General Incidence Function Research Articles

A Mathematical model of Malaria transmission dynamics with general incidence function and maturation delay in a periodic environment

In this paper, we investigate a mathematical model of malaria transmission dynamics with maturation delay of a vector population in a periodic environment. The incidence rate between vector and human hosts is modeled by a general nonlinear incidence function which satisfies a set of conditions. Thus, the model is formulated as a system of retarded functional differential equations. Furthermore, through dynamical systems theory, we rigorously analyze the global behavior of the model. Therefore, we prove that the basic reproduction number of the model denoted by R0 is the threshold between the uniform persistence and the extinction of malaria virus transmission. More precisely, we show that if R0 is less than unity, then the disease-free periodic solution is globally asymptotically stable. Otherwise, the system exhibits at least one positive periodic solution if R0 is greater than unity. Finally, we perform some numerical simulations to illustrate our mathematical results and to analyze the impact of the delay on the disease transmission.

GLOBAL RESULTS FOR AN HIV/AIDS MODEL WITH MULTIPLE SUSCEPTIBLE CLASSES AND NONLINEAR INCIDENCE

In this paper, an HIV/AIDS epidemic model is proposed in which there are two susceptible classes. Two types of general nonlinear incidence functions are employed to depict the scenarios of infection among cautious and incautious individuals. Qualitative analyses are performed, in terms of the basic reproduction number $ \mathcal{R}_0 $, to gain the global dynamics of the model: the disease-free equilibrium is of global asymptotic stability when $ \mathcal{R}_0\leq 1 $; a unique endemic equilibrium exists and globally asymptotically stable $ \mathcal{R}_0> 1 $. The introduction of cautious susceptible and the resulting multiple transmission functions has positive effect on HIV/AIDS prevalence. Numerical simulations are carried out to illustrate and extend the obtained analytical results.

Global stability and Hopf bifurcation of a generalized viral infection model with multi-delays and humoral immunity

The aim of this paper is to develop a mathematical model for viral infection with humoral immunity and two modes of transmission that are the classical virus-to-cell infection and the direct cell-to-cell transmission. These both modes are modeled by two general incidence functions. Also, the model incorporates three delays including two distributed delays in cell infection and virus production, and one discrete delay that models the time needed to activate the immune response. We first prove the well-posedness of the developed model and the biological existence of equilibria. Further, the global stability of equilibria and the existence of Hopf bifurcation are investigated by using the direct and indirect Lyapunov methods. An important number of viral infection models and the corresponding results presented in recent studies are improved and extended.

A numerical method for a diffusive virus model with general incidence function, cell-to-cell transmission and time delay

In this paper, we propose a numerical method for a diffusive virus dynamics model with general incidence function, cell-to-cell transmission and time delay. We justify the wellposedness of the numerical model, and prove that the proposed method preserves the global stability of equilibria by constructing suitable discrete Lyapunov functionals with no constraints on the time and space step sizes. we perform numerical simulation to verify our theoretic results. Epidemiologically, we deduce that the spatial heterogeneity of transmission rate can enlarge the risk of disease outbreaks, and the larger time delay is conductive to cure diseases.

Global stability for a discrete SIR epidemic model with delay in the general incidence function

In this paper, we construct a backward difference scheme for a class of general SIR epidemic model with general incidence function f. We use the step size h > 0, for the discretization. The dynamical properties are investigated (positivity and the boundedness of solution). By constructing the Lyapunov function, under the conditions that function f satisfies some assumptions. The global stabilities of equilibria are obtained. If the basic reproduction number R0<1, the disease-free equilibrium is globally asymptotically stable. If R0>1, the endemic equilibrium is globally asymptotically stable.

Analysis of a reaction-diffusion cholera epidemic model in a spatially heterogeneous environment

In this paper, we derive and analyze a reaction-diffusion cholera model in bounded spatial domain with zero-flux boundary condition and general nonlinear incidence functions. The parameters in the model are space-dependent due to the spatial heterogeneity. By applying the theory of monotone dynamical systems and uniform persistence, we prove that the model admits the global threshold dynamics in terms of the basic reproduction number ℜ0, which is defined by the spectral radius of the next generation operator. When all model parameters are strictly positive constants, we study three types of nonlinear incidence functions to achieve the global stability results on the unique positive cholera-endemic steady state (CESS) whenever it exists. For all these examples, the sharp threshold property based on the basic reproduction number was completely established by using Lyapunov functional techniques under some realistic assumptions. Our numerical results reveal that when ℜ0 > 1, the convergence speed of the solution to the CESS becomes faster as the diffusion coefficient d becomes larger in the spatially homogeneous case. While in the spatially heterogeneous case, cholera can not be controlled by limiting the movement of host individuals, and the spatial heterogeneity does not always enhance the disease persistence.

