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 The Dynamics of the Pulse Birth in an SIR Epidemic Model with Standard Incidence

2009 ◽  
Vol 2009 ◽  
pp. 1-18
Author(s):  
Juping Zhang ◽  
Zhen Jin ◽  
Yakui Xue ◽  
Youwen Li
Keyword(s):  
Periodic Solution ◽  
Numerical Simulations ◽  
Epidemic Model ◽  
Basic Reproductive Number ◽  
Reproductive Number ◽  
Asymptotically Stable ◽  
Sir Epidemic Model ◽  
Sir Epidemic ◽  
Theoretical Results

An SIR epidemic model with pulse birth and standard incidence is presented. The dynamics of the epidemic model is analyzed. The basic reproductive numberR∗is defined. It is proved that the infection-free periodic solution is global asymptotically stable ifR∗<1. The infection-free periodic solution is unstable and the disease is uniform persistent ifR∗>1. Our theoretical results are confirmed by numerical simulations.

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 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 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.

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.

scholarly journals Mathematical modelling for malaria under resistance and population movement

2020 ◽  
Vol 38 (2) ◽  
pp. 133-163
Author(s):  
Cristhian Montoya ◽  
Jhoana P. Romero Leiton
Keyword(s):  
Human Population ◽  
Control Strategies ◽  
Basic Reproductive Number ◽  
Reproductive Number ◽  
Transmission Dynamic ◽  
Equilibrium Solutions ◽  
Population Movements ◽  
Existence And Stability ◽  
Theoretical Results ◽  
Disease Free

In this work, two mathematical models for malaria under resistance are presented. More precisely, the first model shows the interaction between humans and mosquitoes inside a patch under infection of malaria when the human population is resistant to antimalarial drug and mosquitoes population is resistant to insecticides. For the second model, human–mosquitoes population movements in two patches is analyzed under the same malaria transmission dynamic established in a patch. For a single patch, existence and stability conditions for the equilibrium solutions in terms of the local basic reproductive number are developed. These results reveal the existence of a forward bifurcation and the global stability of disease–free equilibrium. In the case of two patches, a theoretical and numerical framework on sensitivity analysis of parameters is presented. After that, the use of antimalarial drugs and insecticides are incorporated as control strategies and an optimal control problem is formulated. Numerical experiments are carried out in both models to show the feasibility of our theoretical results.

scholarly journals Asymptotic Behavior and Stationary Distribution of a Nonlinear Stochastic Epidemic Model with Relapse and Cure

2020 ◽  
Vol 2020 ◽  
pp. 1-12
Author(s):  
Jiying Ma ◽  
Qing Yi
Keyword(s):  
Stationary Distribution ◽  
Stochastic System ◽  
Epidemic Model ◽  
Basic Reproductive Number ◽  
Reproductive Number ◽  
Deterministic System ◽  
Stochastic Epidemic ◽  
Lyapunov Function Method ◽  
Deterministic Framework ◽  
Theoretical Results

In this paper, by introducing environmental perturbation, we extend an epidemic model with graded cure, relapse, and nonlinear incidence rate from a deterministic framework to a stochastic differential one. The existence and uniqueness of positive solution for the stochastic system is verified. Using the Lyapunov function method, we estimate the distance between stochastic solutions and the corresponding deterministic system in the time mean sense. Under some acceptable conditions, the solution of the stochastic system oscillates in the vicinity of the disease-free equilibrium if the basic reproductive number R0≤1, while the random solution oscillates near the endemic equilibrium, and the system has a unique stationary distribution if R0>1. Moreover, numerical simulation is conducted to support our theoretical results.

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.

PULSE VACCINATION IN THE PERIODIC INFECTION RATE SIR EPIDEMIC MODEL

2008 ◽  
Vol 01 (04) ◽  
pp. 409-432 ◽  
Author(s):  
ZHEN JIN ◽  
MAINUL HAQUE ◽  
QUANXING LIU
Keyword(s):  
Periodic Solution ◽  
Infection Rate ◽  
Dynamical Behavior ◽  
Positive Periodic Solution ◽  
Basic Reproductive Number ◽  
Reproductive Number ◽  
Uniform Persistence ◽  
Sir Epidemic Model ◽  
Pulse Vaccination ◽  
Globally Stable

In this paper a pulse vaccination SIR model with periodic infection rate β(t) is studied. The basic reproductive number R0 is defined. The dynamical behavior of the model is analyzed. It is proved that the infection-free periodic solution is globally stable if R0 < 1. The infection-free periodic solution is unstable and the disease will uniform persistence when R0 > 1. We use standard bifurcation theory to show the existence of the positive periodic solution when R0 → 1+. Numerical simulation can give suggestion, the system has a unique positive periodic, and it is globally stable when R0 > 1.

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