Oscillatory Behavior of the Solutions for a Parkinson’s Disease Model with Discrete and Distributed Delays

Abstract

In this paper, the oscillatory behavior of the solutions for a Parkinson’s disease model with discrete and distributed delays is discussed. The distributed delay terms can be changed to new functions such that the original model is equivalent to a system in which it only has discrete delays. Using Taylor’s expansion, the system can be linearized at the equilibrium to obtain both the linearized part and the nonlinearized part. One can see that the nonlinearized part is a disturbed term of the system. Therefore, the instability of the linearized system implies the instability of the whole system. If a system is unstable for a small delay, then the instability of this system will be maintained as the delay increased. By analyzing the linearized system at the smallest delay, some sufficient conditions to guarantee the existence of oscillatory solutions for a delayed Parkinson’s disease system can be obtained. It is found that under suitable conditions on the parameters, time delay affects the stability of the system. The present method does not need to consider a bifurcating equation. Some numerical simulations are provided to illustrate the theoretical result.

1. Introduction

It is known that Parkinson’s disease (PD) is a common progressive neurodegenerative disease. Parkinson’s disease is characterized by tremors and stiffness. Mathematical modeling can help understand such complex multifactorial neurological diseases and help diagnose and treat them. Many mathematical models have been established to discuss Parkinson’s disease mathematically and biologically using the experimental method or analysis method. For example, Tuwairqi and Badrah provided the following model [1]:

$$\left\{\begin{array}{c}{N}^{\prime }\left(t\right)=\sigma -\beta N\left(t\right)\alpha S\left(t\right)-a_{1}N\left(t\right)-a_{2}N\left(t\right)-{\mu }_{1}N\left(t\right),\\ I^{\prime }\left(t\right)=\beta N\left(t\right)\alpha S\left(t\right)-d_{1}I\left(t\right)-a_{1}T\left(t\right)-a_{2}I\left(t\right),\\ \begin{array}{c}\alpha S^{\prime }\left(t\right)=e{d}_{1}I\left(t\right)-a_1\alpha S\left(t\right)-a_2\alpha S\left(t\right)-{\epsilon }_1\alpha \tilde{S}(t, \tau ),\\ M^{\prime }\left(t\right)=a_{1}I\left(t\right)+a_1\alpha S\left(t\right)-{\epsilon }_2\tilde{M}(t, \tau )-{\mu }_{2}M\left(t\right),\\ T^{\prime }\left(t\right)=a_{1}T\left(t\right)+\left(a_1+a_2\right)\alpha S\left(t\right)-{\epsilon }_3\tilde{T}\left(t, \tau \right)-{\mu }_{3}T\left(t\right),\end{array}\end{array}\right.$$

(1)

where $N\left(t\right), I\left(t\right),$ and $\alpha S\left(t\right)$ represent the density of healthy neurons in the brain, the density of infected neurons in the brain, and the density of extracellular $\alpha$-syn in the brain, respectively, $M\left(t\right)$ represents the density of activated microglia, and $T\left(t\right)$ presents the density of the activated $T$ cell; $a_1$, $a_2$, ${\epsilon }_1$, ${\epsilon }_2$, ${\epsilon }_3$, ${\mu }_1$, ${\mu }_2$, ${\mu }_3, \sigma ,$ and $\beta$ are parameters which belong to [0, 1]. The local stability of the free and endemic equilibrium points was established depending on the basic reproduction number. The authors pointed out that the administering time of immunotherapies plays a significant role in hindering the advancement of Parkinson’s disease. Different from traditional viewpoints, Wang et al. provided a delayed model which contains a cortex inhibitory nucleus (INN), a direct inhibitory projection from the subthalamic nucleus (STN), a cortex excitatory nucleus (EXN), and globus pallidus external (GPe). A simplified INN-EXN-STN-GPe resonance mathematical model is the following [2]:

$$\{\begin{array}{c}{\tau }_S{S}^{\prime }\left(t\right)=F_S\left(-W_{GS} G\left(t-T_{GS}\right)+W_{CS} E\left(t-T_{CS}\right)\right)-S\left(t\right),\\ {\tau }_G{G}^{\prime }\left(t\right)=F_G\left(W_{SG} S\left(t-T_{SG}\right)-Str\right)-G\left(t\right),\\ \begin{array}{c}{\tau }_E{E}^{\prime }\left(t\right)=F_E\left(-W_{CC} I\left(t-T_{CC}\right)-W_{SC} S\left(t-T_{SC}\right)+C\right)-E\left(t\right),\\ {\tau }_I{I}^{\prime }\left(t\right)=F_I\left(-W_{CC} E\left(t-T_{CC}\right)\right)-I\left(t\right),\end{array}\end{array}$$

(2)

where $S\left(t\right)$ and $G\left(t\right)$ represent the subthalamic nucleus (STN) and the external segment of the globus pallidus (GPe), $E\left(t\right)$ represents the firing rate of cortical excitatory pyramidal neurons (EXN) and $I\left(t\right)$ represents the firing rate of inhibitory nuclei (INN). T$\left(t\right)$ and W$\left(t\right)$ represent the delay and connection weight in different projections. CIN is a constant excitatory input to the cortex, and Str represents the projection from the striatum. $F_Y\left(x\right)=\frac{M_Y}{1+\left(\frac{M_Y-B_Y}{B_Y}\right)\exp \left(-4x/M_Y\right)} \left(Y=S, G, E, I\right)$ are activation functions.

