. 2023 Jun 21;13:10083. doi: 10.1038/s41598-023-37196-9

Table 1.

The obtained models.

	Treated case	Untreated case
Integer order exponential model	$\frac{dv}{dt} = 0.046 \cdot v (t)$	$\frac{dv}{dt} = 0.109 \cdot v (t)$
Fractional order exponential model	$\frac{{dv}^{0.97}}{dt} = 0.046 \cdot v (t)$	$\frac{{dv}^{0.98}}{dt} = 0.109 \cdot v (t)$
Integer order logistic model	$\frac{dv}{dt} = 0.0735 \cdot v (t) \cdot (1 - {(\frac{v (t)}{2.81})}^{3.35})$	$\frac{d v (t)}{dt} = 0.565 \cdot v (t) \cdot (1 - {(\frac{v (t)}{35.2})}^{0.134})$
Fractional order logistic model	$\frac{d v^{0.94} (t)}{dt} = 0.0735 \cdot v (t) \cdot (1 - {(\frac{v (t)}{2.81})}^{3.35})$	$\frac{d v^{0.88} (t)}{dt} = 0.565 \cdot v (t) \cdot (1 - {(\frac{v (t)}{35.2})}^{0.134})$
Integer order Gompertz model	$\frac{d v (t)}{dt} = 0.204 \ln (\frac{8.81}{v (t) + 5.04})$	$\frac{d v (t)}{dt} = 0.106 \ln (\frac{26.09}{v (t) + 2.24})$
Fractional order Gompertz model	$\frac{d v^{0.95} (t)}{dt} = 0.204 \ln (\frac{8.81}{v (t) + 5.04})$	$\frac{d v^{0.86} (t)}{dt} = 0.106 \ln (\frac{26.09}{v (t) + 2.24})$
Integer order generalized Bertalanffy–Pütter model	$\frac{dv}{dt} = 0.168 \cdot v^{2.18} - 0.1 \cdot v^{2.66}$	$\frac{dv}{dt} = 0.894 \cdot v^{1.38} - 0.741 \cdot v^{1.44}$
Fractional order generalized Bertalanffy–Pütter model	$\frac{d v^{0.96}}{dt} = 0.168 \cdot v^{2.18} - 0.1 \cdot v^{2.66}$	$\frac{d v^{0.96}}{dt} = 0.894 \cdot v^{1.38} - 0.741 \cdot v^{1.44}$
Integer order particularized Bertalanffy–Pütter model	$\frac{dv}{dt} = 0.064 v^{2.41} - \ln (v) \cdot 0.063 \cdot v^{2.41}$	$\frac{dv}{dt} = 0.203 v^{0.74} - \ln (v) \cdot 0.011 \cdot v^{0.74}$
Fractional order particularized Bertalanffy–Pütter models model	$\frac{d v^{0.96}}{dt} = 0.064 v^{2.41} - \ln (v) \cdot 0.063 \cdot v^{2.41}$	$\frac{d v^{0.95}}{dt} = 0.203 v^{0.74} - \ln (v) \cdot 0.011 \cdot v^{0.74}$