scholarly journals The Hopf Bifurcation Analysis and Optimal Control of a Delayed SIR Epidemic Model

2014 ◽  
Vol 2014 ◽  
pp. 1-10 ◽  
Author(s):  
Abdelhadi Abta ◽  
Hassan Laarabi ◽  
Hamad Talibi Alaoui
Keyword(s):  
Optimal Control ◽  
Hopf Bifurcation ◽  
Critical Value ◽  
Control Measures ◽  
Basic Reproductive Number ◽  
Reproductive Number ◽  
Sir Epidemic Model ◽  
Sir Epidemic ◽  
Saturated Incidence Rate ◽  
The Impact

We propose a delayed SIR model with saturated incidence rate. The delay is incorporated into the model in order to model the latent period. The basic reproductive number R0 is obtained. Furthermore, using time delay as a bifurcation parameter, it is proven that there exists a critical value of delay for the stability of diseases prevalence. When the delay exceeds the critical value, the system loses its stability and a Hopf bifurcation occurs. The model is extended to assess the impact of some control measures, by reformulating the model as an optimal control problem with vaccination and treatment. The existence of the optimal control is also proved. Finally, some numerical simulations are performed to verify the theoretical analysis.

THE DYNAMICS OF THE CONSTANT AND PULSE BIRTH IN AN SIR EPIDEMIC MODEL WITH CONSTANT RECRUITMENT

2007 ◽  
Vol 15 (02) ◽  
pp. 203-218 ◽  
Author(s):  
WENJUN CAO ◽  
ZHEN JIN
Keyword(s):  
Epidemic Model ◽  
Disease Model ◽  
Basic Reproductive Number ◽  
Reproductive Number ◽  
Free Solution ◽  
Sir Epidemic Model ◽  
Sir Epidemic ◽  
Stable Period ◽  
Number Formula ◽  
Globally Stable

In this paper, an SIR epidemic model with constant recruitment is considered. The dynamic behavior of this disease model with constant and pulse birth are analyzed. With constant birth, the infection-free equilibrium is locally and globally stable when the basic reproductive number R0 < 1. However, with pulse birth the system converges to a stable period solution with the number of infectious individuals equal to zero. Furthermore, the local and global stability of the periodic infection-free solution is obtained if the basic reproductive number [Formula: see text]. Numerical simulation shows that the periodic infection-free solution is unstable and the disease will persist when [Formula: see text]. The effectiveness of the constant and pulse birth to eliminating the disease are compared.

scholarly journals Stability Analysis and Optimal Control of a Vector-Borne Disease with Nonlinear Incidence

2012 ◽  
Vol 2012 ◽  
pp. 1-21 ◽  
Author(s):  
Muhammad Ozair ◽  
Abid Ali Lashari ◽  
Il Hyo Jung ◽  
Kazeem Oare Okosun
Keyword(s):  
Optimal Control ◽  
Sensitivity Analysis ◽  
Control Measures ◽  
Basic Reproductive Number ◽  
Control Technique ◽  
Global Dynamics ◽  
Nonlinear Incidence ◽  
Vector Borne Disease ◽  
Vector Borne ◽  
The Impact

The paper considers a model for the transmission dynamics of a vector-borne disease with nonlinear incidence rate. It is proved that the global dynamics of the disease are completely determined by the basic reproduction number. In order to assess the effectiveness of disease control measures, the sensitivity analysis of the basic reproductive numberR0and the endemic proportions with respect to epidemiological and demographic parameters are provided. From the results of the sensitivity analysis, the model is modified to assess the impact of three control measures; the preventive control to minimize vector human contacts, the treatment control to the infected human, and the insecticide control to the vector. Analytically the existence of the optimal control is established by the use of an optimal control technique and numerically it is solved by an iterative method. Numerical simulations and optimal analysis of the model show that restricted and proper use of control measures might considerably decrease the number of infected humans in a viable way.

THE IMPACT OF HUMAN LOCATION-SPECIFIC CONTACT PATTERN ON THE SIR EPIDEMIC TRANSMISSION BETWEEN POPULATIONS

2013 ◽  
Vol 23 (05) ◽  
pp. 1350095 ◽  
Author(s):  
LIN WANG ◽  
YAN ZHANG ◽  
ZHEN WANG ◽  
XIANG LI
Keyword(s):  
Population Model ◽  
Disease Incidence ◽  
Basic Reproductive Number ◽  
Reproductive Number ◽  
Contact Pattern ◽  
Structured Population Model ◽  
Epidemic Dynamics ◽  
Sir Epidemic ◽  
Structured Population ◽  
The Impact

