A Fractional SIRC Model For The Spread Of Diseases In Two Interacting Populations
Fractional SIRC models in two interacting populations
Abstract
In this contribution we address the following question: what is the behavior of a disease spreading between two distinct populations that interact, under the premise that both populations have only partial immunity to circulating stains of the disease? Our approach consists of proposing and analyzing a multi-fractional Susceptible (S), Infected (I), Recovered (R) and Cross-immune (C) compartmental model, assuming that the dynamics between the compartments of the same population is governed by a fractional derivative, while the interaction between distinct populations is characterized by the proportion of interaction between susceptible and infected individuals of both populations. We prove the well-posedness of the proposed dynamics, which is complemented with simulated scenarios showing the effects of fractional order derivatives (memory) on the dynamics.
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References
B. Adams and A. Sasaki, “Antigenic distance and cross-immunity, invariability and coexistence of pathogen strains in an epidemiological model with discrete antigenic space”. Theoretical population biology, 76, 157-167, 2009.
V. Andreasen, J. Lin, and S. Levin, “The dynamics of co-circulating influenza strains conferring partial cross-immunity”. Journal of mathematical biology, 35, 825-842, 1997.
R. Casagrandi, et al., “The SIRC model and influenza A”. Mathematical biosciences, 200, 152-169, 2006.
O. Diekmann, J. A. J. Metz, and J. A. P Heesterbeek, “The legacy of Kermack and McKendrick,'', in Epidemic Models: Their Structure and Relation to Data, Mollison, D. E. (ed.), Cambridge University Press, Cambridge, 1995.
K. Diethelm, “The analysis of fractional differential equations: An application-oriented exposition using differential operators of Caputo type'', Springer Science & Business Media, Braunschweig, 2010.
R. Garrappa, “Trapezoidal methods for fractional differential equations: Theoretical and computational aspects”. Mathematics and Computers in Simulation, 110, 96-112, 2015.
A. C. F. N. Gomes, and A. De Cezaro, “Um estudo sobre a memória epidemiológica: modelo SIRC fracionário”. C.Q.D. – Revista Eletrônica Paulista de Matemática, 10, 194-210, 2017.
A. C. F. N. Gomes, and A. De Cezaro, “On a multi-order fractional SIRC model for Influenza”. C.Q.D. – Revista Eletrônica Paulista de Matemática, 22, 1-15, 2022.
M. G. M. Gomes, L.J. White, and G. F. Medley, “Infection, reinfection, and vaccination under suboptimal immune protection: epidemiological perspectives”. Journal of theoretical biology, 228, 539- 549, 2004.
K. Kuszewski, and L. Brydak,, “The epidemiology and history of influenza”. Biomedicine and pharmacotherapy, 54, 188- 195, 2000.
W. Lin, “Global existence theory and chaos control of fractional differential equations”. J.Math. Anal. Appl, 1, 332, 709-726, 2007.
C. M. Pease, “An evolutionary epidemiological mechanism, with applications to type A influenza”. Theoretical population biology, 31, 422- 452, 1987.
J. Piret, and G. Boivin, “Pandemics Throughout History”. Frontiers in Microbiology, 11, 1-16, 2021.
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