Dynamical Model of Epidemic Along with Time Delay; Holling Type II Incidence Rate and Monod–Haldane Type Treatment Rate

2018 ◽  
Vol 27 (1-3) ◽  
pp. 299-312 ◽  
Author(s):  
Abhishek Kumar ◽  
Nilam
Keyword(s):  
Time Delay ◽  
Incidence Rate ◽  
Dynamical Model ◽  
Type Ii ◽  
Treatment Rate

Dynamic Behavior of an SIR Epidemic Model along with Time Delay; Crowley–Martin Type Incidence Rate and Holling Type II Treatment Rate

2019 ◽  
Vol 20 (7-8) ◽  
pp. 757-771 ◽  
Author(s):  
Abhishek Kumar ◽  
Nilam
Keyword(s):  
Incidence Rate ◽  
Time Lag ◽  
Type Ii ◽  
Functional Type ◽  
Sir Epidemic Model ◽  
Treatment Rate ◽  
Nonlinear Incidence Rate ◽  
Sir Epidemic ◽  
The Stability ◽  
Disease Free

Abstract In this article, we propose and analyze a time-delayed susceptible–infected–recovered (SIR) mathematical model with nonlinear incidence rate and nonlinear treatment rate for the control of infectious diseases and epidemics. The incidence rate of infection is considered as Crowley–Martin functional type and the treatment rate is considered as Holling functional type II. The stability of the model is investigated for the disease-free equilibrium (DFE) and endemic equilibrium (EE) points. From the mathematical analysis of the model, we prove that the model is locally asymptotically stable for DFE when the basic reproduction number {R_0} is less than unity ({R_0} \lt 1) and unstable when {R_0} is greater than unity ({R_0} \gt 1) for time lag \tau \ge 0. The stability behavior of the model for DFE at {R_0} = 1 is investigated using Castillo-Chavez and Song theorem, which shows that the model exhibits forward bifurcation at {R_0} = 1. We investigate the stability of the EE for time lag \tau \ge 0. We also discussed the Hopf bifurcation of EE numerically. Global stability of the model equilibria is also discussed. Furthermore, the model has been simulated numerically to exemplify analytical studies.

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

2018 ◽  
Vol 15 (06) ◽  
pp. 1850055 ◽  
Author(s):  
Abhishek Kumar ◽  
Nilam
Keyword(s):  
Incidence Rate ◽  
Basic Reproduction Number ◽  
Reproduction Number ◽  
Type Ii ◽  
Treatment Rate ◽  
Nonlinear Incidence Rate ◽  
Number Formula ◽  
The Stability ◽  
Disease Free ◽  
The Basic Reproduction Number

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.

scholarly journals FT6. Postoperative 2-Day Blood Pressure Management Might Reduce the Incidence Rate of Type II Endoleaks Following Endovascular Aortic Repair

2015 ◽  
Vol 61 (6) ◽  
pp. 15S-16S
Author(s):  
Yoshihiko Kurimoto ◽  
Ryushi Maruyama ◽  
Naritomo Nishioka ◽  
Kousuke Ujihira ◽  
Yutaka Iba ◽  
...  
Keyword(s):  
Blood Pressure ◽  
Incidence Rate ◽  
Endovascular Aortic Repair ◽  
Type Ii ◽  
Pressure Management ◽  
Blood Pressure Management ◽  
Aortic Repair

scholarly journals The Effects of Harvesting and Time Delay on Predator-prey Systems with Holling Type II Functional Response

2009 ◽  
Vol 70 (4) ◽  
pp. 1178-1200 ◽  
Author(s):  
Jing Xia ◽  
Zhihua Liu ◽  
Rong Yuan ◽  
Shigui Ruan
Keyword(s):  
Time Delay ◽  
Functional Response ◽  
Type Ii ◽  
Predator Prey ◽  
Type Ii Functional Response

DYNAMIC COMPLEXITIES OF HOLLING TYPE II FUNCTIONAL RESPONSE PREDATOR–PREY SYSTEM WITH DIGEST DELAY AND IMPULSIVE HARVESTING ON THE PREY

2009 ◽  
Vol 02 (02) ◽  
pp. 229-242 ◽  
Author(s):  
JIANWEN JIA ◽  
HUI CAO
Keyword(s):  
Time Delay ◽  
Functional Response ◽  
Impulsive Differential Equation ◽  
Global Attractivity ◽  
Sufficient Condition ◽  
Type Ii ◽  
Predator Prey ◽  
Prey System ◽  
Impulsive Harvesting ◽  
Type Ii Functional Response

In this paper, we introduce and study Holling type II functional response predator–prey system with digest delay and impulsive harvesting on the prey, which contains with periodically pulsed on the prey and time delay on the predator. We investigate the existence and global attractivity of the predator-extinction periodic solutions of the system. By using the theory on delay functional and impulsive differential equation, we obtain the sufficient condition with time delay and impulsive perturbations for the permanence of the system.

scholarly journals Analysis of a Delayed SIR Model with Nonlinear Incidence Rate

2008 ◽  
Vol 2008 ◽  
pp. 1-16 ◽  
Author(s):  
Jin-Zhu Zhang ◽  
Zhen Jin ◽  
Quan-Xing Liu ◽  
Zhi-Yu Zhang
Keyword(s):  
Time Delay ◽  
Incidence Rate ◽  
Incubation Time ◽  
Complex Dynamics ◽  
Threshold Value ◽  
Global Dynamics ◽  
Sir Epidemic Model ◽  
Globally Asymptotically Stable ◽  
Sir Epidemic ◽  
Saturated Form

An SIR epidemic model with incubation time and saturated incidence rate is formulated, where the susceptibles are assumed to satisfy the logistic equation and the incidence term is of saturated form with the susceptible. The threshold valueℜ0determining whether the disease dies out is found. The results obtained show that the global dynamics are completely determined by the values of the threshold valueℜ0and time delay (i.e., incubation time length). Ifℜ0is less than one, the disease-free equilibrium is globally asymptotically stable and the disease always dies out, while if it exceeds one there will be an endemic. By using the time delay as a bifurcation parameter, the local stability for the endemic equilibrium is investigated, and the conditions with respect to the system to be absolutely stable and conditionally stable are derived. Numerical results demonstrate that the system with time delay exhibits rich complex dynamics, such as quasiperiodic and chaotic patterns.

STABILITY AND HOPF BIFURCATION OF A PREDATOR–PREY SYSTEM WITH TIME DELAY AND HOLLING TYPE-II FUNCTIONAL RESPONSE

2009 ◽  
Vol 02 (02) ◽  
pp. 139-149 ◽  
Author(s):  
LINGSHU WANG ◽  
RUI XU ◽  
GUANGHUI FENG
Keyword(s):  
Time Delay ◽  
Functional Response ◽  
Type Ii ◽  
Normal Form Theory ◽  
Predator Prey ◽  
Center Manifold Theorem ◽  
Predator Prey Model ◽  
Form Theory ◽  
The Stability ◽  
Type Ii Functional Response

A predator–prey model with time delay and Holling type-II functional response is investigated. By choosing time delay as the bifurcation parameter and analyzing the associated characteristic equation of the linearized system, the local stability of the system is investigated and Hopf bifurcations are established. The formulae determining the direction of bifurcations and the stability of bifurcating periodic solutions are given by using the normal form theory and center manifold theorem. Numerical simulations are carried out to illustrate the theoretical results.

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