Asymptotic behavior of a generalization of Bailey's simple epidemic

1971 ◽  
Vol 3 (02) ◽  
pp. 220-221
Author(s):  
George H. Weiss ◽  
Menachem Dishon
Keyword(s):  
Asymptotic Behavior ◽  
Differential Equations ◽  
Rate Constants ◽  
Deterministic Model ◽  
Stochastic Theory ◽  
Expected Number ◽  
Deterministic Theory ◽  
Mean Values ◽  
Image Position ◽  
Fixed Population

It has been shown that for many epidemic models, the stochastic theory leads to essentially the same results as the deterministic theory provided that one identifies mean values with the functions calculated from the deterministic differential equations (cf. [1]). If one considers a generalization of Bailey's simple epidemic for a fixed population of size N, represented schematically by where I refers to an infected, S refers to a susceptible, and α and β are appropriate rate constants, then it is evident that at time t = ∞, the expected number of infected individuals must be zero provided that β > 0. If x(t) denotes the number of infected at time t, then the deterministic model is summarized by

Asymptotic behavior of a generalization of Bailey's simple epidemic

1971 ◽  
Vol 3 (2) ◽  
pp. 220-221 ◽  
Author(s):  
George H. Weiss ◽  
Menachem Dishon
Keyword(s):  
Asymptotic Behavior ◽  
Differential Equations ◽  
Rate Constants ◽  
Deterministic Model ◽  
Stochastic Theory ◽  
Expected Number ◽  
Deterministic Theory ◽  
Mean Values ◽  
Image Position ◽  
Fixed Population

It has been shown that for many epidemic models, the stochastic theory leads to essentially the same results as the deterministic theory provided that one identifies mean values with the functions calculated from the deterministic differential equations (cf. [1]). If one considers a generalization of Bailey's simple epidemic for a fixed population of size N, represented schematically by where I refers to an infected, S refers to a susceptible, and α and β are appropriate rate constants, then it is evident that at time t = ∞, the expected number of infected individuals must be zero provided that β > 0. If x(t) denotes the number of infected at time t, then the deterministic model is summarized by

scholarly journals Asymptotic behavior of nonoscillatory solutions of a higher order functional differential equation

1981 ◽  
Vol 24 (1) ◽  
pp. 85-92 ◽  
Author(s):  
Hiroshi Onose
Keyword(s):  
Differential Equation ◽  
Asymptotic Behavior ◽  
Differential Equations ◽  
Functional Differential Equations ◽  
Sufficient Conditions ◽  
Nonoscillatory Solutions ◽  
Nonlinear Functional ◽  
Functional Differential ◽  
Approach Zero ◽  
Image Position

The asymptotic behavior of nonoscillatory solutions of nth order nonlinear functional differential equationsis investigated. Sufficient conditions are provided which ensure that all nonoscillatory solutions approach zero as t → ∞.

ASYMPTOTIC ANALYSIS OF OPTIMAL NESTED GROUP-TESTING PROCEDURES

2016 ◽  
Vol 30 (4) ◽  
pp. 547-552 ◽  
Author(s):  
Nabil Zaman ◽  
Nicholas Pippenger
Keyword(s):  
Asymptotic Behavior ◽  
Asymptotic Analysis ◽  
Group Testing ◽  
Fractional Part ◽  
Expected Number ◽  
Small Oscillations ◽  
Testing Procedures ◽  
Average Value ◽  
Positive Individual ◽  
Image Position

We analyze a construction for optimal nested group-testing procedures, and show that, when individuals are independently positive with probability p, the expected number of tests per positive individual, F*(p), has, as p→0, the asymptotic behavior $$F^{\ast}(p) = \log_2 {1\over p} + \log_2 \log 2 + 2 + f\left(\log_2 {1\over p} + \log_2 \log 2\right) + O(p),$$ where $$f(z) = 4\times 2^{-2^{1-\{z\}}} - \{z\} - 1,$$ and {z}=z−⌊z⌋ is the fractional part of z. The function f(z) is a periodic function (with period 1) that exhibits small oscillations (with magnitude <0.005) about an even smaller average value (<0.0005).

scholarly journals Asymptotic Solution Of Differential Equations In a Domain Containing a Regular Singular Point