Dynamics of spatially heterogeneous viral model with time delay

A delayed reaction-diffusion virus model with a general incidence function and spatially dependent parameters is investigated. The basic reproduction number for the model is derived, and the uniform persistence of solutions and global attractively of the equilibria are proved. We also show the global attractivity of the positive equilibria via constructing Lyapunov functional, in case that all the parameters are spatially independent. Numerical simulations are finally conducted to illustrate these analytical results.

Bifurcation Analysis of a Delayed Infection Model with General Incidence Function.

In this paper, an infection model with delay and general incidence function is formulated and analyzed. Theoretical results reveal that positive equilibrium may lose its stability, and Hopf bifurcation occurs when choosing delay as the bifurcation parameter. The direction of Hopf bifurcation and the stability of the periodic solutions are also discussed. Furthermore, to illustrate the numerous changes in the local stability and instability of the positive equilibrium, we conduct numerical simulations by using four different types of functional incidence, i.e., bilinear incidence, saturation incidence, Beddington–DeAngelis response, and Hattaf–Yousfi response. Rich dynamics of the model, such as Hopf bifurcations and chaotic solutions, are presented numerically.

Stationary distribution of an HIV model with general nonlinear incidence rate and stochastic perturbations

We present a stochastic HIV infection model with logistic target cell growth, general nonlinear incidence rate, CTL immune response and parameter perturbations. Through a rigorous analysis of the model, we obtain that the solution of the model is positive and global. A critical condition R0s of the model is derived, which depends not only on the general incidence function but also on the noise intensities. Under certain assumptions, by constructing suitable Lyapunov functions, we find that the system has a unique ergodic stationary distribution when R0s>1. We further explore the effect of the noise intensity on model behavior. Our conclusion improves and generalizes the results of the existing HIV stochastic models.

Spatiotemporal Dynamics of a Generalized HBV Infection Model with Capsids and Adaptive Immunity

In this study, we propose and analyze a new mathematical model formulated by partial differential equations in order to better understand the mechanisms and dynamics of hepatitis B virus (HBV) infection in vivo. The proposed model incorporates the intracellular HBV DNA-containing capsids, spatial diffusion in both capsids and viruses, and adaptive immune response exerted by cytotoxic T lymphocytes and antibodies. Further, the infection process is modeled by a general incidence function that includes many cases existing in the literature. We first show the global existence, uniqueness, positivity and boundedness of solutions. The global stability and instability of equilibria are established by means of Lyapunov’s direct and indirect methods. Finally, numerical simulations are presented to illustrate the dynamical behaviors of the model and to support the theoretical results.

Spatiotemporal Dynamics of a Generalized Viral Infection Model with Distributed Delays and CTL Immune Response

In this paper, we propose and investigate a diffusive viral infection model with distributed delays and cytotoxic T lymphocyte (CTL) immune response. Also, both routes of infection that are virus-to-cell infection and cell-to-cell transmission are modeled by two general nonlinear incidence functions. The well-posedness of the proposed model is also proved by establishing the global existence, uniqueness, nonnegativity and boundedness of solutions. Moreover, the threshold parameters and the global asymptotic stability of equilibria are obtained. Furthermore, diffusive and delayed virus dynamics models presented in many previous studies are improved and generalized.

GLOBAL DYNAMICS OF A SEASONAL MATHEMATICAL MODEL OF SCHISTOSOMIASIS TRANSMISSION WITH GENERAL INCIDENCE FUNCTION

In this paper, we investigate a nonautonomous and an autonomous model of schistosomiasis transmission with a general incidence function. Firstly, we formulate the nonautonomous model by taking into account the effect of climate change on the transmission. Through rigorous analysis via theories and methods of dynamical systems, we show that the nonautonomous model has a globally asymptotically stable disease-free periodic equilibrium when the associated basic reproduction ratio [Formula: see text] is less than unity. Otherwise, the system admits at least one positive periodic solutions if [Formula: see text] is greater than unity. Secondly, using the average of periodic functions, we further derive the autonomous model associated with the nonautonomous model. Therefore, we show that the disease-free equilibrium of the autonomous model is locally and globally asymptotically stable when the associated reproduction ratio [Formula: see text] is less than unity. When [Formula: see text] is greater than unity, the existence and global asymptotic stability of the endemic equilibrium is established under certain conditions. Finally, using linear and nonlinear specific incidence function, we perform some numerical simulations to illustrate our theoretical results.

Spatiotemporal Distributions and Dynamics of Human Infections with the A H7N9 Avian Influenza Virus.

In 2013 in mainland China, a novel avian influenza virus H7N9 began to infect humans and had aroused severe fatality in the infected humans, followed by the annual outbreaks. By methods of GIS and kriging interpolation, we get the geographical distributions. We obtain the longitudinal characteristics of these outbreaks based on statistics and diagrams. After these spatiotemporal distributions, an eco-epidemiological model is established and analyzed. In this model, the general incidence functions, the factor of fully killed infected poultry, and the virus in environment are taken into account. Theoretical analysis shows that the endemic will be formed to a large extent once the H7N9 avian influenza virus exists in poultry. On the basis of dynamics, we explore the possible disease control measures by numerical simulations. Simulations indicate that measures of vaccination in poultry and stopping live poultry transactions are the primary choices for disease control in humans, and strengthened inhibition effects and environmental disinfections can effectively control the outbreak.