It can be assumed that ${\tau }_S={\tau }_G={\tau }_E={\tau }_I, T_{SG}=T_{GS}=T_{CC}=T_{SC}=T,$ the Hopf bifurcation of system (2) is considered. A modified model of the system (2) is as follows:

$$\{\begin{array}{c}{\tau }_S{S}^{\prime }\left(t\right)=F_S\left(-W_{GS} G\left(t-T_{GS}\right)+W_{CS} E\left(t-T_{CS}\right)\right)-S\left(t\right),\\ {\tau }_G{G}^{\prime }\left(t\right)=F_G\left(W_{SG} S\left(t-T_{SG}\right)-Str\right)-G\left(t\right),\\ \begin{array}{c}{\tau }_E{E}^{\prime }\left(t\right)=F_E\left(-W_{CC} I\left(t-T_{CC}\right)+W_{EE} E\left(t-T_{EE}\right)+C\right)-E\left(t\right),\\ {\tau }_I{I}^{\prime }\left(t\right)=F_I\left(W_{CC} E\left(t-T_{CC}\right)-W_{II} E\left(t-T_{II}\right)\right)-I\left(t\right).\end{array}\end{array}$$

(3)

Assume that

$${\tau }_S={\tau }_G={\tau }_E={\tau }_I=10, T_{SG}=T_{GS}=T_1, T_{CC}=T_{II}=T_{CS}=T_2.$$

(4)

The Hopf bifurcation critical condition of the system (3) was provided in [3]. However, in models (1) to (3), the delays are discrete, and distributed delays are rarely introduced into neuron models with biological backgrounds. Recently, Kaslik et al. applied the bifurcation and stability theory of distributed delays to the interaction of the Wilson–Cowan model of excitatory and inhibitory mean–field interactions in neuronal populations [4]. Indeed, distributed delays can more truly describe the delay effects of signal transmissions between different neurons. Wang et al. also considered a Parkinson’s model with distributed delays [5]:

$$\{\begin{array}{c}{\tau }_S{S}^{\prime }\left(t\right)=F_S\left(-W_{GS} G\left(t-T_{GS}\right)+W_{CS} {\int }_{-\infty }^t{K}_1\left(t-s\right)E\left(s\right)ds\right)-S\left(t\right),\\ {\tau }_G{G}^{\prime }\left(t\right)=F_G\left(W_{SG} S\left(t-T_{SG}\right)-W_{GG}{\int }_{-\infty }^t{K}_2\left(t-s\right)E\left(s\right)ds-Str\right)-G\left(t\right),\\ {\tau }_E{E}^{\prime }\left(t\right)=F_E\left(-W_{SC} {\int }_{-\infty }^t{K}_3\left(t-s\right)S\left(s\right)ds-INN+C\right)-E\left(t\right),\end{array}$$

(5)

where $K_1, K_2,$ and $K_3$ represent the weak or strong gamma functions. The authors studied the stable, conditional stable, conditional oscillation, and absolute oscillation for model (5), which can explain different mechanisms of oscillation origin. Agiza et al. used the Taylor series transform to discuss two-delay differential equations for the Parkinson’s disease models in [6]. The dynamic behavior of innate immune response to Parkinson’s disease with a therapeutic approach was modeled in [7]. Some authors investigate Parkinson’s disease models via electrical activity rhythms [8], activity patterns [9], the emergence of beta oscillations [10], the intra-operative characterization of subthalamic oscillations [11], the Bayesian adaptive dual control of deep brain stimulation [12]. Hu et al. investigated a bidirectional Hopf bifurcation of Parkinson’s oscillation in a simplified basal ganglia model in [13]. Darcy et al. considered the spectral and spatial distribution of subthalamic beta peak activity [14]. Lang and Espay summarized the current approaches, challenges, and future considerations in Parkinson’s disease in [15]. For other research results on Parkinson’s disease, one can see [16,17,18,19,20,21,22,23,24,25,26,27,28,29,30].

In this paper, we extend the model (3) to the following system which includes not only discrete delays but also distributed delays:

$$\left\{\begin{array}{c}{\tau }_S{S}^{\prime }\left(t\right)=F_S\left(-W_{GS} G\left(t-T_{GS}\right)+W_{CS} {\int }_{-\infty }^t{K}_1(t-s)E\left(s\right)ds\right)-S\left(t\right), \\ {\tau }_G{G}^{\prime }\left(t\right)=F_G\left(W_{SG} S\left(t-T_{SG}\right)-W_{GG}{\int }_{-\infty }^t{K}_2\left(t-s\right)G\left(s\right)ds-Str\right)-G\left(t\right)\\ \begin{array}{c}{\tau }_E{E}^{\prime }\left(t\right)=F_E\left(-W_{SC} I\left(t-T_{ES}\right)+W_{EE} {\int }_{-\infty }^t{K}_3\left(t-s\right)S\left(s\right)ds-INN+C\right)-E\left(t\right).\\ {\tau }_I{I}^{\prime }\left(t\right)=F_I\left(W_{CC} E\left(t-T_{CC}\right)-W_{II} {\int }_{-\infty }^t{K}_4\left(t-s\right)I\left(s\right)ds\right)-I\left(t\right).\end{array}\end{array}\right.$$

(6)

According to the simulation result in [3], the parameters are ${\tau }_S=12.80 \mathrm{ms}$, ${\tau }_G$= 20 ms, ${\tau }_E=$ 10–20 ms, and ${\tau }_I=$ 10–20 ms. Therefore, the results in [3] are only for special parameters. Note that model (6) has four discrete delays. If the four delays are different real numbers, then the bifurcation method is hard to deal with in model (6) due to the complexity of the bifurcating equation. In this paper, using the method of mathematical analysis, the oscillatory behavior of the solutions for model (6) could be obtained. Our result indicated that the four discrete delays can be different real numbers, and condition (4) has been extended.

