This paper presents a positive L 1 state-bounding observer which can estimate the state all the time for nonlinear positive Markovian jump systems. The considered system is represented by Takagi–Sugeno(T–S) fuzzy system. First of all, by constructing a linear co-positive Lyapunov function, the sufficient condition is obtained for stochastic stability and L 1 performance analysis respectively. Furthermore, a positive L 1 state-bounding observer for positive T–S fuzzy Markovian jump systems is designed. The conditions can be solved by linear programming. Finally, the results obtained are applied to the epidemiological model and illustrated to be better than the existing results by a numerical example.