Global dynamics of an SIRS epidemic model with saturation incidence

In this paper, the dynamical behavior of an SIRS epidemic model with birth pulse, pulse vaccination, and saturation incidence is studied. By using a discrete map, the existence and stability of the infection-free periodic solution and the endemic periodic solution are investigated. The conditions required for the existence of supercritical bifurcation are derived. A threshold for a disease to be extinct or endemic is established. The Poincaré map and center manifold theorem are used to discuss flip bifurcation of the endemic periodic solution. Moreover, numerical simulations for bifurcation diagrams, phase portraits and periodic solutions, which are illustrated with an example, are in good agreement with the theoretical analysis.     

Analysis of an SEIRS epidemic model with two delays

 A disease transmission model of SEIRS type with exponential demographic structure is formulated. All newborns are assumed susceptible, there is a natural death rate constant, and an excess death rate constant for infective individuals. Latent and immune periods are assumed to be constants, and the force of infection is assumed to be of the standard form, namely proportional to I(t)/N(t) where N(t) is the total (variable) population size and I(t) is the size of the infective population. The model consists of a set of integro-differential equations. Stability of the disease free proportion equilibrium, and existence, uniqueness, and stability of an endemic proportion equilibrium, are investigated. The stability results are stated in terms of a key threshold parameter. More detailed analyses are given for two cases, the SEIS model (with no immune period), and the SIRS model (with no latent period). Several threshold parameters quantify the two ways that the disease can be controlled, by forcing the number or the proportion of infectives to zero. Received 8 May 1995; received in revised form 7 November 1995     

A periodic SEIRS epidemic model with a time-dependent latent period

Journal of Mathematical Biology - Many infectious diseases have seasonal trends and exhibit variable periods of peak seasonality. Understanding the population dynamics due to seasonal changes...     

Travelling waves of a delayed SIR epidemic model with nonlinear incidence rate and spatial diffusion

This paper is concerned with the existence of travelling waves to a SIR epidemic model with nonlinear incidence rate, spatial diffusion and time delay. By analyzing the corresponding characteristic equations, the local stability of a disease-free steady state and an endemic steady state to this system under homogeneous Neumann boundary conditions is discussed. By using the cross iteration method and the Schauder's fixed point theorem, we reduce the existence of travelling waves to the existence of a pair of upper-lower solutions. By constructing a pair of upper-lower solutions, we derive the existence of a travelling wave connecting the disease-free steady state and the endemic steady state. Numerical simulations are carried out to illustrate the main results.     

Parameter identification for a stochastic SEIRS epidemic model: case study influenza

Journal of Mathematical Biology - A recent parameter identification technique, the local lagged adapted generalized method of moments, is used to identify the time-dependent disease transmission...     

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.     

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.     

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.     

An age-structured epidemic model of rotavirus with vaccination

The recent approval of a rotavirus vaccine in Mexico motivates this study on the potential impact of the use of such a vaccine on rotavirus prevention and control. An age-structured model that describes the rotavirus transmission dynamics of infections is introduced. Conditions that guarantee the local and global stability analysis of the disease-free steady state distribution as well as the existence of an endemic steady state distribution are established. The impact of maternal antibodies on the implementation of vaccine is evaluated. Model results are used to identify optimal age-dependent vaccination strategies. A convergent numerical scheme for the model is introduced but not implemented. This paper is dedicated to Prof. K. P. Hadeler, who continues to push the frontier of knowledge in mathematical biology.     

An SIRVS epidemic model with pulse vaccination strategy

The aim of this paper is to analyze an SIRVS epidemic model in which pulse vaccination strategy (PVS) is included. We are interested in finding the basic reproductive number of the model which determine whether or not the disease dies out. The global attractivity of the disease-free periodic solution (DFPS for short) is obtained when the basic reproductive number is less than unity. The disease is permanent when the basic reproductive number is greater than unity, i.e., the epidemic will turn out to endemic. Our results indicate that the disease will go to extinction when the vaccination rate reaches some critical value.     

An age-dependent epidemic model with application to measles

A combined epidemic-demographic model is developed which models the spread of an infectious disease throughout a population of constant size. The model allows for births, deaths, temporary or permanent immunity, and immunization. The relationship of this model to previously studied epidemic and demographic models is illustrated. An advantage of this model is that all epidemic and demographic parameters may be estimated. The stability of the equilibrium point corresponding to the elimination of the disease is studied and a threshold value is found which indicates whether the disease will die out or remain endemic in the population. The application of the model to measles indicates that immunization levels needed to reduce the incidence to near zero may not be as high as previously predicted.     

SIR-SVS epidemic models with continuous and impulsive vaccination strategies

Vaccination is important for the control of some infectious diseases. This paper considers two SIR-SVS epidemic models with vaccination, where it is assumed that the vaccination for the newborns is continuous in the two models, and that the vaccination for the susceptible individuals is continuous and impulsive, respectively. The basic reproduction numbers of two models, determining whether the disease dies out or persists eventually, are all obtained. For the model with continuous vaccination for the susceptibles, the global stability is proved by using the Lyapunov function. Especially for the endemic equilibrium, to prove the negative definiteness of the derivative of the Lyapunov function for all the feasible values of parameters, it is expressed in three different forms for all the feasible values of parameters. For the model with pulse vaccination for the susceptibles, the global stability of the disease free periodic solution is proved by the comparison theorem of impulsive differential equations. At last, the effect of vaccination strategies on the control of the disease transmission is discussed, and two types of vaccination strategies for the susceptible individuals are also compared.     

