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