Abstract
In this work (Part I), we reinvestigate the study of the stability of the Covid-19 mathematical model constructed by Shah et al. (2020) [1]. In their paper, the transmission of the virus under different control strategies is modeled thanks to a generalized SEIR model. This model is characterized by a five dimensional nonlinear dynamical system, where the basic reproduction number can be established by using the next generation matrix method. In this work (Part I), it is established that the disease free equilibrium point is locally as well as globally asymptotically stable when . When , the local and global asymptotic stability of the equilibrium are determined employing the second additive compound matrix approach and the Li-Wang’s (1998) stability criterion for real matrices [2]. In the second paper (Part II), some control parameters with uncertainties will be added to stabilize the five-dimensional Covid-19 system studied here, in order to force the trajectories to go to the equilibria. The stability of the Covid-19 system with these new parameters will also be assessed in Intissar (2020) [3] applying the Li-Wang criterion and compound matrices theory. All sophisticated technical calculations including those in part I will be provided in appendices of the part II.
Highlights
The evolution of epidemics is one of the most dangerous problems for a society
Definition 3.7 The basic reproduction number R0 is obtained as the spectral radius of matrix FV−1 at disease free equilibrium point
By using the generation matrix method, the basic reproduction number R0 is obtained as the spectral radius of matrix (−FV−1) at disease free equilibrium point where F and V are as below : β1Λ μ
Summary
The evolution of epidemics is one of the most dangerous problems for a society. As mankind already faced severe pandemics such as the Spanish flu in 1917, the Honk Kong flu (H3N2) in 1968 and the swine flu (H1N1) in 2009, the forecast of epidemics evolution appears to be one of the most critical topics for our societies. The work is organized as follows: In section 1, the mathematical Covid-19 model and its parameters are presented, alongside with some preliminary results on linear stability analysis for systems of ordinary differential equations. If the eigenvalues of the Jacobian matrix all have real parts < 0, the steady state is asymptotically stable. Remark 1.7 A hyperbolic equilibrium point x∗ is asymptotically stable if the eigenvalues of the Jacobian matrix all have real parts < 0 or otherwise it is unstable. We outline the Li-Wang’s stability criterion for real matrices and we recall of some spectral properties of M −matrices
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