Why Are Periodic Erythrocytic Diseases so Rare in Humans?

Abstract

Many studies have shown that periodic erythrocytic (red blood cell linked) diseases are extremely rare in humans. To explain this observation, we develop here a simple model of erythropoiesis in mammals and investigate its stability in the parameter space. A bifurcation analysis enables us to sketch stability diagrams in the plane of key parameters. Contrary to some other mammal species such as rabbits, mice or dogs, we show that human-specific parameter values prevent periodic oscillations of red blood cells levels. In other words, human erythropoiesis seems to lie in a region of parameter space where oscillations exclusively concerning red blood cells cannot appear. Further mathematical analysis show that periodic oscillations of red blood cells levels are highly unusual and if exist, might only be due to an abnormally high erythrocytes destruction rate or to an abnormal hematopoietic stem cell commitment into the erythrocytic lineage. We also propose numerical results only for an improved version of our approach in order to give a more realistic but more complex approach of our problem.

Appendices

Appendix 1: Sufficient Stability Condition

In this “Appendix”, we use the theorem established by Hayes (1950) to formulate a sufficient stability condition for the equilibrium $E^*$. Consider a set of model parameters A, $\delta _{me}$, q, $\beta$, $K_c$, $\alpha$, $\delta _e$ and $\tau$. If $\tau = 0$ then the characteristic equation (12) admits a unique root $\uplambda = - \delta _e - L$ which is real and negative because $L > 0$. Now, we consider that $\tau >0$. Eq. (12) can be written as

$$\begin{aligned} \left( \uplambda \tau + \delta _e \tau \right) e^{\uplambda \tau } + L \tau =0, \;\; \uplambda \in {\mathbb {C}}. \end{aligned}$$

(23)

The latter equation is adapted to the theorem formulated by Hayes (1950). Hence, if $\uplambda \in {\mathbb {C}}$ verifies Eq. (23) then

$$\begin{aligned} \mathrm {Re}\, \uplambda< 0 \;\; \Leftrightarrow \;\; {\left\{ \begin{array}{ll} 0< L \tau < \mu \sin (\mu ) -\delta _e \tau \cos (\mu ), \\ \mu = -\delta _e \tau \tan (\mu ), \;\; \mu \in \left( \frac{\pi }{2}, \, \pi \right) . \end{array}\right. } \end{aligned}$$

Which can also be written as

$$\begin{aligned} \mathrm {Re} \, \uplambda< 0 \;\; \Leftrightarrow \;\; {\left\{ \begin{array}{ll} 0< - \cos (\mu ) < \frac{\delta _e}{L },\\ \mu = -\delta _e \tau \tan (\mu ), \;\; \mu \in \left( \frac{\pi }{2}, \;\; \pi \right) . \\ \end{array}\right. } \end{aligned}$$

Given that $0< -\cos (\mu ) < 1$ holds for all $\mu \in \left( \frac{\pi }{2}\, , \, \pi \right)$, we obtain Proposition 2.

Appendix 2: Continuity, Differentiability and Variations with Respect to Model Parameters

Derivatives with respect to E. Given the form of $K(\cdot )$ in Eq. (2), trivial derivation rules and simple computations lead, for $E > 0$, to

$$\begin{aligned} \frac{\partial K}{\partial E}(E)= - q \frac{1}{\beta } \left( \frac{E}{\beta }\right) ^{q-1} \frac{\alpha }{\left( 1+\left( \frac{E}{\beta }\right) ^{q}\right) ^2}>0, \end{aligned}$$

and

$$\begin{aligned} \frac{\partial ^2 K}{\partial E^2}(E)=q\alpha \frac{ \left( q+1 \right) \left( \frac{E}{\beta }\right) ^{2q-2} - \left( q-1 \right) \left( \frac{E}{\beta }\right) ^{q-2}}{\beta ^2 \left( 1+\left( \frac{E}{\beta }\right) ^{q} \right) ^3}. \end{aligned}$$

Continuity and variations with respect to $\tau$. Remembering the fact that L is strictly positive for all $\tau \ge 0$, one can establish from the implicit function theorem that, all other parameters being fixed, $\tau \mapsto E^*(\tau )$ is continuously differentiable on $[0,+\infty )$ and

$$\begin{aligned} \frac{\partial E^*}{\partial \tau }(\tau ) = \frac{- \delta _{me} \delta _e E^*(\tau )}{\delta _e + L(\tau )}>0. \end{aligned}$$

