Global dynamics of epidemic network models via construction of Lyapunov functions

In this paper, we study the global dynamics of epidemic network models with standard incidence or mass-action transmission mechanism, when the dispersal of either the susceptible or the infected people is controlled. The connectivity matrix of the model is not assumed to be symmetric. Our main technique to study the global dynamics is to construct novel Lyapunov type functions

A Duality Theorem for Quantitative Semantics

AbstractThis paper mainly studies quantitative possibility theory in the framework of domain. Using Sugeno's integral and the notion of module a duality theorem is obtained between the extended possibilistic powerdomain over a continuous domain X and the extended fuzzy predicates on X. This duality provides a reassuring link between the spaces of quantitative meaning and the corresponding Scott-topological space

On the dynamics of an epidemic patch model with mass-action transmission mechanism and asymmetric dispersal patterns

This paper examines an epidemic patch model with mass-action transmission mechanism and asymmetric connectivity matrix. Results on the global dynamics of solutions and the spatial structures of endemic solutions are obtained. In particular, we show that when the basic reproduction number $\mathcal{R}_0$ is less than one and the dispersal rate of the susceptible population $d_S$ is large, the population would eventually stabilize at the disease-free equilibrium. However, the disease may persist if $d_S$ is small, even if $\mathcal{R}_0<1$. In such a scenario, explicit conditions on the model parameters that lead to the existence of multiple endemic equilibria are identified. These results provide new insights into the dynamics of infectious diseases in multi-patch environments. Moreover, results in [27], which is for the same model but with symmetric connectivity matrix, are generalized and improved

On a Vector-host Epidemic Model with Spatial Structure

In this paper, we study a reaction-diffusion vector-host epidemic model. We define the basic reproduction number $R_0$ and show that $R_0$ is a threshold parameter: if $R_0\le 1$ the disease free steady state is globally stable; if $R_0>1$ the model has a unique globally stable positive steady state. We then write $R_0$ as the spectral radius of the product of one multiplicative operator $R(x)$ and two compact operators with spectral radius equalling one. Here $R(x)$ corresponds to the basic reproduction number of the model without diffusion and is thus called local basic reproduction number. We study the relationship between $R_0$ and $R(x)$ as the diffusion rates vary