. Hern´andez, Approximations of the solutions of a stochastic differential equation using Dirichlet process mixtures and gaussian mixtures, Statistics Optimization and Information Computing, vol. 4, pp. 289–307, 2016.
S. Infante, L. S´anchez and F. Cedeno, Filtros para predecir incertidumbre de lluvia y clima, Revista de Climatolog´ıa, vol. 12, pp.33–48, 2012.
E. Ionides, Inference and filtering for partially observed diffusion processes via sequential Monte Carlo, Working paper available from http://www.stat.lsa.umich.edu/?ionides/pubs/WorkingPaper-filters.pdf, 2003.
E. Ionides, C. Bret´o and A. King, Inference for Nonlinear Dynamical Systems, Proceedings of the National Academy of Sciences of the United States of America, vol. 103, pp. 18438–18443, 2006.
N. Kirupaharan and L.J.S. Allen, Coexistence of multiple pathogen strains in stochastic epidemic models with density-dependent mortality, Bulletin of Mathematical Biology, vol. 66, pp. 841–864, 2004.
V. Kostitzin, Mathematical biology, Harrap, 1939.
E. Lazkano, B. Sierra, A. Astigarraga and J. Martnez, On the use of bayesian networks to develop behaviours for mobile robots,Robotics and Autonomous Systems, vol. 55, pp. 253–265, 2007.
J. S. Liu, Monte Carlo Strategies in Scientific Computing, Springer Science & Business Media, 2008.
F. Lindsten, M. Jordan and T. Schon, Particle gibbs with ancestor sampling, The Journal of Machine Learning Research, vol. 15,pp. 2145–2184, 2014.
A. Lotka, Elements of physical biology, Williams and Wilkins Company, Baltimore, 1925.
R. Malthus, An Essay on the Principle of Population, Penguin, Harmondsworth, England, 1798.
R.K. McCormack and L.J.S. Allen, Disease emergence in deterministic and stochastic models for host and pathogen, Applied Mathematics and Computation, vol. 168, pp. 1281–1305, 2005.
N. Metropolis, A. Rosenbluth, M. Rosenbluth, A. Teller and E. Teller, Equations of state calculations by fast computing machines,J. Chem. Phys., vol. 21, pp. 1087–1091, 1953.
B. ∅ksendal, Stochastic Differential Equations: An Introduction with Applications, Springer, New York, 6th edition, 2003.
Y. Pokern, A. Stuart and P. Wiberg, Parameter estimation for partially observed hypoelliptic diffusions, Journal of the Royal Statistical Society Serie B., vol. 71, pp. 49–73, 2009.
C. Robert and G. Casella, Monte Carlo Statistical Methods, 2nd ed. Springer-Verlag, New York, 2004.
C. Robert, The Metropolis-Hastings algorithm, Statistics Reference Online, John Wiley & Sons, Ltd., pp. 1–15, 2016.
L. S´anchez, S. Infante, V. Grifin and D. Rey, Spatio-temporal dynamic model and parallelized ensemble kalman filter for precipitation data, Brazilian Journal of Probability and Statistics, vol. 30, pp. 653–675, 2016.
L. S´anchez, S. Infante, J. Marcano and V. Grifin, Polynomial chaos based on the parallelized ensemble kalman filter to estimate precipitation states, Statistics Optimization and Information Computing, vol. 3, pp. 79–95, 2015.
F. Sigrist, H. Kunsch andW. Stahel, A dynamic nonstationary spatio-temporal model for short term prediction of precipitation, The Annals of Applied Statistics, vol. 6, pp. 1452–1477, 2012.
P. Verhulst, Notice sur la loi que la population suit dans son accroissement, Corres. Math. et Physique, vol. 10, pp. 113–121, 1838.
P. Verhulst, Recherches math´ematiques sur la loi d’accroissement de la population, Nouveaux m´emoires de l’Academic Royale des Science et Belles-Lettres de Bruxelles, vol. 18, pp. 1–41, 1845.
V. Volterra, Fluctuations in the abundance of a species considered mathematically, Nature, vol. 118, pp. 558–560, 1926.
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