Periodic solutions in a tumor-immune competition system with time-delay and chemotherapy effects

TL;DR

This work models tumor-immune dynamics with two time delays and periodic chemotherapy via $\dot{T}(t)=T(t) f(t,T(t))-\gamma E(t)T(t)$ and $\dot{E}(t)=\sigma+ \frac{pE(t)T(t-\tau_1)}{g+a T(t-\tau_1)}-\frac{mE(t)T(t-\tau_2)}{g+a T(t-\tau_2)}-\eta E(t)$. It develops a comprehensive stability and bifurcation analysis using the characteristic equation $\mathcal{P}(\lambda,\tau_1,\tau_2)=0$, identifies stability-switching curves in delay space, and applies a continuation/implicit-function framework to establish the existence of $\omega$-periodic solutions under small perturbations. The main contributions include explicit stability criteria for the tumor-free equilibrium via $\Delta=\gamma\sigma-rb\eta$, a detailed description of interior equilibria and their stability under small and large delays, Hopf bifurcation results with crossing curves, and the construction of periodic-solution branches through continuation; these results are complemented by numerical validations. The findings illuminate how chemotherapy scheduling and intrinsic delays shape sustained oscillations in tumor-immune dynamics, with potential implications for optimizing treatment protocols.

Abstract

The main purpose of this paper is to analyze the dynamics of the system of time-delay differential equations (DDEs) \begin{equation*} \begin{split} \dot{T}(t)&=T(t) f(t,T(t))-γE(t)T(t),\\ \dot{E}(t)&=σ+ \frac{pE(t)T(t-τ_1)}{g+a T(t-τ_1)}-\frac{mE(t)T(t-τ_2)}{g+a T(t-τ_2)}-ηE(t), \end{split} \end{equation*} where $T=T(t)$ and $E=E(t)$ represent the concentrations of tumor and effector cells at the time $t$. The coefficients $σ$, $μ$, $γ$, and $η$ are all positive, and $f(t, T)$ represents the relative growth rate of tumor cells, corresponding to a generalized logistic growth function that describes periodic time chemotherapeutic effects. The parameter $τ_1 \in \mathbb{R}_{\ge 0}$ is the response time delay of the immune system (mediated by effector cells) to an invasion of tumor cells, while $τ_2 \in \mathbb{R}_{\ge 0}$ represents the time delay of tumor cells in response to the appearance of effector cells.

Figures (4)

• Figure 1: Graph of $h_{\mu}(T)$ with parameters $\mu=-1.5$, $b=0.35$, and $\beta=0.5$. For these values, $h_{\mu}^{\prime}(b^{1/\beta}) \approx 1.17$, therefore $b^{1/\beta}<T_{\Delta}$. If $0<h_{0}\leq b$ there exists exactly one solution of $(\blacktriangle)$ on $]0,b^{1/\beta}]$ and no solutions if $h_{0}>b.$
• Figure 2: Graph of $h^{\prime}_{\mu}(T)$ with parameters $b=0.8$, and $\beta=0.5$ for $T\in ]0,0.7]$. Three different values of $\mu$ are considered: $\mu_{n}=3.68+n$, $n \in \left\{0,1,2\right\}.$ Notice that $\mu_{1}\approx \mu_{c}=75/16$. For $\mu<\mu_c$, $h^{\prime}_{\mu}(T)>0$ for all $T>0$. Instead, if $\mu>\mu_c$, $h^{\prime}_{\mu}(T)$ have exactly two positive zeros.
• Figure 3: Graph of $h_{\mu}(T)$ with parameters $b=0.8$, and $\beta=0.5$. (Left) $T\in \, ]0,b^{1/\beta}]$. (Right) $T\in \,]0,1.2]$. Three different values of $\mu$ are considered: $\mu_{n}=3.68+n$, $n \in \left\{0,1,2\right\}.$ Notice that $\mu_{1}\approx \mu_{c}=75/16$. For $\mu\leq \mu_c$, there is exactly one solution of ($\blacktriangle$) if $0 \leq h_{0}\leq b$ lying on $[0,b^{1/\beta}]$.
• Figure 4: Graph of $h_{\mu}(T)$ with parameters $b=0.8$, and $\beta=0.5$. for $T\in \, ]0,5b^{1/\beta}/8]$. Three different values of $\tilde{\mu}$ are considered: $\tilde{\mu}_{n}=5.85+0.4 n$, $n \in \left\{0,1,2\right\}.$ Notice that $T_{bif}=4/25$ and $\tilde{\mu}_{1}= \mu_{bif}=25/4$. There are no solutions of $h_{\tilde{\mu}_{0}}(T)=0$ and exactly two solutions of $h_{\tilde{\mu}_{2}}(T)=0$.

Theorems & Definitions (42)

• Lemma 1
• Lemma 2
• Theorem 3
• Lemma 4
• Theorem 5
• proof
• Remark 1
• Lemma 6
• proof
• Remark 2