Abstract
Infectious diseases are now widely analyzed by compartmental models. This paper introduces a SIR model coupled with a social mobility model (SMM). After discretization by a forward Euler Method, and a mixed type Euler method (structured with both forward and backward Euler elements), we obtained a difference equations model for our social mobility model. We calculate the basic reproduction number \(R_{0}\) using the next-generation matrix method. When \(R_{0}<1\), there will be a disease-free equilibrium (DFE), and \(R_{0}<1\) implies DFE will be locally asymptotically stable, while \(R_{0}>1\) implies DFE is unstable. When \(R_{0}=1\), DFE may stable or unstable. Then we obtain a hyperbolic forward Kolmogorov equation corresponding to the SIR epidemic model. We also generate the hyperbolic forward Kolmogorov equations for the SIR model with SMM between 2 locations.
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Acknowledgments
I would like to thank Professor Joseph D. Skufca for his support and encouragement. He kindly read my paper and offered invaluable detailed advice on grammar, organization, and the theme of the paper. I sincerely thank my parents and friends, who provide pieces of advice and financial support. The product of this research paper would not be possible without all of them.
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LI, Y., Xu, J. (2020). Population Motivated Discrete-Time Disease Models. In: Baigent, S., Bohner, M., Elaydi, S. (eds) Progress on Difference Equations and Discrete Dynamical Systems. ICDEA 2019. Springer Proceedings in Mathematics & Statistics, vol 341. Springer, Cham. https://doi.org/10.1007/978-3-030-60107-2_15
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