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A multigroup approach to delayed prion production

  • *Corresponding author: Mattia Sensi

    *Corresponding author: Mattia Sensi
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  • We generalize the model proposed in [Adimy, Babin, Pujo-Menjouet, SIAM Journal on Applied Dynamical Systems (2022)] for prion infection to a network of neurons. We do so by applying a so-called multigroup approach to the system of Delay Differential Equations (DDEs) proposed in the aforementioned paper. We derive the classical threshold quantity $ \mathcal{R}_0 $, i.e. the basic reproduction number, exploiting the fact that the DDEs of our model qualitatively behave like Ordinary Differential Equations (ODEs) when evaluated at the Disease Free Equilibrium. We prove analytically that the disease naturally goes extinct when $ \mathcal{R}_0<1 $, whereas it persists when $ \mathcal{R}_0>1 $. We conclude with some selected numerical simulations of the system, to illustrate our analytical results.

    Mathematics Subject Classification: Primary: 34K18, 34K20, 37G99, 37N25; Secondary: 92B05, 92C20.

    Citation:

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  • Figure 1.  schematic view of the PrP$ ^C $ protein production (in blue) by two neurons (green). The PrP$ ^C $ protein can aggregate and form pathological PrP$ ^{Sc} $ (pink and orange). The PrP$ ^{Sc} $ proteins diffuse and a certain amount can reach the neighbourhood of another neuron (the orange ones can reach the neighbourhood of neuron 1, while the pink ones can reach neuron 2. We refer to Section 2 for a complete description of this case and of the parameters and variables involved

    Figure 2.  schematic representation of a neuron (neuron 1) under Unfolded Protein Response (UPR). Stressed by the overcrowded amount of PrP$ ^{Sc} $ in its neighbourhood, neuron 1 shuts down its activities (except the vital ones). No PrP$ ^C $ protein is then produced, and the population of PrP$ ^{Sc} $ pathological proteins diffuse out the neuron surroundings

    Figure 3.  The three networks we consider in the sections 3.2, 3.3 and 3.4. We only show the forward case from Section 3.4, as the backward case would look exactly the same but with each arrow reversed

    Figure 4.  The four networks we consider in our numerical simulations. Notice that only the network with $ n = 3 $ has double arrows on each edge, representing the fully connected network. The three remaining networks all have unidirectional edges

    Figure 5.  the fully connected network of $ n = 3 $ neurons (recall Fig. 4a), showing the stability of disease free equilibrium when $ \mathcal{R}_0 = \rho(F) = 0.8194<1 $. In this case $ R_{0i} = 0.6944<1 $ for $ i = 1, 2, 3 $. The parameters are: $ \alpha_{i\rightarrow j} = 0.9 $, $ \kappa_{ij} = 0.1 $, $ p = 5 $, $ d = 0.015 $, $ y_c = 60 $, $ T = 0.17 $, $ K_i = 1500 $ and $ \mu_i = 18 $

    Figure 6.  the fully connected network of $ n = 3 $ neurons (depicted by Fig. 4a), showing the stability of endemic equilibrium when $ \mathcal{R}_0 = \rho(F) = 1.1346>1 $. In this case $ R_{0i} = 0.9615<1 $ for $ i = 1, 2, 3 $. The parameters are: $ \alpha_{i\rightarrow j} = 0.9 $, $ \kappa_{ij} = 0.1 $, $ p = 5 $, $ d = 0.015 $, $ y_c = 60 $, $ T = 0.17 $, $ K_i = 1500 $ and $ \mu_i = 13 $

    Figure 7.  line network of $ n = 5 $ neurons (depicted in Fig. 4b). This case shows that cutting the connection showed stabilization. The parameters are: $ \alpha_{i\rightarrow j} = 2.5 $, $ \kappa_{ij} = 0.17 $ (when considered), $ p = 10 $, $ d = 0.15 $, $ y_c = 50 $, $ K_i = 1500 $, $ \mu_i = 20 $ and $ T = 0.15 $

