Nonlinear Incidence Rate Research Articles

TRAVELING WAVES OF A NONLOCAL DIFFUSION SIRS EPIDEMIC MODEL WITH A CLASS OF NONLINEAR INCIDENCE RATES AND TIME DELAY

In this paper, we study the traveling waves of a delayed SIRS epidemic model with nonlocal diffusion and a class of nonlinear incidence rates. When the basic reproduction ratio $ \mathscr{R}_0>1 $, by using the Schauder's fixed point theorem associated with upper-lower solutions, we reduce the existence of traveling waves to the existence of a pair of upper-lower solutions. By constructing a pair of upper-lower solutions, we derive the existence of traveling wave solutions connecting the disease-free steady state and the endemic steady state. When $ \mathscr{R}_0<1 $, the nonexistence of traveling waves is obtained by the comparison principle.

Dynamic Analysis for a SIQR Epidemic Model with Specific Nonlinear Incidence Rate

The article investigates a SIQR epidemic model with specific nonlinear incidence rate and stochastic model based on the former, respectively. For deterministic model, we study the existence and stability of the equilibrium points by controlling threshold parameter R0 which determines whether the disease disappears or prevails. Then by using Routh-Hurwitz criteria and constructing suitable Lyapunov function, we get that the disease-free equilibrium is globally asymptotically stable if R0 or unstable if R0>1. In addition, the endemic equilibrium point is globally asymptotically stable in certain region when R0>1. For the corresponding stochastic model, the existence and uniqueness of the global positive solution are discussed and some sufficient conditions for the extinction of the disease and the persistence in the mean are established by defining its related stochastic threshold R0s. Moreover, our analytical results show that the introduction of random fluctuations can suppress disease outbreak. And numerical simulations are used to confirm the theoretical results.

Global stability of an age-structured epidemic model with general Lyapunov functional.

In this paper, we focus on the study of the dynamics of a certain age structured epidemic model. Our aim is to investigate the proposed model, which is based on the classical SIR epidemic model, with a general class of nonlinear incidence rate with some other generalization. We are interested to the asymptotic behavior of the system. For this, we have introduced the basic reproduction number R₀ of model and we prove that this threshold shows completely the stability of each steady state. Our approach is the use of general constructed Lyapunov functional with some results on the persistence theory. The conclusion is that the system has a trivial disease-free equilibrium which is globally asymptotically stable for R₀ < 1 and that the system has only a unique positive endemic equilibrium which is globally asymptotically stable whenever R₀ > 1. Several numerical simulations are given to illustrate our results.

Global stability of a distributed delayed viral model with general incidence rate

AbstractIn this paper, we discussed a infinitely distributed delayed viral infection model with nonlinear immune response and general incidence rate. We proved the existence and uniqueness of the equilibria. By using the Lyapunov functional and LaSalle invariance principle, we obtained the conditions of global stabilities of the infection-free equilibrium, the immune-exhausted equilibrium and the endemic equilibrium. Numerical simulations are given to verify the analytical results.

Global dynamics for a class of infection-age model with nonlinear incidence

In this paper, we propose an HBV viral infection model with continuous age structure and nonlinear incidence rate. Asymptotic smoothness of the semi-flow generated by the model is studied. Then we caculate the basic reproduction number and prove that it is a sharp threshold determining whether the infection dies out or not. We give a rigorous mathematical analysis on uniform persistence by reformulating the system as a system of Volterra integral equations. The global dynamics of the model is established by using suitable Lyapunov functionals and LaSalle's invariance principle. We further investigate the global behaviors of the HBV viral infection model with saturation incidence through numerical simulations.

Multiple Equilibria in a Non-smooth Epidemic Model with Medical-Resource Constraints.

The issue of medical-resource constraints has the potential to dramatically affect disease management, especially in developing countries. We analyze a non-smooth epidemic model with nonlinear incidence rate and resource constraints, which defines a vaccination program with vaccination rate proportional to the number of susceptible individuals when this number is below the threshold level and constant otherwise. To better understand the impact of this non-smooth vaccination policy, we provide a comprehensive qualitative analysis of global dynamics for the whole parameter space. As the threshold value varies, the target model admits multistability of three regular equilibria, bistability of two regular equilibria, that of one disease-free equilibrium and one generalized endemic equilibria, and that of one disease-free equilibrium and one crossing cycle. The steady-state regimes include healthy, low epidemic and high epidemic. This suggests the key role of the threshold value, as well as the initial infection condition in disease control. Our findings demonstrate that the case number can be contained at a satisfactorily controllable level or range if eradicating it proves to be impossible.

