Qualitative properties and hopf bifurcation in haematopoietic disease model with chemotherapy

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.

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Stability and Hopf Bifurcation Analysis of a Vector-Borne Disease Model with Two Delays and Reinfection

Computational and Mathematical Methods in Medicine ◽

10.1155/2021/6648959 ◽

2021 ◽

Vol 2021 ◽

pp. 1-18

Author(s):

Yanxia Zhang ◽

Long Li ◽

Junjian Huang ◽

Yanjun Liu

Keyword(s):

Hopf Bifurcation ◽

Disease Model ◽

Endemic Equilibrium ◽

Partial Protection ◽

Center Manifold Theorem ◽

Vector Borne Disease ◽

Two Delays ◽

Normal Form Method ◽

Vector Borne ◽

Delay Differential

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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Modeling and Predicting the Influence of PM2.5 on Children’s Respiratory Diseases

International Journal of Bifurcation and Chaos ◽

10.1142/s0218127420502351 ◽

2020 ◽

Vol 30 (15) ◽

pp. 2050235

Author(s):

Lei Shi ◽

Xiaoliang Feng ◽

Longxing Qi ◽

Yanlong Xu ◽

Sulan Zhai

Keyword(s):

Air Pollution ◽

Hopf Bifurcation ◽

Respiratory Diseases ◽

Disease Model ◽

Anhui Province ◽

Main Research ◽

Number Of Children ◽

Related Disease ◽

Number Of Patients ◽

Lag Effect

In this paper, the influence of PM[Formula: see text] on children’s respiratory diseases is taken as the main research focus. Based on the real monitoring data of children’s respiratory diseases in Anhui province, the traditional model is modified substantially, leading to the establishment of two mathematical models. First of all, considering that the PM[Formula: see text] changes over time, a nonautonomous air pollution-related disease model is constructed to study its permanence and extinction. Furthermore, regarding lag days of PM[Formula: see text] exposure, an air pollution-related disease model with the lag effect is installed and its local and global stabilities and Hopf bifurcation are investigated. Meanwhile, the above two models are numerically simulated, respectively. Our study demonstrates that the threshold conditions of permanence and extinction are obtained by the nonautonomous air pollution-related disease model, and the optimal parameters are obtained through the annual revision of the data by integrating the mathematical model, such that the number of children with respiratory diseases in the future can be checked and predicted. Also our study finds that the lag days of PM[Formula: see text] exposure have little effect on children with respiratory diseases in the air pollution-related disease model with a lag effect, but the PM[Formula: see text] has a tremendous influence on the number of patients. Once the lag days are combined with the effect of the PM[Formula: see text], it can have a significant impact on the patients’ number, e.g. an emergence of periodic oscillations, with an approximate period of 11 days in Anhui Province, due to the Hopf bifurcation.

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Hopf bifurcation and its stability for a vector-borne disease model with delay and reinfection

Applied Mathematical Modelling ◽

10.1016/j.apm.2015.09.007 ◽

2016 ◽

Vol 40 (3) ◽

pp. 1685-1702 ◽

Cited By ~ 9

Author(s):

Jinhu Xu ◽

Yicang Zhou

Keyword(s):

Hopf Bifurcation ◽

Disease Model ◽

Vector Borne Disease ◽

Vector Borne

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Hopf and Generalized Hopf Bifurcations in a Recurrent Autoimmune Disease Model

International Journal of Bifurcation and Chaos ◽

10.1142/s0218127416500796 ◽

2016 ◽

Vol 26 (05) ◽

pp. 1650079 ◽

Cited By ~ 1

Author(s):

Wenjing Zhang ◽

Pei Yu

Keyword(s):

Hopf Bifurcation ◽

Normal Form ◽

Limit Cycles ◽

Dynamical Behavior ◽

Disease Model ◽

Equilibrium Solutions ◽

Normal Form Theory ◽

Form Theory ◽

Manifold Theory ◽

Disease Free

This paper is concerned with bifurcation and stability in an autoimmune model, which was established to study an important phenomenon — blips arising from such models. This model has two equilibrium solutions, disease-free equilibrium and disease equilibrium. The positivity of the solutions of the model and the global stability of the disease-free equilibrium have been proved. In this paper, we particularly focus on Hopf bifurcation which occurs from the disease equilibrium. We present a detailed study on the use of center manifold theory and normal form theory, and derive the normal form associated with Hopf bifurcation, from which the approximate amplitude of the bifurcating limit cycles and their stability conditions are obtained. Particular attention is also paid to the bifurcation of multiple limit cycles arising from generalized Hopf bifurcation, which may yield bistable phenomenon involving equilibrium and oscillating motion. This result may explain some complex dynamical behavior in real biological systems. Numerical simulations are compared with the analytical predictions to show a very good agreement.

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