Mathematical Analysis and Stochastic Stability of Nonlinear Epidemic Model SIQ with Incidence Rate

This work takes into account a nonlinear epidemic model SIQ with temporary immunity and a saturated incidence rate.

At any given time, t, there are three subclasses for the size N (t).

N(t) is defined as the sum of S(t)+I(t)+Q(t); where S(t), I(t), and Q(t) denote the populations that are susceptible to disease, infectious, and quarantined with temporary immunity, respectively.

We have made contributions as follows:

Both the infection-free equilibrium and the endemic equilibrium are analyzed for their local stabilities. The basic reproductive number ratio is used to determine the stability of a disease-free equilibrium and the existence of other nontrivial equilibria.
The reduce model that substitutes S with N is the subject of this paper, and it doesn't possess nontrivial periodic orbits with conditions.
Under certain conditions, the endemic disease point is globally asymptotically stable, and we will explore some properties of equilibrium with theorems.
The stochastic stabilities are finally confirmed through the use of certain theorems and their proofs with Ω and S(t), which are almost surely.
We have utilized diverse references from various studies, particularly the writing of the non-linear epidemic mathematical model with (Abta et al., 2012; El Mroufy et al., 2011; Anderson et al., 1986; Øksendal, 2000; Lefschetz & LaSalle, 1961; Li & Hyman, 2009; Gao et al., 2024; Xiao & Chen, 2001; Zhang & Luo, 2024).

To study the various stability and other sections, we have utilized other sources, such as (Bailey, 1977; Batiha et al., 2008; Billard, 1976; Xiao & Ruan, 2007; Jin et al., 2006; Jinliang & Tian, 2013; Lahrouz et al., 2011; Lakshmikantham et al., 1989); and (Lounes & De Arazoza, 2002; Steele, 2001; Sandip & Omar, 2010; Perko, 2001; Zou et al., 2009; Pathak et al., 2010; Ma et al., 2002; Wang, 2002; Watson, 1980; Luo & Mao, 2007; Wen & Yang, 2008); and sometimes the previous references.