Dynamics of a predator-prey model with nonlinear incidence rate, Crowley-Martin type functional response and disease in prey population

Abstract In this paper, a delayed predator-prey model has been developed. Here, on the basis of infectious disease, the prey population has been divided into two sub-populations such as (i) susceptible prey and (ii) infected prey population. It is also assumed that a predator may consume both susceptible prey as well as infected prey. Also here, the Crowley-Martin type functional form has been taken to consume the prey (both susceptible and infected) population by the predator. It also considers two types of time delays in this model, (i) one is disease transmission delay when susceptible prey moves to infected prey stage and (ii) other is predator maturation delay. As the predator consumes both susceptible and infected prey, so we have incorporated the fact in this model that disease in the prey species also effect on predator population growth. Here, positivity and boundedness of solutions of our proposed model have been discussed. Then calculating different equilibrium points, the stability of the system have been discussed around those. In this paper, the Hopf bifurcation analysis has been done for non-delayed system with respect to the crowding coefficient. Finally, some numerical simulations have been presented to validate our theoretical findings.

Threshold Dynamics of an SIR Epidemic Model with Nonlinear Incidence Rate and Non-Local Delay Effect

In this paper, we are concerned with a reaction- diffusion SIR epidemic model with nonlinear incidence rate and non-local delay effect in a continuous bounded spatial domain. We introduce the basic reproduction number R0 of the model by the idea of next generation operator. By means of the theory of dynamical systems and uniform persistence, we investigate the global dynamics of the model in terms of R0. Finally, we implement numerical simulations to show the feasibility of our results and explore some epidemiological insights.

The global stability of two epidemic models with nonlinear recovery incidence rate

In this paper, we present two epidemic models with a nonlinear incidence and transfer from infectious to recovery. For epidemic models, the basic reproductive number is calculated. A dynamic system based on threshold, using LaSalle’s invariance principle and Lyapunov function, is structured completely by the basic reproductive number. By studying the SIR and SIRS models under the nonlinear condition, the general validity of the method is verified.

Threshold Dynamics of an SIR Model with Nonlinear Incidence Rate and Age-Dependent Susceptibility

We propose an SIR epidemic model with different susceptibilities and nonlinear incidence rate. First, we obtain the existence and uniqueness of the system and the regularity of the solution semiflow based on some assumptions for the parameters. Then, we calculate the basic reproduction number, which is the spectral radius of the next-generation operator. Second, we investigate the existence and local stability of the steady states. Finally, we construct suitable Lyapunov functionals to strictly prove the global stability of the system, which are determined by the basic reproduction number ℛ0 and some assumptions for the incidence rate.

Persistence and extinction for an age-structured stochastic SVIR epidemic model with generalized nonlinear incidence rate

We formulate an epidemic model with age of vaccination and generalized nonlinear incidence rate, where the total population consists of the susceptible, the vaccinated, the infected and the removed. We then reach a stochastic SVIR model when the fluctuation is introduced into the transmission rate. By using Itô’s formula and Lyapunov methods, we first show that the stochastic epidemic model admits a unique global positive solution with the positive initial value. We then obtain the sufficient conditions of the stochastic epidemic model. Moreover, the threshold tells the disease spreads or not is derived. If the intensity of the white noise is small enough and R˜0<1, then the disease eventually becomes extinct with negative exponential rate. If R˜0>1, then the disease is weakly permanent. The persistence in the mean of the infected is also obtained when the indicator Rˆ0>1, which means the disease will prevail in a long run. As a consequence, several illustrative examples are separately carried out with numerical simulations to support the main results of this paper.

Stability of a Time Delayed SIR Epidemic Model Along with Nonlinear Incidence Rate and Holling Type-II Treatment Rate

In this paper, we present a mathematical study of a deterministic model for the transmission and control of epidemics. The incidence rate of susceptible being infected is very crucial in the spread of disease. The delay in the incidence rate is proved fatal. In the present study, we propose an SIR mathematical model with the delay in the infected population. We are taking nonlinear incidence rate for epidemics along with Holling type II treatment rate for understanding the dynamics of the epidemics. Model stability has been done by the basic reproduction number [Formula: see text]. The model is locally asymptotically stable for disease-free equilibrium [Formula: see text] when the basic reproduction number [Formula: see text] is less than one ([Formula: see text]). We investigated the stability of the model for disease-free equilibrium at [Formula: see text] equals to one using center manifold theory. We also investigated the stability for endemic equilibrium [Formula: see text] at [Formula: see text]. Further, numerical simulations are presented to exemplify the analytical studies.

Dynamical behaviors of a stochastic SIRS epidemic model

In this paper, we study the dynamical behavior of a stochastic SIRS epidemic model with specific nonlinear incidence rate and vaccination. We show the existence and positivity of the solution of the SIRS stochastic differential equation. We defined a number $\mathcal{R}$ and we prove the disease free equilibrium is almost sure exponentially stable if $\mathcal{R}<1$. We studying the behavior around the endemic equilibrium E*. Numerical simulations presented our theoretical results.

