Zhen Jin

Pattern formation of a spatial predator–prey system

Abstract There are random and directed movements of predator and prey populations in many natural systems which are strongly influenced and modified by spatial factors. To investigate how these migration (directed movement) and diffusion (random movement) affect predator–prey systems, we have studied the spatiotemporal complexity in a predator–prey system with Holling–Tanner form. A theoretical analysis of emerging spatial pattern is presented and wavelength and pattern speed are calculated. At the same time, we present the properties of pattern solutions. The results of numerical simulations show that migration has prominent effect on the pattern formation of the population, i.e., changing Turing pattern into traveling pattern. This study suggests that modelling by migration and diffusion in predator–prey systems can account for the dynamical complexity of ecosystems.

Pattern formation in a spatial S–I model with non-linear incidence rates

In this paper, we investigate a spatial S–I model with non-linear incidence rates βSpIq, and obtain the conditions for transcritical bifurcation, Hopf bifurcation and Turing bifurcation. In particular, the exact Turing domain is found in the parameter space. For parameters are in that domain, a series of numerical simulations reveal that the model has rich dynamics. In our previous paper (Liu and Jin, 2007xa0J.xa0Stat.xa0Mech.xa0P05002), we considered an epidemic model with constant rate of removal of the infectives and the mass action incidence βSI, in which only a stripe pattern was observed. However, in this paper, we obtain not only a stripe-like pattern but also a spot pattern, or coexistence of the two. The results obtained extend well the finding of pattern formation in the epidemic model and may well explain the field observed in some areas.

Modeling Seasonal Rabies Epidemics in China

Human rabies, an infection of the nervous system, is a major public-health problem in China. In the last 60 years (1950–2010) there had been 124,255 reported human rabies cases, an average of 2,037 cases per year. However, the factors and mechanisms behind the persistence and prevalence of human rabies have not become well understood. The monthly data of human rabies cases reported by the Chinese Ministry of Health exhibits a periodic pattern on an annual base. The cases in the summer and autumn are significantly higher than in the spring and winter. Based on this observation, we propose a susceptible, exposed, infectious, and recovered (SEIRS) model with periodic transmission rates to investigate the seasonal rabies epidemics. We evaluate the basic reproduction number R0, analyze the dynamical behavior of the model, and use the model to simulate the monthly data of human rabies cases reported by the Chinese Ministry of Health. We also carry out some sensitivity analysis of the basic reproduction number R0 in terms of various model parameters. Moreover, we demonstrate that it is more reasonable to regard R0 rather than the average basic reproduction number

The analysis of an epidemic model on networks

bar{R}_{0}

Influence of infection rate and migration on extinction of disease in spatial epidemics

or the basic reproduction number

The role of noise in a predator–prey model with Allee effect

hat{R}_{0}

Chaos induced by breakup of waves in a spatial epidemic model with nonlinear incidence rate

of the corresponding autonomous system as a threshold for the disease. Finally, our studies show that human rabies in China can be controlled by reducing the birth rate of dogs, increasing the immunization rate of dogs, enhancing public education and awareness about rabies, and strengthening supervision of pupils and children in the summer and autumn.

Modeling the transmission dynamics of sheep brucellosis in Inner Mongolia Autonomous Region, China

Abstract The paper consider an epidemic model with birth and death on networks. We derive the epidemic threshold R0 dependent on birth rate b, death rate d (natural death) and μ from the infectious disease and natural death, and cure rate γ. And the stability of the equilibriums (the disease-free equilibrium and endemic equilibrium) are analysed. Finally, the effects of various immunization schemes are studied and compared. We show that both targeted, and acquaintance immunization strategies compare favorably to a proportional scheme in terms of effectiveness. For active immunization, the threshold is easier to apply practically. To illustrate our theoretical analysis, some numerical simulations are also included.

Influence of removable devices on computer worms: Dynamic analysis and control strategies

Extinction of disease can be explained by the patterns of epidemic spreading, yet the underlying causes of extinction are far from being well understood. To reveal a mechanism of disease extinction, a cellular automata model with both birth, death rate and migration is presented. We find that, in single patch, when the infection rate is small or large enough, the disease will disappear for a long time. When the invasion form is in the coexistence of stable spiral and turbulent wave state, the disease will persist. Also, we find that the migration has dual effects on the epidemic spreading. On one hand, in the extinction region of single patch, if the migration rate is large enough, there is a phase transition from the disease free to endemic state in two patches. On the other hand, migration will induce extinction in the regime, which can ensure the persistence of the disease in single patch, due to emergence of anti-phase synchrony. The results obtained well reveal the effect of infection rate and migration on the extinction of the disease, which enriches the finding in the filed of epidemiology and may provide some new ideas to control the disease in the real world.

Nonlinear dynamic and pattern bifurcations in a model for spatial patterns in young mussel beds

The existence and implications of alternative stable states in ecological systems have been investigated extensively within deterministic models. However, it is known that natural systems are undeniably subject to random fluctuations, arising from either environmental variability or internal effects. Thus, in this paper, we study the role of noise on the pattern formation of a spatial predator–prey model with Allee effect. The obtained results show that the spatially extended system exhibits rich dynamic behavior. More specifically, the stationary pattern can be induced to be a stable target wave when the noise intensity is small. As the noise intensity is increased, patchy invasion emerges. These results indicate that the dynamic behavior of predator–prey models may be partly due to stochastic factors instead of deterministic factors, which may also help us to understand the effects arising from the undeniable susceptibility to random fluctuations of real ecosystems.