Vaccination based control of infections in SIRS models with reinfection: special reference to pertussis

Safan, Muntaser; Kretzschmar, Mirjam; Hadeler, K. P.

doi:10.1007/s00285-012-0582-1

Cited by 25 publications

(22 citation statements)

References 34 publications

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“…Hence, model (12) does not have a basic reproduction number. Moreover, m is the critical proliferation rate [25,26] above which endemic equilibria do exist, while below which no endemic equilibrium exists, see Fig. 5.…”

Section: Equilibria and Stabilitymentioning

confidence: 99%

A simple mathematical model for Guillain–Barré syndrome

2019

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In this paper, four different forms of a model to describe the dynamics of autoimmune diseases (with emphasis on Guillain-Barré syndrome) are proposed. In the first two cases, the immune response is supposed to be linear, while in the other two cases it is supposed to be in the Holling type III form. In case of linear immune response, the model has a basic reproduction number and shows forward bifurcation. However, in the nonlinear Holling type III immune response cases, the model does not have a basic reproduction number and two positive equilibria do exist for a range of the parameters. The stability analysis of the model's steady states has been established. Our analytical results have been illustrated by numerical simulations.

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Section: Equilibria and Stabilitymentioning

confidence: 99%

A simple mathematical model for Guillain–Barré syndrome

2019

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“…These mechanisms are the re-stocking of a large enough pool of susceptible individuals, a process directly tied in to the study of the role of age-structure on disease dynamics. Additional mechanisms include seasonality, delays from various natural epidemiological starts (asymptomatic or latent stages), behavioral changes in disease dynamics and delays driven by interventions (quarantine/isolation) [5,4,20,19,28,9,25,24]. In this paper we have systematically derived the system of integro-differential equations model given by (7) and showed that system (7) is equivalent to the system of delay differential equations given by (8) for non-exponential waiting times in the Q class.…”

Section: )mentioning

confidence: 99%

A delay differential equations model for disease transmisión dynamics

Erdem

Safan

Castillo‐Chavez

2019

RMTA

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A delay differential equations epidemic model of SIQR (SusceptibleInfective-Quarantined-Recovered) type, with arbitrarily distributed periods in the isolation or quarantine class, is proposed. Its essential mathematical features are analyzed. In addition, conditions that support the existence of periodic solutions via Hopf bifurcation are identified. Nonexponential waiting times in the quarantine/isolation class lead not only to oscillations but can also support stability switches.

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“…Thus, the classical requirement of R 0 < 1 is, although necessary, no longer sufficient for disease elimination [19,38,39,40]. In some epidemiological models, it has been shown that the backward bifurcation phenomenon is caused by factors such as nonlinear incidence (the infection force), disease-induced death or imperfect vaccine [19,40,41,42,43,44]. To confirm whether or not the backward bifurcation phenomenon occurs in this case, one could use the approach developed in [35,43,45], which is based on the general centre manifold theorem [46].…”

Section: Global Stabilty Of the Disease-free Equilibriummentioning

confidence: 99%

Backward bifurcation and control in transmission dynamics of arboviral diseases

Abboubakar

Kamgang

Tieudjo

2016

Mathematical Biosciences

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In this paper, we derive and analyze a compartmental model for the control of arboviral diseases which takes into account an imperfect vaccine combined with individual protection and some vector control strategies already studied in the literature. After the formulation of the model, a qualitative study based on stability analysis and bifurcation theory reveals that the phenomenon of backward bifurcation may occur. The stable disease-free equilibrium of the model coexists with a stable endemic equilibrium when the reproduction number, R 0 , is less than unity. Using Lyapunov function theory, we prove that the trivial equilibrium is globally asymptotically stable; When the diseaseinduced death is not considered, or/and, when the standard incidence is replaced by the mass action incidence, the backward bifurcation does not occur. Under a certain condition, we establish the global asymptotic stability of the disease-free equilibrium of the full model. Through sensitivity analysis, we determine the relative importance of model parameters for disease transmission. Numerical simulations show that the combination of several control mechanisms would significantly reduce the spread of the disease, if we maintain the level of each control high, and this, over a long period.

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Vaccination based control of infections in SIRS models with reinfection: special reference to pertussis

Cited by 25 publications

References 34 publications

A simple mathematical model for Guillain–Barré syndrome

A simple mathematical model for Guillain–Barré syndrome

A delay differential equations model for disease transmisión dynamics

Backward bifurcation and control in transmission dynamics of arboviral diseases

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