Reaction Diffusion Epidemic Model Research Articles

Turing patterns in a reaction–diffusion epidemic model

In this paper, we investigate the spatiotemporal dynamics of a reaction–diffusion epidemic model with zero-flux boundary conditions. The value of our study lies in two aspects: mathematically, by using maximum principle and the linearized stability theory, a priori estimates of the steady state system and the local asymptotic stability of positive constant solution are given. By using the implicit function theorem, the existence and nonexistence of nonconstant positive steady states are shown. Applying the bifurcation theory, the global bifurcation structure of nonconstant positive steady states is established. Epidemiologically, through numerical simulations, under the conditions of the existence of nonconstant positive steady states, we find that the smaller the space, the easier the pattern formation; the bigger the diffusion, the easier the pattern formation. These results are beneficial to disease control, that is, we must do our best to control the diffusion of the infectious to avoid disease outbreak.

An SIS epidemic reaction–diffusion model with spontaneous infection in a spatially heterogeneous environment

This paper concerns an SIS epidemic reaction–diffusion model, which includes the possibility for spontaneous infection, in addition to disease transmission. We prove the global asymptotic stability of the unique endemic equilibrium when the spatial environment is homogeneous. When the spatial environment is heterogeneous, we show the existence of endemic equilibria and then analyze their asymptotic profile if the movement (migration) rate of the susceptible or infected population is small or large. Compared to the case that spontaneous infection is ignored, our theoretical study here suggests that spontaneous infection can enhance persistence of infectious disease and therefore gives new insight into disease control and prediction.

On a Diffusive Susceptible-Infected-Susceptible Epidemic Model with Mass Action Mechanism and Birth-Death Effect: Analysis, Simulations, and Comparison with Other Mechanisms

In the present paper, we are concerned with a susceptible-infected-susceptible epidemic reaction-diffusion model governed by a mass action infection mechanism and linear birth-death growth with no ...

Pattern formations of an epidemic model with Allee effect and time delay

Allee effect widely exists for endangered plants and animals in ecosystem, which indicates that the minimum population density or size is necessary for population survival, namely, Allee threshold. In this paper, a delayed reaction-diffusion epidemic model with respect to Allee effect is investigated. The instability of the positive constant steady state is induced by two mechanisms, one is diffusion-induced instability, the other is delay-induced instability. The first case gives rise to Turing patterns. Moreover, Turing region becomes narrow as incubation delay being increased. We further observe that the range of Turing mode is enlarged with the increase of Allee threshold. The numerical simulations verify our theoretical results. The combined effects of Allee effect and disease on the spatial distributions of endangered species are studied, which provides new insights for human intervention in conservation management of these species.

Analysis on a diffusive SIS epidemic model with logistic source

In this paper, we are concerned with an SIS epidemic reaction–diffusion model with logistic source in spatially heterogeneous environment. We first discuss some basic properties of the parabolic system, including the uniform upper bound of solutions and global stability of the endemic equilibrium when spatial environment is homogeneous. Our primary focus is to determine the asymptotic profile of endemic equilibria (when exist) if the diffusion (migration) rate of the susceptible or infected population is small or large. Combined with the results of Li et al. (J Differ Equ 262:885–913, 2017) where the case of linear source is studied, our analysis suggests that varying total population enhances persistence of infectious disease.

A periodic reaction–diffusion system modelling man–environment–man epidemics

The purpose of this work is to study the spatial dynamics of a periodic reaction–diffusion epidemic model arising from the spread of oral–faecal transmitted diseases. We first show that the disease spreading speed is coincident with the minimal wave speed for monotone periodic travelling waves. Then we obtain a threshold result on the global attractivity of either zero or the positive periodic solution in a bounded spatial domain.

Threshold dynamics of a reaction-diffusion epidemic model with stage structure

A time-delayed reaction-diffusion epidemic model with stage structure and spatial heterogeneity is investigated, which describes the dynamics of disease spread only proceeding in the adult population. We establish the basic reproduction number $\mathcal{R}_0$ for the model system, which gives the threshold dynamics in the sense that the disease will die out if $\mathcal{R}_0＜1$ and the disease will be uniformly persistent if $\mathcal{R}_0>1.$ Furthermore, it is shown that there is at least one positive steady state when $\mathcal{R}_0>1.$ Finally, in terms of general birth function for adult individuals, through introducing two numbers $\check{\mathcal{R}}_0$ and $\hat{\mathcal{R}}_0$ , we establish sufficient conditions for the persistence and global extinction of the disease, respectively.

