Mathematical analysis of an ``SIR'' epidemic model in a continuous reactor - deterministic and probabilistic approaches
J. Korean Math. Soc. 2021 Vol. 58, No. 1, 45-67
Published online November 24, 2020
Printed January 1, 2021
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
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