For convenience, we considered $K_i$ (i = 1, 2, 3, 4) as the weak gamma functions, setting $K_i\left(t-s\right)={\alpha }_i\exp ({\alpha }_i\left(t-s\right)) ({\alpha }_i>0, i=1, 2, 3, 4),$ and let

$$Y_1\left(t\right)={\int }_{-\infty }^t{K}_1\left(t-s\right)E\left(s\right)ds={\int }_{-\infty }^t{\alpha }_1\exp \left({\alpha }_1\left(t-s\right)\right)E\left(s\right)ds,$$

$$Y_2\left(t\right)={\int }_{-\infty }^t{K}_2\left(t-s\right)G\left(s\right)ds={\int }_{-\infty }^t{\alpha }_2\exp \left({\alpha }_2\left(t-s\right)\right)G\left(s\right)ds,$$

$$Y_3\left(t\right)={\int }_{-\infty }^t{K}_3\left(t-s\right)S\left(s\right)ds={\int }_{-\infty }^t{\alpha }_3\exp \left({\alpha }_3\left(t-s\right)\right)S\left(s\right)ds,$$

$$\mathrm{and} Y_4\left(t\right)={\int }_{-\infty }^t{K}_4\left(t-s\right)I\left(s\right)ds={\int }_{-\infty }^t{\alpha }_4\exp \left({\alpha }_4\left(t-s\right)\right)I\left(s\right)ds.$$

Then, using the fundamental theorem of calculus, we obtained

$$Y_1^{\prime }\left(t\right)=-{\alpha }_1{\int }_{-\infty }^t{\alpha }_1\exp \left({\alpha }_1\left(t-s\right)\right)E\left(s\right)ds+{\alpha }_{1}E\left(t\right)=-{\alpha }_1{Y}_1\left(t\right)+{\alpha }_{1}E\left(t\right),$$

$$Y_2^{\prime }\left(t\right)=-{\alpha }_2{\int }_{-\infty }^t{\alpha }_2\exp \left({\alpha }_2\left(t-s\right)\right)G\left(s\right)ds+{\alpha }_{2}G\left(t\right)=-{\alpha }_2{Y}_2\left(t\right)+{\alpha }_{2}G\left(t\right),$$

$$Y_3^{\prime }\left(t\right)=-{\alpha }_3{\int }_{-\infty }^t{\alpha }_3\exp \left({\alpha }_3\left(t-s\right)\right)S\left(s\right)ds+{\alpha }_{3}S\left(t\right)=-{\alpha }_3{Y}_3\left(t\right)+{\alpha }_{3}S\left(t\right),$$

$$Y_4^{\prime }\left(t\right)=-{\alpha }_4{\int }_{-\infty }^t{\alpha }_4\exp \left({\alpha }_4\left(t-s\right)\right)I\left(s\right)ds+{\alpha }_{4}I\left(t\right)=-{\alpha }_4{Y}_4\left(t\right)+{\alpha }_{4}I\left(t\right).$$

Thus, we can rewrite model (6) as the following equivalent system:

$$\left\{\begin{array}{c}{S}^{\prime }\left(t\right)=-r_{1}S\left(t\right)+{r_{1}F}_S\left(-W_{GS} G\left(t-T_{GS}\right)+W_{CS} Y_1\left(t\right)\right),\\ G^{\prime }\left(t\right)={-r_{2}G\left(t\right)+r_{2}F}_G\left(W_{SG} S\left(t-T_{SG}\right)-W_{GG}{Y}_2\left(t\right)-Str\right),\\ \begin{array}{c}{E}^{\prime }\left(t\right)={-r_{3}E\left(t\right)+r_{3}F}_E\left(-W_{SC} I\left(t-T_{ES}\right)+W_{EE} Y_3\left(t\right)-INN+C\right),\\ I^{\prime }\left(t\right)=-r_{4}I\left(t\right)+r_4{F}_I\left(W_{CC} E\left(t-T_{CC}\right)-W_{II} Y_4\left(t\right)\right),\\ \begin{array}{c}{Y}_1^{\prime }\left(t\right)=-{\alpha }_1{Y}_1\left(t\right)+{\alpha }_{1}E\left(t\right),\\ Y_2^{\prime }\left(t\right)=-{\alpha }_2{Y}_2\left(t\right)+{\alpha }_{2}G\left(t\right),\\ \begin{array}{c}{Y}_3^{\prime }\left(t\right)=-{\alpha }_3{Y}_3\left(t\right)+{\alpha }_{3}S\left(t\right),\\ Y_4^{\prime }\left(t\right)=-{\alpha }_4{Y}_4\left(t\right)+{\alpha }_{4}I\left(t\right),\end{array}\end{array}\end{array}\end{array}\right.$$

(7)

where $r_1=\frac{1}{{\tau }_S}, r_2=\frac{1}{{\tau }_G}, r_3=\frac{1}{{\tau }_E},r_4=\frac{1}{{\tau }_I}$. From $F_Y\left(x\right)=\frac{M_Y}{1+\left(\frac{M_Y-B_Y}{B_Y}\right)\exp \left(-4x/M_Y\right)}$ $\left(Y=S, G, E, I\right)$ we know that $F_S<M_S$, $F_G<M_G$, $F_E<M_E$, and $F_I<M_I$, so we can obtain

$$\{\begin{array}{c}{S}^{\prime }\left(t\right)<-r_{1}S\left(t\right)+r_1{M}_S,\\ G^{\prime }\left(t\right)<-r_{2}G\left(t\right)+r_2{M}_G,\\ \begin{array}{c}{E}^{\prime }\left(t\right)<-r_{3}E\left(t\right)+r_3{M}_E,\\ I^{\prime }\left(t\right)<-r_{4}I\left(t\right)+r_4{M}_I,\\ \begin{array}{c}{Y}_1^{\prime }\left(t\right)=-{\alpha }_1{Y}_1\left(t\right)+{\alpha }_{1}E\left(t\right),\\ Y_2^{\prime }\left(t\right)=-{\alpha }_2{Y}_2\left(t\right)+{\alpha }_{2}G\left(t\right),\\ \begin{array}{c}{Y}_3^{\prime }\left(t\right)=-{\alpha }_3{Y}_3\left(t\right)+{\alpha }_{3}S\left(t\right),\\ Y_4^{\prime }\left(t\right)=-{\alpha }_4{Y}_4\left(t\right)+{\alpha }_{4}I\left(t\right).\end{array}\end{array}\end{array}\end{array}$$

(8)

