Abstract
Typically stochastic differential equations (SDEs) involve an additive or multiplicative noise term. Here, we are interested in stochastic differential equations for which the white noise is nonlinearly integrated into the corresponding evolution term, typically termed as random ordinary differential equations (RODEs). The classical averaging methods fail to treat such RODEs. Therefore, we introduce a novel averaging method appropriate to be applied to a specific class of RODEs. To exemplify the importance of our method, we apply it to an important biomedical problem, in particular, we implement the method to the assessment of intratumoral heterogeneity impact on tumor dynamics. Precisely, we model gliomas according to a well-known Go or Grow (GoG) model, and tumor heterogeneity is modeled as a stochastic process. It has been shown that the corresponding deterministic GoG model exhibits an emerging Allee effect (bistability). In contrast, we analytically and computationally show that the introduction of white noise, as a model of intratumoral heterogeneity, leads to monostable tumor growth. This monostability behavior is also derived even when spatial cell diffusion is taken into account.
Highlights
We firstly focus on an averaging principle related to random ordinary differential equations (RODEs), and we extend it to the context of random partial differential equations (RPDEs)
We have introduced a novel averaging method to deal with the study of the dynamics of a class of RODEs and RPDEs since classical averaging methods fail to treat this kind of problem
We consider the development of gliomas according to a well-known Go or Grow (GoG) model, where intratumoral heterogeneity is modeled as a stochastic process
Summary
Publisher’s Note: MDPI stays neutral with regard to jurisdictional claims in published maps and institutional affiliations. We assume that the worst Gaussian perturbation can occur on such a parameter In this framework, the well-posedness is not given and we will prove it using the theory of averaging. In the first case scenario, if k(t) is a real valued process system (1) is well defined and its mathematical study can be delivered through a well established theory, see [6]. The idea presented in the current work consists of substituting the white noise ξ t with a proper approximation, ξ tN , and study the asymptotic behavior for N → ∞, through the averaging theory. The following section focuses on the demonstration and the proof of our main mathematical result, the novel averaging principle: first, an averaging principle for ordinary differential equations is presented, see Theorem 1, the result is extended to partial differential equations, see Theorem 2.
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