Document Type : Research Article
Authors
Department of Mathematics, Payame Noor University (PNU), P.O. BOX 19395-4697, Tehran, Iran.
Abstract
This paper presents a novel approach using artificial neural networks to solve the SEIR (Susceptible, Exposed, Infected, and Recovered) model of infectious diseases based on dynamical systems. Optimal control techniques are employed to determine a vaccination schedule for a standard SEIR epidemic model. The multilayer perceptron is utilized to approximate the state and co-state functions of the SEIR model and to solve the optimal control problem by utilizing a nonlinear programming approach. By constructing a loss function and using Pontryagin's Minimum Principle (PMP) for the SEIR model, a minimization problem is defined, a minimization problem is defined, and the approximate solution of the Hamiltonian system is computed. This method is compared with the fourth-order Runge-Kutta method. The proposed approach's effectiveness is demonstrated through illustrative examples.
Keywords
[1] Acar E., Ozerdem M.S., Akpolat V. (2001). “Diabetes mellitus forecast using various types of artificial neural network”, International Advanced Technologies Symposium, 11.
[2] Afshar S., Abdolrahmani F. (2011). “Recognition and prediction of leukemia with artificial neural network (ANN)”, Medical Journal of Islamic Republic of Iran, 25(1), 35-39.
[3] Avila E., Rivero E., Emilio G. (2017). “Global dynamics of a periodic SEIRS model with general incidence rate”, International Journal of Differential Equations, 2017, 1-14.
[4] Boni M.F., Manh B.H., Thai P.Q., Farrar J., Hien N.T., Kinh N.V., Horby B. (2009). “Modelling the progression of pandemic influenza a (H1N1) in Vietnam and the opportunities for re-assortment with other influenza viruses”, BMC Medicine, 7(43).
[5] Cheng T., Lewis F.L., Abu-Khalaf M. (2007). “Fixed-final-time constrained optimal control of nonlinear systems using neural network HJB approach”, IEEE Transaction Neural Network, 18, 1725-1737.
[6] Chowdhury D.R., Chatterjee M., Samanta R.K. (2011). “An artificial neural network model for neonatal disease diagnosis”, International Journal of Artificial Intelligence and Expert Systems (IJAE), 2(3).
[7] De la Sen M., Alonso-Quesada S. (2011). “Vaccination strategies based on feedback control techniques for a general SEIR epidemic model”, Applied Mathematics and Computation, 218(7), 2637-2658.
[8] De la Sen M., Ibeas A., Alonso-Quesada S. (2012). “On vaccination controls SEIR epidemic model”, Communications in Nonlinear Science Numerical Simulation, 17(6), 3888-3904.
[9] Effati S., Pakdaman M. (2013). “Optimal control problem via neural networks”, Neural Computing and Application, 23, 2093-2100.
[10] Greenhalgh D. (1992). “Some results for an SEIR epidemic model with density dependence in the death rate”, IMA journal of mathematics applied in medicine and biology, 9(2), 67-106.
[11] Greenhalgh D. (1997). “Hopf bifurcation in epidemic models with a latent period and nonpermanent immunity”, Mathematical and Computer Modelling, 25, 85-93.
[12] Greenhalgh D., Moneim I.A. (2003). “SIRS epidemic model and simulations using different types of seasonal contact rate”, Systems Analysis Modelling Simulation, 43(5), 573-600.
[13] Haykin S. (2007). “Neural networks: A comprehensive foundation”, 3rd editions Prentice-Hall, Upper Saddle River.
[14] Jan R., Xiao Y. (2019). “Effect of pulse vaccination on dynamics of dengue with periodic transmission functions”, Advances in Difference Equations, 219(1), 368.
[15] Keckman V. (2001). “Learning and soft computing”, MIT press, Cambridge, MA.
[16] Khemphila A., Boonjing V. (2011). “Heart disease classification using neural network and feature selection”, 21st International Conference on Systems Engineering.
[17] Kirk D. (2004). “Optimal control theory an introduction”, Dover Publications, Mineola, NY.
[18] Kumar M., Yadav N. (2011). “Multilayer perceptrons and radial basis function neural network methods for the solution of differential equations: a survey”, Computers & Mathematics with Applications, 62, 3796-3811.
[19] Lagaris I.E., Likas A. (2012). “Hamilton–Jacobi theory over time scales and applications to linear-quadratic problems”, IEEE Transactions on Neural Network, 9(5), 987-1000.
[20] Lenhart S., Workman J. (2007). “Optimal control applied to biological models”, Boca Raton, Chapman Hall/CRC.
[21] Li X., Fang B. (2009). “Stability of an age-structured SEIR epidemic model with infectivity in latent period”, Applications and Applied Mathematics, 4(1), 218-236.
[22] Li G., Jin Z. (2005). “Global stability of a SEIR epidemic model with infectious force in latent, infected and immune period”, Chaos, Solitons and Fractals, 25(2), 1177-1184.
[23] Li M.Y., Muldowney J.S. (2001). “Global dynamics of a SEIR epidemic model with vertical transmission”, SIAM Journal on Applied Mathematics, 62(1), 58-69.
