Stationary distribution and ergodicity of a stochastic cholera model with multiple pathways of transmission. (English) Zbl 1450.92075
Summary: A stochastic compartmental model for cholera is investigated, which incorporates the effect of two transmission pathways on the diseases via contaminated water. Multiple stages of infection and multiple states of pathogen and environmental white noises are considered, which include or extend the cholera models in some existing articles. The existence of a unique stationary distribution and ergodicity of the model are obtained by using a suitable Lyapunov function, which determines a critical value \(R_0^*\) corresponding to a basic reproduction number \(R_0\) of the ordinary differential equation. The results mean if \(R_0^*>1\), the system still retains the stability and all individuals are persistent in a long term. In addition, the sufficient conditions for extinction of diseases are derived. What’s more, we provide a new method in constructing a Lyapunov function, which can be successfully applied to other complex high-dimensional systems.
MSC:
| 92D30 | Epidemiology |
| 34F05 | Ordinary differential equations and systems with randomness |
| 60H10 | Stochastic ordinary differential equations (aspects of stochastic analysis) |
Keywords:
stochastic compartmental model of cholera; unique stationary distribution; ergodicity; stabilityReferences:
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