Closed form solution of the SIR model for the COVID-19 outbreak in Italy

Giubilei, R.

doi:10.1101/2020.06.06.20124313

Cited by 5 publications

(8 citation statements)

References 5 publications

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“…with the relation between the parameter u and time t is given by, (15) In this study, we use a method similar to the paper [24] which is a reasonably rigorous method.…”

Section: Exact Solution Of the Sirvn Equationmentioning

confidence: 99%

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On the Exact analytical solution and Van der Pol like equation of infectious diseases model with the time-dependent total population

Zakia,

Nurmala,

Wijayanto

et al. 2023

Preprint

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In this paper, we derive the exact analytical solution in the parametric form of the infectious diseases (SIR) model, taking into account the population migration and vaccines SIRVN. By applying derivatives and substitutions, we convert the SIRVN equation into nonlinear third-order differential equation, and get an approximate semi-analytical solution in the form of a parametric function. The long-time oscillatory behavior of SIRVN model studies reduces to Van der Pol like equation with nonlinear damping. An analytic solution is obtained by multi-scale analysis and the Laplace transform methods. The result shows the comparison between the exact solution and the Jakarta outbreak data correlate of about R2 = 0.99. We also found that the vaccine effectively reduces the outbreak’s peak, and the asymptotic stability implies that Jakarta will change from the pandemic to the endemic. Finally, the solutions of Van der Pol-like equation show that the existence of multiple outbreak waves can be explained by this model.

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“…with the relation between the parameter u and time t is given by, (15) In this study, we use a method similar to the paper [24] which is a reasonably rigorous method.…”

Section: Exact Solution Of the Sirvn Equationmentioning

confidence: 99%

“…Analytical solutions are still needed to get the problem frame in a form of easy to understand and get an exact solution. Some researchers try to find a closed-form solution to the SIR equation [14,15,16]. They show an exact solution through analytic geometry and the series form.…”

Section: Introductionmentioning

confidence: 99%

On the Exact analytical solution and Van der Pol like equation of infectious diseases model with the time-dependent total population

Zakia,

Nurmala,

Wijayanto

et al. 2023

Preprint

View full text Add to dashboard Cite

show abstract

“…is the basic reproduction number. Despite its long history, there continues to be many efforts to try and solve it analytically Brauer (2005); Khan et al (2009); Harko et al (2014); Bohner et al (2019); Barlow & Weinstein (2020); Giubilei (2020); Kröger & Schlickeiser (2020); Mieghem (2020); Prodanov (2020); de Souza (2021); Kudryashov et al (2021). In their seminal paper, Kermack and McKendrick offered an approximate solution in the limit of a "small epidemic", i.e.…”

Section: Introductionmentioning

confidence: 99%

Asymptotic solutions of the SIR and SEIR models well above the epidemic threshold

Kozyreff

2022

IMA Journal of Applied Mathematics

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A simple and explicit expression of the solution of the SIR epidemiological model of Kermack and McKendrick is constructed in the asymptotic limit of large basic reproduction numbers ${\mathsf R_0}$. The proposed formula yields good qualitative agreement already when ${\mathsf R_0}\geq 3$ and rapidly becomes quantitatively accurate as larger values of ${\mathsf R_0}$ are assumed. The derivation is based on the method of matched asymptotic expansions, which exploits the fact that the exponential growing phase and the eventual recession of the outbreak occur on distinct time scales. From the newly derived solution, an analytical estimate of the time separating the first inflexion point of the epidemic curve from the peak of infections is given. Finally, we use the same method on the SEIR model and find that the inclusion of the ‘exposed’ population in the model can dramatically alter the time scales of the outbreak.

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“…Above, β is the contact rate, γ is the recovery rate, and their ratio is the basic reproduction number. Despite its long history [15], there continues to be many efforts to try and solve it analytically [16, 17, 18, 19, 20, 21, 22, 23, 24, 25]. In their seminal paper, Kermack and McKendrick offered an approximate solution in the limit of a “small epidemic”, i .…”

Section: Introductionmentioning

confidence: 99%

“…Above, β is the contact rate, γ is the recovery rate, and their ratio is the basic reproduction number. Despite its long history, there continues to be many efforts to try and solve it analytically Brauer (2005); Khan et al (2009); Harko et al (2014); Bohner et al (2019); Barlow & Weinstein (2020); Giubilei (2020); Kröger & Schlickeiser (2020); Mieghem (2020); Prodanov (2020); de Souza (2021); Kudryashov et al (2021). In their seminal paper, Kermack and McKendrick offered an approximate solution in the limit of a “small epidemic”, i .…”

Section: Introductionmentioning

confidence: 99%

Asymptotic solutions of the SIR and SEIR models well above the epidemic threshold

Kozyreff

2021

Preprint

View full text Add to dashboard Cite

A simple and explicit expression of the solution of the SIR epidemiological model of Kermack and McKendrick is constructed in the asymptotic limit of large basic reproduction numbers $\ro$. The proposed formula yields good qualitative agreement already when $\ro\geq3$ and rapidly becomes quantitatively accurate as larger values of $\ro$ are assumed. The derivation is based on the method of matched asymptotic expansions, which exploits the fact that the exponential growing phase and the eventual recession of the outbreak occur on distinct time scales. From the newly derived solution, an analytical estimate of the time separating the first inflexion point of the epidemic curve from the peak of infections is given.

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Closed form solution of the SIR model for the COVID-19 outbreak in Italy

Cited by 5 publications

References 5 publications

On the Exact analytical solution and Van der Pol like equation of infectious diseases model with the time-dependent total population

On the Exact analytical solution and Van der Pol like equation of infectious diseases model with the time-dependent total population

Asymptotic solutions of the SIR and SEIR models well above the epidemic threshold

Asymptotic solutions of the SIR and SEIR models well above the epidemic threshold

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