Stationary distribution and threshold dynamics of a stochastic SIRS model with a general incidence

Fatini, Mohamed El; Khalifi, Mohamed El; Gerlach, Richard; Laaribi, Aziz; Taki, Regragui

doi:10.1016/j.physa.2019.03.061

Cited by 27 publications

(26 citation statements)

References 21 publications

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“…The conditions of the theorems in the two articles [1,2] are also different, especially the thresholds R 0 and R 0 S . The thresholds R 0 and R 0 S in article [2] are…”

Section: The Different Conditions Of the Theorems And Resultsmentioning

confidence: 99%

“…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].…”

Section: The Different Conditions Of the Theorems And Resultsmentioning

confidence: 99%

“…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].…”

Section: The Different Conditions Of the Theorems And Resultsmentioning

confidence: 99%

“…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 βS f (I) the incidence rate considered in Reference 14 cannot include the BD saturated incidence rate in [1] (1 − p) β 2 SI 1+mS+nI , because 1 + mS + nI, the denominator of BD saturated incidence rate contains S and I. (ii) we also consider two different kinds of white noise interference, namely, pσ 1 SIdB 1 (t) and (1 − p) σ 2 SI 1+mS+nI dB 2 , which are different from one white noise interference in [2], σS f (I)dB(t).…”

Section: Different Modelsmentioning

confidence: 99%

“…(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]."…”

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confidence: 99%

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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

Liu

Meng

2020

Symmetry

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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: LV a (I, R) should be corrected to LV 2 (I, R), and 1 − R 0 S should be corrected to(3) We explain the differences between articles [1,2] which has been cited in [1] as Reference 14. Different ModelsThe model of article [1] is:(1)The model of article [2] is: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 βS f (I) the incidence rate considered in Reference 14 cannot include the BD saturated incidence rate in [1] (1 − p) β 2 SI 1+mS+nI , because 1 + mS + nI, the denominator of BD saturated incidence rate contains S and I. (ii) we also consider two different kinds of white noise interference, namely, pσ 1 SIdB 1 (t) and (1 − p) σ 2 SI 1+mS+nI dB 2 , which are different from one white noise interference in [2], σS f (I)dB(t).

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“…The conditions of the theorems in the two articles [1,2] are also different, especially the thresholds R 0 and R 0 S . The thresholds R 0 and R 0 S in article [2] are…”

Section: The Different Conditions Of the Theorems And Resultsmentioning

confidence: 99%

Section: The Different Conditions Of the Theorems And Resultsmentioning

confidence: 99%

“…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].…”

Section: The Different Conditions Of the Theorems And Resultsmentioning

confidence: 99%

Section: Different Modelsmentioning

confidence: 99%

mentioning

confidence: 99%

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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

Liu

Meng

2020

Symmetry

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The effect of stochasticity with respect to reinfection and nonlinear transition states for some diseases with relapse

Fatini

Khalifi

Lahrouz

et al. 2020

Math Methods in App Sciences

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In this paper, we consider a stochastic epidemic model with relapse, reinfection, and a general incidence function. Using stochastic tools, we establish a stochastic threshold  0 s and prove the extinction of the disease when its value is equal or less than unity. We also show the persistence in mean of the disease when  0 s > 1. Moreover, we prove the existence and uniqueness of a stationary distribution. Finally, numerical simulations are presented to show the effectiveness of theoretical results.

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Dynamics of stochastic HTLV‐I infection model with nonlinear CTL immune response

Kuang

Yin

2021

Math Methods in App Sciences

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In this paper, the dynamics of stochastic human T-cell leukemia virus type I (HTLV-I) infection model with cytotoxic T lymphocyte (CTL) immune response is investigated. First, we show that the stochastic model exists as a unique positive global solution originating from the positive initial value. Second, we demonstrate that the stochastic model is stochastically permanent and stochastically ultimately bounded for any positive initial value. Third, we establish sufficient conditions for the existence of ergodic stationary distribution of the stochastic model. Fourth, the threshold R * 0 between extinction and persistence of the virus is obtained. Finally, numerical simulations are carried out to illustrate the theoretical results.

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Stationary distribution and threshold dynamics of a stochastic SIRS model with a general incidence

Cited by 27 publications

References 21 publications

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

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

The effect of stochasticity with respect to reinfection and nonlinear transition states for some diseases with relapse

Dynamics of stochastic HTLV‐I infection model with nonlinear CTL immune response

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