Threshold Analysis and Stationary Distribution of a Stochastic Model with Relapse and Temporary Immunity

Abstract

In this paper, a stochastic model with relapse and temporary immunity is formulated. The main purpose of this model is to investigate the stochastic properties. For two incidence rate terms, we apply the ideas of a symmetric method to obtain the results. First, by constructing suitable stochastic Lyapunov functions, we establish sufficient conditions for the extinction and persistence of this system. Then, we investigate the existence of a stationary distribution for this model by employing the theory of an integral Markov semigroup. Finally, the numerical examples are presented to illustrate the analytical findings.

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

“…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…”

“…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).…”

“…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.…”

“…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.…”

“…(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]."…”