$\tau \mapsto E^*(\tau )$ is thus a strictly decreasing function with $\lim \limits _{\tau \rightarrow \infty }E^*(\tau )=0$. After few lines of computations not mentioned here, one also obtains

$$\begin{aligned} \frac{\partial L}{\partial \tau }(\tau ) = - \frac{q\delta _{me} L(\tau )}{\left( \delta _e + L(\tau )\right) \left( 1+\left( \frac{E^*(\tau )}{\beta }\right) ^{q} \right) } \left[ \delta _e - \frac{AK_c}{\beta }\left( \frac{E^*(\tau )}{\beta }\right) ^{q-1} e^{-\tau \delta _{me}} \right] . \end{aligned}$$

This expression enables us to study the variation of L with respect to $\tau$. From

$$\begin{aligned} \frac{A K_c}{\delta _e} \le E^*(\tau = 0)\le \frac{A(K_c+\alpha )}{\delta _e}\, , \end{aligned}$$

we establish that if model parameters are such that Condition (14) holds then $\tau \mapsto L(\tau )$ admits a unique maximum at

$$\begin{aligned} {\tilde{\tau }} = \frac{1}{\delta _{me}} \left[ \frac{q-1}{2q}\ln \left( 1+ \frac{\alpha }{K_c}\right) + \ln \left( \frac{AK_c}{\beta \delta _e} \right) \right] , \end{aligned}$$

and

$$\begin{aligned} L({\tilde{\tau }}) = \frac{q\delta _e \alpha }{2K_c \left( 1+\sqrt{1 + \frac{\alpha }{K_c}}\right) +\alpha }. \end{aligned}$$

Continuity and variations with respect to $\delta _e$. Remembering the fact L is strictly positive for all $\delta _e >0$, one can establish that $\delta _e \mapsto E^*(\delta _e)$ is continuously differentiable on $(0,+\infty )$ and

$$\begin{aligned} \frac{\partial E^*}{\partial \delta _e}(\delta _e) = -\frac{E^*(\delta _e)}{\delta _e + L(\delta _e)}. \end{aligned}$$

$\delta _e \mapsto E^*(\delta _e)$ is thus a strictly decreasing ranging from $+\infty$ (when $\delta _e\rightarrow 0$) to 0. One also obtains

$$\begin{aligned} \frac{\partial L}{\partial \delta _e}(\delta _e) = L(\delta _e) \frac{\left( q+1\right) \left( \frac{E^*(\delta _e)}{\beta }\right) ^q - \left( q-1\right) }{\left( \delta _e + L(\delta _e)\right) \left( 1+\left( \frac{E^*(\delta _e)}{\beta }\right) ^q \right) }. \end{aligned}$$

We establish after a few tedious computations that the function $\delta _e \mapsto L(\delta _e)$ admits a unique maximum

$$\begin{aligned} L(\tilde{\delta _e}) = \frac{A \alpha e^{ -\tau \delta _{me} }}{4\beta }\frac{ \left( q-1\right) ^{\frac{q-1}{q}} \left( q+1\right) ^{\frac{q+1}{q}} }{q}, \end{aligned}$$

with

$$\begin{aligned} \tilde{\delta _e} = \frac{Ae^{- \tau \delta _{me}}}{\beta } \left( \frac{q+1}{q-1}\right) ^{\frac{1}{q}} \left( K_c + \frac{3 \alpha }{5} \right) . \end{aligned}$$

Continuity and variations with respect to $K_c$. As previously done, one can establish that the function $K_c \mapsto E^*(K_c)$ is continuously differentiable and

$$\begin{aligned} \frac{\partial E^*}{\partial K_c}(K_c) = \frac{\delta _e E^*(K_c)}{\left( \delta _e + L(K_c) \right) \left( 1+\left( \frac{E^*(K_c)}{\beta } \right) ^{q}\right) }. \end{aligned}$$

