Zhisheng Shuai

A graph-theoretic approach to the method of global Lyapunov functions

A class of global Lyapunov functions is revisited and used to resolve a long-standing open problem on the uniqueness and global stability of the endemic equilibrium of a class of multi-group models in mathematical epidemiology. We show how the group structure of the models, as manifested in the derivatives of the Lyapunov function, can be completely described using graph theory.

Optimal impulsive harvesting policy for single population

In this paper, we established the exploitation of impulsive harvesting single autonomous population model by Logistic equation. By some special methods, we analysis the impulsive harvesting population equation and obtain existence, the explicit expression and global attractiveness of impulsive periodic solutions for constant yield harvest and proportional harvest. Then, we choose the maximum sustainable yield as management objective, and investigate the optimal impulsive harvesting policies respectively. The optimal harvest effort that maximizes the sustainable yield per unit time, the corresponding optimal population levels are determined. At last, we point out that the continuous harvesting policy is superior to the impulsive harvesting policy, however, the latter is more beneficial in realistic operation.

Global Stability of Infectious Disease Models Using Lyapunov Functions

Two systematic methods are presented to guide the construction of Lyapunov functions for general infectious disease models and are thus applicable to establish their global dynamics. Specifically, a matrix-theoretic method using the Perron eigenvector is applied to prove the global stability of the disease-free equilibrium, while a graph-theoretic method based on Kirchhoffs matrix tree theorem and two new combinatorial identities are used to prove the global stability of the endemic equilibrium. Several disease models in the literature and two new cholera models are used to demonstrate the applications of these methods.

Global Dynamics of a General Class of Multistage Models for Infectious Diseases

We propose a general class of multistage epidemiological models that allow possible deterioration and amelioration between any two infected stages. The models can describe disease progression through multiple latent or infectious stages as in the case of HIV and tuberculosis. Amelioration is incorporated into the models to account for the effects of antiretroviral or antibiotic treatment. The models also incorporate general nonlinear incidences and general nonlinear forms of population transfer among stages. Under biologically motivated assumptions, we derive the basic reproduction number

Global dynamics of cholera models with differential infectivity

R_0

Reproduction numbers for infections with free-living pathogens growing in the environment

and show that the global dynamics are completely determined by

Dynamics of an age-of-infection cholera model.

R_0

A cholera model in a patchy environment with water and human movement.

: if

Extending the type reproduction number to infectious disease control targeting contacts between types

R_0\leq 1

Cholera Models with Hyperinfectivity and Temporary Immunity

, the disease-free equilibrium is globally asymptotically stable, and the disease dies out; if