Stability analysis of a novel epidemics model with vaccination and nonlinear infectious rate

Liu, Defang; Wang, Bochu; Guo, Songtao

doi:10.1016/j.amc.2013.07.016

Cited by 3 publications

(6 citation statements)

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“…Vaccination strategy is an efective way to prevent the outbreak and spread of infectious diseases. Tere are also many studies on the model of infectious diseases under the action of vaccination [14][15][16]. Karand and Batabyal [14] focused on the study of a nonlinear mathematical SIR epidemic model with a vaccination program and discussed the existence and the stability of both the diseasefree and endemic equilibrium.…”

Section: Introductionmentioning

confidence: 99%

Stability Analysis of SIRS Model considering Pulse Vaccination and Elimination Disturbance

Ma,

Zuo

2024

Journal of Mathematics

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It is well known that many natural phenomena and human activities do exhibit impulsive effects in the fields of epidemiology. At the same time, compared with a single control strategy, it is obvious that the multiple control strategies are more beneficial to restrain the spread of infectious diseases. In this paper, we consider pulse vaccination and pulse elimination strategies at the same time and establish an SIRS epidemic model with standard incidence. Firstly, according to the stroboscopic mapping method of the discrete dynamical system, the disease-free T periodic solution of the model under the condition of pulse vaccination and pulse elimination is obtained. Secondly, the basic reproductive number R0 is defined, and the local asymptotic stability of the disease-free T periodic solution is proved by Floquet theory for R0<1. Finally, based on the impulsive differential inequality theory, the global asymptotic stability of the disease-free T periodic solution is given for R0<1, and the disease dies out eventually. The results show that in order to stop the disease epidemic, it is necessary to choose the appropriate vaccination rate and elimination rate and the appropriate impulsive period.

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Section: Introductionmentioning

confidence: 99%

Stability Analysis of SIRS Model considering Pulse Vaccination and Elimination Disturbance

Ma,

Zuo

2024

Journal of Mathematics

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“…It has been used to tackle diseases such as measles, mumps, rubella, diphtheria, tetanus, influenza, polio, etc. Recently, the epidemiological models with vaccination strategy have been analyzed by many authors in [19][20][21][22][23][24][25][26][27]. For example, Li et al [19] discussed the global analysis of SIS epidemic model with a simple vaccination and multiple endemic equilibria; Liu et al [20] established two SVIR models by considering the time for them to obtain immunity and the possibility for them to be infected before this; Trawicki [21] proposes a new SEIRS model with vital dynamics (birth and death rates), vaccination, and temporary immunity provides a mathematical description of infectious diseases and corresponding spread in biology; T.K.…”

Section: Introductionmentioning

confidence: 99%

Global Dynamics of an SIQR Model with Vaccination and Elimination Hybrid Strategies

Liu

2018

Mathematics

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In this paper, an SIQR (Susceptible, Infected, Quarantined, Recovered) epidemic model with vaccination, elimination, and quarantine hybrid strategies is proposed, and the dynamics of this model are analyzed by both theoretical and numerical means. Firstly, the basic reproduction number R 0 , which determines whether the disease is extinct or not, is derived. Secondly, by LaSalles invariance principle, it is proved that the disease-free equilibrium is globally asymptotically stable when R 0 < 1 , and the disease dies out. By Routh-Hurwitz criterion theory, we also prove that the disease-free equilibrium is unstable and the unique endemic equilibrium is locally asymptotically stable when R 0 > 1 . Thirdly, by constructing a suitable Lyapunov function, we obtain that the unique endemic equilibrium is globally asymptotically stable and the disease persists at this endemic equilibrium if it initially exists when R 0 > 1 . Finally, some numerical simulations are presented to illustrate the analysis results.

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“…Epidemiological mathematical models can be used in predicting and analyzing the emergence, spread and control of infectious diseases. As per Liu et al (2013), these models are very critical to the studying of virus spreading dynamics which can state clearly the origination, evolution and effects of viruses. Also, they can help in figuring out decisions (policy-making) that are of significant importance in a way that human reasoning cannot before time when an outbreak is forecasted (Tumwiine et al, 2007), which results to implementation of mitigation strategies for early outbreak control.…”

Section: Rationale/justification Of the Researchmentioning

confidence: 99%

“…Increasing migration, accelerating urbanization, and improved travel infrastructure are global trends that increase the risk of YF spreading to parts of the world where the disease had disappeared. Liu et al (2013) argued that, mathematical analysis and modelling operation of infectious diseases are critical to the studying of virus spreading dynamics which can state clearly the origination and evolution of viruses. Also, according to Tumwiine et al (2007), mathematical modelling can help in figuring out decisions that are of significant importance and increase influence the theory and practice of disease management and control.…”

Section: Introductionmentioning

confidence: 99%

“…Mathematical and epidemiological models have been developed in the literature to address vector-borne diseases transmission dynamics and control (Garske et al, 2014;Liu et al, 2013;Lashari et al, 2012;Johansson et al, 2010). Many of these models have a disease-free equilibrium (DFE) at which the population remains unaffected by the disease (absence of disease)…”

Section: Introductionmentioning

confidence: 99%

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Mathematical Modelling of Intra and Inter Dynamics and Control of Yellow Fever in Primate and Humna Populations

Kung’aro¹

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A deterministic mathematical model was formulated using non-linear ordinary differential equations to gain an insight of dynamics of yellow fever (YF) between primates, human beings and Aedes mosquito for the purpose of controlling the disease. Basic reproduction number, R0, was computed and its sensitivity analysis with respect to epidemiological parameters was performed to study the effect of model parameters to R0. Results showed that R0 is most sensitive to daily biting rate of mosquitoes, recruitment rate of vectors, probability of transmission of infection, recruitment of unvaccinated immigrants and the incubation period for both vector and humans. Thus, for the minimization of YF transmission, these parameters should closely be monitored. Stability analysis of disease-free equilibrium (DFE) and endemic equilibrium (EE) points were performed to study perseverance and conditionnecessaryfordiseaseinterruptionandcontrol. ResultsshowedthattheDFEislocally asymptotically stable if the rate of new infection from infected monkey to vector is less than unity, and is globally asymptotically stable if the rate of new infection from infected vector to human is less than unity. Lyapunov stability theory and LaSalles Invariant Principle were used to investigate stability of EE. Results show that EE is globally asymptotically stable whenever R0 > 1. To assess the impact of control measures on YF dynamics, we derived and analysed the necessary conditions for optimal control using optimal control theory. Results show that multipleoptimalcontrolstrategyisthemosteffectivetobringastabledisease-freeequilibrium compared to single and two controls. However, spray of insecticides alone was not effective without personal protection, and optimal use of personal protection alone is beneﬁcial to minimize transmission of the infection to the community. Furthermore, cost-effectiveness analysis of the optimal control measures was considered. We used incremental cost-effectiveness ratio to investigate and compare the costs required against the health beneﬁts achieved between two or more alternative intervention strategies that compete for the same resource. Results showed that combination of all strategies is the most cost-effective compared to others.

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Stability analysis of a novel epidemics model with vaccination and nonlinear infectious rate

Cited by 3 publications

References 30 publications

Stability Analysis of SIRS Model considering Pulse Vaccination and Elimination Disturbance

Stability Analysis of SIRS Model considering Pulse Vaccination and Elimination Disturbance

Global Dynamics of an SIQR Model with Vaccination and Elimination Hybrid Strategies

Mathematical Modelling of Intra and Inter Dynamics and Control of Yellow Fever in Primate and Humna Populations

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