Threshold of Stochastic SIRS Epidemic Model from Infectious to Susceptible Class with Saturated Incidence Rate Using Spectral Method

Abstract

Stochastic SIRS models play a key role in formulating and analyzing the transmission of infectious diseases. These models reflect the environmental changes of the diseases and their biological mechanisms. Therefore, it is very important to study the uniqueness and existence of the global positive solution to investigate the asymptotic properties of the model. In this article, we investigate the dynamics of the stochastic SIRS epidemic model with a saturated incidence rate. The effects of both deterministic and stochastic distribution from infectious to susceptible are analyzed. Our findings show that the occurrence of symmetry breaking as a function of the stochastic noise has a significant advantage over the deterministic one to prevent the spread of the infectious diseases. The larger stochastic noise will guarantee the control of epidemic diseases with symmetric Brownian motion. Periodic outbreaks and re-infection may occur due to the existence of feedback memory. It is shown that the endemic equilibrium is stable under some suitable initial conditions, taking advantage of the symmetry of the large amount of contact structure. A numerical method based on Legendre polynomials that converts the given stochastic SIRS model into a nonlinear algebraic system is used for the approximate solution. Finally, some numerical experiments are performed to verify the theoretical results and clearly show the sharpness of the obtained conditions and thresholds.

An orthogonal class of $p$-Legendre polynomials on variable interval

“…The third (infinitesimal) body is taken to be located at some point (x, y, z) in the rotating frame. Now, if r 1 is the distance between the first primary and third body and r 2 is the distance between the second primary and third body, then [21, p. 8] r 2 1 = (x + µ) 2 + y 2 + z 2 ; r 2 2 = (x − 1 + µ) 2 + y 2 + z 2 . In order to expand the nonlinear term: 1−µ r1 + µ r2 , the following formula is used [21, p. 146].…”

“…Moreover, this polynomial is used in a spectral collocation method for solving the system of nonlinear Fredholm integral equations of second kind [20]. Besides this, the Legendre polynomial occurs in a Stochastic SIRS epidemic model [2] and in determining the mean values [13]. There is a good collection of multivariate orthogonal polynomials in [16].…”

Dynamical Analysis of Two-Dimensional Fractional-Order-in-Time Biological Population Model Using Chebyshev Spectral Method

“…This property significantly minimizes the maximum error between the numerical solution and the true solution. The efficiency of this approach is further increased by the employment of Chebyshev nodes, or the roots of these polynomials, which assist prevent the Runge phenomenon and guarantee stability and dependability in numerical approximations [32][33][34][35][36][37][38][39][40]. Several studies have applied fractional calculus to model biological systems.…”

A new design and analysis of optimal control problems arising from COVID‐19 outbreak

“…In addition, public health organizations can use these optimal control models to take better steps and emergency actions to prevent infectious diseases. There are many research articles available in literature that provide basics and fundamentals of optimal control theory to dynamical systems particularly in the field of epidemiology [35][36][37][38][39][40][41][42][43][44][45][46][47][48].…”