Dynamics of a diffusive virus model with general incidence function, cell-to-cell transmission and time delay

In this paper, we revisit a diffusive virus dynamics model with general incidence function and time delay. One novelty of our model is that we introduce cell-to-cell transmission via formation of virological synapses to reflect the fact that it may play a more important role in virus spreading in addition to virus-to-cell infection. We justify the well-posedness of the model and identify the basic reproduction number ℜ0 for the model to be a sharp threshold value. The global stability of equilibria is determined by constructing suitable Lyapunov functionals in the sense that: the infection-free equilibrium is globally asymptotically stable if ℜ0≤1, and when ℜ0>1, the global asymptotic stability of infection equilibrium implies that the infection will persist. A significant impact of the cell-to-cell transmission is that they increase the basic reproduction number. If one neglects either the cell-to-cell transmission or virus-to-cell infection, the basic reproduction number of the model that is under-evaluated. Last, we perform numerical simulation to support our theoretic results. We set the domain of the viruses to be a two-dimensional square domain with the homogeneous Neumann boundary conditions to reflect the spatial spreading.

Global dynamics of an age-structured viral infection model with general incidence function and absorption

In this paper, we propose an age-structured viral infection model with general incidence function that takes account of the loss of viral particles due to their absorption into susceptible cells. The proposed model is described by partial differential and ordinary differential equations. We first show that the model is mathematically and biologically well-posed. Furthermore, the uniform persistence and the global behavior of the model are investigated. Moreover, the age-structured models and results presented in many previous studies are improved and generalized.

Global dynamics of a latent HIV infection model with general incidence function and multiple delays

In this paper, we propose a latent HIV infection model with general incidence function and multiple delays. We derive the positivity and boundedness of solutions, as well as the existence and local stability of the infection-free and infected equilibria. By constructing Lyapunov functionals, we establish the global stability of the equilibria based on the basic reproduction number. We further study the global dynamics of this model with Holling type-Ⅱ incidence function through numerical simulations. Our results improve and generalize some existing ones. The results show that the prolonged time delay period of the maturation of the newly produced viruses may lead to the elimination of the viruses.

Optimal control of a SIR epidemic model with general incidence function and a time delays

In this paper, we introduce an optimal control for a SIR model governed by an ODE system with time delay. We extend the stability studies of model (2.2) in section 2, by incorporating suitable controls. We consider two control strategies in the optimal control model, namely: the vaccination and treatment strategies. The model has a time delays that represent the incubation period. We derive the first-order necessary conditions for the optimal control and perform numerical simulations to show the effectiveness as well as the applicability of the model for different values of the time delays. These numerical simulations show that the model is sensitive to the delays representing the incubation period.

A remark on stochastic stability of a novel hybrid system

A stochastic switching Susceptible–Exposed–Infective–Removed ( S E I R ) epidemic model with continuous and impulsive control schemes is proposed and investigated, in which a general incidence function is considered. The existence and uniqueness of the global positive solution is achieved. By using the Lyapunov–Razumikhin approach, sufficient conditions are developed which guarantee the hybrid system is stochastically stable. The results of simulating our new S E I R model indicate that the hybrid control schemes could be an effective strategy to eradicate the disease.

Stability analysis of [formula omitted] rumor spreading model in complex networks

In this paper, we propose I2S2R rumor spreading model with general incidence function in homogeneous networks and I2S2R rumor spreading model in heterogeneous network. In the first model, we establish a dynamic model for rumor propagation called I2S2R, in which we consider two general incidence functions. Then the basic reproduction number, the locally stability and the global stability of rumor-free equilibrium are discussed. In the second model, we mainly consider another network structure, i.e. heterogeneous network. Thus, we propose a new dynamic model for rumor spreading. By deriving the mean-field equation, the basic reproduction number is obtained. Moreover, the global stability of rumor-free equilibrium is established by comparison principle. By using a monotone iterative algorithm, the global attractivity of the positive equilibrium is proved. Finally, we obtain some important conclusions with numerical simulations.

Stability Analysis for a Discrete SIR Epidemic Model with Delay and General Nonlinear Incidence Function

In this paper, we construct a backward difference scheme for a class of SIR epidemic model with general incidence f . The step sizeτ used in our discretization is one. The dynamical properties are investigated (positivity and the boundedness of solution). By constructing the Lyapunov function, the general incidence function f must satisfy certain assumptions, under which, we establish the global stability of endemic equilibrium when R0 >1. The global stability of diseases-free equilibrium is also established when R0 ≤1. In addition we present numerical results of the continuous and discrete model of the different class according to the value of basic reproduction number R0.