Analysis and Simulation of Epidemic COVID-19 Curves with the Verhulst Model Applied to Statistical Inhomogeneous Age Groups
Pandemic curves, such as COVID-19, often show multiple and unpredictable contamination peaks, often called second, third and fourth waves, which are separated by wide plateaus. Here, by considering the statistical inhomogeneity of age groups, we show a quantitative understanding of the different beh...
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MDPI AG
2021-05-01
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| author | Lode K. J. Vandamme Paulo R. F. Rocha |
| author_facet | Lode K. J. Vandamme Paulo R. F. Rocha |
| author_sort | Lode K. J. Vandamme |
| collection | DOAJ |
| description | Pandemic curves, such as COVID-19, often show multiple and unpredictable contamination peaks, often called second, third and fourth waves, which are separated by wide plateaus. Here, by considering the statistical inhomogeneity of age groups, we show a quantitative understanding of the different behaviour rules to flatten a pandemic COVID-19 curve and concomitant multi-peak recurrence. The simulations are based on the Verhulst model with analytical generalized logistic equations for the limited growth. From the log–lin plot, we observe an early exponential growth proportional to <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msup><mi>e</mi><mrow><mi>t</mi><mo>/</mo><msub><mi>τ</mi><mrow><mi>g</mi><mi>r</mi><mi>o</mi><mi>w</mi></mrow></msub></mrow></msup></mrow></semantics></math></inline-formula>. The first peak is often τ<sub>grow</sub> ≅ 5 <i>d</i>. The exponential growth is followed by a recovery phase with an exponential decay proportional to <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msup><mi>e</mi><mrow><mo>−</mo><mi>t</mi><mo>/</mo><msub><mi>τ</mi><mrow><mi>r</mi><mi>e</mi><mi>c</mi><mi>o</mi><mi>v</mi></mrow></msub></mrow></msup></mrow></semantics></math></inline-formula>. For the characteristic time holds: <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msub><mi>τ</mi><mrow><mi>g</mi><mi>r</mi><mi>o</mi><mi>w</mi></mrow></msub><mo><</mo></mrow></semantics></math></inline-formula> <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msub><mi>τ</mi><mrow><mi>r</mi><mi>e</mi><mi>c</mi><mi>o</mi><mi>v</mi></mrow></msub></mrow></semantics></math></inline-formula>. Even with isolation, outbreaks due to returning travellers can result in a recurrence of multi-peaks visible on log–lin scales. The exponential growth for the first wave is faster than for the succeeding waves, with characteristic times, τ of about 10 <i>d</i>. Our analysis ascertains that isolation is an efficient method in preventing contamination and enables an improved strategy for scientists, governments and the general public to timely balance between medical burdens, mental health, socio-economic and educational interests. |
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| spelling | doaj.art-878e3006a3c94668a7474ce32163f51c2023-11-21T18:10:38ZengMDPI AGApplied Sciences2076-34172021-05-01119415910.3390/app11094159Analysis and Simulation of Epidemic COVID-19 Curves with the Verhulst Model Applied to Statistical Inhomogeneous Age GroupsLode K. J. Vandamme0Paulo R. F. Rocha1Faculty of Electrical Engineering, Eindhoven University of Technology, 5600 MB Eindhoven, The NetherlandsCentre for Functional Ecology (CFE), Department of Life Sciences, University of Coimbra, 3000-456 Coimbra, PortugalPandemic curves, such as COVID-19, often show multiple and unpredictable contamination peaks, often called second, third and fourth waves, which are separated by wide plateaus. Here, by considering the statistical inhomogeneity of age groups, we show a quantitative understanding of the different behaviour rules to flatten a pandemic COVID-19 curve and concomitant multi-peak recurrence. The simulations are based on the Verhulst model with analytical generalized logistic equations for the limited growth. From the log–lin plot, we observe an early exponential growth proportional to <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msup><mi>e</mi><mrow><mi>t</mi><mo>/</mo><msub><mi>τ</mi><mrow><mi>g</mi><mi>r</mi><mi>o</mi><mi>w</mi></mrow></msub></mrow></msup></mrow></semantics></math></inline-formula>. The first peak is often τ<sub>grow</sub> ≅ 5 <i>d</i>. The exponential growth is followed by a recovery phase with an exponential decay proportional to <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msup><mi>e</mi><mrow><mo>−</mo><mi>t</mi><mo>/</mo><msub><mi>τ</mi><mrow><mi>r</mi><mi>e</mi><mi>c</mi><mi>o</mi><mi>v</mi></mrow></msub></mrow></msup></mrow></semantics></math></inline-formula>. For the characteristic time holds: <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msub><mi>τ</mi><mrow><mi>g</mi><mi>r</mi><mi>o</mi><mi>w</mi></mrow></msub><mo><</mo></mrow></semantics></math></inline-formula> <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msub><mi>τ</mi><mrow><mi>r</mi><mi>e</mi><mi>c</mi><mi>o</mi><mi>v</mi></mrow></msub></mrow></semantics></math></inline-formula>. Even with isolation, outbreaks due to returning travellers can result in a recurrence of multi-peaks visible on log–lin scales. The exponential growth for the first wave is faster than for the succeeding waves, with characteristic times, τ of about 10 <i>d</i>. Our analysis ascertains that isolation is an efficient method in preventing contamination and enables an improved strategy for scientists, governments and the general public to timely balance between medical burdens, mental health, socio-economic and educational interests.https://www.mdpi.com/2076-3417/11/9/4159COVID-19pandemicscontamination peakslimited growth modelsVerhulst modellogistic equations |
| spellingShingle | Lode K. J. Vandamme Paulo R. F. Rocha Analysis and Simulation of Epidemic COVID-19 Curves with the Verhulst Model Applied to Statistical Inhomogeneous Age Groups Applied Sciences COVID-19 pandemics contamination peaks limited growth models Verhulst model logistic equations |
| title | Analysis and Simulation of Epidemic COVID-19 Curves with the Verhulst Model Applied to Statistical Inhomogeneous Age Groups |
| title_full | Analysis and Simulation of Epidemic COVID-19 Curves with the Verhulst Model Applied to Statistical Inhomogeneous Age Groups |
| title_fullStr | Analysis and Simulation of Epidemic COVID-19 Curves with the Verhulst Model Applied to Statistical Inhomogeneous Age Groups |
| title_full_unstemmed | Analysis and Simulation of Epidemic COVID-19 Curves with the Verhulst Model Applied to Statistical Inhomogeneous Age Groups |
| title_short | Analysis and Simulation of Epidemic COVID-19 Curves with the Verhulst Model Applied to Statistical Inhomogeneous Age Groups |
| title_sort | analysis and simulation of epidemic covid 19 curves with the verhulst model applied to statistical inhomogeneous age groups |
| topic | COVID-19 pandemics contamination peaks limited growth models Verhulst model logistic equations |
| url | https://www.mdpi.com/2076-3417/11/9/4159 |
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