We derive a discretized SIRS epidemic model with time delay by applying a nonstandard finite difference scheme. Sufficient conditions for the global dynamics of the solution are obtained by improvements in discretization and applying proofs for continuous epidemic models. These conditions for our discretized model are the same as for the original continuous model.
In this paper, we consider the following logistic equation with piecewise constant arguments:where r > 0, a 0 , a 1 , . . . , a m 0, m j =0 a j > 0, and [x] means the maximal integer not greater than x. The sequence {N n } ∞ n=0 , where N n = N(n), n = 0, 1, 2, . . . , satisfies the difference equationUnder the condition that the first term a 0 dominates the other m coefficients a i , 1 i m, we establish new sufficient conditions of the global asymptotic stability for the positive equilibrium N * = 1/( m j =0 a j ).
The discrete hungry Lotka-Volterra (dhLV) system is a generalization of the discrete Lotka-Volterra (dLV) system which stands for a prey-predator model in mathematical biology. In this paper, we show that (1) some invariants exist which are expressed by dhLV variables and are independent from the discrete time and (2) a dhLV variable converges to some positive constant or zero as the discrete time becomes sufficiently large. Some characteristic polynomial is then factorized with the help of the dhLV system. The asymptotic behaviour of the dhLV system enables us to design an algorithm for computing complex eigenvalues of a certain band matrix.
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