The structured-population model is extensively used to study the complexity of epidemic dynamics. In many seminal researches, the impact of human mobility on the outbreak threshold has been profoundly studied, with the general assumption that the human contact pattern is mixing homogeneously. As the individual contact is assumed uniform among different subpopulations, the basic reproductive number, R0, which relates to the stability at the disease-free equilibrium, is equal to the same constant on separate locations. However, recent studies have shown that there may exist location-related factors driving the variance of disease incidence between populations, in reality. Therefore, in this study, the location-specific heterogeneous contact pattern has been introduced into a famous phenomenological structured-population model, where bidirectional recurrent commuting flows couple two typical subpopulations, to study the complex dynamics behaviors of spatial transmission of epidemics. Besides the usual SIR epidemic dynamics with birth and death processes, we take into account the contact process by assigning each member from a given subpopulation with a characteristic contact rate. Through theoretical arguments and agent-based computer simulations, we unveil that the stressed element dramatically affects the epidemic threshold of the system.

scholarly journals Dynamical analysis of the spread of African swine fever with the live pig price in China

2021 ◽  
Vol 18 (6) ◽  
pp. 8123-8148
Author(s):  
Yihao Huang ◽  
◽  
Jing Li ◽  
Juan Zhang ◽  
Zhen Jin ◽  
...  
Keyword(s):  
Supply And Demand ◽  
African Swine Fever ◽  
Control Measures ◽  
Basic Reproductive Number ◽  
Dynamical Analysis ◽  
Critical Conditions ◽  
Reproductive Number ◽  
Price Function ◽  
The Government ◽  
The Impact

<abstract><p>Pork makes up the highest proportion of household expenditure on meat in China and supply and demand have been basically stable in the past decade. However, the catastrophic outbreak of African swine fever (ASF) in August 2018 disrupted the balance and reduced the national herd by half within six months. The consequence was a gross lack of supply to the market and consumer demand was unable to be met. Accordingly, live pig prices rose sharply from 2019. In order to assess the influence of ASF on the price of the live pigs, we use a price function to characterize the relationship between price of the live pigs and the nation's pig stock, and then establish a time delay ASF epidemic dynamical model with the price function. By analyzing the dynamical behaviors of the model, we calculate the basic reproductive number, discuss the stability of equilibrium, and obtain the critical conditions for Hopf bifurcation. The model reasonableness is confirmed by carrying out data fitting and parameter estimation based on price data of the live pigs, the pig stock data and the outbreak data of ASF. By performing sensitivity analysis, we intuitively show the impact of ASF on the price of live pigs and the pig stocks, and assess the key factors affecting the outbreak of ASF. The conclusion is drawn that, with the control measures adopted by related government department in China, the basic reproductive number ($ R_0 = 0.6005 $) means that the ASF epidemic has been controlled. Moreover, the price of the live pig increases linearly with $ R_0 $, while the effect of the number of infected pigs on the subsequent price is non-linear related. Our findings suggest that society and the government should pay more attention to the prevention of animal disease epidemics.</p></abstract>

scholarly journals Behavior of a Discrete Fractional Order SIR Epidemic Model

2018 ◽  
Vol 7 (4.10) ◽  
pp. 675 ◽  
Author(s):  
A. George Maria Selvam ◽  
D. Abraham Vianny
Keyword(s):  
Equilibrium Point ◽  
Fractional Order ◽  
Epidemic Model ◽  
Dynamical Behavior ◽  
Basic Reproductive Number ◽  
Phase Portraits ◽  
Reproductive Number ◽  
Sir Epidemic Model ◽  
Sir Epidemic ◽  
Parameter Values

In this paper we investigate the dynamical behavior of a SIR epidemic model of fractional order. Disease Free Equilibrium point, Endemic Equilibrium point and basic reproductive number are obtained. Time series plots, phase portraits and bifurcation diagrams are presented for suitable parameter values. Also some numerical examples are provided to illustrate the dynamics of the system.  

scholarly journals Rich Dynamics of an Epidemic Model with Saturation Recovery

2013 ◽  
Vol 2013 ◽  
pp. 1-9 ◽  
Author(s):  
Hui Wan ◽  
Jing-an Cui
Keyword(s):  
Basic Reproduction Number ◽  
Epidemic Model ◽  
Disease Transmission ◽  
Reproduction Number ◽  
Basic Reproductive Number ◽  
Reproductive Number ◽  
Medical Resource ◽  
Sir Epidemic Model ◽  
The Impact ◽  
The Basic Reproduction Number

A SIR epidemic model is proposed to understand the impact of limited medical resource on infectious disease transmission. The basic reproduction number is identified. Existence and stability of equilibria are obtained under different conditions. Bifurcations, including backward bifurcation and Hopf bifurcation, are analyzed. Our results suggest that the model considering the impact of limited medical resource may exhibit vital dynamics, such as bistability and periodicity when the basic reproduction numberℝ0is less than unity, which implies that the basic reproductive number itself is not enough to describe whether the disease will prevail or not and a subthreshold number is needed. It is also shown that a sufficient number of sickbeds and other medical resources are very important for disease control and eradication. Considering the costs, we provide a method to estimate a suitable treatment capacity for a disease in a region.

scholarly journals Optimal Control and Cost Effectiveness Analysis of SIRS Malaria Disease Model with Temperature Variability Factor

