Journal of the
Korean Mathematical Society
JKMS

ISSN(Print) 0304-9914 ISSN(Online) 2234-3008

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J. Korean Math. Soc. 2021; 58(1): 45-67

Online first article November 24, 2020      Printed January 1, 2021

https://doi.org/10.4134/JKMS.j190788

Copyright © The Korean Mathematical Society.

Mathematical analysis of an ``SIR'' epidemic model in a continuous reactor - deterministic and probabilistic approaches

Miled El Hajji, Sayed Sayari, Abdelhamid Zaghdani

ENIT-LAMSIN; Isteub; Northern Border University

Abstract

In this paper, a mathematical dynamical system involving both deterministic (with or without delay) and stochastic ``SIR'' epidemic model with nonlinear incidence rate in a continuous reactor is considered. A profound qualitative analysis is given. It is proved that, for both deterministic models, if $\R_d > 1$, then the endemic equilibrium is globally asymptotically stable. However, if $\R_d \leq 1$, then the disease-free equilibrium is globally asymptotically stable. Concerning the stochastic model, the Feller's test combined with the canonical probability method were used in order to conclude on the long-time dynamics of the stochastic model. The results improve and extend the results obtained for the deterministic model in its both forms. It is proved that if $\R_s > 1$, the disease is stochastically permanent with full probability. However, if $\R_s \leq 1$, then the disease dies out with full probability. Finally, some numerical tests are done in order to validate the obtained results.

Keywords: ``SIR" models, deterministic, time delay, stochastic, nonlinear incidence rate, equilibrium points, local and global stability, Lyapunov functions, Feller's test, stochastically permanent, chemostat

MSC numbers: Primary 34D23, 35N25, 37B25, 49K40, 60H10, 65C30, 91B70