Corrigendum to a novel covid-19 mathematical model with fractional derivatives: Singular and nonsingular kernels [Chaos Solitons & Fractals 139 (2020) 110060]

The author regrets to observe some misprint typo mistake in the paper. The following formulas were published by they should be.

For endemic-equilibrium, (3.2) should be

$$\begin{array}{c}{\displaystyle -{\alpha }_e{S}^{*}{E}^{*}-{\alpha }_i{S}^{*}{I}^{*}+r{R}^{*}+\Lambda =0}\hfill \\ {\displaystyle {\alpha }_e{S}^{*}{E}^{*}+{\alpha }_i{S}^{*}{I}^{*}-K{E}^{*}-\phi E^{*}=0}\hfill \\ {\displaystyle K{E}^{*}-\beta I^{*}-\mu I^{*}=0}\hfill \\ {\displaystyle \beta I^{*}+\phi E^{*}-r{R}^{*}=0}\hfill \end{array}$$ (0.1)

System (3.3) should be

$$\begin{array}{ccc}\hfill I^{*}& =& \frac{K}{\beta +\mu }{E}^{*}\hfill \\ \hfill R^{*}& =& \frac{1}{r}(\frac{\beta K}{\beta +\mu }+\phi )E^{*}\hfill \\ \hfill S^{*}& =& \frac{K+\phi }{{\alpha }_e+\frac{{\alpha }_{i}K}{\beta +\mu }}\hfill \end{array}$$ (0.2)

Equation (3.4) should be

$$E^{*}=\frac{\Lambda }{({\alpha }_i+{\alpha }_e)\left(\frac{K+\phi }{{\alpha }_e+\frac{{\alpha }_{i}K}{\beta +\mu }}\right)-(\frac{\beta K}{\beta +\mu }+\phi )}$$ (0.3)

Equation (3.6) should be

$$R^{*}=\frac{1}{r}(\frac{\beta K}{\beta +\mu }+\phi )\frac{\Lambda }{({\alpha }_i+{\alpha }_e)\left(\frac{K+\phi }{{\alpha }_e+\frac{{\alpha }_{i}K}{\beta +\mu }}\right)-(\frac{\beta K}{\beta +\mu }+\phi )}$$ (0.4)

Also equation (3.5), (3.7) and (3.8) the term $\left(\frac{K+\phi }{{\alpha }_i+\frac{{\alpha }_{i}K}{\beta +\mu }}\right)$ should be $\left(\frac{K+\phi }{{\alpha }_e+\frac{{\alpha }_{i}K}{\beta +\mu }}\right)$

In the second part

$$V=\left(\begin{array}{cc}K+\phi & 0\\ -K& \beta +\mu \end{array}\right)$$ (0.5)

Then

$$F{V}^{-1}=\left(\begin{array}{cc}{\alpha }_{e}S& {\alpha }_{i}S\\ 0& 0\end{array}\right)\left(\begin{array}{cc}{\displaystyle \frac{\beta +\mu }{(\beta +\mu )(K+\phi )}}& 0\\ {\displaystyle \frac{K}{(\beta +\mu )(K+\phi )}}& {\displaystyle \frac{K+\phi }{(\beta +\mu )(K+\phi )}}\end{array}\right)$$ (0.6)

Equation (3.23) is R ₀ > 1.

Equation (3.24) should be

$$J=\left(\begin{array}{cccc}-{\alpha }_{e}E-{\alpha }_{i}I& -{\alpha }_{e}S& -{\alpha }_{i}S& r\\ {\alpha }_{e}E+{\alpha }_{i}I& -K-\phi +{\alpha }_{e}S& {\alpha }_{i}S& 0\\ 0& K& -\beta -\mu & 0\\ 0& \phi & \beta & -r\end{array}\right)$$ (0.7)

such that Equation (3.25) is

$$J\left(E_0\right)=\left(\begin{array}{cccc}0& 0& 0& r\\ 0& -K-\phi & 0& 0\\ 0& K& -\beta -\mu & 0\\ 0& \phi & \beta & -r\end{array}\right)$$ (0.8)

In section 5 equation (5.1) should be

$$J=\left(\begin{array}{cccc}{\displaystyle -{\alpha }_e{E}^{*}-{\alpha }_i{I}^{*}}& {\displaystyle -{\alpha }_e{S}^{*}}& {\displaystyle -{\alpha }_i{S}^{*}}& {\displaystyle r}\\ {\displaystyle {\alpha }_e{E}^{*}+{\alpha }_i{I}^{*}}& {\displaystyle -K-\phi +{\alpha }_e{S}^{*}}& {\displaystyle {\alpha }_i{S}^{*}}& {\displaystyle 0}\\ {\displaystyle 0}& {\displaystyle K}& {\displaystyle -\beta -\mu }& {\displaystyle 0}\\ {\displaystyle 0}& {\displaystyle \phi }& {\displaystyle \beta }& {\displaystyle -r}\end{array}\right)$$ (0.9)

Equation (5.3) should be

$$det\left(\begin{array}{cccc}{\displaystyle \lambda +{\delta }_1}& {\displaystyle {\alpha }_e{S}^{*}}& {\displaystyle {\alpha }_i{S}^{*}}& {\displaystyle -r}\\ {\displaystyle -{\delta }_1}& {\displaystyle \lambda +{\delta }_2}& {\displaystyle -{\alpha }_i{S}^{*}}& {\displaystyle 0}\\ {\displaystyle 0}& {\displaystyle -K}& {\displaystyle \lambda +{\delta }_3}& {\displaystyle 0}\\ {\displaystyle 0}& {\displaystyle -\phi }& {\displaystyle -\beta }& {\displaystyle \lambda +r}\end{array}\right)=0$$ (0.10)

where

$${\delta }_1={\alpha }_e{E}^{*}+{\alpha }_i{I}^{*}$$

$${\delta }_2=K+\phi -{\alpha }_e{S}^{*}$$

$${\delta }_3=\beta +\mu$$

Thus the characteristic polynomial should be

$$P_{J{E}^{*}}={\lambda }_4+a_1{\lambda }_3+a_2{\lambda }_2+a_3{\lambda }_1+a_4$$

The descriptions of parameters for proposed model are follows:

α_e= is the rate at which susceptible population becomes exposed

α_i= rate at which susceptible population become infected

Λ=present the recruitment rate

K=rate at which exposed become infected

β=recovered rate

μ=death rate due to covid-19

r= relapsed rate

φ=recovered rate for exposed class.

The author would be like to apologize to readers for any inconvenience caused.

Declaration of Competing Interest

The authors declare that they do not have any financial or non- financial conflict of interests.