A time-periodic reaction–diffusion epidemic model with infection period

In this paper, we propose a time-periodic and diffusive SIR epidemic model with constant infection period. By introducing the basic reproduction number $${\mathcal{R}_0}$$ via a next generation operator for this model, we show that the disease goes extinction if $${\mathcal{R}_0 < 1}$$ ; while the disease is uniformly persistent if $${\mathcal{R}_0 > 1}$$ .

A spatial SIS model in advective heterogeneous environments

We study the effects of diffusion and advection for a susceptible-infected-susceptible epidemic reaction–diffusion model in heterogeneous environments. The definition of the basic reproduction number R0 is given. If R0<1, the unique disease-free equilibrium (DFE) is globally asymptotically stable. Asymptotic behaviors of R0 for advection rate and mobility of the infected individuals (denoted by dI) are established, and the existence of the endemic equilibrium when R0>1 is studied. The effects of diffusion and advection rates on the stability of the DFE are further investigated. Among other things, we find that if the habitat is a low-risk domain, there may exist one critical value for the advection rate, under which the DFE changes its stability at least twice as dI varies from zero to infinity, while the DFE is unstable for any dI when the advection rate is larger than the critical value. These results are in strong contrast with the case of no advection, where the DFE changes its stability at most once as dI varies from zero to infinity.

A reaction–diffusion SIS epidemic model in an almost periodic environment

A susceptible–infected–susceptible almost periodic reaction–diffusion epidemic model is studied by means of establishing the theories and properties of the basic reproduction ratio \({R_{0}}\). Particularly, the asymptotic behaviors of \({R_{0}}\) with respect to the diffusion rate \({D_{I}}\) of the infected individuals are obtained. Furthermore, the uniform persistence, extinction and global attractivity are presented in terms of \({R_{0}}\). Our results indicate that the interaction of spatial heterogeneity and temporal almost periodicity tends to enhance the persistence of the disease.

Spatiotemporal Dynamics in a Reaction–Diffusion Epidemic Model with a Time-Delay in Transmission

In this paper, we investigate the effects of time-delay and diffusion on the disease dynamics in an epidemic model analytically and numerically. We give the conditions of Hopf and Turing bifurcations in a spatial domain. From the results of mathematical analysis and numerical simulations, we find that for unequal diffusive coefficients, time-delay and diffusion may induce that Turing instability results in stationary Turing patterns, Hopf instability results in spiral wave patterns, and Hopf–Turing instability results in chaotic wave patterns. Our results well extend the findings of spatiotemporal dynamics in the delayed reaction–diffusion epidemic model, and show that time-delay has a strong impact on the pattern formation of the reaction–diffusion epidemic model.

Solving the inverse problem of an SIS epidemic reaction–diffusion model by optimal control methods

In this contribution, a novel inverse problem on an SIS epidemic reaction–diffusion model is investigated by employing the optimal control methods. By setting a proper regular cost functional and choosing the coefficients to be identified as control variables, we embed the original inverse problem which is very difficult to solve into an optimal control problem. The existence of the optimal controls and the first order necessary optimality condition satisfied by the optimal controls are established, respectively. In particular, the local uniqueness and stability of the identification of the coefficients are proved by using analytical techniques.

Patch invasion in a spatial epidemic model

Understanding of patterns in disease spreading is an issue of significant current interest in epidemiology. In this paper, an epidemic model with spatial diffusion is considered. It was found that this model has stationary patterns including spotted and stripe patterns. Moreover, patch invasion was obtained in this reaction–diffusion epidemic model. It was well known that the spreading of disease takes place via irregular movement of separated patches due to environmental stochasticity. However, we show that pattern transition from stationary pattern to patch invasion appears to be possible in a fully deterministic parasite-host model, which may provide new insights to control the disease.