[24] Li M.Y., Muldowney J.S., Wang L.C., Karsai J. (1999). “Global dynamics of an SEIR epidemic model with a varying total population size”, Mathematical Biosciences, 160(2), 191-213.
[25] Mansoori A., Eshaghnezhad M., Effati S. (2019). “Recurrent neural network model: A new strategy to solve fuzzy matrix games”, IEEE Transactions on Neural Networks and Learning Systems, 30(8), 2538-2547.
[26] McAsey M., Mou L., Weimin W. (2012). “Convergence of the forward-backward sweep method in optimal control”, Computational Optimization and Applications, 53, 207-226.
[27] Mohammadi M., Mansoori A. (2018). “A projection neural network for identifying copy number variants”, IEEE Journal of Biomedical and Health Informatics, 23(5), 2182-2188.
[28] Moneim I.A. (2011). “Different vaccination strategies for measles diseases: A simulation study”, Journal of Informatics and Mathematical Sciences, 3, 227-236.
[29] Moneim I.A. (2013). “Efficiency of different vaccination strategies for childhood diseases: A simulation study”, Advances in Bioscience and Biotechnology, 4(2), 193-205.
[30] Moneim I.A. (2016). “Modeling and simulation of the spread of H1N1 flu with periodic vaccination”, International Journal of Biomathematics, 9(1).
[31] Moneim I.A., Greenhalgh D. (2005). “Threshold and stability results for an SIRS epidemic model with a general periodic vaccination strategy”, Journal of Biological Systems, 13(2), 131-150.
[32] Moneim I.A., Khalil H.A. (2015). “Modelling and simulation of the spread of HBV disease with infectious latent”, Applied Mathematics, 6(5), 745-753.
[33] Muller B., Reinhardt J., Strickland M.T. (2002). “Neural networks: An introduction”, 2nd editions Springer, Berlin.
[34] Nazemi A.R., Effati S. (2013). “An application of a merit function for solving convex programming problems”, Computers & Industrial Engineering, 66, 212-221.
[35] Nazemi A., Karami R. (2017). “A neural network approach for solving optimal control problems with inequality constraints and some applications”, Neural Processing Letters, 45(3), 995-1023.
[36] Nazemi A.R., Omidi F. (2012). “A capable neural network model for solving the maximum flow problem”, Journal of Computational and Applied Mathematics, 236, 3498-3513.
[37] Neilan R.M., Lenhart S. (2010). “An introduction to optimal control with an application in disease modeling”, DIMACS Series in Discrete Mathematics, 78, 67-81.
[38] Nishiur H., Inaba H. (2011). “Estimation of the incubation period of influenza a (H1N1) among imported cases: Addressing censoring using outbreak data origin of importation”, Journal of Theoretical Biology, 272(1), 123-130.
[39] Pontryagin L.S., Boltyanskii V.G., Gamkrelize R.V., Mishchenko E.F. (1962). “The mathematical theory of optimal processes”, New York, Wiley.
[40] Rausanu S., Grosan C. (2014). “A hierarchical network model for epidemic spreading. Analysis of a H1N1 virus spreading in Romania”, Applied Mathematics and Computation, 233, 39-54.
[41] Samanta G.P., Sen P., Maiti A. (2016). “A delayed epidemic model of diseases through droplet infection and direct contact with saturation incidence and pulse vaccination”, Systems Science and Control Engineering: An Open Access Journal, 4(1), 320-333.
[42] Sasuzzoha M., Singh M., Lucy D. (2013). “Parameter estimation of influenza epidemic model”, Applied Mathematics and Computation, 220, 616-629.
[43] Shi J., Chui D.M. (2012). “Extract knowledge from site-sampled data sets and fused hierarchical neural networks for detecting cardiovascular diseases”, International Conference on Biomedical Engineering and Biotechnology.
[44] Shirvany Y., Hayati M., Moradian R. (2009). “Multilayer perceptron neural networks with novel unsupervised training method for numerical solution of the partial differential equations”, Applied Soft Computing, 9, 20-29.
[45] Shulgin B., Stone L., Agur Z. (1998). “Pulse vaccination strategy in the SIR epidemic model”, Bulletin of Mathematical Biology, 60, 1123-1148.
[46] Stone L., Shulgin B., Agur Z. (2000). “Theoretical examination of the pulse vaccination policy in the SIR epidemic model”, Mathematical and Computer Modelling, 31(4-5), 207-215.
[47] Vrabie D., Lewis F.L. (2009). “Neural network approach to continuous-time direct adaptive optimal control for partially unknown nonlinear systems”, Neural Network, 22, 237-246.
[48] Yu X., Wu L., Xu C., Hu Y., Ma C. (2019). “A novel neural network for solving nonsmooth nonconvex optimization problems”, IEEE Transactions on Neural Networks and Learning Systems, 31(5), 1475-1488.
[49] Zhang J., Ma Z. (2003). “Global stability of SEIR model with saturating contact rate”, Mathematical Biosciences, 185(1), 15-32.
)