Open AccessCorrection

Correction: Liu, P., et al. Threshold Analysis and Stationary Distribution of a Stochastic Model with Relapse and Temporary Immunity. Symmetry 2020, 12, 331

Peng Liu

Xinzhu Meng

and

Haokun Qi

College of Mathematics and Systems Science, Shandong University of Science and Technology, Qingdao 266590, China

Author to whom correspondence should be addressed.

Symmetry 2020, 12(4), 629; https://doi.org/10.3390/sym12040629

Submission received: 7 April 2020 / Accepted: 8 April 2020 / Published: 15 April 2020

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The authors wish to make the following corrections and explanations to this paper [1]:

(1): The authors in [2] used the integral Markov semigroup theory to prove the existence of a stationary distribution of a stochastic system, thus paper [1] should cite [2] on page 11, lines 5–6, although the article [2] has been previously cited in [1] as Reference 14. Consequently, the authors wish to correct “To handle this situation, we apply the integral Markov semigroup theory, presented in [24,25].” to “To handle this situation, we apply the integral Markov semigroup theory, presented in [14,24,25].”
(2): We have found two inadvertent errors on page 15, lines 2 and 5 in the paper [1]. We would like to make the correction: $L V_{a} (I, R)$ should be corrected to $L V_{2} (I, R)$ , and $1 - R_{S}^{0}$ should be corrected to $1 - \frac{1}{R_{S}^{0}}$ .
(3): We explain the differences between articles [1,2] which has been cited in [1] as Reference 14.

1. Different Models

The model of article [1] is:

{\begin{cases} d S = [μ - μ S - p β_{1} S I - (1 - p) \frac{β_{2} S I}{1 + m S + n I} + γ_{2} R] d t - p σ_{1} S I d B_{1} (t) - (1 - p) \frac{σ_{2} S I}{1 + m S + n I} d B_{2} (t), \\ d I = [p β_{1} S I + (1 - p) \frac{β_{2} S I}{1 + m S + n I} - (μ + η) I + γ_{1} R] d t + p σ_{1} S I d B_{1} (t) + (1 - p) \frac{σ_{2} S I}{1 + m S + n I} d B_{2} (t), \\ d R = [η I - (μ + γ_{1} + γ_{2}) R] d t . \end{cases}

(1)

The model of article [2] is:

{\begin{cases} d S = (μ - μ_{1} S - β S f (I) + ε I + γ R) d t - σ S f (I) d B (t), \\ d I = [β S f (I) - (μ_{2} + ε + λ) I + σ R] d t + σ S f (I) d B (t), \\ d R = (λ I - (μ_{3} + γ + σ) R) d t . \end{cases}

(2)

The main difference between the two models is that (i) the model (1) mainly considers two different infectious forms, namely bilinear incidence rate and Beddington–DeAngelis (BD) saturated incidence rate, and the BD incidence rate here is different from the general incidence rate in [2], and

β S f (I)

the incidence rate considered in Reference 14 cannot include the BD saturated incidence rate in [1]

(1 - p) \frac{β_{2} S I}{1 + m S + n I}

, because

1 + m S + n I,

the denominator of BD saturated incidence rate contains

S

and

I .

(ii) we also consider two different kinds of white noise interference, namely,

p σ_{1} S I d B_{1} (t)

and

(1 - p) \frac{σ_{2} S I}{1 + m S + n I} d B_{2}

, which are different from one white noise interference in [2],

σ S f (I) d B (t)

2. The Different Purposes of Researches

The main purpose of our article [1] is to study the comprehensive effects of two different infection rate functions and two different kinds of white noise interference on stochastic dynamics. Therefore, it also brings some difficulties to the proof and analysis of our theorems, which is also the main purpose of our article.

3. The Different Specific Techniques for the Proof of Theorems

The specific research techniques of the two papers are different due to the different models. Especially in the proving process of theorems 1–7, the constructed function

V

and the differential of the

V

function

d V

are different. Moreover, in the mathematical derivation process, in order to deal with the nonlinear terms of BD saturated incidence rate and white noise interference, some differential inequality techniques are required.