    Figure 8.  circle (ring) network of $ n = 5 $ neurons (depicted in Fig. 4c). This case shows that linking the connection in Fig. 7 showed destabilization of the system. The parameters are: $ \alpha_{i\rightarrow j} = 2.5 $, $ \kappa_{ij} = 0.17 $ (when considered), $ p = 10 $, $ d = 0.15 $, $ y_c = 50 $, $ K_i = 1500 $, $ \mu_i = 20 $ and $ T = 0.15 $. Similar results can be obtained by varying the time delay $ T $ and fixing all other parameters

    Figure 9.  line network of $ n = 5 $ neurons (depicted in Fig. 4b). For this figure showing the oscillation of only some neurons ($ n = 4, 5 $, the last ones), we took $ \kappa = 0.071 $. Parameters are: $ p = 10 $, $ d = 0.15 $, $ y_c = 60 $, $ K_i = 1800 $, $ \mu_i = 50 $, $ \alpha_{i\rightarrow j} = 0.9 $ ($ \alpha_i = 3.6 $) and $ T = 0.15 $. Similar results can be produced by changing the delay $ T $

    Figure 10.  line network of $ n = 5 $ neurons (depicted in Fig. 4b). Our starting point was the case where the system is stable and we have increased the bifurcation parameter $ \kappa $ and noted each value for which a neuron is destabilized. The value of bifurcations are $ 0.07 $, $ 0.0709 $, $ 0.728 $ and $ 0.768 $. The neurons are destabilized one by one from the last and going up to the second. The last neurons always look destabilized almost at the same time (approximately the same bifurcation values). Parameters are: $ p = 10 $, $ d = 0.15 $, $ y_c = 60 $, $ K_i = 1800 $, $ \mu_i = 50 $, $ \alpha_{i\rightarrow j} = 0.9 $ ($ \alpha_i = 3.6 $) and $ T = 0.15 $. A curve similar to this one can be obtained as a function of the time delay $ T $

    Figure 11.  line network of $ n = 5 $ neurons (recall Fig. 4b). The amplitude of the oscillations as a function of $ \kappa\in[0.055, 0.1] $. The amplitude of the oscillations becomes almost the same for higher values of $ \kappa $. Parameters are: $ p = 10 $, $ d = 0.15 $, $ y_c = 60 $, $ K_i = 1800 $, $ \mu_i = 50 $, $ \alpha_{i\rightarrow j} = 0.9 $ ($ \alpha_i = 3.6 $) and $ T = 0.15 $. The red, blue, green and yellow curves are associated respectively with $ (x_2, y_2) $, $ (x_3, y_3) $, $ (x_4, y_4) $ and $ (x_5, y_5) $. The same behavior can be obtained as a function of time delay $ T $, see Figure 12

    Figure 12.  line network of $ n = 5 $ neurons (recall Fig. 4b). The amplitude of the oscillations as a function of $ T \in[0.15, 0.4] $. All the cycles except for the blue one are almost overlapping for every value of $ T $. Parameters are: $ p = 10 $, $ d = 0.15 $, $ y_c = 60 $, $ K_i = 1800 $, $ \mu_i = 50 $, $ \alpha_{i\rightarrow j} = 0.9 $ ($ \alpha_i = 3.6 $) and $ \kappa = 0.07 $. The blue, red, green, purple and yellow curves are associated respectively with $ (x_1, y_1) $, $ (x_2, y_2) $, $ (x_3, y_3) $, $ (x_4, y_4) $ and $ (x_5, y_5) $

    Figure 13.  line network of $ n = 9 $ neurons (recall Fig. 4d). We only plot the time series of $ (x_i, y_i) $ with $ i = 1, 2, 5, 8, 9 $, respectively the first two, the central one and the last two neurons in the line network. By increasing the number $ n $ of neurons from $ 5 $ to $ 9 $ with the same parameters as in Fig. 10 and 11 and with $ \kappa = 0.125 $, the system becomes stable. Recall parameters: $ p = 10 $, $ d = 0.15 $, $ y_c = 60 $, $ K_i = 1800 $, $ \mu_i = 50 $, $ \alpha_{i\rightarrow j} = 0.45 $ ($ \alpha_i = 3.6 $) and $ T = 0.15 $

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