67,680 research outputs found
New approximations, and policy implications, from a delayed dynamic model of a fast pandemic
We study an SEIQR (Susceptible-Exposed-Infectious-Quarantined-Recovered)
model for an infectious disease, with time delays for latency and an
asymptomatic phase. For fast pandemics where nobody has prior immunity and
everyone has immunity after recovery, the SEIQR model decouples into two
nonlinear delay differential equations (DDEs) with five parameters. One
parameter is set to unity by scaling time. The subcase of perfect quarantining
and zero self-recovery before quarantine, with two free parameters, is examined
first. The method of multiple scales yields a hyperbolic tangent solution; and
a long-wave approximation yields a first order ordinary differential equation
(ODE). With imperfect quarantining and nonzero self-recovery, the long-wave
approximation is a second order ODE. These three approximations each capture
the full outbreak, from infinitesimal initiation to final saturation.
Low-dimensional dynamics in the DDEs is demonstrated using a six state
non-delayed reduced order model obtained by Galerkin projection. Numerical
solutions from the reduced order model match the DDE over a range of parameter
choices and initial conditions. Finally, stability analysis and numerics show
how correctly executed time-varying social distancing, within the present
model, can cut the number of affected people by almost half. Alternatively,
faster detection followed by near-certain quarantining can potentially be even
more effective
Stability implications of delay distribution for first-order and second-order systems
Kiss, G., & Krauskopf, B. (2009). Stability implications of delay distribution for first-order and second-order systems. Early version, also known as pre-print Link to publication record in Explore Bristol Research PDF-documen
Fast generation of stability charts for time-delay systems using continuation of characteristic roots
Many dynamic processes involve time delays, thus their dynamics are governed
by delay differential equations (DDEs). Studying the stability of dynamic
systems is critical, but analyzing the stability of time-delay systems is
challenging because DDEs are infinite-dimensional. We propose a new approach to
quickly generate stability charts for DDEs using continuation of characteristic
roots (CCR). In our CCR method, the roots of the characteristic equation of a
DDE are written as implicit functions of the parameters of interest, and the
continuation equations are derived in the form of ordinary differential
equations (ODEs). Numerical continuation is then employed to determine the
characteristic roots at all points in a parametric space; the stability of the
original DDE can then be easily determined. A key advantage of the proposed
method is that a system of linearly independent ODEs is solved rather than the
typical strategy of solving a large eigenvalue problem at each grid point in
the domain. Thus, the CCR method significantly reduces the computational effort
required to determine the stability of DDEs. As we demonstrate with several
examples, the CCR method generates highly accurate stability charts, and does
so up to 10 times faster than the Galerkin approximation method.Comment: 12 pages, 6 figure
Optimal linear stability condition for scalar differential equations with distributed delay
Linear scalar differential equations with distributed delays appear in the
study of the local stability of nonlinear differential equations with feedback,
which are common in biology and physics. Negative feedback loops tend to
promote oscillations around steady states, and their stability depends on the
particular shape of the delay distribution. Since in applications the mean
delay is often the only reliable information available about the distribution,
it is desirable to find conditions for stability that are independent from the
shape of the distribution. We show here that for a given mean delay, the linear
equation with distributed delay is asymptotically stable if the associated
differential equation with a discrete delay is asymptotically stable. We
illustrate this criterion on a compartment model of hematopoietic cell dynamics
to obtain sufficient conditions for stability
Analyzing stability of a delay differential equation involving two delays
Analysis of the systems involving delay is a popular topic among applied
scientists. In the present work, we analyze the generalized equation
involving two delays
viz. and . We use the the stability conditions to
propose the critical values of delays. Using examples, we show that the chaotic
oscillations are observed in the unstable region only. We also propose a
numerical scheme to solve such equations.Comment: 10 pages, 7 figure
Lp-calculus approach to the random autonomous linear differential equation with discrete delay
[EN] In this paper, we provide a full probabilistic study of the random autonomous linear differential equation with discrete delay , with initial condition x(t)=g(t), -t0. The coefficients a and b are assumed to be random variables, while the initial condition g(t) is taken as a stochastic process. Using Lp-calculus, we prove that, under certain conditions, the deterministic solution constructed with the method of steps that involves the delayed exponential function is an Lp-solution too. An analysis of Lp-convergence when the delay tends to 0 is also performed in detail.This work has been supported by the Spanish Ministerio de Economia y Competitividad Grant MTM2017-89664-P. The author Marc Jornet acknowledges the doctorate scholarship granted by Programa de Ayudas de Investigacion y Desarrollo (PAID), Universitat Politecnica de Valencia.Calatayud-Gregori, J.; Cortés, J.; Jornet-Sanz, M. (2019). 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