2015/05/11 by C. N. Angstmann, B. I. Henry, A. V. McGann +2 · 57 citations
Biochemistry, Genetics and Molecular Biology · Mathematics · Medicine · #Applied mathematics #COVID-19 epidemiological studies #Convergence (economics) #Differential equation #Discrete time and continuous time #Economics #Epidemic model #Fractional Differential Equations Solutions #Fractional calculus #Mathematical analysis #Mathematical and Theoretical Epidemiology and Ecology Models #Mathematical optimization #Mathematics #Order (exchange) #Ordinary differential equation #Statistics #Stochastic differential equation #Stochastic modelling #Stochastic process #math.DS #msc:26A33 #msc:37M05 #msc:92D30 #q-bio.PE
paper · pdf · doi:10.1007/s11538-016-0151-7
published in arXiv (Cornell University) 78(3), 468-499 (Cornell University) · 32 pages, 3 figures
openalex publication_date 2015/05/11 · arxiv created 2015/09/30 · arxiv updated 2016/03/31 · openalex created_date 2025/10/10 · openalex updated_date 2026/08/05
Over the past several decades there has been a proliferation of epidemiological models with ordinary derivatives replaced by fractional derivatives in an an-hoc manner. These models may be mathematically interesting but their relevance is uncertain. Here we develop an SIR model for an epidemic, including vital dynamics, from an underlying stochastic process. We show how fractional differential operators arise naturally in these models whenever the recovery time from the disease is power law distributed. This can provide a model for a chronic disease process where individuals who are infected for a long time are unlikely to recover. The fractional order recovery model is shown to be consistent with the Kermack-McKendrick age-structured SIR model and it reduces to the Hethcote-Tudor integral equation SIR model. The derivation from a stochastic process is extended to discrete time, providing a stable numerical method for solving the model equations. We have carried out simulations of the fractional order recovery model showing convergence to equilibrium states. The number of infecteds in the endemic equilibrium state increases as the fractional order of the derivative tends to zero.