Epidemiological Models and Process Engineering
Keywords:epidemiology, reaction engineering, deterministic models, stochastic models, model ﬁtting, measles case study
AbstractThe paper discusses the principles behind epidemiology models, with examples taken from the classic SIR (suspectible-infected-recovered) and SEIR (S-exposed-IR) models. Both continuous time deterministic and stochastic models are treated, where the stochastic models are based on Poisson-distributed events/reactions. These models use real approximations to the integers representing the number of people in each of the classes S, (E,) I, R. An alternative stochastic representation is the ﬁrst reaction time description, where the variables are kept as integers, and where one instead computes the time between each event. The models are presented in a form compatible with standard chemical engineering models. Based on the model description, the SIR and SEIR models are ﬁtted to a measles case study using the Markov Chain Monte Carlo approach. For the given data, the SIR model appears to give much smaller uncertainty in predicitons. The continuous time stochastic description and the ﬁrt reaction time approaches give similar variation in the models. An important measure of the state of epidemics is the reproduction number, R, which tells whether the infection is growing or decreasing from an initial infection. The development of an expression for Ris indicated both from eigenvalues and from the Next-Generation Approach, and it is shown that the expression for R is identical for the SIR and the SEIR model. The principles of epidemiology model development discussed in the paper are used in models ranging from HIV/AIDS to COVID-19.
Fred Brauer, Carlos Castillo-Chavez, and Zhilan Feng. Mathematical Models in Epidemiology. Number 69 in Texts in Applied Mathematics. Springer, New York, 2019. ISBN 978-1-4939-9826-5.
Tom Britton and Etienne Pardoux, editors. Stochastic Epidemic Models with Inference. Number 2255 in Lecture Notes in Mathematics. Springer, Springer Nature, Switzerland, 2019. ISBN 978-3030308995.
Geir Evensen. Data Assimilation. The Ensemble Kalman Filter. Springer, Berlin, 2nd edition, 2009. ISBN 978-3-642-03710-8.
Daniel T. Gillespie. A general method for numerically simulating the stochastic time evolution of coupled chemical reactions. Journal of Computational Physics, 22(4):403–434, 1976. doi:https://doi.org/10.1016/0021-9991(76)90041-3.
Daniel T. Gillespie. Exact stochastic simulation of coupled chemical reactions. The Journal of Physical Chemistry, 81 (25):2340–2361, 1977. doi:10.1021/j100540a008.
Matt J. Keeling and Pejman Rohani. Modeling Infectious Diseases in Humans and Animals. Princeton University Press, 41William Street, Princeton, New Jersey 08540, 2008. ISBN 978-0-691-11617-4.
W.O. Kermack and A.G. McKendrick. A contribution to the mathematical theory of epidemics. Proceedings of the Royal Society A., 115(772):700–721, 1927. doi:https://royalsocietypublishing.org/doi/10.1098/rspa.1927.0118.
Maia Martcheva. An Introduction to Mathematical Epidemiology, volume 61 of Texts in Applied Mathematics. Springer, New York, 2015. ISBN 978-1-4899-7611-6.
Christopher Rackauckas and Qing Nie. DifferentialEquations.jl — A Performant and Feature-Rich Ecosystem for Solving Differential Equations in Julia. Journal of Open Research Software, 5(15), 2017a. doi:10.5334/jors.151.
Christopher Rackauckas and Qing Nie. Adaptive methods for stochastic differential equations via natural embeddings and rejection sampling with memory. Discrete and continuous dynamical systems. Series B, 22(7):2731, 2017b.
Christopher Rackauckas and Qing Nie. Stability-Optimized High Order Methods and Stiffness Detection for Pathwise Stiff Stochastic Differential Equations. arXiv:1804.04344 [math], 2018. URL http://arxiv.org/abs/1804. 04344.
C.W. Schwabe, H.P. Riemann, and C.E. Franti. Epidemiology in Veterinary Practice. Lea & Febiger, 1977. pp. 258–260.
Pauline van den Driessche. Reproduction numbers of infectious disease models. Infectious Disease Modelling, 2(3):288–303, aug 2017. doi:10.1016/j.idm.2017.06.002.
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