Search

We introduce stochasticity into a multigroup SIR model with nonlinear incidence. We prove that when the intensity of white noise is small, the solution of stochastic system converges weakly to a singular measure (i.e., a distribution) ifℛ0≤1and there exists an invariant distribution which is ergodic ifℛ0>1. This is the same situation as the corresponding deterministic case. When the intensity of white noise is large, white noise controls this system. This means that the disease will extinct exponentially regardless of the magnitude ofℛ0.

We investigate a class of multigroup epidemic models with general exposed distribution and nonlinear incidence rates. For a simpler case that assumes an identical natural death rate for all groups, and with a gamma distribution for exposed distribution is considered. Some sufficient conditions are obtained to ensure that the global dynamics are completely determined by the basic production numberR0. The proofs of the main results exploit the method of constructing Lyapunov functionals and a graph-theoretical technique in estimating the derivatives of Lyapunov functionals.

A dengue disease epidemic model with nonlinear incidence is formulated and analyzed. The equilibria and threshold of the model are found. The stability of the system is analyzed through a geometric approach to stability. The proposed model also exhibits backward bifurcation under suitable conditions on parameters. Our results imply that a nonlinear incidence produces rich dynamics and they should be studied carefully in order to analyze the spread of disease more accurately. Finally, numerical simulations are presented to illustrate the analytical findings.

A delayed SIRS infectious disease model with nonlocal diffusion and nonlinear incidence is investigated. By constructing a pair of upper-lower solutions and using Schauder's fixed point theorem, we derive the existence of a traveling wave solution connecting the disease-free steady state and the endemic steady state.

The SEIR model with nonlinear incidence rates in epidemiology is studied. Global stability of the endemic equilibrium is proved using a general criterion for the orbital stability of periodic orbits associated with higher-dimensional nonlinear autonomous systems as well as the theory of competitive systems of differential equations.

Considering hybrid impulsive and switching control and the case that many networks relevant to the epidemic spreading are homogeneous, including ‘small-world’ networks, random networks and lattice models networks, an improved SIR epidemic model with impulsive control on homogeneous complex networks is presented. The dynamic behaviours of the model are considered, the conditions and threshold to the existence of the disease-free T-Periodic solution are established and the globally asymptotic stability is proved in the paper.

In this paper, by constructing Lyapunov functionals, we consider the global dynamics of an SIRS epidemic model with a wide class of nonlinear incidence rates and distributed delays \(\int^{h}_{0} p(\tau)f(S(t),I(t-\tau)) \mathrm{d}\tau\) under the condition that the total population converges to 1. By using a technical lemma which is derived from strong condition of strict monotonicity of functions f(S,I) and f(S,I)/I with respect to S≥0 and I>0, we extend the global stability result for an SIR epidemic model if R0>1, where R0 is the basic reproduction number. By using a limit system of the model, we also show that the disease-free equilibrium is globally asymptotically stable if R0=1.

The paper considers a model for the transmission dynamics of a vector-borne disease with nonlinear incidence rate. It is proved that the global dynamics of the disease are completely determined by the basic reproduction number. In order to assess the effectiveness of disease control measures, the sensitivity analysis of the basic reproductive numberR0and the endemic proportions with respect to epidemiological and demographic parameters are provided. From the results of the sensitivity analysis, the model is modified to assess the impact of three control measures; the preventive control to minimize vector human contacts, the treatment control to the infected human, and the insecticide control to the vector. Analytically the existence of the optimal control is established by the use of an optimal control technique and numerically it is solved by an iterative method. Numerical simulations and optimal analysis of the model show that restricted and proper use of control measures might considerably decrease the number of infected humans in a viable way.

In this article, we establish the global asymptotic stability of a disease-free equilibrium and an endemic equilibrium of an SIRS epidemic model with a class of nonlinear incidence rates and distributed delays. By using strict monotonicity of the incidence function and constructing a Lyapunov functional, we obtain sufficient conditions under which the endemic equilibrium is globally asymptotically stable. When the nonlinear incidence rate is a saturated incidence rate, our result provides a new global stability condition for a small rate of immunity loss.

There has been a constant barrage of worms over the internet during the recent past. To develop appropriate tools for thwarting quick spread of worms, researchers are trying to understand the behavior of the worm propagation with the aid of epidemiological models. In this study, the bifurcation dynamics of a worm propagation model with nonlinear incidence rates is analyzed in detail. Numerical simulations confirmed the theoretical results.

