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# Analysis of a model for hepatitis C virus transmission that includes the effects of vaccination with waning immunity

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### Abstract

This paper considers a mathematical model based on the transmission dynamics of hepatitis C virus (HCV) infection. In addition to the usual compartments for susceptible, exposed, and infected individuals, this model includes compartments for individuals who are under treatment and those who have had vaccination against HCV infection. It is assumed that the immunity provided by the vaccine fades with time. The basic reproduction number, $$R_0$$, and the equilibrium solutions of the model are determined. The model exhibits the phenomenon of backward bifurcation where a stable disease-free equilibrium co-exists with a stable endemic equilibrium whenever $$R_0$$ is less than unity. It is shown that the use of only a perfect vaccine can eliminate backward bifurcation completely. Furthermore, a unique endemic equilibrium of the model is proved to be globally asymptotically stable under certain restrictions on the parameter values. Numerical simulation results are given to support the theoretical predictions. [epidemiological model; equilibrium solutions; backward bifurcation; global asymptotic stability; Lyapunov function.]

### Most cited references6

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### Can antiviral therapy for hepatitis C reduce the prevalence of HCV among injecting drug user populations? A modeling analysis of its prevention utility.

(2011)
Hepatitis C virus antiviral treatment is effective for individual patients but few active injecting drug users are treated. We considered the utility of antiviral treatment for primary prevention of hepatitis C. A hepatitis C transmission model among injecting drug users was developed, incorporating treatment (62.5% average sustained viral response) with no retreatment after initial treatment failure, potential re-infection for those cured, equal genotype setting (genotype 1:genotype 2/3), and no immunity. In addition, we examined scenarios with varied treatment response rates, immunity, or retreatment of treatment failures. In the baseline scenario, annually treating 10 infections per 1000 injecting drug users results in a relative decrease in hepatitis C prevalence over 10 years of 31%, 13%, or 7% for baseline (untreated endemic chronic infection) prevalences of 20%, 40%, or 60%, respectively. Sensitivity analyses show that including the potential for immunity has minimal effect on the predictions; prevalence reductions remain even if SVR is assumed to be 25% lower among active IDU than current evidence suggests; retreatment of treatment failures does not alter the short-term (<5 years) projections, but does increase treatment gains within 20 years; hepatitis C free life years gained from treating active injecting drug users are projected to be higher than from treating non-injecting drug users for prevalences below 60%. Despite the possibility of re-infection, modest rates of hepatitis C treatment among active injecting drug users could effectively reduce transmission. Evaluating and extending strategies to treat hepatitis C among active injectors are warranted. Copyright © 2010 European Association for the Study of the Liver. Published by Elsevier B.V. All rights reserved.
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### Backward bifurcations in dengue transmission dynamics.

(2008)
A deterministic model for the transmission dynamics of a strain of dengue disease, which allows transmission by exposed humans and mosquitoes, is developed and rigorously analysed. The model, consisting of seven mutually-exclusive compartments representing the human and vector dynamics, has a locally-asymptotically stable disease-free equilibrium (DFE) whenever a certain epidemiological threshold, known as the basic reproduction number(R(0)) is less than unity. Further, the model exhibits the phenomenon of backward bifurcation, where the stable DFE coexists with a stable endemic equilibrium. The epidemiological consequence of this phenomenon is that the classical epidemiological requirement of making R(0) less than unity is no longer sufficient, although necessary, for effectively controlling the spread of dengue in a community. The model is extended to incorporate an imperfect vaccine against the strain of dengue. Using the theory of centre manifold, the extended model is also shown to undergo backward bifurcation. In both the original and the extended models, it is shown, using Lyapunov function theory and LaSalle Invariance Principle, that the backward bifurcation phenomenon can be removed by substituting the associated standard incidence function with a mass action incidence. In other words, in addition to establishing the presence of backward bifurcation in models of dengue transmission, this study shows that the use of standard incidence in modelling dengue disease causes the backward bifurcation phenomenon of dengue disease.
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### Global stability of an epidemic model with latent stage and vaccination

(2011)
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### Author and article information

###### Journal
22 December 2017
###### Article
1712.08548