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In this paper we extend the classical SIS epidemic model from a deterministic framework to a stochastic one. We also study the long time behavior of the stochastic system. We mainly establish conditions for the extinction of disease from the population as well as the persistence of disease under different conditions. In the case of persistence, we show the existence of a stationary distribution. we found that the introduction of stochastic noise changes the basic reproduction number. The presented results are demonstrated by numerical simulations.

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In this paper, we propose parametric and nonparametric locally andasymptotically optimal tests for regression models with superdiagonal bilinear time series errors in short panel data (large n, small T). We establish a local asymptotic normality property– with respect to intercept μ, regression coefficient β, the scale parameter σ of the error, and the parameter b of panel superdiagonal bilinear model (which is the parameter of interest)– for a given density f1 of the error terms. Rank-based versions of optimal parametric tests are provided. This result, which allows, by Hájek’s representation theorem, the construction of locally asymptotically optimal rank-based tests for the null hypothesis b = 0 (absence of panel superdiagonal bilinear model). These tests –at specified innovation densities f1– are optimal (most stringent), but remain valid under any actual underlying density. From contiguity, we obtain the limiting distribution of our test statistics under the null and local sequences of alternatives. The asymptotic relative efficiencies, with respect to the pseudo-Gaussian parametric tests, are derived. A Monte Carlo study confirms the good performance of the proposed tests.

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