Estimation of parameters based on artificial neural networks and threshold of HIV/AIDS epidemic system in Cuba

Abstract In this paper, a method of parameter estimation, based upon Hopfield neural networks, is applied to the identification of a model of the HIV/AIDS epidemic in Cuba. This estimation technique presents a number of features that make it especially suitable for the identification of such epidemic models. In particular, its remarkable ability to estimate time variant parameters allows us to outline the evolution of parameters, thus assessing the efficiency of health policies. A complementary aim is the validation of the model, which stems from basic concepts and reasoning in disease modelling rather than from specific physical laws, thus it is not supported by rigorous mathematical results. Indeed, the experimental results show a significant adjustment of simulated data to actual ones, which provides considerable support to model validity. A sequence of real data, corresponding to the HIV and AIDS affected populations in Cuba from 1986 to 2004, has been compared to both simulations of the epidemic model and used to test the estimation method. A detection threshold, which plays a similar role to the basic reproduction number, has been calculated to complement the information about the current situation of the epidemic and the possible evolution perspectives. The obtained results are plausible and coherent according to health experts.

[1]  P. Olver.  Nonlinear Systems , 2013 .

[2]  Gonzalo Joya,et al.  Hopfield neural networks for optimization: study of the different dynamics , 2002 .

[3]  S. Abe.  Theories on the Hopfield neural networks , 1989, International 1989 Joint Conference on Neural Networks.

[4]  John J. Hopfield,et al.  Neural networks and physical systems with emergent collective computational abilities , 1999 .

[5]  N. Ling.  The Mathematical Theory of Infectious Diseases and its applications , 1978 .

[6]  H. Engl,et al.  Regularization of Inverse Problems , 1996 .

[7]  O. Diekmann,et al.  Mathematical Epidemiology of Infectious Diseases: Model Building, Analysis and Interpretation , 2000 .

[8]  Alexander Grey,et al.  The Mathematical Theory of Infectious Diseases and Its Applications , 1977 .

[9]  R. Anderson,et al.  Mathematical and statistical studies of the epidemiology of HIV. , 1989, AIDS.

[10]  Philipp Kügler,et al.  Online parameter identification in time-dependent differential equations as a non-linear inverse problem , 2008, European Journal of Applied Mathematics.

[11]  J. J. Hopfield,et al.  “Neural” computation of decisions in optimization problems , 1985, Biological Cybernetics.

[12]  Bruno Lara,et al.  Parameter Estimation of Some Epidemic Models. The Case of Recurrent Epidemics Caused by Respiratory Syncytial Virus , 2009, Bulletin of mathematical biology.

[13]  Gonzalo Joya,et al.  Modelling the HIV-AIDS Cuban Epidemics with Hopfield Neural Networks , 2009 .

[14]  A. Bos.  Parameter Estimation for Scientists and Engineers , 2007 .

[15]  M. Hirsch,et al.  Differential Equations, Dynamical Systems, and Linear Algebra , 1974 .

[16]  Biao Huang,et al.  System Identification , 2000, Control Theory for Physicists.

[17]  Lennart Ljung,et al.  System Identification: Theory for the User , 1987 .

[18]  Ivan Kramer,et al.  Accurately simulating the growth in the size of the HIV infected population in any AIDS epidemic country: Computing the U.S.A. HIV infection curve , 1994 .

[19]  H. Arazoza,et al.  A non-linear model for a sexually transmitted disease with contact tracing. , 2002, IMA journal of mathematics applied in medicine and biology.

[20]  Carlos Castillo-Chavez,et al.  Modeling TB and HIV co-infections. , 2009, Mathematical biosciences and engineering : MBE.

[21]  Giuseppe Schinaia,et al.  Estimating the Size of the HIV/AIDS Epidemic: Complementary Use of the Empirical Bayesian Back-Calculation and the Mover-Stayer Model for Gathering the Largest Amount of Information , 1998, Simul..

[22]  David Starer,et al.  Artificial Neural Nets , 1995 .

[23]  Teresa Mendonça,et al.  Hopfield neural networks for on-line parameter estimation , 2009, Neural Networks.

[24]  Ying-Hen Hsieh,et al.  A Class of Methods for HIV Contact Tracing in Cuba: Implications for Intervention and Treatment , 2004 .

[25]  A. Roddam.  Mathematical Epidemiology of Infectious Diseases: Model Building, Analysis and Interpretation O Diekmann and JAP Heesterbeek, 2000, Chichester: John Wiley pp. 303, £39.95. ISBN 0-471-49241-8 , 2001 .

[26]  Weiping Li,et al.  Applied Nonlinear Control , 1991 .

[27]  F. Doyle,et al.  A benchmark for methods in reverse engineering and model discrimination: problem formulation and solutions. , 2004, Genome research.

[28]  Gesham Magombedze,et al.  Optimal control of a sex‐structured HIV/AIDS model with condom use , 2009 .

[29]  Francisco Sandoval Hernández,et al.  Estimation of the Rate of Detection of Infected Individuals in an Epidemiological Model , 2007, IWANN.

[30]  Katsuyuki Hagiwara,et al.  Regularization learning, early stopping and biased estimator , 2002, Neurocomputing.

[31]  Klaus Schittkowski,et al.  Numerical Data Fitting in Dynamical Systems , 2002 .

[32]  Gonzalo Joya,et al.  Parametric identification of robotic systems with stable time-varying Hopfield networks , 2004 .

[33]  Francisco Sandoval Hernández,et al.  Hopfield Neural Networks for Parametric Identification of Dynamical Systems , 2004, Neural Processing Letters.

[34]  Dieter Fensel,et al.  Problem-Solving Methods , 2001, Lecture Notes in Computer Science.

[35]  Gonzalo Joya,et al.  Dynamical analysis of continuous higher-order hopfield networks for combinatorial optimization. , 2005, Neural computation.

[36]  Agraj Tripathi,et al.  Modelling and analysis of the spread of AIDS epidemic with immigration of HIV infectives , 2009, Math. Comput. Model..

[37]  David Greenhalgh,et al.  Asymmetry and multiple endemic equilibria in a model for HIV transmission in a heterosexual population , 1999 .