Abstract
In this paper, we discuss the basic reproduction number of stochastic epidemic models with random perturbations. We define the basic reproduction number in epidemic models by using the integral of a function or survival function. We study the systems of stochastic differential equations for SIR, SIS, and SEIR models and their stability analysis. Some results on deterministic epidemic models are also obtained. We give the numerical conditions for which the disease-free equilibrium point is asymptotically stable.
Highlights
Pandemics can cause sudden and drastic increases in mortality and morbidity rates as well as social, political, and economic disruptions
Our objective is to find the smallest value of RS0,IES such that RS0,IES < 1 and for which the SIS model with random perturbation is asymptotically stable on we search for the smallest value of is asymptotically stable on the SEIR model with random perturbations
We focus our attention on the simulations of the stability for the SEIR model with random perturbations which are shown for determining the numeric conditions under which the point is asymptotically stable on the model with random perturbations, for example, the values of are verified numerically
Summary
Pandemics can cause sudden and drastic increases in mortality and morbidity rates as well as social, political, and economic disruptions. We define the basic reproduction number in epidemic models by using the survival function and demonstrate the numerical conditions under which the disease-free equilibrium point is asymptotically stable. In the deterministic epidemic models, the disease-free equilibrium points are locally asymptotically stable if the reproduction number is less than unity. We establish numerical conditions for which some deterministic epidemic models are asymptotically stable on the disease-free equilibrium points (for more details, see Appendix B). For observing the stability in SIS and SEIR models with random perturbations, using adequate Lyapunov functions, we state the following theorem given in [22] without proof. We prove the following theorem by constructing a Lyapunov function and give the sufficient conditions at which the point stable in model with random perturbations. 2.2, that model with random perturbations is asymptotically stable, and it is necessary that μ + υ > 1 and inequality (2.9) hold and can be written as σ 2η2υ2N2 υβ μ(γ + μ)(υ + μ) + 2μ2(γ + μ)(υ + μ) < 1
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