Analysis and Simulation of Epidemic COVID-19 Curves with the Verhulst Model Applied to Statistical Inhomogeneous Age Groups

• Main Authors: Lode K. J. Vandamme, Paulo R. F. Rocha
• Format: Article
• Language: English
• Published: MDPI AG 2021-05-01
• Series: Applied Sciences
• Subjects: COVID-19; pandemics; contamination peaks; limited growth models; Verhulst model; logistic equations
• Online Access: https://www.mdpi.com/2076-3417/11/9/4159

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_grow ≅ 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.