System (8) implies that $S\left(t\right)<\frac{r_1}{r_1} M_S=M_S, G\left(t\right)<\frac{r_2}{r_2} M_G=M_G,$ $E\left(t\right)<\frac{r_3}{r_3} M_E=M_E,$ $I\left(t\right)<\frac{r_4}{r_4} M_I=M_I, Y_1\left(t\right)<M_E, Y_2\left(t\right)<M_G, Y_3\left(t\right)<M_S,$ and $Y_4\left(t\right)<M_I$. In other words, all of the solutions of system (7) are boundedness. According to the parameter values in [3]: $M_S=$ 300 spk/s, $B_S=$ 17 spk/s, $M_G=$ 400 spk/s, $B_G=$ 75 spk/s, $M_E=$ 71.77 spk/s, $B_E=$ 3.62 spk/s, $M_I=$ 276 spk/s, $B_I=$ 7.18 spk/s, we know that $F_Y\left(x\right)$ are monotone increasing functions for $Y=S, G, E,$ and I. Therefore, system (6) has a unique equilibrium point ${\left(S^{*}, G^{*}, E^{*}, I^{*}\right)}^T.$ Equivalently, system (7) has a unique equilibrium point ${\left(S^{*}, G^{*}, E^{*}, I^{*}, Y_1^{*}, Y_2^{*}, Y_3^{*}, Y_4^{*}\right)}^T,$ where $Y_1^{*}=E^{*}, Y_2^{*}=G^{*}, Y_3^{*}=S^{*},$ and $Y_4^{*}=I^{*}.$ If we make the change in variables $S\left(t\right)$ → $S\left(t\right)$−$S^{*},G\left(t\right)$ → $G\left(t\right)-G^{*}$, $E\left(t\right)$ → $E\left(t\right)$−$E^{*}, I\left(t\right)$ → $I\left(t\right)$−$I^{*}, Y_i\left(t\right)\to Y_i\left(t\right)-Y_i^{*}$ (i = 1, 2, 3, 4), the Taylor expansion of system (7) at the equilibrium point is the following:

$$\left\{\begin{array}{c}{S}^{\prime }\left(t\right)=-r_{1}S\left(t\right)+a_{12}G\left(t-T_{GS}\right)+a_{15}{Y}_1\left(t\right)\\ +\sum _{i+j\ge 2}\frac{{\left[G\left(t-T_{GS}\right)\right]}^i}{i! } \frac{{\left[Y_1\left(t\right)\right]}^j}{j !} ·{\left.\frac{{\partial }^{i+j}{F}_S}{\partial G^i\partial {Y_1}^j}\right|}_{\left(G^{*}, Y_1^{*}\right)},\\ \begin{array}{c}{G}^{\prime }\left(t\right)=-r_{2}G\left(t\right)+a_{21}S\left(t-T_{SG}\right)+a_{26}{Y}_2\left(t\right)\\ +\sum _{i+j\ge 2}\frac{{\left[S\left(t-T_{SG}\right)\right]}^i}{i! } \frac{{\left[Y_2\left(t\right)\right]}^j}{j !} ·{\left.\frac{{\partial }^{i+j}{F}_G}{\partial S^i\partial {Y_2}^j}\right|}_{\left(S^{*}, Y_2^{*}\right)},\\ \begin{array}{c}{E}^{\prime }\left(t\right)=-r_{3}E\left(t\right)+a_{34}I\left(t-T_{ES}\right)+a_{37}{Y}_3\left(t\right)\\ +\sum _{i+j\ge 2}\frac{{\left[I\left(t-T_{CC}\right)\right]}^i}{i! } \frac{{\left[Y_3\left(t\right)\right]}^j}{j !} ·{\left.\frac{{\partial }^{i+j}{F}_E}{\partial I^i\partial {Y_3}^j}\right|}_{\left(I^{*}, Y_3^{*}\right)},\\ \begin{array}{c}{I}^{\prime }\left(t\right)=-r_{4}I\left(t\right)+a_{43}E\left(t-T_{CC}\right)+a_{48}{Y}_4\left(t\right)\\ +\sum _{i+j\ge 2}\frac{{\left[E\left(t-T_{CC}\right)\right]}^i}{i! } \frac{{\left[Y_4\left(t\right)\right]}^j}{j !} ·{\left.\frac{{\partial }^{i+j}{F}_I}{\partial E^i\partial {Y_4}^j}\right|}_{\left(E^{*}, Y_4^{*}\right)}\\ \begin{array}{c}{Y}_1^{\prime }\left(t\right)=-{\alpha }_1{Y}_1\left(t\right)+{\alpha }_{1}E\left(t\right),\\ Y_2^{\prime }\left(t\right)=-{\alpha }_2{Y}_2\left(t\right)+{\alpha }_{2}G\left(t\right),\\ \begin{array}{c}{Y}_3^{\prime }\left(t\right)=-{\alpha }_3{Y}_3\left(t\right)+{\alpha }_{3}S\left(t\right),\\ Y_4^{\prime }\left(t\right)=-{\alpha }_4{Y}_4\left(t\right)+{\alpha }_{4}I\left(t\right),\end{array}\end{array}\end{array}\end{array}\end{array}\end{array}\right.$$

(9)

where $a_{12}=r_{1 }{\frac{\partial F_S}{\partial G}|}_{\left(G^{*}, Y_1^{*}\right)}$, $a_{15}=r_1{\left.\frac{\partial F_S}{\partial Y_1}\right|}_{(G^{\mathrm{*}},Y_1^{\mathrm{*}})}$, $a_{21}=r_2{\left.\frac{\partial F_G}{\partial S}\right|}_{(S^{\mathrm{*}},Y_2^{\mathrm{*}})}$, $a_{26}=r_2{\left.\frac{\partial F_S}{\partial Y_2}\right|}_{(S^{\mathrm{*}},Y_2^{\mathrm{*}})}$, $a_{34}=r_3{\left.\frac{\partial F_E}{\partial I}\right|}_{(I^{\mathrm{*}},Y_3^{\mathrm{*}})}$, $a_{37}=r_3{\left.\frac{\partial F_E}{\partial Y_3}\right|}_{(I^{\mathrm{*}},Y_3^{\mathrm{*}})}$, $a_{43}=r_4{\left.\frac{\partial F_I}{\partial E}\right|}_{(E^{\mathrm{*}},Y_4^{\mathrm{*}})}$, $a_{48}=r_4{\left.\frac{\partial F_I}{\partial Y_4}\right|}_{(E^{\mathrm{*}},Y_4^{\mathrm{*}})}.$ The linearized system of (9) is the following:

$$\left\{\begin{array}{c}{S}^{\prime }\left(t\right)=-r_{1}S\left(t\right)+a_{12}G\left(t-T_{GS}\right)+a_{15}{Y}_1\left(t\right),\\ G^{\prime }\left(t\right)=-r_{2}G\left(t\right)+a_{21}S\left(t-T_{SG}\right)+a_{26}{Y}_2\left(t\right),\\ \begin{array}{c}{E}^{\prime }\left(t\right)=-r_{3}E\left(t\right)+a_{34}I\left(t-T_{ES}\right)+a_{37}{Y}_3\left(t\right),\\ I^{\prime }\left(t\right)=-r_{4}I\left(t\right)+a_{43}E\left(t-T_{CC}\right)+a_{48}{Y}_4\left(t\right),\\ \begin{array}{c}{Y}_1^{\prime }\left(t\right)=-{\alpha }_1{Y}_1\left(t\right)+{\alpha }_{1}E\left(t\right),\\ Y_2^{\prime }\left(t\right)=-{\alpha }_2{Y}_2\left(t\right)+{\alpha }_{2}G\left(t\right),\\ \begin{array}{c}{Y}_3^{\prime }\left(t\right)=-{\alpha }_3{Y}_3\left(t\right)+{\alpha }_{3}S\left(t\right),\\ Y_4^{\prime }\left(t\right)=-{\alpha }_4{Y}_4\left(t\right)+{\alpha }_{4}I\left(t\right).\end{array}\end{array}\end{array}\end{array}\right.$$

(10)

Let s = min{$T_{GS}$, $T_{SG}$, $T_{ES}$, $T_{CC}$}. We consider a special case of the system (10):

$$\left\{\begin{array}{c}{S}^{\prime }\left(t\right)=-r_{1}S\left(t\right)+a_{12}G\left(t-s\right)+a_{15}{Y}_1\left(t\right),\\ G^{\prime }\left(t\right)=-r_{2}G\left(t\right)+a_{21}S\left(t-s\right)+a_{26}{Y}_2\left(t\right),\\ \begin{array}{c}{E}^{\prime }\left(t\right)=-r_{3}E\left(t\right)+a_{34}I\left(t-s\right)+a_{37}{Y}_3\left(t\right),\\ I^{\prime }\left(t\right)=-r_{4}I\left(t\right)+a_{43}E\left(t-s\right)+a_{48}{Y}_4\left(t\right),\\ \begin{array}{c}{Y}_1^{\prime }\left(t\right)=-{\alpha }_1{Y}_1\left(t\right)+{\alpha }_{1}E\left(t\right),\\ Y_2^{\prime }\left(t\right)=-{\alpha }_2{Y}_2\left(t\right)+{\alpha }_{2}G\left(t\right),\\ \begin{array}{c}{Y}_3^{\prime }\left(t\right)=-{\alpha }_3{Y}_3\left(t\right)+{\alpha }_{3}S\left(t\right),\\ Y_4^{\prime }\left(t\right)=-{\alpha }_4{Y}_4\left(t\right)+{\alpha }_{4}I\left(t\right).\end{array}\end{array}\end{array}\end{array}\right.$$

(11)

The matrix form of the system (11) is as follows:

$$u^{\prime }\left(t\right)=Cu\left(t\right)+Au\left(t-s\right)$$

(12)

where $u\left(t\right)={\left[S\left(t\right), G\left(t\right), E\left(t\right), I\left(t\right), Y_1\left(t\right), Y_2\left(t\right), Y_3\left(t\right), Y_4\left(t\right)\right]}^T,$ $u\left(t-s\right)=$ ${\left[S\left(t-s\right), G\left(t-s\right), E\left(t-s\right), I\left(t-s\right), 0, 0, 0, 0\right]}^T,$

$$A={\left(a_{ij}\right)}_{8\times 8}=\left(\begin{array}{cccccccc}0& a_{12}& 0& 0& 0& 0& 0& 0\\ a_{21}& 0& 0& 0& 0& 0& 0& 0\\ 0& 0& 0& a_{34}& 0& 0& 0& 0\\ 0& 0& a_{43}& 0& 0& 0& 0& 0\\ 0& 0& 0& 0& 0& 0& 0& 0\\ 0& 0& 0& 0& 0& 0& 0& 0\\ 0& 0& 0& 0& 0& 0& 0& 0\\ 0& 0& 0& 0& 0& 0& 0& 0\end{array}\right),$$

and

$$C={\left(c_{ij}\right)}_{8\times 8}=\left(\begin{array}{cccccccc}-r_1& 0& 0& 0& a_{15}& 0& 0& 0\\ 0& -r_2& 0& 0& 0& a_{26}& 0& 0\\ 0& 0& -r_3& 0& 0& 0& a_{37}& 0\\ 0& 0& 0& -r_4& 0& 0& 0& a_{48}\\ 0& 0& {\alpha }_1& 0& -{\alpha }_1& 0& 0& 0\\ 0& {\alpha }_2& 0& 0& 0& -{\alpha }_2& 0& 0\\ {\alpha }_3& 0& 0& 0& 0& 0& -{\alpha }_3& 0\\ 0& 0& 0& {\alpha }_4& 0& 0& 0& -{\alpha }_4\end{array}\right).$$

2. The Existence of Oscillatory Solutions

To discuss the existence of oscillatory solutions for system (7) including four-time delays, we first provide the following lemma.

Lemma 1. 

