Mathematical Analysis and Stochastic Stability of Nonlinear Epidemic Model SIQ with Incidence Rate
1Mathematical Analysis and Stochastic Stability of Nonlinear Epidemic Model SIQ with Incidence Rate
Layman Abstract : This study examines a mathematical model for the spread of a disease, called the SIQ model, which includes temporary immunity and a saturated infection rate (meaning the disease spreads at a controlled rate when cases get too high). The total population at any time consists of three groups:
S(t): People who are susceptible to infection.
I(t): People who are infected and can spread the disease.
Q(t): People who are quarantined and have temporary immunity after recovery.
The study makes several key contributions:
It examines two main scenarios: one where the disease eventually disappears (infection-free equilibrium) and one where it continues to exist in the population (endemic equilibrium). Stability is determined using the basic reproduction number, which measures how easily the disease spreads.
A simplified version of the model is explored, which removes unnecessary complexities while still giving accurate results.
It proves that under certain conditions, the disease will eventually settle into a predictable state rather than fluctuating wildly.
The study confirms the model’s accuracy using stochastic stability tests, ensuring that the results hold even when real-world randomness is factored in.
By analyzing the stability of different disease scenarios, this study provides insights into how diseases behave over time, which can help in public health planning and disease control strategies.
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Original Abstract : 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:
1. 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.
2. The reduce model that substitutes S with N is the subject of this paper, and it doesn't possess nontrivial periodic orbits with conditions.
3. Under certain conditions, the endemic disease point is globally asymptotically stable, and we will explore some properties of equilibrium with theorems.
4. 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.
View Book: https://doi.org/10.9734/bpi/mcsru/v3/3548
#Local_and_global_stability #stochastic_stability #incidence_rate #nonlinear_epidemic_model