A General Discrete Time Model of Population Dynamics in the Presence of an Infection

Abstract

We present a set of difference equations which generalizes that proposed in the work of G. Izzo and A. Vecchio (2007) and represents the discrete counterpart of a larger class of continuous model concerning the dynamics of an infection in an organism or in a host population. The limiting behavior of this new discrete model is studied and a threshold parameter playing the role of the basic reproduction number is derived.

Global stability for a class of discrete SIR epidemic models

“…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).…”

“…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].…”

Bifurcations and chaos in a discrete SI epidemic model with fractional order

“…There are many discretization methods that have been used to construct the discretetime model using continuous-time methods such as explicit and implicit Euler's method, Runge-Kutta method, predictor-corrector method and nonstandard finite difference methods [33][34][35][36][37]. Some of them are approximation for the derivative and some for the integral.…”

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].…”