A compartmental epidemic model, introduced by Gumel and Moghadas [1], is considered. The model incorporates a nonlinear incidence rate and an imperfect preventive vaccine given to susceptible individuals. A bifurcation analysis is performed by applying the bifurcation method introduced in [2], which is based on the use of the center manifold theory. Conditions ensuring the occurrence of backward bifurcation are derived. The obtained results are numerically validated and then discussed from both the mathematical and the epidemiological perspective.

In this paper, we studied an epidemic model with nonlinear incidence and treatment. We described and analyzed by elementary means of the model, a limited resource for treatment is proposed to understand the effect of the capacity for treatment. It is shown that a backward bifurcation will take place if the capacity is small. The dynamical behaviors of the SIR epidemic model with nonlinear incidence and treatment were also studied.

In this paper, for SIRS epidemic models with a class of nonlinear incidence rates and distributed delays of the forms βS(t)∫0hk(τ)G(I(t-τ) )dτ, we establish the global asymptotic stability of the disease-free equilibrium E0 for R0

In this paper, we have observed the global behaviour of an epidemiological SEIRS model with nonlinear incidence rates @lI^pS^q by constructing a suitable Liapunov function. Moreover, we have found the conditions for global existence of a SEIS model by using Bendixon-Dulac criterion.

The interplay of seasonality, the system's nonlinearities and intrinsic stochasticity is studied for a seasonally forced susceptible-exposed-infective-recovered stochastic model. The model is explored in the parameter region that corresponds to childhood infectious diseases such as measles. The power spectrum of the stochastic fluctuations around the attractors of the deterministic system that describes the model in the thermodynamic limit is computed analytically and validated by stochastic simulations for large system sizes. Size effects are studied through additional simulations. Other effects such as switching between coexisting attractors induced by stochasticity often mentioned in the literature as playing an important role in the dynamics of childhood infectious diseases are also investigated. The main conclusion is that stochastic amplification, rather than these effects, is the key ingredient to understand the observed incidence patterns., 13 pages, 9 figures, 3 tables

There has been a constant barrage of worms over the internet during the recent past. To develop appropriate tools for thwarting quick spread of worms, researchers are trying to understand the behavior of the worm propagation with the aid of epidemiological models. In this study, a SIR epidemic model with dynamic input-output property and nonlinear incidence rates is presented. The dynamics of this model under pulse control are analyzed. We established the conditions and threshold to the existence of the disease-free T-Periodic solution and proved that the disease-free T-Periodic solution is globally asymptotically stable.

According to biological strategy for pest control, we investigate the dynamic behavior of a pest management SEI model with nonlinear incidence concerning impulsive strategy-periodic releasing infected pests at fixed times. We prove that all solutions of the system are uniformly ultimately bounded and there exists a globally asymptotically attractive pest-eradication periodic solution when the impulsive period satisfies A1. When the impulsive period satisfies A2, the stability of pest-eradication periodic solution is lost, the system is uniformly permanent. Thus, we can use the stability of the positive periodic solution and its period to control insect pests at acceptably low levels.

A threshold parameter R0 is identified for an SIRS epidemiological model which has nonlinear incidence and a distributed delay for transfer out of the removed class. For R0 < 1, the disease free equilibrium is proved to be the global attractor for all solutions.

In this paper, the propagation of a nonlinear delay SIR epidemic using the double epidemic hypothesis is modeled. In the model, a system of impulsive functional differential equations is studied and the sufficient conditions for the global attractivity of the semi-trivial periodic solution are drawn. By use of new computational techniques for impulsive differential equations with delay, we prove that the system is permanent under appropriate conditions. The results show that time delay, pulse vaccination, and nonlinear incidence have significant effects on the dynamics behaviors of the model. The conditions for the control of the infection caused by viruses A and B are given.

In this paper the spreading of epidemic model on complex networks with a generalized nonlinear incidence rate is presented. Firstly the SIS model on homogeneous networks with nonlinear incidence rate is considered, and the existence conditions about the disease-free equilibrium and the endemic equilibrium are given. And then the model on heterogenous scale-free (SF) networks is considered, where the absence of the threshold on SF networks with nonlinear incidence is demonstrated. At last the stability of the disease-free equilibrium on SF networks is obtained. From this paper, it is shown that, while the number of the equilibria is indeed different from the corresponding model with linear incidence rate, the basic reproductive number, which determinate whether the disease is spreading or not, is independent of the functional form of the nonlinear incidence rate.

