A confidence ellipse analysis for stochastic dynamics model of Alzheimer's disease

Abstract

The Alzheimer’s disease (AD) is a neurodegenerative disease, which is caused by the aggregation of beta-amyloid peptide ($A\beta$) in the patient’s brain and the disorder of ${\text{Ca}}^{2 + }$ homeostasis in neurons. Caluwé and Dupont (Theor Biol 331:12–18, 2013) proposed a deterministic AD model to explore the effect of ${\text{Ca}}^{2 + }$ on AD. They demonstrated the positive feedback loop between $A\beta$ and ${\text{Ca}}^{2 + }:$ and the occurrence of bistability. Based on their results, we further discuss the bistable behaviors. We present two periodically feasible drug strategies to alleviate the AD and screen out more effective one. In this paper, we also formulate a stochastic AD model, analyze the existence and uniqueness of global positive solutions and establish sufficient conditions for the existence of ergodic stationary distribution. Furthermore, the confidence ellipses describing the configurational arrangement of stochastic coexistence equilibria are constructed by stochastic sensitivity function technique, and tipping threshold is estimated as well. Noise-induced stochastic switching between two coexistence equilibria is observed in bistability region. Our results provide a new idea to control noise to alleviate AD through physical therapy.

Appendix

From the second equation of the deterministic system (i.e., $\sigma_{1} = \sigma_{2} = 0$ in (2)), we have

$$ y = \frac{{V_{4} x + V_{2} }}{{k_{2} }}. $$

(A.1)

The bring (A.1) into the first equation of the deterministic system, we get a cubic equation in one variable with respect to $x$

$$ F(x) = T_{3} x^{3} + T_{2} x^{2} + T_{1} x + T_{0} = 0, $$

(A.2)

where

$$ \begin{aligned} T_{3} & = - k_{1} V_{4}^{2} , \\ T_{2} & = - 2V_{2} V_{4} k_{1} + V_{4}^{2} (V_{1} + V_{3} ), \\ T_{1} & = 2V_{2} V_{4} (V_{1} + V_{3} ) - \left( {V_{2}^{2} + k_{2}^{2} k_{3}^{2} } \right)k_{1} , \\ T_{0} & = V_{2}^{2} (V_{1} + V_{3} ) + V_{1} k_{2}^{2} k_{3}^{2} . \\ \end{aligned} $$

We assume that (A.2) has three positive equilibria, $x_{1} ,x_{2}$ and $x_{3}$ ($x_{1} < x_{2} < x_{3}$). $F^{\prime } (x_{1} )$ is the derivative of $F(x)$ at $x_{i}$. We can easily get that $F^{\prime } (x_{1} ) < 0$, $F^{\prime } (x_{2} ) > 0$ and $F^{\prime } (x_{3} ) < 0$. Next, we give the stability analysis of the equilibria.

Calculate the Jacobian matrix $J(E_{i} )$ of the deterministic system at internal equilibrium $E_{i} (x_{i} ,y_{i} )$

$$ J(E_{i} ) = \left( {\begin{array}{*{20}c} { - k_{1} } & {\frac{{2V_{3} k_{3}^{2} y_{i} }}{{\left( {k_{3}^{2} + y_{i}^{2} } \right)^{2} }}} \\ {\frac{{V_{4} }}{\varepsilon }} & { - \frac{{k_{2} }}{\varepsilon }} \\ \end{array} } \right). $$

The eigenvalues of $J(E_{i} )$ satisfy the equation:

$$ \lambda^{2} - {\text{tr}}(J)\lambda + {\text{Det}}(J) = 0, $$

where,

$$ \begin{array}{*{20}c} {{\text{tr}}(J) = - k_{1} - \frac{{k_{2} }}{\varepsilon } < 0,\quad {\text{Det}}(J) = - \frac{{\left( {V_{4} x_{i} + V_{2} } \right)F^{\prime } \left( {x_{i} } \right)}}{{k_{2}^{2} y_{i} \left( {k_{3}^{2} + y_{i}^{2} } \right)}}} \\ \end{array} , $$

because $F^{\prime } (x_{1} ) < 0$, $F^{\prime } (x_{2} ) > 0$ and $F^{\prime } (x_{3} ) < 0$, so $E_{1}$ and $E_{3}$ are asymptotically stable and $E_{2}$ is an unstable saddle point.

In addition, the two equations on the right of the deterministic system are respectively defined as $H_{1}$ and $H_{2}$, then

$$ \frac{{\partial H_{1} }}{\partial x} + \frac{{\partial H_{2} }}{\partial y} = - k_{1} - \frac{{k_{2} }}{\varepsilon } < 0. $$

According to the Bendixson–Dulac criteria, it is clear that there is no closed orbit in the deterministic system, that is, when the deterministic system has a unique equilibrium, it must be globally asymptotically stable.