The equilibrium $E^*$ is thus a strictly increasing function of $K_c$ and we have. One subsequently conclude, by composition, that $K_c \mapsto L(K_c)$ is continuously differentiable on ${\mathbb {R}}_+$, all other model parameters being fixed. With computations similar to the one performed by the reader in the latter paragraph, few lines of computations, not mentioned here, leads to

$$\begin{aligned} \frac{\partial L}{\partial K_c}(K_c) = - \delta _e L(K_c) \dfrac{\left( q+1\right) \left( \frac{E^*(K_c)}{\beta }\right) ^q - \left( q-1\right) }{\left( \delta _e + L(K_c)\right) \left( 1+\left( \frac{E^*(K_c)}{\beta }\right) ^q \right) \left( K_c + \frac{\alpha }{1+\left( \frac{E^*(K_c)}{\beta } \right) ^{q}}\right) }. \end{aligned}$$

From this expression, one can study the variation of L with respect to $K_c$. Using Eq. (9), one sees that

$$\begin{aligned} E^*(K_c=0) \le \frac{A \alpha \mathrm {e}^{-\tau \delta _{me}}}{\delta _e}, \end{aligned}$$

and easily establish that if the sufficient condition (13) holds, then $K_c \mapsto L(K_c)$ admits a maximum at

$$\begin{aligned} \tilde{K_c} = \frac{\beta }{A}\mathrm {\tau \delta _{me}}\left( \frac{q-1}{q+1}\right) - \frac{\alpha (q+1)}{2q}, \end{aligned}$$

given by

$$\begin{aligned} L(\tilde{K_c}) = A\alpha \mathrm {e}^{-\tau \delta _{me}}\frac{(q+1)^2}{4\beta q} \left( \frac{q-1}{q+1} \right) ^{\frac{q-1}{q}}. \end{aligned}$$

Continuity and variations with respect to $\delta _{me}$. Similarly to the previous paragraphs, we have that $\delta _{me} \mapsto E^*(\delta _{me})$ is continuously differentiable and

$$\begin{aligned} \frac{\partial E^*}{\partial \delta _{me}}(\delta _{me}) = - \frac{\tau \delta _e E^*(\delta _{me})}{\delta _e + L(\delta _{me})}<0. \end{aligned}$$

Then, $\delta _{me} \mapsto E^*(\delta _{me})$ is a strictly decreasing with $\lim \limits _{\delta _{me} \rightarrow \infty }E^*(\delta _{me})=0$. Similarly, one also obtains

$$\begin{aligned} \begin{aligned} \frac{\partial L}{\partial \delta _{me}}(\delta _{me}) {}={}&- \frac{q\delta _{me} L(\delta _{me})}{\left( \delta _e + L(\delta _{me})\right) \left( 1+\left( \frac{E^*(\delta _{me})}{\beta }\right) ^{q} \right) } \\ {} {}&\ \ \times \left( \delta _e - \frac{AK_c}{\beta }\left( \frac{E^*(\delta _{me})}{\beta }\right) ^{q-1} e^{-\tau \delta _{me}} \right) . \end{aligned} \end{aligned}$$

With the same arguments previously mentioned, if Condition (14) is satisfied then $\delta _{me} \mapsto L(\delta _{me})$ admits a maximum

$$\begin{aligned} L({\tilde{\delta }}_{me}) = \frac{q\delta _e \alpha }{2K_c \left( 1+\sqrt{1 + \frac{\alpha }{K_c}}\right) +\alpha }, \end{aligned}$$

at

$$\begin{aligned} {\tilde{\delta }}_{me} = \frac{1}{\tau } \left[ \frac{q-1}{2q}\ln \left( 1+ \frac{\alpha }{K_c}\right) + \ln \left( \frac{AK_c}{\beta \delta _e} \right) \right] . \end{aligned}$$

Appendix 3: Proof of Proposition 7

Let $\psi$ be one of the parameter among $\delta _e$, $K_c$, $\delta _{me}$ or $\tau$. First, let us remark that 0 is a root of Eq. (12) if and only if $\delta _e = -L (\psi )$. Yet, we know that $L (\psi )>0$. Consequently, 0 cannot be a root Eq. (12).

