Global dynamics of delay epidemic models with nonlinear incidence rate and relapse (original) (raw)
Stability analysis of delayed SIR epidemic models with a class of nonlinear incidence rates
Applied Mathematics and Computation
We analyze stability of equilibria for a delayed SIR epidemic model, in which population growth is subject to logistic growth in absence of disease, with a nonlinear incidence rate satisfying suitable monotonicity conditions. The model admits a unique endemic equilibrium if and only if the basic reproduction number R 0 exceeds one, while the trivial equilibrium and the disease-free equilibrium always exist. First we show that the disease-free equilibrium is globally asymptotically stable if and only if R 0 ≤ 1. Second we show that the model is permanent and it has a unique endemic equilibrium if and only if R 0 > 1. Moreover, using a threshold parameter R 0 characterized by the nonlinear incidence function, we establish that the endemic equilibrium is locally asymptotically stable for 1 < R 0 ≤ R 0 and it loses stability as the length of the delay increases past a critical value for 1 < R 0 < R 0 . Our result is an extension of the stability results in [J-J. Wang, J-Z. Zhang, Z. Jin, Analysis of an SIR model with bilinear incidence rate, Nonl. Anal. RWA. 11 (2009RWA. 11 ( ) 2390RWA. 11 ( -2402.
An epidemic model with different distributed latencies and nonlinear incidence rate
An SEIR epidemic model with different distributed latencies and general nonlinear incidence is presented and studied. By constructing suitable Lyapunov functionals, the biologically realistic sufficient conditions for threshold dynamics are established. It is shown that the infection-free equilibrium is globally attractive when the basic reproduction number is equal to or less than one, and that the disease becomes globally attractively endemic when the basic reproduction number is larger than one. The criteria in this paper generalize and improve some previous results in the literatures.
Global dynamics of a delayed SIRS epidemic model with a wide class of nonlinear incidence rates
Journal of Applied Mathematics and Computing, 2012
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 h 0 p(τ )f (S(t), I (t − τ ))dτ 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 Y. Enatsu ( )
British Journal of Mathematics & Computer Science, 2017
In this paper, we investigate the global stability and the existence of traveling waves for a delayed diffusive epidemic model. The disease transmission process is modeled by a specific nonlinear function that covers many common types of incidence rates. In addition, the global stability of the disease-free equilibrium and the endemic equilibrium is established by using the direct Lyapunov method. By constructing a pair of upper and lower solutions and applying the Schauder fixed point theorem, the existence of traveling wave solution which connects the two steady states is obtained and characterized by two parameters that are the basic reproduction number and the minimal wave speed. Furthermore, the models and main results studied the existence of traveling waves presented in the literature are extended and generalized.
Bifurcation analysis of a delayed epidemic model
Applied Mathematics and Computation, 2010
In this paper, Hopf bifurcation for a delayed SIS epidemic model with stage structure and nonlinear incidence rate is investigated. Through theoretical analysis, we show the positive equilibrium stability and the conditions that Hopf bifurcation occurs. Applying the normal form theory and the center manifold argument, we derive the explicit formulas determining the properties of the bifurcating periodic solutions. In addition, we also study the effect of the inhibition effect on the properties of the bifurcating periodic solutions. To illustrate our theoretical analysis, some numerical simulations are also included.
A Delayed Sis Epidemic Model with Non-Linear Incidence Rate
2018
Abstract: This paper makes a modest attempt to study an SIS epidemic model with time delay corresponding to the infectious period and non-linear incidence rate, where the growth of susceptible individuals is governed by the logistic equation. The local stability of the diseasefree equilibrium and the endemic equilibrium with and without delay has also been analyzed. Conditions for the existence of Hopf bifurcation of the endemic equilibrium were established from applying time delay as a bifurcation parameter. Further, the analytical results supported by numerical simulations.
Global stability of a diffusive SEIR epidemic model with distributed delay
Elsevier eBooks, 2022
We study the global dynamics of a reaction-diffusion SEIR infection model with distributed delay and nonlinear incidence rate. The well-posedness of the proposed model is proved. By means of Lyapunov functionals, we show that the disease free equilibrium state is globally asymptotically stable when the basic reproduction number is less or equal than one, and that the disease endemic equilibrium is globally asymptotically stable when the basic reproduction number is greater than one. Numerical simulations are provided to illustrate the obtained theoretical results.
An epidemic model with time delays determined by the infectivity and disease durations
Mathematical Biosciences and Engineering, 2023
We propose an epidemiological model with distributed recovery and death rates. It represents an integrodifferential system of equations for susceptible, exposed, infectious, recovered and dead compartments. This model can be reduced to the conventional ODE model under the assumption that recovery and death rates are uniformly distributed in time during disease duration. Another limiting case, where recovery and death rates are given by the delta-function, leads to a new point-wise delay model with two time delays corresponding to the infectivity period and disease duration. Existence and positiveness of solutions for the distributed delay model and point-wise delay model are proved. The basic reproduction number and the final size of the epidemic are determined. Both, the ODE model and the delay models are used to describe COVID-19 epidemic progression. The delay model gives a better approximation of the Omicron data than the conventional ODE model from the point of view of parameter estimation.
A Delayed Sir Epidemic Model with General Incidence Rate
Electronic Journal of Qualitative Theory of Differential Equations, 2013
A delayed SIR epidemic model with a generalized incidence rate is studied. The time delay represents the incubation period. The threshold parameter, R 0 (τ ) is obtained which determines whether the disease is extinct or not. Throughout the paper, we mainly use the technique of Lyapunov functional to establish the global stability of both the disease-free and endemic equilibrium.
Global stability of multi-group epidemic models with distributed delays
We investigate a class of multi-group epidemic models with distributed delays. We establish that the global dynamics are completely determined by the basic reproduction number R 0 . More specifically, we prove that, if R 0 1, then the disease-free equilibrium is globally asymptotically stable; if R 0 > 1, then there exists a unique endemic equilibrium and it is globally asymptotically stable. Our proof of global stability of the endemic equilibrium utilizes a graph-theoretical approach to the method of Lyapunov functionals.