Consider the following delayed differential equations:

$$x^{\prime }\left(t\right)=f\left(x\left(t-{\tau }_{*}\right)\right),$$

(13)

$$x^{\prime }\left(t\right)=f\left(x\left(t-{\tau }^{*}\right)\right),$$

(14)

where ${\tau }^{*}>{\tau }_{*}>0, x\in R^n, f={\left(f_1, f_2, \cdots , f_n\right)}^T, f\left(0\right)=0.$ Assume that the trivial solution of system (13) is unstable, then the trivial solution of system (14) is also unstable.

Proof of Lemma 1. 

Since the trivial solution of system (13) is unstable, this means that for arbitrary $\epsilon >0$, there exists an infinite sequence {$t_k$}, where ${\tau }_{*}<$ $t_1<t_2<t_3<\cdots ,$ such that the trivial solution x(t) of system (13) satisfies $\left|x\left(t_k-{\tau }_{*}\right)\right|>\epsilon$. Noting that ${\left\{t_k\right\}}_{k=1}^{\infty }$ is an infinite sequence, one can select a subsequence {$t_{k_i}$} ⊂ {$t_k$}, such that $t_{k_i}=$ $t_k+\left({\tau }^{*}-{\tau }_{*}\right)$. Thus, for the trivial solution of system (14), we obtained the following:

$$\left|x\left(t_{k_i}-{\tau }^{*}\right)\right|=\left|x\left(t_k+\left({\tau }^{*}-{\tau }_{*}\right)-{\tau }^{*}\right)\right| =\left|x\left(t_k-{\tau }_{*}\right)\right|>\epsilon .$$

(15)

Inequivalent (15) indicates that the trivial solution of system (14) is also unstable. □

Since the system (10) is a linearized system of (9), we can see that system (9) is a disturbed system of (10). If the trivial solution of system (11) is unstable, then the trivial solution of system (10) is also unstable according to the Lemma 1. In what follows, we first consider the instability of the zero equilibrium point of the system (11) (or (12)). Therefore, we considered the following theorems.

Theorem 1. 

Assume that the system (11) has a unique trivial solution and ${\gamma }_1, {\gamma }_2,\cdots ,{\gamma }_8$ are characteristic values of matrix C. ${\rho }_1,{\rho }_2,{\rho }_3,{\rho }_4, 0, 0, 0, 0,$ are characteristic values of matrix $A.$

If there is a characteristic value, say ${\gamma }_k$, satisfying the following:

(i) 

Re (${\gamma }_k$) = 0, Im(${\gamma }_k$) $\ne 0$, and ${\gamma }_k=\omega i;$ or

(ii) 

Re (${\gamma }_k$) > 0, and Re (${\gamma }_k$) > max {|${\rho }_1$|, |${\rho }_2$|, $\left|{\rho }_3\right|$, |${\rho }_4$|}, or

(iii) 

Im (${\gamma }_k$) = 0, ${\gamma }_k$ > 0.

Then, the trivial solution of system (11) (thus system (9)) is unstable, implying that there exists a limit cycle in the system (7); namely, system (7) has a periodic solution.

Proof of Theorem 1. 

We show that the trivial solution of the system (11) is unstable. Since ${\gamma }_1, {\gamma }_2,\cdots ,{\gamma }_8$ are characteristic values of matrix C and ${\rho }_1,{\rho }_2,{\rho }_3,{\rho }_4, 0, 0, 0, 0$ are characteristic values of matrix A, then the characteristic equation of (11) is the following:

$${\prod }_{i=1}^8\lambda -{\gamma }_i-{\rho }_i{e}^{-\lambda s}=0.$$

(16)

When there is a characteristic value ${\gamma }_k$ such that Re (${\gamma }_k$) = 0, Im (${\gamma }_k$) $\ne 0$, and ${\gamma }_k=\omega i,$ then

$$e^{i\omega t}=\cos \omega t+i\sin \omega t.$$

(17)

We know that cos ωt is a periodic function; therefore, the trivial solution of system (11) is unstable. Noting that all characteristic values of matrix A are ${\rho }_i$ or 0, there is a characteristic equation from the system (16) as follows:

$$\mathsf{\lambda }-{\gamma }_k-{\rho }_k e^{-\lambda s}=0.$$

(18)

or

$$\mathsf{\lambda }-{\gamma }_k=0.$$

(19)

If Re (${\gamma }_k$) > 0 and Re (${\gamma }_k$) > max{|${\rho }_1$|, |${\rho }_2$|, $\left|{\rho }_3\right|$, $\left|{\rho }_4\right|\}$, this means that Equation (18) has a positive real part characteristic value. If Im (${\gamma }_k$) = 0, ${\gamma }_k$ > 0, it suggests that there is a positive characteristic value from (19). Thus, the trivial solution of system (11) is unstable. Based on Lemma 1, the trivial solution of system (10) is unstable. This implies that the equilibrium point ${\left(S^{*}, G^{*}, E^{*}, I^{*}, Y_1^{*}, Y_2^{*}, Y_3^{*}, Y_4^{*}\right)}^T$ of system (9) is unstable.

Equivalently, the unique equilibrium point of the system (7) is unstable. This instability of the unique equilibrium point, together with the boundedness of the solutions, forces system (7) to generate a limit cycle, namely, a periodic solution according to the extended Chafee’s criterion [31,32]. The proof is complete. □

Let σ = max{${\alpha }_3-r_1$, ${\alpha }_2-r_2$, ${\alpha }_1-r_3$, ${\alpha }_4-r_4$, $\left|a_{15}\right|-{\alpha }_1, \left|a_{26}\right|-{\alpha }_2$, $\left|a_{37}\right|-{\alpha }_3$, $\left|a_{48}\right|-{\alpha }_4$}, then we have Theorem 2.

Theorem 2. 

Assume that system (11) has a unique trivial solution. If the following condition holds

r + σ > 0,

(20)

where r = max {|$a_{12}$|, |$a_{21}$|, |$a_{34}|$|, |$a_{43}$|}. Then, the trivial solution of system (11) is unstable, implying that there exists a limit cycle of system (9); namely, system (7) has a periodic solution.