Threshold dynamics of a time periodic reaction–diffusion epidemic model with latent period

In this paper, we first propose a time-periodic reaction–diffusion epidemic model which incorporates simple demographic structure and the latent period of infectious disease. Then we introduce the basic reproduction number R0 for this model and prove that the sign of R0−1 determines the local stability of the disease-free periodic solution. By using the comparison arguments and persistence theory, we further show that the disease-free periodic solution is globally attractive if R0<1, while there is an endemic periodic solution and the disease is uniformly persistent if R0>1.

Asymptotic profile of the positive steady state for an SIS epidemic reaction–diffusion model: Effects of epidemic risk and population movement

Identifying the epidemic risk for infectious disease is crucial in order to effectively perform control measures. In a series of our work, from an analytical aspect we study the effects of epidemic risk and population movement on the spatiotemporal transmission of infectious disease via an SIS epidemic reaction–diffusion model proposed by Allen et al. (2008) in [36]. In Allen et al. (2008) [36], Peng (2009) [37], it was assumed that the habitat of the populations consists of only the low and high risk areas. The present paper concerns a more complicated heterogeneous environment where the moderate risk area occurs, and deals with two cases: (i) only the moderate and high risk areas exist; (ii) the low, moderate and high risk areas coexist. In each case, we rigorously determine the asymptotic profile of the positive steady state (i.e., the endemic equilibrium) as the migration rate of either the susceptible or infected population tends to zero. Our results show how epidemic risk and population movement affect the spatial distribution of infectious disease and thereby suggest important implications for predicting the patterns of disease occurrence and designing optimal control strategies. Numerical simulations are carried out to support the theoretical results.

Traveling Wave Solutions in a Reaction-Diffusion Epidemic Model

We investigate the traveling wave solutions in a reaction-diffusion epidemic model. The existence of the wave solutions is derived through monotone iteration of a pair of classical upper and lower solutions. The traveling wave solutions are shown to be unique and strictly monotonic. Furthermore, we determine the critical minimal wave speed.

The Existence of Positive Nonconstant Steady States in a Reaction: Diffusion Epidemic Model

We investigate the disease’s dynamics of a reaction-diffusion epidemic model. We first give a priori estimates of upper and lower bounds for positive solutions to model and then give the conditions of the existence and nonexistence of the positive nonconstant steady states, which guarantees the existence of the stationary patterns.

Traveling wave solutions for a diffusive sis epidemic model

In this paper, we study the traveling wave solutions of an SIS reaction-diffusion epidemic model. The techniques of qualitative analysis has been developed which enable us to show the existence of traveling wave solutions connecting the disease-free equilibrium point and an endemic equilibrium point. In addition, we also find the precise value of the minimum speed that is significant to study the spreading speed of the population towards to the endemic steady state.

Complex dynamics of a reaction–diffusion epidemic model

In this paper, we investigate the complex dynamics of a reaction–diffusion S−I model incorporating demographic and epidemiological processes with zero-flux boundary conditions. By the method of Lyapunov function, the global stability of the disease free equilibrium and the epidemic equilibrium was established. In addition, the conditions of Turing instability were obtained and the Turing space in the parameters space were given. Based on these results, we present the evolutionary processes that involves organism distribution and their interaction of spatially distributed population with local diffusion, and find that the model dynamics exhibits a diffusion-controlled formation growth to “holes, holes–stripes, stripes, spots–stripes and spots” pattern replication. Furthermore, we indicate that the diseases’ spread is getting smaller with R0 increasing, and the increasing the diffusion of infectious will increase the speed of diseases spreading. Our results indicate that the diffusion has a great influence on the spread of the epidemic and extend well the finding of spatiotemporal dynamics in the epidemic model.

Stability analysis of an epidemic model with diffusion and stochastic perturbation

In this paper, we investigate the stability of an epidemic model with diffusion and stochastic perturbation. We first show both the local and global stability of the endemic equilibrium of the deterministic epidemic model by analyzing corresponding characteristic equation and Lyapunov function. Second, for the corresponding reaction–diffusion epidemic model, we present the conditions of the globally asymptotical stability of the endemic equilibrium. And we carry out the analytical study for the stochastic model in details and find out the conditions for asymptotic stability of the endemic equilibrium in the mean sense. Furthermore, we perform a series of numerical simulations to illustrate our mathematical findings.