Global stability for SIRS epidemic models with general incidence rate and transfer from infectious to susceptible

We study a class of SIRS epidemic dynamical models with a general nonlinear incidence rate and transfer from infectious to susceptible. The incidence rate includes a wide range of monotonic, concave incidence rates and some non-monotonic or concave cases. We apply LaSalle’s invariance principle and Lyapunov’s direct method to prove that the disease-free equilibrium is globally asymptotically stable if the basic reproduction number $$R_0\le 1$$ , and the endemic equilibrium is globally asymptotically stable if $$R_0>1$$ , under some conditions imposed on the incidence function f(S, I).

Threshold conditions for a family of epidemic dynamic models for malaria with distributed delays in a non-random environment

A family of deterministic SEIRS epidemic dynamic models for malaria is presented. The family type is determined by a general functional response for the nonlinear incidence rate of the disease. Furthermore, the malaria models exhibit three random delays — the incubation periods of the plasmodium inside the female mosquito and human hosts, and also the period of effective acquired natural immunity against the disease. Insights about the effects of the delays and the nonlinear incidence rate of the disease on (1) eradication and (2) persistence of malaria in the human population are obtained via analyzing and interpreting the global asymptotic stability results of the disease-free and endemic equilibrium of the system. The basic reproduction numbers and other threshold values for malaria are calculated, and superior threshold conditions for the stability of the equilibria are found. Numerical simulation results are presented.

Delay-induced Hopf bifurcation of an SVEIR computer virus model with nonlinear incidence rate

We are concerned with the Hopf bifurcation of an SVEIR computer virus model with time delay and nonlinear incident rate. First of all, by analyzing the associated characteristic equation we obtain sufficient conditions for its local stability and the existence of a Hopf bifurcation. Directly afterward, by means of the normal form theory and the center manifold theorem we derive explicit formulas that determine the direction of the Hopf bifurcation and the stability of the bifurcated periodic solutions. Finally, we carry out numerical simulations to illustrate and verify the theoretical results.

Traveling waves in an SEIR epidemic model with a general nonlinear incidence rate

ABSTRACTWe study the traveling waves of reaction-diffusion equations for a diffusive SEIR model with a general nonlinear incidence. The existence of traveling waves is determined by the basic reproduction number of the corresponding ordinary differential equations and the minimal wave speed. Its proof is showed by introducing an auxiliary system, applying Schauder fixed point theorem and then a limiting argument. The non-existence proof is obtained by two-sided Laplace transform when the speed is less than the critical velocity. Finally, we present some examples to support our theoretical results.

Optimal Control for an SEIR Epidemic Model with Nonlinear Incidence Rate

AbstractThe objective of this paper is to explore possible impacts of nonlinearity of functional responses and a number of compartments of an infection disease model on principal qualitative properties of the optimal controls. To address this issue, we consider optimal controls for an Susceptible‐Exposed‐Infectious‐Removed (SEIR) model of an endemically persisting infectious disease. We assume that the incidence rate is given by an unspecified nonlinear function constrained by a few biologically motivated conditions. For this model, we consider five controls (which comprise all controls that are possible for this model) with a possibility of acting simultaneously, and establish principal qualitative properties of the controls. A comparison with a similar SIR model is provided.

An age-structured epidemic model with waning immunity and general nonlinear incidence rate

In this paper, a susceptible-vaccinated-exposed-infectious-recovered epidemic model with waning immunity and continuous age structures in vaccinated, exposed and infectious classes has been formulated. By using the Fluctuation lemma and the approach of Lyapunov functionals, we establish a threshold dynamics completely determined by the basic reproduction number. When the basic reproduction number is less than one, the disease-free steady state is globally asymptotically stable, and otherwise the endemic steady state is globally asymptotically stable.

Hopf Bifurcation and Hybrid Control of a Delayed Ecoepidemiological Model with Nonlinear Incidence Rate and Holling Type II Functional Response

Hopf bifurcation analysis of a delayed ecoepidemiological model with nonlinear incidence rate and Holling type II functional response is investigated. By analyzing the corresponding characteristic equations, the conditions for the stability and existence of Hopf bifurcation for the system are obtained. In addition, a hybrid control strategy is proposed to postpone the onset of an inherent bifurcation of the system. By utilizing normal form method and center manifold theorem, the explicit formulas that determine the direction of Hopf bifurcation and the stability of bifurcating period solutions of the controlled system are derived. Finally, some numerical simulation examples confirm that the hybrid controller is efficient in controlling Hopf bifurcation.