Proof of Theorem 2. 

To prove the instability of the trivial solution of the system (11), let $z\left(t\right)=S\left(t\right)+G\left(t\right)+E\left(t\right)+I\left(t\right)+{\sum }_{i=1}^4{Y}_i\left(t\right),$ then we have the following:

$$z\left(t\right)\le \sigma z\left(t\right)+r z\left(t-s\right).$$

(21)

Specifically, consider the following scalar equation:

$$v\left(t\right)=\sigma v\left(t\right)+r v\left(t-s\right).$$

(22)

According to the comparison theory of the differential equation, we have $z\left(t\right)\le v\left(t\right).$ We claim that the trivial solution of Equation (22) is unstable. Indeed, the characteristic Equation (22) is as follows:

$$\lambda =\sigma +r{e}^{-\lambda s}.$$

(23)

Consider a function $\varphi \left(\lambda \right)=\lambda -\sigma -r{e}^{-\lambda s}$. Then, $\varphi \left(\lambda \right)$ is a continuous function of $\lambda .$ Noting that $\varphi \left(0\right)=-\sigma -r$ = $-(\sigma +r)$ < 0. Clearly, there exists a real number L > 0 such that $\varphi \left(L\right)=L-\sigma -r{e}^{-Ls}>0.$ Using the Intermediate Value Theorem, there exists ${\lambda }_0\in \left(0, L\right)$ such that $\varphi \left( {\lambda }_0\right)=0$. In other words, there exists a positive characteristic root of Equation (22), which means that the trivial solution of Equation (22) is unstable, implying that the trivial solution of Equation (11) is unstable and that the trivial solution of system (11), thus (9), is unstable. Similar to Theorem 1, system (7) has a periodic solution. The proof is complete. □

3. Computer Simulation Result

This simulation is based on model (7). In model (7), according to the parameters in [3], we set $M_S=$ 300, $B_S=$ 17, $M_G=$ 400, $B_G=$ 75, $M_E=$ 71.77, $B_E=$ 3.62, $M_I=$ 276, $B_I=$ 7.18,$W_{GS}=3, W_{SG}=2.5, W_{SC}=6, W_{CC}=3, W_{EE}=W_{GG}=1, W_{II}=0.1, C=277, Str=40$. When we select ${\alpha }_1=0.8, {\alpha }_2=0.85, {\alpha }_3=0.5, {\alpha }_4=0.55,$ time delay $T_{GS}=15.5, T_{SG}=18, T_{ES}=16.5, T_{CC}=17,$ ${\tau }_S=12.5, {\tau }_G=20, {\tau }_E=10, {\tau }_I=15,$ so $r_1=$ $0.08, r_2=0.05, r_3=0.1, r_4=0.067,$ then the unique positive equilibrium point ${\left(S^{*}, G^{*}, E^{*}, I^{*}, Y_1^{*}, Y_2^{*}, Y_3^{*}, Y_4^{*}\right)}^T={(98.4164, 163.4268, 44.8525, 60.5816, 44.8525, 163.4268, 98.4164, 60.5816)}^T.$ Thus,

$$a_{12}=r_{1 }{\frac{\partial F_S}{\partial G}|}_{\left(G^{*}, Y_1^{*}\right)}=-0.2372, a_{15}=r_{1 }{\frac{\partial F_S}{\partial Y_1}|}_{\left(G^{*}, Y_1^{*}\right)}=0.4744,$$

$$a_{21}=r_{2 }{\frac{\partial F_G}{\partial S}|}_{\left(S^{*}, Y_2^{*}\right)}=0.2256,a_{26}=r_{2 }{\frac{\partial F_S}{\partial Y_2}|}_{\left(S^{*}, Y_2^{*}\right)}=-0.0923,$$

$$a_{34}=r_{3 }{\frac{\partial F_E}{\partial I}|}_{\left(I^{*}, Y_3^{*}\right)}=-0.0092, a_{37}=r_{3 }{\frac{\partial F_E}{\partial Y_3}|}_{\left(I^{*}, Y_3^{*}\right)}=0.0031,$$

$$a_{43}=r_{4 }{\frac{\partial F_I}{\partial E}|}_{\left(E^{*}, Y_4^{*}\right)}=0.1496,a_{48}=r_{4 }{\frac{\partial F_I}{\partial Y_4}|}_{\left(E^{*}, Y_4^{*}\right)}=-0.0053.$$

The characteristic values of the matrix C are 0.0257, −0.0863, −0.5437, −0.5734, $-$0.2661$\pm 0.1496 i, \mathrm{and}-0.2750\pm 0.0994 i$. Since there exists a positive characteristic value of 0.0257, the conditions of Theorem 1 are satisfied. There exists a periodic oscillatory solution (see Figure 1). Figure 2 indicates a case for all parameters are the same as in Figure 1, but time delays are changed. When we select ${\alpha }_1=0.8, {\alpha }_2=0.85, {\alpha }_3=0.75, {\alpha }_4=0.78,$ time delay is $T_{GS}=12.5, T_{SG}=14, T_{ES}=13.5, T_{CC}=13,$ ${\tau }_S=10, {\tau }_G=20, {\tau }_E=16, {\tau }_I=12,$ so $r_1=0.1, r_2=0.05, r_3=0.063, r_4=0.083,$ then the unique positive equilibrium point ${\left(S^{*}, G^{*}, E^{*}, I^{*}, Y_1^{*}, Y_2^{*}, Y_3^{*}, Y_4^{*}\right)}^T=(75.4916, 127.1502, 55.3846, 64.9912,$ $55.3846$,$127.1502, 75.4916, 64.9912{)}^T.$ Thus,

$$a_{12}=r_{1 }{\frac{\partial F_S}{\partial G}|}_{\left(G^{*}, Y_1^{*}\right)}=-0.0349, a_{15}=r_{1 }{\frac{\partial F_S}{\partial Y_1}|}_{\left(G^{*}, Y_1^{*}\right)}=0.0698,$$

