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Session S12 - Delay and functional differential equations and applications

Monday, July 12, 14:15 ~ 14:50 UTC-3

On almost periodic solutions for a model of hematopoiesis with an oscillatory circulation loss rate

Rocio Celeste Balderrama

Universidad de Buenos Aires, Argentina   -   This email address is being protected from spambots. You need JavaScript enabled to view it.

The following nonlinear autonomous delay differential equation was proposed by Mackey and Glass to describe the regulation of hematopoiesis, namely, \begin{equation} \label{MG0} \frac{dP(t)}{dt}=\frac{\lambda \theta^{n}P(t-\tau)}{\theta^n + P^{n}(t-\tau)}-\gamma P(t)\, \hspace{1cm} \hbox{ where } \lambda, \theta, n, \gamma, \tau\in(0,+\infty).\quad (1) \end{equation} Here $P(t)$ is the concentration of cells in the circulating blood and the flux function $f(v)=\frac{\lambda \theta^{n}v }{\theta^n + v^{n}}$ of cells into the blood stream from the stem cell compartment depends on the cell concentration at an earlier time. The delay $\tau$ describes the time between the start of cellular production in the bone marrow and their maturation for release in circulating bloodstream. It is assumed that the cells are lost at a rate proportional to their concentration, namely $\gamma P(t)$.

In the real-world phenomena, the environment varies with time. Thus, the following nonautonomous nonlinear delay differential equation with time-varying coefficients and delays, and oscillatory circulation loss rate is a natural extensions of (1)

\begin{equation} \label{MG} x'(t) = \sum_{k=1}^M \lambda_k r_k(t)\frac{x^m(t-\tau_k(t))}{1+x^n(t-\tau_k(t))}-b(t)x(t), \quad (2) \end{equation} where $m\geq 0$, $n, \lambda_k>0$, $b(t),r_k, \tau_k$ are positive almost periodic functions for $k= 1, 2, \ldots, M$ and $b(t)$ is oscillating. The function $ \lambda_k r_k(t)\frac{x^m(t-\tau_k(t))}{1+x^n(t-\tau_k(t))}$ is the flux of cells into the blood stream from the $i-th$ stem cell compartment.

In this work, we establish and prove a fixed point theorem from which some sufficient conditions are deduced on the existence of positive almost periodic solutions for (2). Some particular conditions under the nonlinearity of the equation have been previously considered by authors as fundamental assumption for the study of almost periodic solutions of the model. The aim of this work is to establish results without such assumption. Some examples are given to illustrate our results.

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