Stability and Hopf bifurcation for an epidemic disease model with delay

2006 ◽  
Vol 30 (1) ◽  
pp. 204-216 ◽  
Author(s):  
Chengjun Sun ◽  
Yiping Lin ◽  
Maoan Han
2020 ◽  
Vol 30 (01) ◽  
pp. 2050011 ◽  
Author(s):  
Peng Yang ◽  
Yuanshi Wang

This paper is devoted to the study of a new delayed eco-epidemiological model with infection-age structure and Holling type II functional response. Firstly, the disease transmission rate function among the predator population is treated as the piecewise function concerning the incubation period [Formula: see text] of the epidemic disease and the model is rewritten as an abstract nondensely defined Cauchy problem. Besides, the prerequisite which guarantees the presence of the coexistence equilibrium is achieved. Secondly, via utilizing the theory of integrated semigroup and the Hopf bifurcation theorem for semilinear equations with nondense domain, it is found that the model exhibits a Hopf bifurcation near the coexistence equilibrium, which suggests that this model has a nontrivial periodic solution that bifurcates from the coexistence equilibrium as the bifurcation parameter [Formula: see text] crosses the bifurcation critical value [Formula: see text]. That is, there is a continuous periodic oscillation phenomenon. Finally, some numerical simulations are shown to support and extend the analytical results and visualize the interesting phenomenon.


2019 ◽  
Vol 2019 ◽  
pp. 1-17 ◽  
Author(s):  
Zhixing Hu ◽  
Shanshan Yin ◽  
Hui Wang

This paper established a delayed vector-borne disease model with saturated infection rate and cure rate. First of all, according to the basic reproductive number R0, we determined the disease-free equilibrium E0 and the endemic equilibrium E1. Through the analysis of the characteristic equation, we consider the stability of two equilibriums. Furthermore, the effect on the stability of the endemic equilibrium E1 by delay was studied, the existence of Hopf bifurcations of this system in E1 was analyzed, and the length of delay to preserve stability was estimated. The direction and stability of the Hopf bifurcation were also been determined. Finally, we performed some numerical simulation to illustrate our main results.


2012 ◽  
Vol 09 ◽  
pp. 373-379
Author(s):  
SARKHOSH SEDDIGHI CHAHARBORJ ◽  
A. B. MOHD RIZAM ◽  
I. FUDZIAH

As we know there are two kind of systems in modeling epidemic disease, deterministic systems and stochastic systems. This two systems relate to deterministic and stochastic epidemic disease models, respectively. Almost we use deterministic model for big population size and stochastic model for small population size. To use stochastic models for epidemic disease models, can obtain good results with less error. Study and solving of full stochastic models has not been yet investigated so more. In this article we use the homotopy analysis method to solve the full stochastic susceptible-infective epidemic disease model in disease-free equilibrium point.


1993 ◽  
Vol 23 (2) ◽  
pp. 323-346 ◽  
Author(s):  
Elizabeth Fee ◽  
Nancy Krieger

In the United States, we see three main phases in the construction of the history of AIDS, with each having very different implications for health and social policy. In the first, AIDS was conceived of as an epidemic disease, a “gay plague,” by analogy to the sudden, devastating epidemics of the past. In the second, it was normalized as a chronic disease, similar in many ways to diseases such as cancer. In the third, the authors propose a new historical model of a slow-moving, long-lasting pandemic, a chronic infectious ailment manifested through myriad specific HIV-related diseases. The new paradigm of AIDS incorporates the positive aspects of both earlier conceptions. It emphasizes, like the plague model, the etiology, transmission, and prevention of disease but rejects its assumption of a time-limited crisis. It takes from the chronic disease model an appropriate time frame and concern with the clinical management of protracted illness but insists on the primacy of prevention. The authors criticize both infectious and chronic disease models for their individualistic conceptions of disease and their narrow strategies for disease prevention. They further argue that the traditional distinction between, and approaches to, infectious and chronic diseases need to be rethought for other diseases as well as for AIDS.


2014 ◽  
Vol 16 ◽  
pp. 10007
Author(s):  
R. Yafia ◽  
M. A. Aziz-Alaoui ◽  
A. Tridane ◽  
F. Rihan

2021 ◽  
Vol 2021 ◽  
pp. 1-18
Author(s):  
Yanxia Zhang ◽  
Long Li ◽  
Junjian Huang ◽  
Yanjun Liu

In this paper, a vector-borne disease model with two delays and reinfection is established and considered. First of all, the existence of the equilibrium of the system, under different cases of two delays, is discussed through analyzing the corresponding characteristic equation of the linear system. Some conditions that the system undergoes Hopf bifurcation at the endemic equilibrium are obtained. Furthermore, by employing the normal form method and the center manifold theorem for delay differential equations, some explicit formulas used to describe the properties of bifurcating periodic solutions are derived. Finally, the numerical examples and simulations are presented to verify our theoretical conclusions. Meanwhile, the influences of the degree of partial protection for recovered people acquired by a primary infection on the endemic equilibrium and the critical values of the two delays are analyzed.


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