Improving Stability Conditions for Equilibria of SIR Epidemic Model with Delay under Stochastic Perturbations

Shaikhet, Leonid

doi:10.3390/math8081302

Cited by 10 publications

(11 citation statements)

References 38 publications

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“…Remark In the scalar case ( n = 1) without loss of generality in the LMIs (16) and (22) one can use P = 1. In the general case these LMIs are successfully investigated using MATLAB 27 …”

Section: Stabilitymentioning

confidence: 99%

“…Equilibria of the system (54) are defined by the system of algebraic equations

b = β S I + μ_{1} S, β S I = (μ_{2} + λ) I, λ I = μ_{3} R,

with two solutions:

E_{0} = (b μ_{1}^{- 1}, 0, 0)

and the positive equilibrium E + = ( S ∗ , I ∗ , R ∗ ), where

S^{*} = \frac{μ_{2} + λ}{β} < \frac{b}{μ_{1}}, I^{*} = \frac{b {(S^{*})}^{- 1} - μ_{1}}{β}, R^{*} = \frac{λ I^{*}}{μ_{3}} .

Previously stability conditions for the SIR epidemic model (54) were investigated in References [1,25]. Stability conditions significantly improved by virtue of the proposed here method, including detail investigation of LMI of the type (16) by virtue of MATLAB, are presented in Reference [27].…”

Section: Application To Known Mathematical Modelsmentioning

confidence: 99%

“…It is shown that the investigated equation includes a lot of different very popular in research mathematical models such as Mackey–Glass model, 2‐4 neoclassical growth model, 5‐11 Nicholson's blowflies equation, 4,12‐17 mosquito 18,19 and glassy‐winged sharpshooter populations, 20,21 e‐commerce model, 22‐24 SIR epidemic model, 1,25‐27 some social epidemics models, 1,28,29 model of viral dynamics, 30,31 and many similar others which can be considered as particular cases of the investigated here nonlinear equation.…”

Section: Introductionmentioning

confidence: 99%

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About one method of stability investigation for nonlinear stochastic delay differential equations

Shaikhet

2021

Intl J Robust & Nonlinear

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Stability of equilibria of a nonlinear delay differential equation is investigated under stochastic perturbations. The use of Stiltjes integrals allows to consider both discrete and distributed delays in the investigated equation. Obtained conditions of stability in probability are formulated in terms of linear matrix inequalities. The proposed method of investigation can be applied to a number of very popular in research mathematical models such as Mackey–Glass model, neoclassical growth model and e‐commerce model, Nicholson's blowflies equation, mosquito and glassy‐winged sharpshooter populations, SIR epidemic model, social epidemic models, and many similar other mathematical models which can be considered as particular cases of the investigated nonlinear equation.

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“…Remark In the scalar case ( n = 1) without loss of generality in the LMIs (16) and (22) one can use P = 1. In the general case these LMIs are successfully investigated using MATLAB 27 …”

Section: Stabilitymentioning

confidence: 99%

“…Equilibria of the system (54) are defined by the system of algebraic equations

b = β S I + μ_{1} S, β S I = (μ_{2} + λ) I, λ I = μ_{3} R,

with two solutions:

E_{0} = (b μ_{1}^{- 1}, 0, 0)

and the positive equilibrium E + = ( S ∗ , I ∗ , R ∗ ), where

S^{*} = \frac{μ_{2} + λ}{β} < \frac{b}{μ_{1}}, I^{*} = \frac{b {(S^{*})}^{- 1} - μ_{1}}{β}, R^{*} = \frac{λ I^{*}}{μ_{3}} .

Section: Application To Known Mathematical Modelsmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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About one method of stability investigation for nonlinear stochastic delay differential equations

Shaikhet

2021

Intl J Robust & Nonlinear

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“…It is of interest to extend the above results in the future to the stochastic context by considering perturbations in the equilibrium points and uncertainties in the data. See, for instance, [31,32].…”

Section: Numerical Examplesmentioning

confidence: 99%

“…It has to be pointed out that the stochastic formalism has been also seen to be useful in studying some concerns in the epidemic models behavior, typically, the influences of perturbations in the equilibrium points and uncertainties in the data. See, for instance, [31,32] and some references therein. Moreover, the relevance of modeling small communities can be useful for interpreting the disease evolution or taking intervention decisions since the total population is an aggregation of smaller communities [33].…”

Section: Introductionmentioning

confidence: 99%

About Partial Reachability Issues in an SEIR Epidemic Model and Related Infectious Disease Tracking in Finite Time under Vaccination and Treatment Controls

Sen

Ibeas

Nistal

2021

Discrete Dynamics in Nature and Society

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This paper studies some basic properties of an SEIR (Susceptible-Exposed-Infectious-Recovered) epidemic model subject to vaccination and treatment controls. Firstly, the basic stability, boundedness, and nonnegativity of the state trajectory solution are investigated. Then, the problem of partial state reachability from a certain state value to a targeted one in finite time is focused on since it turns out that epidemic models are, because of their nature, neither (state) controllable from a given state to the origin nor reachable from a given initial condition. The particular formal statement of the partial reachability is focused on as a problem of output-reachability by defining a measurable output or lower dimension than that of the state. A special case of interest is that when the output is defined as the infectious subpopulation to be step-to-step tracked under suitable amounts being compatible with the required constraints. As a result, and provided that the output-controllability Gramian is nonsingular on a certain time interval of interest, a feedback control effort might be designed so that a prescribed value of the output can be approximately tracked. A linearization approximation is performed to simplify and facilitate the above task which is based on a point-to-point linearization of the solution trajectory. To this end, an “ad hoc” sampled approximate output trajectory is defined as control objective to be targeted through a point-wise calculated Jacobian matrix. A supervised appropriate restatement of the targeted suited sampled output values is redefined, if necessary, to make the initial proposed sampled trajectory compatible with the various needed constraints on nonnegativity and control boundedness. The design can be optionally performed under constant or adaptive sampling rates. Finally, some numerical examples are given to test the theoretical aspects and the design efficiency of the model.

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Global stability and control strategies of a SIQRS epidemic model with time delay

Cui

Wang

2022

Math Methods in App Sciences

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Improving Stability Conditions for Equilibria of SIR Epidemic Model with Delay under Stochastic Perturbations

Cited by 10 publications

References 38 publications

About one method of stability investigation for nonlinear stochastic delay differential equations

About one method of stability investigation for nonlinear stochastic delay differential equations

About Partial Reachability Issues in an SEIR Epidemic Model and Related Infectious Disease Tracking in Finite Time under Vaccination and Treatment Controls

Global stability and control strategies of a SIQRS epidemic model with time delay

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