An epidemic model with infector and exposure dependent severity

A stochastic epidemic model allowing for both mildly and severely infectious individuals is defined, where an individual can become severely infectious directly upon infection or if additionally exposed to infection. It is shown that, assuming a large community, the initial phase of the epidemic may be approximated by a suitable branching process and that the main part of an epidemic that becomes established admits a law of large numbers and a central limit theorem, leading to a normal approximation for the final outcome of such an epidemic. Effects of vaccination prior to an outbreak are studied and the critical vaccination coverage, above which only small outbreaks can occur, is derived. The results are illustrated by simulations that demonstrate that the branching process and normal approximations work well for finite communities, and by numerical examples showing that the final outcome may be close to discontinuous in certain model parameters and that the fraction mildly infected may actually increase as an effect of vaccination.     

Wave propagation of a discrete SIR epidemic model with a saturated incidence rate

This paper is concerned with the traveling wave solutions for a discrete SIR epidemic model with a saturated incidence rate. We show that the existence and non-existence of the traveling wave solutions are determined by the basic reproduction number R0 of the corresponding ordinary differential system and the minimal wave speed c*. More specifically, we first prove the existence of the traveling wave solutions for R0>1 and c>c* via considering a truncated initial value problem and using the Schauder’s fixed point theorem. The existence of the traveling wave solutions with speed c=c? is then proved by using a limiting argument. The main difficulty is to show that the limit of a decreasing sequence of the traveling wave solutions with super-critical speeds is non-trivial. Finally, the non-existence of the traveling wave solutions for R0>1,00 is proved.     

Wave propagation of a discrete SIR epidemic model with a saturated incidence rate

This paper is concerned with the traveling wave solutions for a discrete SIR epidemic model with a saturated incidence rate. We show that the existence and non-existence of the traveling wave solutions are determined by the basic reproduction number R0 of the corresponding ordinary differential system and the minimal wave speed c*. More specifically, we first prove the existence of the traveling wave solutions for R0>1 and c>c* via considering a truncated initial value problem and using the Schauder’s fixed point theorem. The existence of the traveling wave solutions with speed c=c? is then proved by using a limiting argument. The main difficulty is to show that the limit of a decreasing sequence of the traveling wave solutions with super-critical speeds is non-trivial. Finally, the non-existence of the traveling wave solutions for R0>1,00 is proved.     

Threshold dynamics of a non-autonomous SEIRS model with quarantine and isolation

A model for assessing the effect of periodic fluctuations on the transmission dynamics of a communicable disease, subject to quarantine (of asymptomatic cases) and isolation (of individuals with clinical symptoms of the disease), is considered. The model, which is of a form of a non-autonomous system of non-linear differential equations, is analysed qualitatively and numerically. It is shown that the disease-free solution is globally-asymptotically stable whenever the associated basic reproduction ratio of the model is less than unity, and the disease persists in the population when the reproduction ratio exceeds unity. This study shows that adding periodicity to the autonomous quarantine/isolation model developed in Safi and Gumel (Discret Contin Dyn Syst Ser B 14:209–231, 2010) does not alter the threshold dynamics of the autonomous system with respect to the elimination or persistence of the disease in the population.     

Threshold dynamics in an SEIRS model with latency and temporary immunity

A disease transmission model of SEIRS type with distributed delays in latent and temporary immune periods is discussed. With general/particular probability distributions in both of these periods, we address the threshold property of the basic reproduction number \(R_0\) and the dynamical properties of the disease-free/endemic equilibrium points present in the model. More specifically, we 1. show the dependence of \(R_0\) on the probability distribution in the latent period and the independence of \(R_0\) from the distribution of the temporary immunity, 2. prove that the disease free equilibrium is always globally asymptotically stable when \(R_0<1\) , and 3. according to the choice of probability functions in the latent and temporary immune periods, establish that the disease always persists when \(R_0>1\) and an endemic equilibrium exists with different stability properties. In particular, the endemic steady state is at least locally asymptotically stable if the probability distribution in the temporary immunity is a decreasing exponential function when the duration of the latency stage is fixed or exponentially decreasing. It may become oscillatory under certain conditions when there exists a constant delay in the temporary immunity period. Numerical simulations are given to verify the theoretical predictions.     

Mathematical analysis of a delayed stage-structured predator–prey model with impulsive diffusion between two predators territories

In most models of population dynamics, diffusion between two patches is assumed to be either continuous or discrete, but in reality, many species diffuse only during a single period, and diffusion often occurs in regular pulses. Further, in forest habitats, the highest-level predator species are restricted to a specific territory, but prey can impulsively move between territories. Therefore, in this paper, we consider a delayed stage-structured predator–prey model with impulsively diffusive prey between two patches; in the model, patches represent the territories of two different predator populations. Here, we analytically obtain the global attractivity condition of predator-extinction periodic solutions for the system by using the concepts of Hui and Chen (2005); a numerical simulation is also included to illustrate this result. Further, we establish permanence conditions for the coexistence of the species using the theory of impulsive delayed differential equations. Finally, we explore the possibilities of the permanence of the system by using the growth rates of immature predators and the impulse period as critical parameters, and we also obtain the parameters’ threshold limits using numerical experimentation.     

An SIR epidemic model with partial temporary immunity modeled with delay

The SIR epidemic model for disease dynamics considers recovered individuals to be permanently immune, while the SIS epidemic model considers recovered individuals to be immediately resusceptible. We study the case of temporary immunity in an SIR-based model with delayed coupling between the susceptible and removed classes, which results in a coupled set of delay differential equations. We find conditions for which the endemic steady state becomes unstable to periodic outbreaks. We then use analytical and numerical bifurcation analysis to describe how the severity and period of the outbreaks depend on the model parameters.