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Scaling limit of a discrete prion dynamics model. (English) Zbl 1196.34012

The authors consider the following two models desribing an intracellular prion infection
\[ \frac{dv}{dt}=\lambda-\gamma v-v\sum^\infty_{i=n_0}\tau_iu_i+2\sum_{j\geq n_0}\sum_{i< n_0} ik_{i,j}\beta_ju_j, \]
\[ \frac{du_i}{dt}=-\mu_iu_i-\beta_iu_i-v(\tau_iu_i-\tau_{i-1}u_{i-1})+2\sum_{j>i}\beta_jk_{i,j}u_j, \]
\(i=n_0,n_0+1,\dots\), with the convention \(\tau_{n_0-1}u_{n_0-1}=0\).
Here, \(v\) represents the quantity of healthy monomers, \(u_i\) the quantity of infections polymers of size \(i\).
\[ \frac{dv}{dt}=\lambda-\gamma v-v\int^\infty_{x_0}\tau(x){\mathcal U}(t,x)\,dx+2\int^\infty_{x=x_0}\int^{x_0}_{y=0}y k(y,x)\beta(x){\mathcal U}(t,x)\,dx dy, \]
\[ \frac{\partial {\mathcal U}}{\partial t}=-\mu(x){\mathcal U}(t,x)-\beta(x){\mathcal U}(t,x)-V\,\frac{\partial}{\partial x}(\tau{\mathcal U})+2\int^\infty_x\beta(y)k(x,y){\mathcal U}(t,y)\,dy, \]
with appropriate boundary conditions at \(x=x_0\).
The aim of the authors is to investigate the link between both models. They discuss the mathematical assumptions under which the continuous model is the limit of the discrete one, and establish a convergence statement.

MSC:

34A33 Ordinary lattice differential equations
34A35 Ordinary differential equations of infinite order
35F15 Boundary value problems for linear first-order PDEs
35Q92 PDEs in connection with biology, chemistry and other natural sciences
92E99 Chemistry
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