arXiv subjects

Yukihiko Nakata

Publications and source records attributed to Yukihiko Nakata.

8 recordsLinked to original sources

arXiv2025

Period two solution for a class of distributed delay differential equations

We study the existence of a periodic solution for a differential equation with distributed delay. It is shown that, for a class of distributed delay diferential quations, a symmetric period 2 solution, where the period is twice the maximum delay, is given as a periodic solution of a Hamiltonian system of ordinary differential equations. Proof of the idea is based on (Kaplan & Yorke, 1974, J. Math. Anal. Appl.) for a discrete delay differential equation with an odd nonlinear function. To illustrate the results, we present distributed delay differential equations that have periodic solutions expressed in terms of the Jacobi elliptic functions.

math.DS↗

arXiv2021

Delay-induced blow-up in a planar oscillation model

In this paper we study a system of delay differential equations from the viewpoint of a finite time blow-up of the solution. We prove that the system admits a blow-up solution, no matter how small the length of the delay is. In the non-delay system every solution approaches to a stable unit circle in the plane, thus time delay induces blow-up of solutions, which we call "delay-induced blow-up" phenomenon. Furthermore, it is shown that the system has a family of infinitely many periodic solutions, while the non-delay system has only one stable limit cycle. The system studied in this paper is an example that arbitrary small delay can be responsible for a drastic change of the dynamics. We show numerical examples to illustrate our theoretical results.

math.DS↗

arXiv2018

$\mathcal{R}_{0}$ fails to predict the outbreak potential in the presence of natural-boosting immunity

Time varying susceptibility of host at individual level due to waning and boosting immunity is known to induce rich long-term behavior of disease transmission dynamics. Meanwhile, the impact of the time varying heterogeneity of host susceptibility on the shot-term behavior of epidemics is not well-studied, even though the large amount of the available epidemiological data are the short-term epidemics. Here we constructed a parsimonious mathematical model describing the short-term transmission dynamics taking into account natural-boosting immunity by reinfection, and obtained the explicit solution for our model. We found that our system show "the delayed epidemic", the epidemic takes off after negative slope of the epidemic curve at the initial phase of epidemic, in addition to the common classification in the standard SIR model, i.e., "no epidemic" as $\mathcal{R}_{0}\leq1$ or normal epidemic as $\mathcal{R}_{0}>1$. Employing the explicit solution we derived the condition for each classification.

q-bio.PE↗

arXiv2018

A periodic solution of period two of a delay differential equation

In this paper we prove that the following delay differential equation \[ \frac{d}{dt}x(t)=rx(t)\left(1-\int_{0}^{1}x(t-s)ds\right), \] has a periodic solution of period two for $r>\frac{π^{2}}{2}$ (when the steady state, $x=1$, is unstable). In order to find the periodic solution, we study an integrable system of ordinary differential equations, following the idea by Kaplan and Yorke \cite{Kaplan=000026Yorke:1974}. The periodic solution is expressed in terms of the Jacobi elliptic functions.

math.DS↗

arXiv2017

Unbounded and blow-up solutions for a delay logistic equation with positive feedback

We study bounded, unbounded and blow-up solutions of a delay logistic equation without assuming the dominance of the instantaneous feedback. It is shown that there can exist an exponential (thus unbounded) solution for the nonlinear problem, and in this case the positive equilibrium is always unstable. We obtain a necessary and sufficient condition for the existence of blow-up solutions, and characterize a wide class of such solutions. There is a parameter set such that the non-trivial equilibrium is locally stable but not globally stable due to the co-existence with blow-up solutions.

math.DS↗

arXiv2017

Global stability of an SIS epidemic model with a finite infectious period

Assuming a general distribution for the sojourn time in the in- fectious class, we consider an SIS type epidemic model formulated as a scalar integral equation. We prove that the endemic equilibrium of the model is globally asymptotically stable whenever it exists, solving the conjecture of Hethcote and van den Driessche (1995) for the case of nonfatal diseases.

math.DS↗

arXiv2010

Global dynamics of cell mediated immunity in viral infection models with distributed delays

In this paper, we investigate global dynamics for a system of delay differential equations which describes a virus-immune interaction in \textit{vivo}. The model has two distributed time delays describing time needed for infection of cell and virus replication. Our model admits three possible equilibria, an uninfected equilibrium and infected equilibrium with or without immune response depending on the basic reproduction number for viral infection $R_{0}$ and for CTL response $R_{1}$ such that $R_{1} 1$. The immune activation has a positive role in the reduction of the infection cells and the increasing of the uninfected cells if $R_{1}>1$.

math.DS↗

arXiv2010

On a generalization of the global attractivity for a periodically forced Pielou's equation

In this paper, we study the global attractivity for a class of periodic difference equation with delay which has a generalized form of Pielou's difference equation. The global dynamics of the equation is characterized by using a relation between the upper and lower limit of the solution. There are two possible global dynamics: zero solution is globally attractive or there exists a periodic solution which is globally attractive. Recent results in [E. Camouzis, G. Ladas, Periodically forced Pielou's equation, J. Math. Anal. Appl. 333 (1) (2007) 117-127] is generalized. Two examples are given to illustrate our results.

math.DS↗