Quan-Xing Liu

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

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.

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

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.

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

Spatial epidemiology is the study of spatial variation in disease risk or incidence, including the spatial patterns of the population. The spread of diseases in human populations can exhibit large scale patterns, underlining the need for spatially explicit approaches. In this paper, the spatiotemporal complexity of a spatial epidemic model with nonlinear incidence rate, which includes the behavioral changes and crowding effect of the infective individuals, is investigated. Based on both theoretical analysis and computer simulations, we find out when, under the parameters which can guarantee a stable limit cycle in the non-spatial model, spiral and target waves can emerge. Moreover, two different kinds of breakup of waves are shown. Specifically, the breakup of spiral waves is from the core and the breakup of target waves is from the far-field, and both kinds of waves become irregular patterns at last. Our results reveal that the spatiotemporal chaos is induced by the breakup of waves. The results obtained confirm that diffusion can form spiral waves, target waves or spatial chaos of high population density, which enrich the findings of spatiotemporal dynamics in the epidemic model.

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

Young mussel beds on soft sediments can display large-scale regular spatial patterns. This phenomenon can be explained relatively simply by a reaction–diffusion–advection (RDA) model of the interaction between algae and mussel, which includes the diffusive spread of mussel and the advection of algae. We present a detailed analysis of pattern formation in this RDA model. We derived the conditions for differential-flow instability that cause the formation of spatial patterns, and then systematically investigated how these patterns depend on model parameters. We also present a detailed study of the patterned solutions in the full nonlinear model, using numerical bifurcation analysis of the ordinary differential equations, which were obtained from the RDA model. We show that spatial patterns occur for a wide range of algal concentrations, even when algal concentration is much lower than the prediction by linear analysis in the RDA model. That is to say, spatial patterns result from the interaction of nonlinear terms. Moreover, patterns with different wavelength, amplitude and movement speed may coexist. The results obtained are consistent with the previous observation that self-organization allows mussels to persist with algal concentrations that would not permit survival of mussels in a homogeneous bed.

FORMATION OF SPATIAL PATTERNS IN AN EPIDEMIC MODEL WITH CONSTANT REMOVAL RATE OF THE INFECTIVES

This paper addresses the question of how population diffusion affects the formation of the spatial patterns in the spatial epidemic model by Turing mechanisms. In particular, we present a theoretical analysis of results of the numerical simulations in two dimensions. Moreover, there is a critical value for the system within the linear regime. Below the critical value the spatial patterns are impermanent, whereas above it stationary spot and stripe patterns can coexist over time. We have observed the striking formation of spatial patterns during the evolution, but the isolated ordered spot patterns do not emerge in the space.

SPATIAL PATTERN IN AN EPIDEMIC SYSTEM WITH CROSS-DIFFUSION OF THE SUSCEPTIBLE

In this paper, pattern formation of a spatial model with cross diffusion of the susceptible is investigated. We compute Hopf and Turing bifurcations for the model. In particular, the exact Turing domain is delineated in the parameter space. When the parameters are in that domain, a series of numerical simulations reveals that the typical dynamics of the infecteds class typically involves the formation of isolated groups, i.e., striped, spotted or labyrinthine patterns. Furthermore, spatial oscillatory and anti-phase dynamics of different spatial points were also found. These results demonstrate that cross diffusion of susceptibles may have great influence on the spread of the epidemic.

SPATIAL PATTERN IN A PREDATOR-PREY SYSTEM WITH BOTH SELF- AND CROSS-DIFFUSION

The vast majority of models for spatial dynamics of natural populations assume a homogeneous physical environment. However, in practice, dispersing organisms may encounter landscape features that significantly inhibit their movement. And spatial patterns are ubiquitous in nature, which can modify the temporal dynamics and stability properties of population densities at a range of spatial scales. Thus, in this paper, a predator-prey system with Michaelis-Menten-type functional response and self- and cross-diffusion is investigated. Based on the mathematical analysis, we obtain the condition of the emergence of spatial patterns through diffusion instability, i.e., Turing pattern. A series of numerical simulations reveal that the typical dynamics of population density variation is the formation of isolated groups, i.e., stripe-like or spotted or coexistence of both. The obtained results show that the interaction of self-diffusion and cross-diffusion plays an important role on the pattern formation of the predator-prey system.

Persistence, extinction and spatio-temporal synchronization of SIRS spatial models

Spatially explicit models are widely used in todays mathematical ecology and epidemiology to study persistence and extinction of populations as well as their spatial patterns. Here we extend the earlier work on static dispersal between neighboring individuals to the mobility of individuals as well as multi-patch environments. As is commonly found, the basic reproductive ratio is maximized for the evolutionarily stable strategy for disease persistence in mean field theory. This has important implications, as it implies that for a wide range of parameters the infection rate tends to a maximum. This is opposite to the present result obtained from spatially explicit models, which is that the infection rate is limited by an upper bound. We observe the emergence of trade-offs of extinction and persistence for the parameters of the infection period and infection rate, and show the extinction time as having a linear relationship with respect to system size. We further find that higher mobility can pronouncedly promote the persistence of the spread of epidemics, i.e., a phase transition occurs from the extinction domain to the persistence domain, and the wavelength of the spirals increases with the mobility ratio enhancement and will ultimately saturate at a certain value. Furthermore, for the multi-patch case, we find that lower coupling strength leads to anti-phase oscillation of the infected fraction, while higher coupling strength corresponds to in-phase oscillation.

EXISTENCE OF TRAVELLING WAVES IN NONLINEAR SI EPIDEMIC MODELS

In this paper, we investigate a spatially extended SI epidemic system with a nonlinear incidence rate. Using mathematical analysis, we study the existence of a heteroclinic orbit connecting two equilibrium points in R3 which corresponds to a travelling wave solution connecting the disease-free and endemic equilibria for the reaction-diffusion system. In other words, the travelling wave solutions of the model are studied to determine the speed of disease dissemination, form the biological point of view. Moreover, this wave speed is obtained as a function of the models parameters, in order to assess the control strategies. Also, our theoretical results are confirmed by numerical simulations. The obtained results confirm that travelling wave can enhance the spread of the disease, which can provide some insights into controlling the disease.