Abstract
In this paper, we construct a backward difference scheme for a class of general SIR epidemic model with general incidence function f. We use the step size h > 0, for the discretization. The dynamical properties are investigated (positivity and the boundedness of solution). By constructing the Lyapunov function, under the conditions that function f satisfies some assumptions. The global stabilities of equilibria are obtained. If the basic reproduction number R0<1, the disease-free equilibrium is globally asymptotically stable. If R0>1, the endemic equilibrium is globally asymptotically stable.
Highlights
In the theory of epidemic dynamical models there are two kinds of mathematical models, the continuous-time models described by differential equations and the discrete-time models described by difference equations.SIR SIRS Nowadays, in order to study the continuous time and epidemic models, many various discrete dynamical model have been constructed and dynamical properties have been considered in many papers such as ([9, 5, 11, 12, 13])
We study the dynamical properties, especially the global stability of the disease-free equilibrium and endemic equilibrium for this discrete model
ESIRR we study the global stability of the endemic equilibrium given by ∗ = ( ∗, ∗, ∗)
Summary
In the theory of epidemic dynamical models there are two kinds of mathematical models, the continuous-time models described by differential equations and the discrete-time models described by difference equations. SIR SIRS Nowadays, in order to study the continuous time and epidemic models, many various discrete dynamical model have been constructed and dynamical properties have been considered in many papers such as ([9, 5, 11, 12, 13]). We h SIR discretize the continuous-time model studied in [6], by using the backward difference scheme with time step size. We study the dynamical properties, especially the global stability of the disease-free equilibrium and endemic equilibrium for this discrete model. R The paper is organized as followed, in second section we give the discrete mathematical model, the basic reproduction number 0 and the existence and uniqueness of disease-free equilibrium and endemic equilibrium.
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