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In this paper, an SIR patch model with vaccination is formulated to investigate the effect of vaccination coverage and the impact of human mobility on the spread of diseases among patches. The control reproduction number Rv is derived. It shows that the disease-free equilibrium is unique and is globally asymptotically stable if Rv< 1, and unstable if Rv>1. The sufficient condition for the local stability of boundary equilibria and the existence of equilibria are obtained for the case n=2. Numerical simulations indicate that vaccines can control and prevent the outbreak of infectious in all patches while migration may magnify the outbreak size in one patch and shrink the outbreak size in other patch.

In this paper, we discuss the global dynamics of a general susceptible-infected-recovered-susceptible (SIRS) epidemic model. By using LaSalle's invariance principle and Lyapunov direct method, the global stability of equilibria is completely established. If there is no input of infectious individuals, the dynamical behaviors completely depend on the basic reproduction number. If there exists input of infectious individuals, the unique equilibrium of model is endemic equilibrium and is globally asymptotically stable. Once one place has imported a disease case, then it may become outbreak after that. Numerical simulations are presented to expound and complement our theoretical conclusions. [ABSTRACT FROM AUTHOR]

This paper focuses on the global stability of an epidemic model with vaccination, treatment and isolation. The basic reproduction number R 0 is derived. By constructing suitable Lyapunov functions, sufficient conditions for the global asymptotic stability of equilibria are obtained. Numerical simulations are performed to verify and complement the theoretical results. Furthermore, we consider the uncertainty and sensitivity analysis of the basic reproduction number R 0 . The results show that the transmission rate, the fraction of infected receives treatment, vaccination rate, the isolation rate are crucial to prevent the spread of infectious diseases. These suggest that public health workers design the control strategies of disease should consider the influence of vaccination, treatment and isolation. [ABSTRACT FROM AUTHOR]

• We proposed a pulmonary tuberculosis transmission model by considering direct/ indirect infection and the difference between smear positive and negative individuals in transmission capability. • Basic reproduction number R 0 = 2.447 , indicating that the disease has been prevalent in Xinjiang during 2008–2017. • Control measures like early screening, isolating and treating the various types of pulmonary tuberculosis patients, and enhancing the clearance of Mycobacterium tuberculosis in the environment could effectively reduce the disease transmission in Xinjiang. Xinjiang has been one of the high incidence areas of pulmonary tuberculosis (PTB) in China. Besides being infected by direct contacting with active PTB individuals (direct infection), the susceptible would be infected because of the exposure to the environment contaminated by Mycobacterium tuberculosis (indirect infection). Active PTB individuals include not only the smear-positive PTB (PTB+) but also the smear-negative PTB (PTB–) who are infectious due to their ability to release tiny Mycobacterium tuberculosis particles even in the absence of visible Mycobacterium tuberculosis in sputum. By taking account of direct/indirect infection and the difference between PTB+ and PTB– individuals in transmission capability, a periodic dynamical PTB transmission model is proposed. The model is fitted to the newly monthly PTB+ and PTB– cases in Xinjiang from 2008 to 2017 by Markov Chain Monte Carlo algorithm. Moreover, global sensitivity analysis is constructed to address the uncertainty of some key parameters by using Latin hypercube sampling and partial rank correlation coefficient methods. Basic reproduction number R 0 for PTB transmission in Xinjiang is estimated to be 2.447 (95% CrI:(1.203, 3.844)), indicating that PTB has been prevalent in Xinjiang over the study period. Our results suggest that reducing the direct/indirect transmission rates, early screening, isolating and treating the latent, PTB+ and PTB– individuals, and enhancing the clearance of Mycobacterium tuberculosis in the environment could more effectively control PTB transmission in Xinjiang. The model fits the reported PTB data well and achieves acceptable prediction accuracy. We believe that our model can provide heuristic support for controlling PTB transmission in Xinjiang. [ABSTRACT FROM AUTHOR]