Mathematical Analysis of a Fractional-order "SIR" Epidemic Model with a General Nonlinear Saturated Incidence Rate in a Chemostat

Abstract

In the present work, a fractional-order differential equation based on the Susceptible-Infected- Recovered (SIR) model with nonlinear incidence rate in a continuous reactor is proposed. A profound qualitative analysis is given. The analysis of the local and global stability of equilibrium points is carried out. It is proved that if the basic reproduction number R > 1 then the disease-persistence (endemic) equilibrium is globally asymptotically stable. However, if R ≤ 1, then the disease-free equilibrium is globally asymptotically stable. Finally, some numerical tests are done in order to validate the obtained results.

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“…Further, by Lemma 1, Ω 2 is a compact, absorbing subset of R 5 + , and the largest compact invariant set in {(S, E, I, R,W ) ∈ Ω 2 :V 1 = 0} is {Q}. Therefore, by the Lasalle's invariance principle (see, for instance, [15,Theorem 3.1] and [7,8,9,10,11,12,13,14] for other applications), every solution of system (1) with initial conditions in R 5 + converges toQ as t → +∞.…”

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“…R 5 + , the closed non-negative cone in R 5 , is positively invariant by the system (2.6) [22,23,24,25,26,27,28,29,30,31,32,33]. More precisely, let m = min(mS, mV , mE, mI , mR), then I get Proposition 1.…”

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“…+ , the closed non-negative cone in R n+1 , is positively invariant [22,23,24,25,26,15,17,16,27,28,12,29,18,13]…”