Linearizability Problem of Resonant Degenerate Singular Point for Polynomial Differential Systems

The linearizability (or isochronicity) problem is one of the open problems for polynomial differential systems which is far to be solved in general. A progressive way to find necessary conditions for linearizability is to compute period constants. In this paper, we are interested in the linearizability problem of p : −q resonant degenerate singular point for polynomial differential systems. Firstly, we transform degenerate singular point into the origin via a homeomorphism. Moreover, we establish a new recursive algorithm to compute the so-called generalized period constants for the origin of the transformed system. Finally, to illustrate the effectiveness of our algorithm, we discuss the linearizability problems of 1 : −1 resonant degenerate singular point for a septic system. We stress that similar results are hardly seen in published literatures up till now. Our work is completely new and extends existing ones

Seasonality Impact on the Transmission Dynamics of Tuberculosis

The statistical data of monthly pulmonary tuberculosis (TB) incidence cases from January 2004 to December 2012 show the seasonality fluctuations in Shaanxi of China. A seasonality TB epidemic model with periodic varying contact rate, reactivation rate, and disease-induced death rate is proposed to explore the impact of seasonality on the transmission dynamics of TB. Simulations show that the basic reproduction number of time-averaged autonomous systems may underestimate or overestimate infection risks in some cases, which may be up to the value of period. The basic reproduction number of the seasonality model is appropriately given, which determines the extinction and uniform persistence of TB disease. If it is less than one, then the disease-free equilibrium is globally asymptotically stable; if it is greater than one, the system at least has a positive periodic solution and the disease will persist. Moreover, numerical simulations demonstrate these theorem results

Seasonality Impact on the Transmission Dynamics of Tuberculosis

The statistical data of monthly pulmonary tuberculosis (TB) incidence cases from January 2004 to December 2012 show the seasonality fluctuations in Shaanxi of China. A seasonality TB epidemic model with periodic varying contact rate, reactivation rate, and disease-induced death rate is proposed to explore the impact of seasonality on the transmission dynamics of TB. Simulations show that the basic reproduction number of time-averaged autonomous systems may underestimate or overestimate infection risks in some cases, which may be up to the value of period. The basic reproduction number of the seasonality model is appropriately given, which determines the extinction and uniform persistence of TB disease. If it is less than one, then the disease-free equilibrium is globally asymptotically stable; if it is greater than one, the system at least has a positive periodic solution and the disease will persist. Moreover, numerical simulations demonstrate these theorem results

A Mathematical Study of a TB Model with Treatment Interruptions and Two Latent Periods

A TB transmission model which incorporates treatment interruptions and two latent periods is presented. The threshold parameter known as the control reproduction number and the equilibria for the model are determined, and the global asymptotical stabilities of the equilibria are studied by constructing the proper Lyapunov functions. The reproduction numbers and numerical simulations show that treatment of active TB cases always helps to control the TB epidemic, while treatment interruptions may have a negative, positive, or no effect on combating TB epidemic

A Stochastic HIV Infection Model with Latent Infection and Antiretroviral Therapy

Recent studies have demonstrated that the latent infection is a major obstacle to the viral elimination in HIV infection process. In this paper, we formulate a stochastic HIV infection model to include both latent infection and combination drug therapies. We derive that the model solution is unique and positive, and the solution is global. By constructing appropriate stochastic Lyapunov functions, the existence of an ergodic stationary distribution is obtained when the critical condition is greater than one. Furthermore, through rigorous analysis and deduction, the extinction of the virus is established under certain conditions. Numerical simulations are performed to show that small intensity of white noises can maintain the existence of a stationary distribution, while large intensity of white noises is beneficial to the extinction of the virus

Impact of migrant workers on the tuberculosis transmission : general models and a case study for China

With heavy influx of migrant workers into cities, curbing the spread of large-scale tuberculosis (TB) and HIV infection is an immense challenge. A case study, based on the TB epidemiological and other statistical data in China, indicates that the disease spread can be controlled if effective measures are taken to reduce the reactivation rate of exposed/latent migrant workers. Impact of the migration rate and direction, as well as the duration of home visit stay, on the control of disease spread is also examined numerically. The research develops a mathematical model that factors in epidemiological, social and economic features of migrant workers