On the dynamics of SEIRS epidemic model with transport-related infection

Transportation amongst cities is found as one of the main factors which affect the outbreak of diseases. To understand the effect of transport-related infection on disease spread, an SEIRS (Susceptible, Exposed, Infectious, Recovered) epidemic model for two cities is formulated and analyzed. The epidemiological threshold, known as the basic reproduction number, of the model is derived. If the basic reproduction number is below unity, the disease-free equilibrium is locally asymptotically stable. Thus, the disease can be eradicated from the community. There exists an endemic equilibrium which is locally asymptotically stable if the reproduction number is larger than unity. This means that the disease will persist within the community. The results show that transportation among regions will change the disease dynamics and break infection out even if infectious diseases will go to extinction in each isolated region without transport-related infection. In addition, the result shows that transport-related infection intensifies the disease spread if infectious diseases break out to cause an endemic situation in each region, in the sense of that both the absolute and relative size of patients increase. Further, the formulated model is applied to the real data of SARS outbreak in 2003 to study the transmission of disease during the movement between two regions. The results show that the transport-related infection is effected to the number of infected individuals and the duration of outbreak in such the way that the disease becomes more endemic due to the movement between two cities. This study can be helpful in providing the information to public health authorities and policy maker to reduce spreading disease when its occurs.     

Global analysis of an epidemic model with nonmonotone incidence rate

In this paper we study an epidemic model with nonmonotonic incidence rate, which describes the psychological effect of certain serious diseases on the community when the number of infectives is getting larger. By carrying out a global analysis of the model and studying the stability of the disease-free equilibrium and the endemic equilibrium, we show that either the number of infective individuals tends to zero as time evolves or the disease persists.     

一类具饱和传染力和常数输入的SIRS脉冲接种模型研究

利用Floquet乘子理论,研究了一类具饱和传染力和常数输入的SIRS脉冲接种模型,得到了无病周期解全局渐近稳定和系统持久的充分条件.     

Global dynamics of an SEIR epidemic model with saturating contact rate

Heesterbeek and Metz [J. Math. Biol. 31 (1993) 529] derived an expression for the saturating contact rate of individual contacts in an epidemiological model. In this paper, the SEIR model with this saturating contact rate is studied. The basic reproduction number R0 is proved to be a sharp threshold which completely determines the global dynamics and the outcome of the disease. If R0 < or =1, the disease-free equilibrium is globally stable and the disease always dies out. If R0 > 1, there exists a unique endemic equilibrium which is globally stable and the disease persists at an endemic equilibrium state if it initially exists. The contribution of the saturating contact rate to the basic reproduction number and the level of the endemic equilibrium is also analyzed.     

Global stability of an SIR epidemic model with time delays

An SIR disease transmission model is formulated under the assumption that the force of infection at the present time depends on the number of infectives at the past. It is shown that a disease free equilibrium point is globally stable if no endemic equilibrium point exists. Further the endemic point (if it exists) is globally stable with respect to the whole state space except the neighborhood of the disease free state.Research partly supported by the Ministry of Education, Science and Culture, Japan, Grant 05640256     

Global Hopf bifurcation analysis of an susceptible-infective-removed epidemic model incorporating media coverage with time delay

An susceptible-infective-removed epidemic model incorporating media coverage with time delay is proposed. The stability of the disease-free equilibrium and endemic equilibrium is studied. And then, the conditions which guarantee the existence of local Hopf bifurcation are given. Furthermore, we show that the local Hopf bifurcation implies the global Hopf bifurcation after the second critical value of delay. The obtained results show that the time delay in media coverage can not affect the stability of the disease-free equilibrium when the basic reproduction number is less than unity. However, the time delay affects the stability of the endemic equilibrium and produces limit cycle oscillations while the basic reproduction number is greater than unity. Finally, some examples for numerical simulations are included to support the theoretical prediction.     

