Correction to: Studies on the basic reproduction number in stochastic epidemic models with random perturbations

1 Correction

Following publication of the original article [1], some errors in the equations were found in the article due to the typesetting mistakes:

On page 7,

$$\begin{aligned} \mathcal{L} \bigl( W ( t ) \bigr) =f^{T} \frac{\partial V}{\partial x}+ \frac{406}{2}g^{T} \frac{\partial V}{\partial x}g\leq \lambda _{1}a(t)+\lambda _{2}b(t)+\lambda _{3}c(t) \end{aligned}$$

should be

$$\begin{aligned} \mathcal{L} \bigl( W ( t ) \bigr) =f^{T} \frac{\partial V}{\partial x}+ \frac{1}{2}g^{T} \frac{\partial V}{\partial x}g\leq \lambda _{1}a(t)+\lambda _{2}b(t)+\lambda _{3}c(t). \end{aligned}$$

On page 12,

$$\begin{aligned} \lim_{l\rightarrow +\infty } B ( l ) e^{- ( \mu +\gamma ) l}=B ( 0 ) - ( \mu +\gamma ) \int_{0}^{{+\infty } }B ( a ) e^{- ( \mu +\gamma ) a}\,da+ \int_{0}{^{+\infty } }e^{- ( \mu + \gamma ) a}\,dB ( a ) \end{aligned}$$

should be

$$\begin{aligned} \lim_{l\rightarrow +\infty } B ( l ) e^{- ( \mu +\gamma ) l}=B ( 0 ) - ( \mu +\gamma ) \int_{0}^{{+\infty } }B ( a ) e^{- ( \mu +\gamma ) a}\,da+ \int_{0}^{{+\infty } }e^{- ( \mu + \gamma ) a}\,dB ( a ). \end{aligned}$$

On page 15,

$$\begin{aligned} \frac{\sigma Z_{\alpha /2}+\bar{R}\sqrt{2n}\beta \sqrt{ ( \mu +\upsilon ) ( \mu +\gamma ) }}{\bar{R}\sqrt{2n}\beta \sqrt{ ( \mu +\upsilon ) ( \mu +\gamma ) }}< \frac{945}{R_{0}^{\mathrm{SIR}}}< \frac{\sigma Z_{1-\alpha /2}+\bar{R}\sqrt{2n}\beta \sqrt{ ( \mu +\upsilon ) ( \mu +\gamma ) }}{\bar{R}\sqrt{2n}\beta \sqrt{ ( \mu +\upsilon ) ( \mu +\gamma ) }} \end{aligned}$$

should be

$$\begin{aligned} \frac{\sigma Z_{\alpha /2}+\bar{R}\sqrt{2n}\beta \sqrt{ ( \mu +\upsilon ) ( \mu +\gamma ) }}{\bar{R}\sqrt{2n}\beta \sqrt{ ( \mu +\upsilon ) ( \mu +\gamma ) }}< \frac{1 }{R_{0}^{\mathrm{SIR}}}< \frac{\sigma Z_{1-\alpha /2}+\bar{R}\sqrt{2n}\beta \sqrt{ ( \mu +\upsilon ) ( \mu +\gamma ) }}{\bar{R}\sqrt{2n}\beta \sqrt{ ( \mu +\upsilon ) ( \mu +\gamma ) }}. \end{aligned}$$

On page 18,

$$\begin{aligned} \frac{}{ ( \mu +\upsilon ) ( \mu +\gamma )} \biggl( \frac{\eta N}{2 \mu } + \frac{1 }{\sigma g } \biggr) \leq E \bigl( R_{0,v}^{\mathrm{SEIRS}} \bigr) \leq \frac{}{ ( \mu +\upsilon ) ( \mu +\gamma )} \biggl( \frac{\eta N}{2 \mu } + \frac{1 }{\sigma l } \biggr) \end{aligned}$$

(4.4)

should be

$$\begin{aligned} \frac{\upsilon \beta }{ ( \mu +\upsilon ) ( \mu +\gamma )} \biggl( \frac{\eta N}{2 \mu } + \frac{1 }{\sigma g } \biggr) \leq E \bigl( R_{0,v}^{\mathrm{SEIRS}} \bigr) \leq \frac{\upsilon \beta }{ ( \mu +\upsilon ) ( \mu +\gamma )} \biggl( \frac{\eta N}{2 \mu } + \frac{1 }{\sigma l } \biggr). \end{aligned}$$

(4.4)

On page 21,

$$\begin{aligned} & \frac{\partial W}{\partial I ( t )}=\lambda _{2} \bigl( \upsilon E+ ( \mu +\upsilon ) I \bigr) y \frac{\partial W}{\partial R ( t )}= \lambda _{4}R \end{aligned}$$

should be

$$\begin{aligned} & \frac{\partial W}{\partial I ( t )}=\lambda _{2} \bigl( \upsilon E+ ( \mu +\upsilon ) I \bigr),\qquad \frac{\partial W}{\partial R ( t )}= \lambda _{4}R. \end{aligned}$$

On page 23,

$$\begin{aligned} V \bigl( S ( t ),I ( t ),R ( t ) \bigr):=\lambda _{1} \biggl( \frac{\eta }{\mu }N-S ( t ) \biggr) ^{2}+\lambda _{2} \frac{1 }{2}I^{2} ( t ) +\lambda _{3} \frac{1428}{2}R^{2} ( t ) \end{aligned}$$

should be

$$\begin{aligned} V \bigl( S ( t ),I ( t ),R ( t ) \bigr):=\lambda _{1} \biggl( \frac{\eta }{\mu }N-S ( t ) \biggr) ^{2}+\lambda _{2} \frac{1 }{2}I^{2} ( t ) +\lambda _{3} \frac{1}{2}R^{2} ( t ). \end{aligned}$$

The publisher apologizes for the errors caused. The original paper has been updated.