4. The Different Conditions of the Theorems and Results

The conditions of the theorems in the two articles [1,2] are also different, especially the thresholds

R_{0}

and

R_{S}^{0}

The thresholds

R_{0}

and

R_{S}^{0}

in article [2] are

R_{0} = \frac{β μ f^{'} (0)}{μ_{1} (μ_{2} + ε + λ - \frac{σ λ}{μ_{3} + γ + σ})}

(3)

R_{S}^{0} = R_{0} - \frac{ρ^{2} {(\frac{μ}{μ_{1}} f^{'} (0))}^{2}}{2 (μ_{2} + ε + λ - \frac{σ λ}{μ_{3} + γ + σ})} .

(4)

The thresholds

R_{0}

and

R_{S}^{0}

in article [1] are

R_{0} = \frac{β_{1} p + β_{2} (1 - p)}{μ + η + \frac{1}{2} σ_{1}^{2} p^{2} + \frac{1}{2} σ_{2}^{2} {(1 - p)}^{2} - \frac{γ_{1} η}{μ + γ_{1} + γ_{2}}},

(5)

R_{S}^{0} = \frac{p β_{1} + \frac{(1 - p) β_{2}}{1 + m + n}}{μ + η + \frac{1}{2} σ_{1}^{2} p^{2} + \frac{1}{2} σ_{2}^{2} {(1 - p)}^{2} - \frac{γ_{1} η}{μ + γ_{1} + γ_{2}}} .

(6)

The thresholds

R_{0}

and

R_{S}^{0}

in article [1] imply the comprehensive effects of two different infection rate functions and two different kinds of white noise interference on stochastic dynamics.

Both articles [1,2] used Ito’s formula and semigroup theory, then the lemma, some mathematical symbols, mathematical formulas and some steps may be similar and identical. Therefore, we gained some knowledge from [2] as Reference 14 of [1], and cited it. Here, we thank the authors of all References, especially [2].

The authors would like to apologize for any inconvenience caused to the readers by these changes.

References

Liu, P.; Meng, X.; Qi, H. Threshold Analysis and Stationary Distribution of a Stochastic Model with Relapse and Temporary Immunity. Symmetry 2020, 12, 331. [Google Scholar] [CrossRef] [Green Version]
Fatini, M.E.; Khalifi, M.E.; Gerlach, R.; Laaribi, A.; Taki, R. Stationary distribution and threshold dynamics of a stochastic SIRS model with a general incidence. Phys. A 2019, 534, 120696. [Google Scholar] [CrossRef]

© 2020 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (http://creativecommons.org/licenses/by/4.0/).

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MDPI and ACS Style

Liu, P.; Meng, X.; Qi, H. Correction: Liu, P., et al. Threshold Analysis and Stationary Distribution of a Stochastic Model with Relapse and Temporary Immunity. Symmetry 2020, 12, 331. Symmetry 2020, 12, 629. https://doi.org/10.3390/sym12040629

AMA Style

Liu P, Meng X, Qi H. Correction: Liu, P., et al. Threshold Analysis and Stationary Distribution of a Stochastic Model with Relapse and Temporary Immunity. Symmetry 2020, 12, 331. Symmetry. 2020; 12(4):629. https://doi.org/10.3390/sym12040629

Chicago/Turabian Style

Liu, Peng, Xinzhu Meng, and Haokun Qi. 2020. "Correction: Liu, P., et al. Threshold Analysis and Stationary Distribution of a Stochastic Model with Relapse and Temporary Immunity. Symmetry 2020, 12, 331" Symmetry 12, no. 4: 629. https://doi.org/10.3390/sym12040629

APA Style

Liu, P., Meng, X., & Qi, H. (2020). Correction: Liu, P., et al. Threshold Analysis and Stationary Distribution of a Stochastic Model with Relapse and Temporary Immunity. Symmetry 2020, 12, 331. Symmetry, 12(4), 629. https://doi.org/10.3390/sym12040629

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Correction: Liu, P., et al. Threshold Analysis and Stationary Distribution of a Stochastic Model with Relapse and Temporary Immunity. Symmetry 2020, 12, 331

1. Different Models

2. The Different Purposes of Researches

3. The Different Specific Techniques for the Proof of Theorems

4. The Different Conditions of the Theorems and Results

References

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