## Journals

- Advances in Mathematics of Communications
- Big Data & Information Analytics
- Communications on Pure & Applied Analysis
- Discrete & Continuous Dynamical Systems - A
- Discrete & Continuous Dynamical Systems - B
- Discrete & Continuous Dynamical Systems - S
- Evolution Equations & Control Theory
- Inverse Problems & Imaging
- Journal of Computational Dynamics
- Journal of Dynamics & Games
- Journal of Geometric Mechanics
- Journal of Industrial & Management Optimization
- Journal of Modern Dynamics
- Kinetic & Related Models
- Mathematical Biosciences & Engineering
- Mathematical Control & Related Fields
- Mathematical Foundations of Computing
- Networks & Heterogeneous Media
- Numerical Algebra, Control & Optimization
- AIMS Mathematics
- Conference Publications
- Electronic Research Announcements
- Mathematics in Engineering

### Open Access Journals

In this paper, a stochastic model is formulated to describe the transmission dynamics of tuberculosis. The model incorporates vaccination and treatment in the intervention strategies. Firstly, sufficient conditions for persistence in mean and extinction of tuberculosis are provided. In addition, sufficient conditions are obtained for the existence of stationary distribution and ergodicity. Moreover, numerical simulations are given to illustrate these analytical results. The theoretical and numerical results show that large environmental disturbances can suppress the spread of tuberculosis.

The competitive exclusion principle means that the strain with the largest reproduction number persists while eliminating all other strains with suboptimal reproduction numbers. In this paper, we extend the competitive exclusion principle to a multi-strain vector-borne epidemic model with age-since-infection. The model includes both incubation age of the exposed hosts and infection age of the infectious hosts, both of which describe the different removal rates in the latent period and the variable infectiousness in the infectious period, respectively. The formulas for the reproduction numbers $\mathcal R^j_0$ of strain $j,j=1,2,···, n$, are obtained from the biological meanings of the model. The strain $j$ can not invade the system if $\mathcal R^j_0<1$, and the disease free equilibrium is globally asymptotically stable if $\max_j\{\mathcal R^j_0\}<1$. If $\mathcal R^{j_0}_0>1$, then a single-strain equilibrium $\mathcal{E}_{j_0}$ exists, and the single strain equilibrium is locally asymptotically stable when $\mathcal R^{j_0}_0>1$ and $\mathcal R^{j_0}_0>\mathcal R^{j}_0,j≠ j_0$. Finally, by using a Lyapunov function, sufficient conditions are further established for the global asymptotical stability of the single-strain equilibrium corresponding to strain $j_0$, which means strain $j_0$ eliminates all other stains as long as $\mathcal R^{j}_0/\mathcal R^{j_0}_0<b_j/b_{j_0}<1,j≠ j_0$, where $b_j$ denotes the probability of a given susceptible vector being transmitted by an infected host with strain $j$.

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 $\mathfrak{R}_v$ is derived. It shows that the disease-free equilibrium is unique and is globally asymptotically stable if $\mathfrak{R}_v < 1$, and unstable if $\mathfrak{R}_v>1$. 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.

## Year of publication

## Related Authors

## Related Keywords

[Back to Top]