Moreover, considering the complex conjugate of Eq. (12), we obtain that if $\uplambda \in i{\mathbb {R}}$ is a root of Eq. (12) then $-\uplambda$ is also a root of this equation. From this, we know that if pure imaginary roots of Eq. (12) exist, then they will be given by $\uplambda = \pm \mathrm {i} \omega (\psi )$, $\omega (\psi )>0$.

Let $\psi ^*\in {\mathcal {P}}$ be a chosen parameter. Thanks to Proposition 2 and to Corollary 1, we know that if $\psi ^* \notin \varPi$ then pure imaginary roots of Eq. (12) cannot exist. It is thus necessary that $\psi ^* \in \varPi$.

We begin to show the first part of Proposition 7. From what we just mentioned above, we assume, without loss of generality, that $\mathrm {i}\omega (\psi ^*)$, $\omega (\psi ^*) >0$, is a root of Eq. (12). In such situation, separating the real from the imaginary part, we obtain that $\omega (\psi ^*)$ must verify

$$\begin{aligned} {\left\{ \begin{array}{ll} \begin{aligned} \delta _e {}={}&{} - L (\psi ^*) \cos (\omega (\psi ^*) \tau ), \\ \omega (\psi ^*) {}={}&{} L (\psi ^*) \sin (\omega (\psi ^*) \tau ). \end{aligned} \end{array}\right. } \end{aligned}$$

(24)

Considering that $-1 \le \cos (x) \le 1$ and that $\delta _e > 0$, we verify that $\psi ^*\in \varPi$ is a necessary condition for the definition of System (24). The latter system also reads

$$\begin{aligned} {\left\{ \begin{array}{ll} \begin{aligned} \cos \left( \omega (\psi ^*) \tau \right) {}={}&{} -\frac{\delta _e}{L(\psi ^*)},\\ \sin \left( \omega (\psi ^*) \tau \right) {}={}&{} \frac{\omega (\psi ^*)}{L(\psi ^*)}, \end{aligned} \end{array}\right. } \end{aligned}$$

and we obtain

$$\begin{aligned} 1 = \cos (\omega (\psi ^*) \tau )^2 +\sin (\omega (\psi ^*) \tau )^2 = \left( \frac{\omega (\psi ^*)}{L(\psi ^*)}\right) ^2+\left( \frac{\delta _e}{L(\psi ^*)} \right) ^2. \end{aligned}$$

Consequently,

$$\begin{aligned} \omega (\psi ^*)^2 = L(\psi ^*)^2 - \delta _e^2, \end{aligned}$$

and we get that $\omega (\psi ^*)$ is expressed by Eq. (16).

Remark 4

It is necessary that $\psi ^*\in \varPi$ otherwise $\omega (\psi ^*)$ is not defined. This condition also guarantee, a posteriori, that the function $\psi \mapsto \omega (\psi )$ is well defined on the set $\varPi$.

Finally, from the signs involved in Eq. (24), it is necessary that

$$\begin{aligned} \omega (\psi ^*) \tau \in \underset{k\in {\mathbb {N}}}{\cup } \left( \frac{(4k+1)}{2}\pi \, , \, (2k+1)\pi \right) , \end{aligned}$$

from which it follows that there exists $k\in {\mathbb {N}}$ such that $\psi ^* \in \varPi$ verifies

$$\begin{aligned} \omega (\psi ^*) \tau = \arctan \left( -\frac{\omega (\psi ^*)}{\delta _e} \right) + (2k+1)\pi . \end{aligned}$$

We reformulate the latter condition using the function z defined in Eq. (17) and obtain the first part of Proposition 7: if $\mathrm {i}\omega (\psi ^*)$ for $\psi ^*\in \varPi$ is a root of (12) then there exists $k\in {\mathbb {N}}$ such that $\psi ^*\in \varPi$ is a zero of $z(\cdot ,k)$.

Remark 5

For all $k \in {\mathbb {N}}$, $\psi \mapsto z(\psi ,k)$ is well defined and continuously differentiable on $\varPi$.

The proof of this remark directly originates from the continuous differentiability of $\psi \mapsto L(\psi )$ and $\psi \mapsto \omega (\psi )$ on $\varPi$.

The reciprocal of Proposition 7 is simply established by separating real and imaginary parts and by doing similar computations to the one above. The proof of the end of this proposition is given by Beretta and Kuang (2002).