Epidemiological models with nonlinear incidence rates can have very different dynamic behaviors than those with the usual bilinear incidence rate. The first model considered here includes vital dynamics and a disease process where susceptibles become exposed, then infectious, then removed with temporary immunity and then susceptible again. When the equilibria and stability are investigated, it is found that multiple equilibria exist for some parameter values and periodic solutions can arise by Hopf bifurcation from the larger endemic equilibrium. Many results analogous to those in the first model are obtained for the second model which has a delay in the removed class but no exposed class.

A heroin model with nonlinear incidence rate and age structure is investigated. The basic reproduction number is determined whether or not a heroin epidemic breaks out. By employing the Lyapunov functionals, the drug-free equilibrium is globally asymptotically stable if [Formula: see text]; while the drug spread equilibrium is also globally asymptotically stable if [Formula: see text]. Our results imply that improving detected rates and drawing up the efficient prevention play more important role than increasing the treatment for drug users.

Lyapunov functions for two-dimension SIR and SIRS compartmental epidemic models with non-linear transmission rate of a very general form f(S, I) constrained by a few biologically feasible conditions are constructed. Global properties of these models including these with vertical and horizontal transmission, are thereby established. It is proved that, under the constant population size assumption, the concavity of the function f(S, I) with respect to the number of the infective hosts I ensures the uniqueness and the global stability of the positive endemic equilibrium state.

An epidemic model with a class of nonlinear incidence rates and distributed delay is analyzed. The nonlinear incidence is used to describe the saturated or the psychological effect of certain serious epidemics on the community when the number of infectives is getting larger. The distributed delay is derived to describe the dynamics of infectious diseases with varying immunity. Lyapunov functionals are used to show that the disease-free equilibrium state is globally asymptotically stable when the basic reproduction number is less than or equal to one. Moreover, it is shown that the disease is permanent if the basic reproduction number is greater than one. Furthermore, the sufficient conditions under which the endemic equilibrium is locally and globally asymptotically stable are obtained.

In this paper, an SIRS epidemic model with time delay and vaccination is investigated. By analyzing the corresponding characteristic equation, the local stability of disease-free equilibrium of the model is established. By constructing Lyapunov functional, sufficient conditions are established for the local stability of an endemic equilibrium of the model. Further, a threshold value is obtained. By using comparison arguments, it is proved when the threshold value is less than unity, the disease-free equilibrium is globally asymptotically stable. When the threshold value is greater than unity, by using an iteration scheme and by constructing appropriate Lyapunov functional, respectively, sufficient conditions are derived for the global stability of the endemic equilibrium of the model. Numerical simulations are carried out to illustrate the theoretical results.

Epidemiological models with nonlinear incidence rates lambda IpSq show a much wider range of dynamical behaviors than do those with bilinear incidence rates lambda IS. These behaviors are determined mainly by p and lambda, and secondarily by q. For such models, there may exist multiple attractive basins in phase space; thus whether or not the disease will eventually die out may depend not only upon the parameters, but also upon the initial conditions. In some cases, periodic solutions may appear by Hopf bifurcation at critical parameter values.

When the traditional assumption that the incidence rate is proportional to the product of the numbers of infectives and susceptibles is dropped, the SIRS model can exhibit qualitatively different dynamical behaviors, including Hopf bifurcations, saddle-node bifurcations, and homoclinic loop bifurcations. These may be important epidemiologically in that they demonstrate the possibility of infection outbreak and collapse, or autonomous periodic coexistence of disease and host. The possible mechanisms leading to nonlinear incidence rates are discussed. Finally, a modified general criterion for supercritical or subcritical Hopf bifurcation of 2-dimensional systems is presented.

In this paper, we study an SIRI epidemic model with nonlinear incidence rate and latent period, namely k I ( t − τ ) S 1 + α I h ( t − τ ) , which describes the psychological effect of certain serious diseases on the community when the size of the set of infective individuals is getting larger. We first obtain the threshold dynamics on the global stability of the equilibria for the model without latent period, and then we analyze the stability and Hopf bifurcation for the model with the latent period. The results show the influence of nonlinear incidence rate and latent period on the dynamical behaviors of the SIRI model. The examples and its simulations are given to illustrate the obtained results.