A discrete time version for models of population dynamics in the presence of an infection

Abstract

We present a set of difference equations which represents the discrete counterpart of a large class of continuous model concerning the dynamics of an infection in an organism or in a host population. The limiting behavior of the discrete model is studied and a threshold parameter playing the role of the basic reproduction number is derived

Stability in a system subject to noise with regulated periodicity

“…In this section, we present a model of a source of pulse noise that can be used as a multiplicative noise source for a linear system model. The corresponding stochastic linear differential equation is obtained starting from a discrete linear model, sometimes used in the literature [37].…”

Global stability for a class of discrete SIR epidemic models

“…For a discrete epidemic model with immigration of infectives, Jang and Elaydi [10] showed the global asymptotic stability of the disease-free equilibrium, the local asymptotic stability of the endemic equilibrium and the strong persistence of susceptible class by means of the nonstandard discretization method. Recently, using a discretization called "mixed type" formula in Izzo and Vecchio [8] and Izzo et al [9], Sekiguchi [16] obtained the permanence of a class of SIR discrete epidemic models with one delay and SEIRS discrete epidemic models with two delays if an endemic equilibrium of each model exists. For the detailed property for a class of discrete epidemic models, we refer to [3, 4, 8-11, 16, 18].…”

“…To prove the positivity of s(p), i(p) and r(p) for any p ≥ 0, we need to use the backward Euler discretization instead of the forward Euler discretization (see, e.g., [8,9]). Moreover, to apply a discrete time analogue of the Lyapunov function proposed by McCluskey [14,15], we adopt a variation of the backward Euler method which is different from that of Sekiguchi [16].…”

“…Various mathematical models have been proposed in the literature of population dynamics, ecology and epidemiology. Many authors have studied the epidemic models, which displays the dynamical behavior of the transmission of infectious diseases (see also [1][2][3][4][5][6][7][8][9][10][11][12][13][14][15][16][17][18] and the references therein).…”

Qualitative behavior of a discrete SIR epidemic model

“…Moreover, in case of discrete-time models, we can use statistical data for numerical simulations because infection data are computed at discrete-time. For some interesting results related to qualitative behavior of discrete-time epidemic models, we refer interested reader to [1,[16][17][18]20].…”