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Global attractivity of a discrete SIRS epidemic model with standard incidence rate. (English) Zbl 1278.92032

The authors use the forward Euler scheme for a straightforward discretization of a continuous-time SIRS model with a standard incidence of the form

$\begin{array}{cc}\hfill \stackrel{˙}{S}& =A-dS-\lambda \frac{SI}{S+I+R}+\sigma R,\hfill \\ \hfill \stackrel{˙}{I}& =\lambda \frac{SI}{S+I+R}-\left(d+\gamma +\alpha \right)I,\hfill \\ \hfill \stackrel{˙}{R}& =\gamma I-\left(d+\sigma \right)R\hfill \end{array}\phantom{\rule{2.em}{0ex}}\left(1\right)$

obtaining a discrete system

$\begin{array}{cc}\hfill {S}_{t+1}& ={S}_{t}+h\left[A-d{S}_{t}-\lambda \frac{{S}_{t}{I}_{t}}{{S}_{t}+{I}_{t}+{R}_{t}}+\sigma {R}_{t}\right],\hfill \\ \hfill {I}_{t+1}& ={I}_{t}+h\left[\lambda \frac{{S}_{t}{I}_{t}}{{S}_{t}+{I}_{t}+{R}_{t}}-\left(d+\gamma +\alpha \right){I}_{t}\right],\hfill \\ \hfill {R}_{t+1}& ={R}_{t}+h\left[\gamma {I}_{t}-\left(d+\sigma \right){R}_{t}\right]·\hfill \end{array}\phantom{\rule{2.em}{0ex}}\left(2\right)$

Stating that “for model (2), from the preceding process of discretization, we can imagine that, when the time step size $h>0$ is small enough, the properties of stability of model (2) should be accordant with model (1)”, the authors study global attractivity of the endemic equilibrium of model (2) concluding that “the dynamical behavior of the discrete-time epidemic model (2) is more complicated than the corresponding continuous-time epidemic model (1)”.

It should be emphasized at this point that even for a simple logistic equation, a direct discretization leads to an equation whose solutions with the growth of the intrinsic growth rate $r$ exhibit a completely different dynamics as compared to that of the solutions to the original differential equation. Therefore, it is not clear whether equations (1) and (2) model same epidemiological phenomena.

##### MSC:
 92C60 Medical epidemiology 92D30 Epidemiology
##### Keywords:
SIRS model; discretization; endemic equilibrium; attractivity