Stochastic dynamics of an SIS epidemiological model with media coverage. (English) Zbl 1540.92238
Summary: In this paper, we establish a stochastic SIS epidemic model with general transmission function and media coverage, and prove that two thresholds \(\mathcal{R}_1^s\) and \(\mathcal{R}_2^s (\mathcal{R}_2^s < \mathcal{R}_1^s)\) can be used to govern the stochastic dynamics of the stochastic SIS model. If \(\mathcal{R}_1^s < 1\), the disease will die out with probability one and the distribution of the susceptible converges weakly to a boundary distribution; while if \(\mathcal{R}_2^s > 1\), the disease is persistent almost surely and there exists a unique stationary distribution. Furthermore, we study the disease dynamics when \(\mathcal{R}_2^s < 1 < \mathcal{R}_1^s\) numerically, and find that the disease dynamics in this range is rich and complex, i.e., the disease may persist or extinct. Epidemiologically, we find that the smaller the intensity of random disturbance, the smaller the oscillation amplitude of the solution, and as the intensities of random disturbance increase, the mean of the infectious decreases and the distribution of them becomes more and more right-skewed, which provide a theoretical basis for disease control.
MSC:
| 92D30 | Epidemiology |
| 60H10 | Stochastic ordinary differential equations (aspects of stochastic analysis) |
| 34C60 | Qualitative investigation and simulation of ordinary differential equation models |
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