$$a_{21}=r_{2 }{\frac{\partial F_G}{\partial S}|}_{\left(S^{*}, Y_2^{*}\right)}=0.1042, a_{26}=r_{2 }{\frac{\partial F_S}{\partial Y_2}|}_{\left(S^{*}, Y_2^{*}\right)}=-0.0417,$$

$$a_{34}=r_{3 }{\frac{\partial F_E}{\partial I}|}_{\left(I^{*}, Y_3^{*}\right)}=-0.0051, a_{37}=r_{3 }{\frac{\partial F_E}{\partial Y_3}|}_{\left(I^{*}, Y_3^{*}\right)}=0.0017,$$

$$a_{43}=r_{4 }{\frac{\partial F_I}{\partial E}|}_{\left(E^{*}, Y_4^{*}\right)}=0.2025, a_{48}=r_{4 }{\frac{\partial F_I}{\partial Y_4}|}_{\left(E^{*}, Y_4^{*}\right)}=-0.0067.$$

Thus, σ = max{${\alpha }_3-r_1$, ${\alpha }_2-r_2$, ${\alpha }_1-r_3$, ${\alpha }_4-r_4$, $\left|a_{15}\right|-{\alpha }_1, \left|a_{26}\right|-{\alpha }_2$, $\left|a_{37}\right|-{\alpha }_3$, $\left|a_{48}\right|-{\alpha }_4$} = 0.8, r = max{|${\mathrm{a}}_{12}$|, |${\mathrm{a}}_{21}$|, |${\mathrm{a}}_{34}|$, |${\mathrm{a}}_{43}$|} = 0.1042, and σ + r = 0.9042 > 0. The condition of Theorem 2 is satisfied. The system (7) has an oscillatory solution (see Figure 3). In Figure 4, we changed the time delays and kept all parameters the same as in Figure 3. When we selected ${\alpha }_1=1.5, {\alpha }_2=1.6, {\alpha }_3=1.8, {\alpha }_4=1.75,$ time delay is $T_{GS}=18.5, T_{SG}=17.5, T_{ES}=18, T_{CC}=17,$ ${\tau }_S=12.8,$ ${\tau }_G=16, {\tau }_E=20, {\tau }_I=14,$ so $r_1=0.078, r_2=0.063, r_3=0.05, r_4=0.07,$ then the unique positive equilibrium point ${\left(S^{*}, G^{*}, E^{*}, I^{*}, Y_1^{*}, Y_2^{*}, Y_3^{*}, Y_4^{*}\right)}^T={(98.5443, 151.1364, 52.1243, 63.4582, 52.1243 151.1364, 98.5443, 63.4582)}^T$.

Therefore, we can obtain the following:

$$a_{12}=r_{1 }{\frac{\partial F_S}{\partial G}|}_{\left(G^{*}, Y_1^{*}\right)}=-0.0108, a_{15}=r_{1 }{\frac{\partial F_S}{\partial Y_1}|}_{\left(G^{*}, Y_1^{*}\right)}=0.0216,$$

$$a_{21} =r_{2 }{\frac{\partial F_G}{\partial S}|}_{\left(S^{*}, Y_2^{*}\right)}=0.1471, a_{26}=r_{2 }{\frac{\partial F_S}{\partial Y_2}|}_{\left(S^{*}, Y_2^{*}\right)}=-0.0588,$$

$$a_{34}=r_{3 }{\frac{\partial F_E}{\partial I}|}_{\left(I^{*}, Y_3^{*}\right)}=-0.0062, a_{37}=r_{3 }{\frac{\partial F_E}{\partial Y_3}|}_{\left(I^{*}, Y_3^{*}\right)}=0.0021,$$

$$a_{43}=r_{4 }{\frac{\partial F_I}{\partial E}|}_{\left(E^{*}, Y_4^{*}\right)}=0.1904,a_{48}=r_{4 }{\frac{\partial F_I}{\partial Y_4}|}_{\left(E^{*}, Y_4^{*}\right)}=-0.0063.$$

It is easy to see that the condition of Theorem 2 holds. Therefore, system (7) has an oscillatory solution (see Figure 5). In Figure 6, we reduced the time delays. It can be seen that both the oscillatory frequency and amplitude were changed.

4. Discussion

Our result provides a criterion to determine whether or not there exists an oscillatory solution for model (6), which includes discrete and distributed delays. The oscillatory solution of the model (6) corresponds to the tremor of Parkinson’s disease. Therefore, our main result is very significant. We also point out that the bifurcation method is hard to deal with in the present model. Because the bifurcation equation about model (10) is as follows:

$$\begin{gathered} p_{\mathbf{1}}\left(\mathit{\lambda }\right)\exp \left(-\mathit{\lambda }{T}_{GS}\right)+p_{\mathbf{2}}\left(\mathit{\lambda }\right)\exp \left(-\mathit{\lambda }{T}_{SG}\right) \\ +p_{\mathbf{3}}\left(\mathit{\lambda }\right)\exp \left(-\mathit{\lambda }{T}_{ES}\right)+p_{\mathbf{4}}\left(\mathit{\lambda }\right)\exp \left(-\mathit{\lambda }{T}_{CC}\right)+p_{\mathbf{5}}\left(\mathit{\lambda }\right)=0. \end{gathered}$$

(24)

It can be noted that if $T_{GS}$, $T_{SG}$, $T_{ES}$, $T_{CC}$ are different positive numbers, Equation (24) is a transcendental equation with four variables. Solving Equation (24) and finding the bifurcating points are hard work.

5. Conclusions

In this paper, we discussed the oscillatory behavior of the solutions for a Parkinson’s disease model with discrete and distributed delays. Two theorems were provided to determine the existence of oscillatory solutions, which were easy to inspect to compare the method of bifurcation. We made the change in distributed delay terms as new functions such that the original system became only a discrete system. This method can be used to deal with all distributed systems. We point out that the present criteria are only sufficient conditions.