Periodicity in an epidemic model with a generalized non-linear incidence

We develop and analyze a simple SIV epidemic model including susceptible, infected and perfectly vaccinated classes, with a generalized non-linear incidence rate subject only to a few general conditions. These conditions are satisfied by many models appearing in the literature. The detailed dynamics analysis of the model, using the Poincaré index theory, shows that non-linearity of the incidence rate leads to vital dynamics, such as bistability and periodicity, without seasonal forcing or being cyclic. Furthermore, it is shown that the basic reproductive number is independent of the functional form of the non-linear incidence rate. Under certain, well-defined conditions, the model undergoes a Hopf bifurcation. Using the normal form of the model, the first Lyapunov coefficient is computed to determine the various types of Hopf bifurcation the model undergoes. These general results are applied to two examples: unbounded and saturated contact rates; in both cases, forward or backward Hopf bifurcations occur for two distinct values of the contact parameter. It is also shown that the model may undergo a subcritical Hopf bifurcation leading to the appearance of two concentric limit cycles. The results are illustrated by numerical simulations with realistic model parameters estimated for some infectious diseases of childhood.     

一类具有标准发生率的SIS型传染病模型的全局稳定性

研究一类具有标准发生率的SIS传染病模型,讨论了各类平衡点存在的条件;运用微分方程的定性理论,得到了无病平衡点E_1和地方病平衡点E_2的全局渐近稳定的充分条件.     

Modelling diseases with relapse and nonlinear incidence of infection: a multi-group epidemic model

In this paper, we introduce a basic reproduction number for a multi-group SIR model with general relapse distribution and nonlinear incidence rate. We find that basic reproduction number plays the role of a key threshold in establishing the global dynamics of the model. By means of appropriate Lyapunov functionals, a subtle grouping technique in estimating the derivatives of Lyapunov functionals guided by graph-theoretical approach and LaSalle invariance principle, it is proven that if it is less than or equal to one, the disease-free equilibrium is globally stable and the disease dies out; whereas if it is larger than one, some sufficient condition is obtained in ensuring that there is a unique endemic equilibrium which is globally stable and thus the disease persists in the population. Furthermore, our results suggest that general relapse distribution are not the reason of sustained oscillations. Biologically, our model might be realistic for sexually transmitted diseases, such as Herpes, Condyloma acuminatum, etc.     

Competitive exclusion and coexistence for pathogens in an epidemic model with variable population size

We study an SIR epidemic model with a variable host population size. We prove that if the model parameters satisfy certain inequalities then competition between n pathogens for a single host leads to exclusion of all pathogens except the one with the largest basic reproduction number. It is shown that a knowledge of the basic reproduction numbers is necessary but not sufficient for determining competitive exclusion. Numerical results illustrate that these inequalities are sufficient but not necessary for competitive exclusion to occur. In addition, an example is given which shows that if such inequalities are not satisfied then coexistence may occur.     

具有饱和发生率的病毒感染模型的全局稳定性分析

讨论了一类具有饱和发生率的病毒感染数学模型,分析得到了无病平衡点和持续带毒平衡点的全局稳定性条件.当病毒感染的基本再生数R_01时,无病平衡点全局渐近稳定;当R_01时,持续带毒平衡点全局渐近稳定.     

Dynamics of an SIQS epidemic model with transport-related infection and exit-entry screenings

Population dispersal, as a common phenomenon in human society, may cause the spreading of many diseases such as influenza, SARS, etc. which are easily transmitted from one region to other regions. Exit and entry screenings at the border are considered as effective ways for controlling the spread of disease. In this paper, the dynamics of an SIQS model are analyzed and the combined effects of transport-related infection enhancing and exit-entry screenings suppressing on disease spread are discussed. The basic reproduction number is computed and proved to be a threshold for disease control. If it is not greater than the unity, the disease free equilibrium is globally asymptotically stable. And there exists an endemic equilibrium which is locally asymptotically stable if the reproduction number is greater than unity. It is shown that the disease is endemic in the sense of permanence if and only if the endemic equilibrium exists. Exit screening and entry screening are shown to be helpful for disease eradication since they can always have the possibility to eradicate the disease endemic led by transport-related infection and furthermore have the possibility to eradicate disease even when the isolated cites are disease endemic.     