2021 ◽  
Vol 53 (1) ◽  
pp. 134-163
Author(s):  
Temesgen Duressa Keno ◽  
Oluwole Daniel Makinde ◽  
Legesse Lemecha Obsu
Keyword(s):  
Optimal Control ◽  
Cost Effectiveness ◽  
Initial Conditions ◽  
Control Strategies ◽  
Disease Model ◽  
Temperature Variability ◽  
Control Measures ◽  
Basic Reproductive Number ◽  
Bed Nets ◽  
Reproductive Number

In this study, we proposed and analyzed the optimal control and cost-effectiveness strategies for malaria epidemics model with impact of temperature variability. Temperature variability strongly determines the transmission of malaria. Firstly, we proved that all solutions of the model are positive and bounded within a certain set with initial conditions. Using the next-generation matrix method, the basic reproductive number at the present malaria-free equilibrium point was computed. The local stability and global stability of the malaria-free equilibrium were depicted applying the Jacobian matrix and Lyapunov function respectively when the basic reproductive number is smaller than one. However, the positive endemic equilibrium occurs when the basic reproductive number is greater than unity. A sensitivity analysis of the parameters was conducted; the model showed forward and backward bifurcation. Secondly, using Pontryagin’s maximum principle, optimal control interventions for malaria disease reduction are described involving three control measures, namely use of insecticide-treated bed nets, treatment of infected humans using anti-malarial drugs, and indoor residual insecticide spraying. An analysis of cost-effectiveness was also conducted. Finally, based on the simulation of different control strategies, the combination of treatment of infected humans and insecticide spraying was proved to be the most efficient and least costly strategy to eradicate the disease.

scholarly journals An SIR Epidemic Model with Time Delay and General Nonlinear Incidence Rate

2014 ◽  
Vol 2014 ◽  
pp. 1-7 ◽  
Author(s):  
Mingming Li ◽  
Xianning Liu
Keyword(s):  
Time Delay ◽  
Incidence Rate ◽  
Epidemic Model ◽  
Transmission Function ◽  
Basic Reproductive Number ◽  
Reproductive Number ◽  
Sir Epidemic Model ◽  
Nonlinear Incidence Rate ◽  
Nonlinear Incidence ◽  
Sir Epidemic

An SIR epidemic model with nonlinear incidence rate and time delay is investigated. The disease transmission function and the rate that infected individuals recovered from the infected compartment are assumed to be governed by general functionsF(S,I)andG(I), respectively. By constructing Lyapunov functionals and using the Lyapunov-LaSalle invariance principle, the global asymptotic stability of the disease-free equilibrium and the endemic equilibrium is obtained. It is shown that the global properties of the system depend on both the properties of these general functions and the basic reproductive numberR0.

DIFFERENTIAL SUSCEPTIBILITY TIME-DEPENDENT SIR EPIDEMIC MODEL

2008 ◽  
Vol 01 (01) ◽  
pp. 45-64 ◽  
Author(s):  
TAILEI ZHANG ◽  
JUNLI LIU ◽  
ZHIDONG TENG
Keyword(s):  
Differential Susceptibility ◽  
Time Dependent ◽  
Basic Reproductive Number ◽  
Reproductive Number ◽  
Threshold Values ◽  
Sir Epidemic Model ◽  
Sir Epidemic ◽  
Threshold Conditions ◽  
Theoretical Results ◽  
General Time

A non-autonomous epidemic dynamical system, in which we include variable susceptibility, is proposed. Some threshold conditions are derived which determine whether or not the disease will go to extinction. Some new threshold values, [Formula: see text], [Formula: see text] and [Formula: see text], are deduced for this general time-dependent system such that when [Formula: see text] is greater than 0, the disease is endemic in the sense of permanence and when one of the threshold values [Formula: see text] and [Formula: see text] is less than 0, the disease will die out. As an application of these results, the basic reproductive number ℛ0 will be given if all the coefficients are periodic with common period. In addition, ℛ0 < 1 implies the global stability of the disease-free periodic solution. Some corollaries are given for periodic and almost-periodic cases. The theoretical results are confirmed by a special example and numerical simulations.

scholarly journals Optimal control of an epidemic model with a saturated incidence rate

2012 ◽  
Vol 17 (4) ◽  
pp. 448-459 ◽  
Author(s):  
Hassan Laarabi ◽  
El Houssine Labriji ◽  
Mostafa Rachik ◽  
Abdelilah Kaddar
Keyword(s):  
Optimal Control ◽  
Incidence Rate ◽  
Epidemic Model ◽  
Sir Epidemic Model ◽  
Sir Epidemic ◽  
Control Framework ◽  
Adjoint Variables ◽  
Saturated Incidence Rate ◽  
Saturated Incidence

In this study we consider a mathematical model of an SIR epidemic model with a saturated incidence rate. We used the optimal vaccination strategies to minimize the susceptible and infected individuals and to maximize the number of recovered individuals. We work in the nonlinear optimal control framework. The existence result was discussed. A characterization of the optimal control via adjoint variables was established. We obtained an optimality system that we sought to solve numerically by a competitive Gauss–Seidel like implicit difference method.

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