1956 ◽  
Vol 8 ◽  
pp. 97-104 ◽  
Author(s):  
N. D. Kazarinoff ◽  
R. McKelvey
Keyword(s):  
Differential Equation ◽  
Asymptotic Behavior ◽  
Singular Point ◽  
Differential Equations ◽  
Asymptotic Solution ◽  
Connected Domain ◽  
Regular Singular Point ◽  
Simply Connected Domain ◽  
Image Position ◽  
Simply Connected

1. Introduction. In this paper we study the asymptotic behavior in λ of the solutions about the origin in the z-plane of the differential equation.Both the variable z and the parameter λ are complex. The coefficient P(z, λ) is assumed to be analytic and single-valued in λ at infinity and in z throughout a bounded, closed, simply connected domain D containing z = 0.

Infinite Systems of Differential Equations

1976 ◽  
Vol 28 (6) ◽  
pp. 1132-1145 ◽  
Author(s):  
J. P. McClure ◽  
R. Wong
Keyword(s):  
Asymptotic Behavior ◽  
Differential Equations ◽  
Nonlinear Equations ◽  
Asymptotic Behavior Of Solutions ◽  
Systems Of Nonlinear Equations ◽  
Infinite Systems ◽  
Systems Of Differential Equations ◽  
Constant Coefficients ◽  
Linear Differential ◽  
Image Position

In an earlier paper [7], we have studied the existence, uniqueness and asymptotic behavior of solutions to certain infinite systems of linear differential equations with constant coefficients. In the present paper we are interested in systems of nonlinear equations whose coefficients are not necessarily constants; more specifically, we are concerned with infinite systems of the form

Asymptotic behavior of solutions of half-linear differential equations and generalized Karamata functions

2020 ◽  
Vol 0 (0) ◽  
Author(s):  
Kusano Takaŝi ◽  
Jelena V. Manojlović
Keyword(s):  
Differential Equation ◽  
Asymptotic Behavior ◽  
Continuous Function ◽  
Differential Equations ◽  
Linear Differential Equation ◽  
Regular Variation ◽  
Asymptotic Formulas ◽  
Asymptotic Behavior Of Solutions ◽  
Positive Continuous Function ◽  
Linear Differential

AbstractWe study the asymptotic behavior of eventually positive solutions of the second-order half-linear differential equation(p(t)\lvert x^{\prime}\rvert^{\alpha}\operatorname{sgn}x^{\prime})^{\prime}+q(% t)\lvert x\rvert^{\alpha}\operatorname{sgn}x=0,where q is a continuous function which may take both positive and negative values in any neighborhood of infinity and p is a positive continuous function satisfying one of the conditions\int_{a}^{\infty}\frac{ds}{p(s)^{1/\alpha}}=\infty\quad\text{or}\quad\int_{a}^% {\infty}\frac{ds}{p(s)^{1/\alpha}}<\infty.The asymptotic formulas for generalized regularly varying solutions are established using the Karamata theory of regular variation.

scholarly journals New Results on Qualitative Behavior of Second Order Nonlinear Neutral Impulsive Differential Systems with Canonical and Non-Canonical Conditions

Symmetry ◽  
2021 ◽  
Vol 13 (6) ◽  
pp. 934
Author(s):  
Shyam Sundar Santra ◽  
Khaled Mohamed Khedher ◽  
Kamsing Nonlaopon ◽  
Hijaz Ahmad
Keyword(s):  
Asymptotic Behavior ◽  
Differential Equations ◽  
Sufficient Conditions ◽  
Impulsive Differential Equations ◽  
Oscillatory Behavior ◽  
Second Order ◽  
Discrete Equation ◽  
Qualitative Behavior ◽  
Differential Systems ◽  
Impulsive Differential Systems

The oscillation of impulsive differential equations plays an important role in many applications in physics, biology and engineering. The symmetry helps to deciding the right way to study oscillatory behavior of solutions of impulsive differential equations. In this work, several sufficient conditions are established for oscillatory or asymptotic behavior of second-order neutral impulsive differential systems for various ranges of the bounded neutral coefficient under the canonical and non-canonical conditions. Here, one can see that if the differential equations is oscillatory (or converges to zero asymptotically), then the discrete equation of similar type do not disturb the oscillatory or asymptotic behavior of the impulsive system, when impulse satisfies the discrete equation. Further, some illustrative examples showing applicability of the new results are included.

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