Stationary distribution and density function expression for a stochastic SIQRS epidemic model with temporary immunity

Abstract

Recently, considering the temporary immunity of individuals who have recovered from certain infectious diseases, Liu et al. (Phys A Stat Mech Appl 551:124152, 2020 ) proposed and studied a stochastic susceptible-infected-recovered-susceptible model with logistic growth. For a more realistic situation, the effects of quarantine strategies and stochasticity should be taken into account. Hence, our paper focuses on a stochastic susceptible-infected-quarantined-recovered-susceptible epidemic model with temporary immunity. First, by means of the Khas’minskii theory and Lyapunov function approach, we construct a critical value corresponding to the basic reproduction number of the deterministic system. Moreover, we prove that there is a unique ergodic stationary distribution if . Focusing on the results of Zhou et al. (Chaos Soliton Fractals 137:109865, 2020), we develop some suitable solving theories for the general four-dimensional Fokker–Planck equation. The key aim of the present study is to obtain the explicit density function expression of the stationary distribution under . It should be noted that the existence of an ergodic stationary distribution together with the unique exact probability density function can reveal all the dynamical properties of disease persistence in both epidemiological and statistical aspects. Next, some numerical simulations together with parameter analyses are shown to support our theoretical results. Last, through comparison with other articles, results are discussed and the main conclusions are highlighted.

Analysis of Stochastic SIRC Model with Cross Immunity Based on Ornstein–Uhlenbeck Process

“… If , based on the method given in the references [ 36 ], let , where the standardized transformation matrix then we have where So with conditions ( 15 ) and ( 16 ), we verify that Since the value of has no influence on the results of this paper, we omit the value of here. Equation ( 18 ) can be equivalently transformed into Let , where , it can be simplified as By solving the equation above, we get where From the reference [ 40 ], we can conclude that is a semi-positive definite matrix. In the same way, we also get that is a positive definite matrix.…”

“…Let be the minimum eigenvalue of the matrix , then and so, If , let , where the standardized transformation matrix is Using the same method, we can get where By direct calculation, we have Since the value of and has no influence on the results of this paper, we omit the value of and here. Meanwhile, equation ( 18 ) can be equivalently transformed into Let , where , we similarly obtain By solving the equation above, we have From the reference [ 40 ], we can conclude that is a semi-positive definite matrix. Hence, It is also a semi-positive definite matrix.…”

Acute threshold dynamics of an epidemic system with quarantine strategy driven by correlated white noises and Lévy jumps associated with infinite measure

“… If , then the illness dies out. Predominantly, can be rewritten as the basic reproduction number and we can compare this ratio with the number 1 to assort the large time behavior of the deterministic system ( 1.1 ) [ 6 ].…”

A SIQRB delayed model for cholera and optimal control treatment

“…With the purpose of avoiding cholera outbreaks in China, the researchers suggest to increase the immunization coverage rate and to make efforts for improving environmental management, mainly for drinking water [9]. Another important topic to have a full overview of mathematical modeling, within the scope of SIR models in biomathematics, concerns uncertainty quantification using randomized models, as, e.g., in [26,27]. With respect to cholera, one can find, e.g., the paper [22], where the authors propose and analyze a stochastic epidemic model to study such disease.…”