The identification of the periodic behaviour in an epidemic model

In this paper we study a method for the identification of the unknown parameter of the periodic function and also the first component of the state vector, in a mathematical model which describes the evolution of some diseases with an oro-fecal transmission.To solve the identification problem we use a numerical method to integrate the differential equations system, which reproduces the stability properties of the above mentioned continuous system.The numerical methods which we propose can be applied also to a spatial semi discretization of the reaction-diffusion model which is a diffusive generalization of the system that we consider in this paper.Finally, through an analysis on both the continuous and the discrete system we also obtain a necessary condition on the experimental data in order that a periodic trajectory of the system exists.Work supported by: Progetto Finalizzato Controllo Malattie da Infezione-CNR and by Ministero Pubblica Istruzione     

To switch or taper off: the dynamics of saturation

Many situations in population biology involve a rate--typically a contact or recruitment rate--which increases linearly for small populations but reaches a maximum value (saturates) for large populations. Models for populations of variable size need to incorporate both characteristics to predict behavior accurately. This can be done by defining the rate as a continuous, piecewise linear function with a switch point, or via a Verhulst-type (smooth) saturation function. This paper presents several examples of both approaches and draws some conclusions about the differences from a modeling perspective.     

Threshold and stability results for an age-structured epidemic model

We study a mathematical model for an epidemic spreading in an age-structured population with age-dependent transmission coefficient. We formulate the model as an abstract Cauchy problem on a Banach space and show the existence and uniqueness of solutions. Next we derive some conditions which guarantee the existence and uniqueness for non-trivial steady states of the model. Finally the local and global stability for the steady states are examined.     

An analysis of a dendritic neuron model with an active membrane site

We formulate and analyze a mathematical model that couples an idealized dendrite to an active boundary site to investigate the nonlinear interaction between these passive and active membrane patches. The active site is represented mathematically as a nonlinear boundary condition to a passive cable equation in the form of a space-clamped FitzHugh-Nagumo (FHN) equation. We perform a bifurcation analysis for both steady and periodic perturbation at the active site. We first investigate the uncoupled space-clamped FHN equation alone and find that for periodic perturbation a transition from phase locked (periodic) to phase pulling (quasiperiodic) solutions exist. For the model coupling a passive cable with a FHN active site at the boundary, we show for steady perturbation that the interval for repetitive firing is a subset of the interval for the space-clamped case and shrinks to zero for strong coupling. The firing rate at the active site decreases as the coupling strength increases. For periodic perturbation we show that the transition from phase locked to phase pulling solutions is also dependent on the coupling strength.This work was supported in part by NSF Grants MCS 83-00562 and MDS 85-01535     

The dynamics of two viral infections in a single host population with applications to hantavirus

An SI epidemic model for a host with two viral infections circulating within the population is developed, analyzed, and numerically simulated. The model is a system of four differential equations which includes a state for susceptible individuals, two states for individuals infected with a single virus, one which is vertically transmitted and the other which is horizontally transmitted, and a fourth state for individuals infected with both viruses. A general growth function with density-dependent mortality is assumed. A special case of this model, where there is no coinfection and total cross immunity, is thoroughly analyzed. Several threshold values are defined which determine establishment of the disease and persistence at equilibrium for one or both of the infections within the host population. The model has applications to a hantavirus and an arenavirus that infect cotton rats. The hantavirus is transmitted horizontally whereas the arenavirus is transmitted vertically. It is shown through analysis and numerical simulations that both diseases can be maintained within a single host population, where individuals can be either infected with both viruses or with a single virus.     

An SEIS epidemic model with transport-related infection

In this paper, an SEIS epidemic model is proposed to study the effect of transport-related infection on the spread and control of infectious disease. New result implies that traveling of the exposed (means exposed but not yet infectious) individuals can bring disease from one region to other regions even if the infectious individuals are inhibited from traveling among regions. It is shown that transportation among regions will change the disease dynamics and break infection out even if infectious diseases will go to extinction in each isolated region without transport-related infection. In addition, our analysis shows that transport-related infection intensifies the disease spread if infectious diseases break out to cause an endemic situation in each region, in the sense of that both the absolute and relative size of patients increase. This suggests that it is very essential to strengthen restrictions of passengers once we know infectious diseases appeared.