SIR Model with Vaccination: Bifurcation Analysis
There are few adapted SIR models in the literature that combine vaccination and logistic growth. In this article, we study bifurcations of a SIR model where the class of Susceptible individuals grows logistically and has been subject to constant vaccination. We explicitly prove that the endemic equi...
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| Published in | Qualitative theory of dynamical systems Vol. 22; no. 3 |
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| Main Authors | , |
| Format | Journal Article |
| Language | English |
| Published |
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01.09.2023
Springer Nature B.V |
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| Abstract | There are few adapted SIR models in the literature that combine vaccination and logistic growth. In this article, we study bifurcations of a SIR model where the class of
Susceptible
individuals grows logistically and has been subject to constant vaccination. We explicitly prove that the endemic equilibrium is a
codimension two singularity
in the parameter space
(
R
0
,
p
)
, where
R
0
is the
basic reproduction number
and
p
is the proportion of
Susceptible
individuals successfully vaccinated at birth. We exhibit explicitly the
Hopf
,
transcritical
,
Belyakov
,
heteroclinic
and
saddle-node bifurcation
curves unfolding the
singularity
. The two parameters
(
R
0
,
p
)
are written in a useful way to evaluate the proportion of vaccinated individuals necessary to eliminate the disease and to conclude how the vaccination may affect the outcome of the epidemic. We also exhibit the region in the parameter space where the disease persists and we illustrate our main result with numerical simulations, emphasizing the role of the parameters. |
|---|---|
| AbstractList | There are few adapted SIR models in the literature that combine vaccination and logistic growth. In this article, we study bifurcations of a SIR model where the class of Susceptible individuals grows logistically and has been subject to constant vaccination. We explicitly prove that the endemic equilibrium is a codimension two singularity in the parameter space (R0,p), where R0 is the basic reproduction number and p is the proportion of Susceptible individuals successfully vaccinated at birth. We exhibit explicitly the Hopf, transcritical, Belyakov, heteroclinic and saddle-node bifurcation curves unfolding the singularity. The two parameters (R0,p) are written in a useful way to evaluate the proportion of vaccinated individuals necessary to eliminate the disease and to conclude how the vaccination may affect the outcome of the epidemic. We also exhibit the region in the parameter space where the disease persists and we illustrate our main result with numerical simulations, emphasizing the role of the parameters. There are few adapted SIR models in the literature that combine vaccination and logistic growth. In this article, we study bifurcations of a SIR model where the class of Susceptible individuals grows logistically and has been subject to constant vaccination. We explicitly prove that the endemic equilibrium is a codimension two singularity in the parameter space $$(\mathcal {R}_0, p)$$ ( R 0 , p ) , where $$\mathcal {R}_0$$ R 0 is the basic reproduction number and p is the proportion of Susceptible individuals successfully vaccinated at birth. We exhibit explicitly the Hopf , transcritical , Belyakov , heteroclinic and saddle-node bifurcation curves unfolding the singularity . The two parameters $$(\mathcal {R}_0, p)$$ ( R 0 , p ) are written in a useful way to evaluate the proportion of vaccinated individuals necessary to eliminate the disease and to conclude how the vaccination may affect the outcome of the epidemic. We also exhibit the region in the parameter space where the disease persists and we illustrate our main result with numerical simulations, emphasizing the role of the parameters. There are few adapted SIR models in the literature that combine vaccination and logistic growth. In this article, we study bifurcations of a SIR model where the class of Susceptible individuals grows logistically and has been subject to constant vaccination. We explicitly prove that the endemic equilibrium is a codimension two singularity in the parameter space ( R 0 , p ) , where R 0 is the basic reproduction number and p is the proportion of Susceptible individuals successfully vaccinated at birth. We exhibit explicitly the Hopf , transcritical , Belyakov , heteroclinic and saddle-node bifurcation curves unfolding the singularity . The two parameters ( R 0 , p ) are written in a useful way to evaluate the proportion of vaccinated individuals necessary to eliminate the disease and to conclude how the vaccination may affect the outcome of the epidemic. We also exhibit the region in the parameter space where the disease persists and we illustrate our main result with numerical simulations, emphasizing the role of the parameters. |
| ArticleNumber | 105 |
| Author | Maurício de Carvalho, João P. S. Rodrigues, Alexandre A. |
| Author_xml | – sequence: 1 givenname: João P. S. orcidid: 0000-0001-7709-1631 surname: Maurício de Carvalho fullname: Maurício de Carvalho, João P. S. email: jocarvalho@fc.up.pt organization: Faculty of Sciences, University of Porto, Centre for Mathematics, University of Porto – sequence: 2 givenname: Alexandre A. orcidid: 0000-0001-8182-9889 surname: Rodrigues fullname: Rodrigues, Alexandre A. organization: Faculty of Sciences, University of Porto, Centre for Mathematics, University of Porto, Lisbon School of Economics and Management |
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| Keywords | Double-zero singularity 34C23 92B05 SIR model Vaccination Unfoldings Bifurcations 37G10 37D05 37G15 |
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| Snippet | There are few adapted SIR models in the literature that combine vaccination and logistic growth. In this article, we study bifurcations of a SIR model where... |
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| SubjectTerms | Bifurcations Difference and Functional Equations Dynamical Systems and Ergodic Theory Immunization Mathematical models Mathematics Mathematics and Statistics Parameters Singularities |
| Title | SIR Model with Vaccination: Bifurcation Analysis |
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