This article proposes a stochastic SIR model with general nonlinear incidence and Lévy jumps, which is used to describe diseases spreading in human populations. The model takes into account the randomness and sublinearity of diseases and can more accurately describe the disease transmission process. Firstly, we prove that this stochastic SIR model has a unique global positive solution. Then, sufficient conditions for the extinction of the disease are given. We also discuss the case that the disease persists in the model. In addition, we study the asymptotic behavior of the solution of the stochastic SIR model relative to the equilibrium points of the deterministic SIR model. These results allow us to understand the trends and dynamic changes of diseases in human populations, providing theoretical support for developing more scientific and effective disease control strategies and prevention measures. Finally, we give some examples and numerical simulations to demonstrate the effectiveness and feasibility of the theoretical results.

Under-dispersed count data often appear in clinical trials, medical studies, demography, actuarial science, ecology, biology, industry and engineering. Although the generalized Poisson (GP) distribution possesses the twin properties of under- and over-dispersion, in the past 50 years, many authors only treat the GP distribution as an alternative to the negative binomial distribution for modeling over-dispersed count data. To our best knowledge, the issues of calculating maximum likelihood estimates (MLEs) of parameters in GP model without covariates and with covariates for the case of under-dispersion were not solved up to now. In this paper, we first develop a new minimization–maximization (MM) algorithm to calculate the MLEs of parameters in the GP distribution with under-dispersion, and then we develop another new MM algorithm to compute the MLEs of the vector of regression coefficients for the GP mean regression model for the case of under-dispersion. Three hypothesis tests (i.e., the likelihood ratio, Wald and score tests) are provided. Some simulations are conducted. The Bangladesh demographic and health surveys dataset is analyzed to illustrate the proposed methods and comparisons with the existing Conway–Maxwell–Poisson regression model are also presented.